0
- + -r = 1, and esti-
ciiv"«ni,Hi£ < 1 ( I I V » « U £ . ) J + - ^ (chwiy' = »7^-||Vm«|U. + ^ f - c m / ^ | | u | | L . . Taking 77 = ™E the corollary follows. D The density of Cg°(R n ) in Wm'P(M.n), 1 < p < 00, gives immediately
2.1 Second Order Elliptic Differential Operators
21
Corollary 2.1.8. A. The estimate (2.16) holds for all u G Wm
1 <
(2.17)
B. On Wm'p(M.n) an equivalent norm is given by ||Vmti||Lp + |M| L p.
(2.18)
Moreover, we have the following important commutator estimate. Recall that for two operators on a linear space their commutator is defined by [A,B} = AB-BA.
(2.19)
In particular we have for a function
(2.20)
Corollary 2.1.9. Let ip G C6°°(R") and u G Wm'p(M.n). Further let a G NJJ, \a\ < m, and e > 0 be given. Then we have for 0 < k < m — \a\ \\{
(2.21)
implying ||[y,5 a ]u|| wfc , P < £r||w||v^|e.i+fc,„+c(y>,e)||w||x-P.
(2.22)
Proof: By Corollary 2.1.8.A it is obvious that (2.21) gives (2.22). Now, by the definition of [ip, da]u we have
[
J2
(„Va~/W
which implies (2.21). D Now let G C R n be any open set. By definition we have Wm'P(G) := C£°(G) "w Obviously, Wm'v{G)
.
(2.23)
C Wm'P(G) and it is a Banach space. Note that for a
bounded set G C E " we always have Wm*(G)
^ Wm'p(G).
This follows from
Chapter 2
22
Generators of Feller and Sub-Markovian Semigroups
the fact that the function I H I belongs for any bounded set to Wm'p(G), but it is not possible to approximate this function in the norm || • ||w>, P) m > 1, by test functions. In fact, in this situation we have Wm,p(G) C WAl,1(G) and for Hwi.i = / \l-
/
V
JG
j=1
_d_ dxi
Moreover, since f(x)
=
fXl
d
-z—
we find, see also Theorem 2.1.10,
ML,
JG
'
[l^-dx, dx
(2.24)
i
which yields
Y}6*^ dx
||l-y|ki.i > f ldx- f \
JG
dx.
JG J^[
^ I dX^(G)-(c(G)-l) [J2\ dx JG
J=I
>AW(G)-(c(G)- 1)111-^11^, thus we have |l-vllwi.i >
A(G) c(G)
which tells us that we can not approximate x i-> 1 by test functions in the norm || • ||vyi,i. We want to give a more general version of (2.24). Theorem 2.1.10 (Poincare's inequality). Let G c Rn be a bounded open o
set and suppose that 1 < p < oo. Then, for all u € W 1'P(G) \\u\\LP
(2.25)
2.1 Second Order Elliptic Differential Operators
23
Proof: It is enough to prove (2.25) for all u € C£°(G) which we consider as usual as subspace of Cg°(M.n). Since G is bounded we may find I > 0 such that G lies between the two hyperplanes x\ = a and x\ = a + l. Without loss of generality we may choose a — 0. For x e G we find with - + -V = 1 that
Kx)l
=
fXl
d
a
I
~a~ u ( y i ' x 2 ,
/
1
...,xn)dyi\
d
P
g—u(yi,x 2 , ...,xn)
\
VP
dyA
/ ll/p .
Thus we get
fl
f' \
3
d
/ |u(xi,...,a:„)| '(ixi < F / —— u(yi, x2, ••-, xn) Jo Jo {c,yi which yields / \u\pdx
dyi,
D
JG
There are several generalizations of Poincare's inequality which we will not discuss here, but we refer to the monograph [80] of D.E.Edmunds and W.D.Evans, the book [81] of Yu.Egorov and V.Kondratiev, or the monograph [106] of D.Gilbarg and N.S.Trudinger. We need a further embedding theorem for the space Wm'p(R.n), namely Sobolev's embedding theorem. Our presentation is oriented at the treatment of the subject by Chr.Sogge [253], and we need the following L e m m a 2.1.11. For s > 0 define Ks(x) :=
(2TT)-"/2
/
e ^ ( l + K| 2 )-* /2 <^-
(2.26)
JR"
It follows that K3 is a function and ifQ<s
i#.(*)i
we have for every N S N the
(2.27)
Proof: First note that since s > 0 the function £ ^ (1 + |£| 2 ) s^2 is bounded and measurable, thus (2.26) make sense at least in S'(Rn). Suppose first that
24
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
Ixl > 1. For m G N observe that (2ir)n/2Ks(x)
= f
\x\-2m\x\2meix-t(l
+ |£| 2 ) _ ' / 2 d£
= \*\~2m I ((-A ? ) m e i x '0 (1 + | ^ | 2 ) _ s / 2 ^ = \x\~2m [ e ix '«(-A) m (l + |£| 2 )- s/2 d£. JRn
Since
-Ar(l + |^| 2 )- s / 2 |
(2.28)
is an integrable function for m > ^ , it follows that (2.27) holds for |x| > 1. Next we prove (2.27) for |a:| < 1. For this take ip 6 C£°(]Rn) such that 0 < tp < 1, supp ip c Bro(0) and ip = 1. For R > 0 we find Br0/2(0)
(27r)-"/2^e»-«v ( | ) (i + iei 2 r s/2 ^
•xun—a (l + |e| 2 )-«^ cR
l\t\ ^ we get
(27r) n/2 eixi
~ L ^ ~Kl))(i+^~,/2^\
= (27r)-"/ 2 ^Jxr 2m ((-A c re^) (l-V;(|))(l + |e|2)-s/2^|
|x|-2m|^|-s-2m^ <
c~'\x\-2mRn-s-2m,
AO^R where we used once again (2.28). Finally, taking R = jxj ^, we obtain (2.27) for all x, which also implies that Ks is a function. (Compare also [256], p. 132.) • Theorem 2.1.12 (Sobolev's embedding theorem). Let m - k > -, p > 1. Then Wm'p(M.n) is continuously embedded into the space C£,(]Rn). More precisely, each element of Wm>p(Rn) has a representation by an element of C^,(R™) and we have the estimate \\u\\nk
Wm'P •
(2.29)
2.1 Second Order Elliptic Differential Operators
25
Proof: We prove (2.29) first for k = 0. Since C£°(R n ) is dense in W m ' p ( R n ) , it is sufficient to show (2.29) for all u G C£°(K n ). Defining v := ( i d - A ) m / 2 u , it is sufficient to prove H/JTm + wHoo
(2.30)
In fact we have Km*v =
F-\F(Km*v))
= (2Kyl2F-\Kmv) = (2ir)n/2F-1
((1 + | • | 2 ) - m / 2 ( l + | • | 2 ) m / 2 ^
= (2TT)"/ 2 W,
and further we have H i , = | | ( i d - A ) m / 2 w | | L P < c\\u\\Wm,P.
(2.31)
Now, if m > ^ it follows that — np' + mp' > —1 and (2.27) implies that Km G Lp (R n ). Prom Lemma 1.2.3.15, Young's inequality, we deduce finally (2.30) and therefore (2.29) for k = 0. The general case follows by induction. • Note that (2.31) used in the proof is trivial for m = 21, I e N, or p = 2. For general m € N and p > 1 this is just the content of (2.2) and is proved using some Fourier multiplier results, we will discuss this result in a more general context in a later section. Note that (2.3) combined with Theorem 1.3.11.12 yields also \L,np/(i-p)
< c\\u\\wi,P
(2.32)
for p < n and all u G W1'p{Rn). As noted after Corollary 1.3.11.13, it is possible to show that for m - k > % elements of Wm'p(W) = H™(W) do not only belong to C ^ ( R n ) , but their fcth partial derivatives are also Holder continuous with some Holder exponent 7 6 (0,1] such that (m — k — 7) > ^ . Since we need this embedding result explicitly for the case m = 1, we will give a proof which is essentially borrowed from W.Ziemer [290]. Theorem 2.1.13. Ifu e W 1 , P(R n ), p > n, then we have with 7 = 1 - | that u G C°'T(Mn) and \u{x)-u{y)\ \x-y\i ~ W^1""
. (2-33)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
26
Proof: Since Cg°(R n ) is dense in W^W1) it is sufficient to prove (2.33) n n for u G C°°(R ). Let x 6 R™ and y G R fixed. Put y - x = r e tf , r G R+, and e,, G S" 1-1 . It follows that u(y) — u(x) = / (j/ — x) • grad w(x + t(y — x))dt Jo — / re#- gradu(x + £r e#)dt Jo = / r e^ • gradu(x + ir e^)xg^j{x
+ tr e#)dt.
JO
Integrating over SP1'1 we find with w„ = A^" -1 ) (S" 1-1 ) wn(u(x)-u(y)) / re* •grad^x + i r e ^ ) * " - 1 - ^ ^
= / =
C l
/
5
( g r a d . ) ( z ) ^ ^ ,
Using Holder's inequality we get u{y)-u(x)
|gradu(z)|
Z XBA*J( ) Br(x)
\JB7U)
\X - z "p'-p'
Since
dZ =C5rl f=C yll f UIBT&J T ~ ^- ~ \~ B-^\X-z\^-r>' X Z
we finally arrive at \u(x)-u{y)\ \x-vYH-?
< c\\u\ wi.p
proving the theorem.
D
dz
Ix-zl"-1
JBr(x)
;
dS{d)dt
2.1 Second Order Elliptic Differential Operators
27
In order to construct sub-Markovian semigroups and Dirichlet form we will discuss now some special properties of the space W1'p(M.n). We refer to D.Gilbarg and N.S.Trudinger [106] or to D.Kinderlehrer and G.Stampacchia [175] as our main reference. First we prove the chain rule for functions in W^R"). Lemma 2.1.14. Let f £ C ^ R ) with f £ Cb(R) and u £ W ^ R " ) . Then f o u 6 W1'p(Rn) and we have for 1 < j < n the chain rule g f r ( / o it) =
Proof: We know that u £ Wl'p(Rn) may be approximated in the norm II • IIWLPCR") by a sequence (um)meN of functions um £ Co°(R n ). Without loss of generality we may assume that um —> u almost everywhere. Since / ' is bounded, we have for all functions v, w £ W 1 , p (R n )
\f(v(x))-f(u,(z))\
< ll/looK*)-^*)!.
Thus we find ||/(«m)-/(«)||LP < ll/'lloollt^-uHi,,
and
f>rn)^Um-f(u)£-U LP
LP
+
(f'(um)-f'{u))—u LP
+ (/ l/'K . ( * ) . ) - / ' ( t i ( x ) ) | *
du(x) dxi
\P
dx
) •
n
By our assumption we have um —> u in LP(R ) and almost everywhere as well as gf-Wm —* w~u m £ p (R n )- Moreover, since / ' is continuous we have f'(um(x)) —> f'(u(x)) almost everywhere and
\f'(um(x))-f'(u{x))\.\~(x) \dxi implying that f(um) -» / ( « ) and / > find g § - / ( u m ) —• f'(u)-§^:,
<2||/'|| 0 O m
£-(x)
dxj
in D>(Rn). Thus we
) f ^ -> f'(u)^
which proves the lemma.
•
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
28
Corollary 2.1.15. Let u € W 1 , p (R n ;R). Then u+,u~ and \u\ are also elements in W1'p(Rn) and we have for 1 < j < n (2.34)
dxj
|o,
d —u-
fo, = \ '
ifu<0' ifu>0 ,
, 2.35
and
'£-u, — I
0
ifu>0 ifu = 0.
(2.36)
Recall that derivatives of Wl,p -functions are distributional derivatives belonging to Lp(M.n). Proof: For s > 0 consider the function f(A Mt)
_j(t2 + s2)i-e,
-\o,
ift>0 ift
Applying Lemma 2.1.14 we find for any ip € Cfi°(M.n)
•/M"
"Xj
Ju>0
Oxj
and for e —• 0 we obtain u T -—
/ n
JR
Vxj
(pu-—ax Ju>0
Vxj
which gives (2.34). Since u~ — (— u)+ and |u| = u+ + u~ we also get (2.35) and (2.36). • Corollary 2.1.16. Let u s W^1,P(R"). Then we have gradu = 0 on any open set where u is constant. Proof: First note that if u is constant on a set with infinite measure, this constant must be zero. Now, if u = c on a bounded open set G, we may
2.1 Second Order Elliptic Differential Operators
29
multiply u by a test function I/J being equal to 1 in a neighbourhood of G. Taking another test function which is equal to c on the support of •)/>, we find by subtracting that we may assume that c = 0 in any case. But now the corollary follows from gradtz = g r a d u + — g r a d u - and Corollary 2.1.15. • Theorem 2.1.17. Let f : R —» R be a continuous function with a continuous and bounded first order derivative except of finite numbers of points where the graph off may have corners. The set of these corners is denoted by E. Further let u S W1'P(Wl, R). Then we have foue W^iW1, R) and
»(,„,_ few'*?- *<*.*»** 9xj
\0,
(2.3T)
if(x,u(x))eE
Proof: Clearly we may assume that E consists only of the origin. Take h, h G Cl(Rn) such that /i(0) = / 2 (0) and /i(u) = f(u) for u > 0 and fiiu) = f(u) for u < 0. Since now f(u) = fi(u+) + f2{—u~), the result follows from Lemma 2.1.14 and Corollary 2.1.15. • Corollary 2.1.18. A. Let u € ^ 1 - P ( R " , R ) . Then the function belongs to W 1 ' P (R",R) and we have
A((ov«)Al) = R '
*»<«<\
OXJ
otherwise
[ 0,
B. Moreover, for u 6 W1,p(Rn,R) W^1-P(Rn,R), and
-* („-!)+ = / £ ' K
dxj
|o,
ifU>
\
(OVw)Al
(2.38)
the function (u — 1) + is an element of
(2.39)
ifu<0
holds. Proof: A. The function 11-> f(t) = t + | t | + 2 ~ | t + | t | ~ 2 1 fulfills the requirements of Theorem 2.1.17, but f(t) = (0 V t) A 1. B. Since t \-^> g(t) := t - 1 "^ satisfies the assumptions of Theorem 2.1.17 and (t — 1) + = g(t), we may apply once more Theorem 2.1.17. • In Section 1.4.6 we introduce the concept of a Dirichlet operator in the space L (R",R). Our next aim is to determine classes of second order differential P
30
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
operators which might have a closure being a Dirichlet operator. For this let us consider the operator n
L(X,D)= k,l=l Y, M
x
n d2 d )dxk ^ 7 7dxi^ + & ( x ) ^ r :J+ c ( a 0
*
*
(2.40)
j=l
with coefficients a^, bj,cG C&(Rn, R). Lemma 2.1.19. Let L(x, D) be as in (2.40) with akl, bj,c€ Cb(Rn, R). Then L(x,D) maps W2'p(Rn, R) continuously to Lp{Rn, R), 1 < p < oo, and it maps CI^)(R", R) continuously to Coo(Rn, R). Moreover we have the estimates \\L(x,D)u\\LP
(2.41)
and (2.42)
||L(a;,£>)u||oo < c\\u\\c • n
Proof: It is sufficient to prove (2.41) and (2.42) for all u e C£°(R ,R). For 1 < p < oo we find \\L{x,D)u\\
LP
< 5 3 ll°wlloo k,l=l <
d2u dxk dxi
LP
+ £IMI< j=\
du dxj
LP
+ ||c||oo||w||tP
C||xt||v^2.p,
and (2.42) follows analogously.
•
Now, let us consider L(x,D) as operator on LP(R™, R), 1 < p < oo, with domain W2
f
(L(x,JD)W)((u-l)+)p-1dx<0
for all u S W2'p(Rn, get
R). First we observe that for c(x) < 0 for all x £ Rn, we
f c(x)u(x)((u(x)-l)+)p-1dx
= f
JR"
c(x)w(x)(w(a;)-l) p - 1 dx < 0.
Ju>l
Next we find by Corollary 2.1.18.B that
9u dxj
: ( ( W -m
p
-= X { u > 1 } fci)(.-i r l 1 d(u - l ) p = X{u>i}p
dxj
'
2.1 Second Order Elliptic Differential Operators
31
and therefore we have for bj £ C^M", E) and u £ W 2 ' P (E", E)
f
JRn
bj(x)^loxiu{x)_1)+r-idx
=
_I
j
Thus, if E | ^ - > 0, we have for u £ W2'p(Rn,
I
du(x) bj{x)^^{{u(x)~l)+Y-1dx
ox
j
E)
< 0. ~
OXj
.
\u{x)_l)PJLh.{x)dx,
Ju>l P
Finally we get for akl £ C?(R n , E)
/
S««(x)gg((»-irwr'^
JRK" £j=x
oxk oxi
=[i{»i}(*-i)((«-i)(>r' l
kC\ S i
k
X„>1,M^'(I)9("':1)W)V,
where we used once again Corollary 2.1.18.B. Now it follows T h e o r e m 2.1.20. Suppose that L(x,D) has coefficients aki £ C^(R n , E), bj £ C£(R n , R) and c £ C b (R",R) such that c < 0, aki = aik and £ £ i=i aki(x)£k£i > 0 for all x £ E n and £ £ E n . /fin addition we assume that £7-i ^ J ( W 1 < /> < oo, /
- £Li % ? )
> 0- then we have for all u £ W2
(L(x,D)u(x))((u-l)+(x))p~~ldx<0.
(2.43)
Note that we need only the regularity assumptions aki £ C^(E n ; E) and b i - £ L i fe e C ^ R " ; R). This leads immediately to
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
32
Corollary 2.1.21. Suppose that L(x,D) is given by L{x,D)u(x)=±^lKaH(x)^y±bj(X)^+^)u(x),
with aM = o;fc e 0 for alx e R n and £ £ R n , and c{x) < 0. T/ien (2.43) holds provided that Y^=i" STJ — 0For p = 2we may weaken the sign condition on
3
j=i
k=\
in the following way. Consider the form
B(u,v)=Y, J^W +£ j=
/
l
(biix) - £
JRn
v
Qxi
dxk
dx
%ux ^ ) ^x
fc=1
k
'
«(*) dx+f
Vj
c{x)u{x)v{x) dx. ,/R"
Suppose that B satisfies the sector condition, i.e. (1.4.333) in Definition 1.4.7.12. If we add to h(x) := c(x) - £ ? = 1 gf- (b3{x) - £ L i % ^ ) a function x t-* d(x) such that /i(x) + d(x) > 0, we find by Theorem 1.4.7.20 that the contraction condition (1.4.347) is fulfilled, compare Example 1.4.7.35. Thus B(-, •) + (d(-)-, -)o extends to a semi-Dirichlet form with domain Hl(Hn, M) = W1''2(Rn, R) and L(x, D) — d(x) • id extends to a Dirichlet operator. In order to identify (L(x,D), W2'p(Rn, R)) as a Dirichlet operator on Lp(Rn, R), 1 < p < oo, we have to prove closedness. Thus we have to show that || • \\W2
D)U\\LP
+
\\U\\LP)
•
(2.44)
This leads to the concept of ellipticity. Definition 2.1.22. Let L(x,D) be given by (2.40) with coefficients aki = &lk £ Cfc(Rn; R). We call L{x, D) uniformly elliptic if for some Ao > 0 n 2
Ao|£| < £ fc,;=i
akl(xMi
(2.45)
2.1 Second Order Elliptic Differential Operators
33
holds for all x £ Rn and £ 6 Rn. We will prove (2.44) for uniformly elliptic differential operator of second order. Our presentation follows mainly the paper [37] of F.E.Browder, we refer also to the papers [2], [3] of S.Agmon, A.Douglis and L.Nirenberg, as well as to the work [252] of Chr.Simader. Let us start with some preparatory material. Given e > 0 and let (V'Ofcez™ be the partition of unity from the second part of Lemma 1.2.1.3.B. The family (V4)/cez™, I > 0, defined by #(*):=
tf(£Z*)
(2.46)
fulfills diam(suppV>L)< {l+e)l-n1/2, \daij}lk{x)\ < ( | ) ~ a
M
'
H<m
(2.47) and M = M(n,m)
(2.48)
and £ ( 4 ( * ) ) 2 = 1. fceZ"
(2.49)
Moreover, there exists N = N(e, n, I) such that the intersection of N + l of the supports of the functions iplk is empty. In fact, we may choose N independent of I. L e m m a 2.1.23. Given u £ Wm'P(M.n) and for e > 0, / > 0 let (Vi)fceZ" &e given by (2.46). Then we have for all j G No, j < m,
fceZ" and
£ ii(vi)3*i,P < ^ £ (|)~°~ Jl)p \\urwil,p < oo. Proof: We prove first (2.51). For this observe that
E iiWfc)2*;.- = £ £ / H(4)2«) fe|<JC
p
|fc|<^"|a|
E
E
|fc|<JC | a | < j |/3|+| 7 | + | « | = | a | '
/" i^Ln^in^v^.
(2.5i)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
34
where we used
c^s{d^lk){d^lk){d6u)^
Y l/3|+l7l + l<5| = l«|
Y
i^in^n^r-
I/3|+|7|+I*I=I«|
Since supp d^iplk C supp ip\, it follows from the conditions on the supports that
Y |aMll^Ll<(^+i)M2(f)H/5H71, \k\
implying that Co
E
E
/ E l^il'WiHaV**
±nyu-jl)p[ Ji=0
Y i«v«k"K
|«|<j
In particular we have that
E UWi)2*!* I*I
has a finite bound independent of if, hence as K —> oo it converges to a finite limit and (2.51) is proved. Now (2.50) follows easily since
/
\dau\"dx= f
Jut"
\ff*YWk)2u\Pdx>
JM-
fc€Z„
1
and for each x e W at most JV + 1 terms of the integral on the right-hand side are different from zero. Hence the integral is less than 2 ( W )(P-I)
£ kezn
r is-((^)»„)p4fa<2^+i)&-i)5:ii(^)»«|iu.
K
"
fc€Z"
Adding up over all a e N J , |CK| < m, gives (2.50). D We will prove (2.44) for uniformly elliptic operator in several steps. First we observe
2.1 Second Order Elliptic Differential Operators
35
Proposition 2.1.24. Suppose that for the second order differential operator Lpr(x,D) = Y2ia\=2aa(x)Da we have the estimate \M\w2-p < cp(\\Lpr(x,D)u\\LP
+ \\u\\LP)
(2.52)
0
for all u £ W2'P(G), G c E " being an open set. Suppose further that bj £ Cb(G), 1 < j < n, and c £ Cf,(G) are given functions. Then the differential operator p.
n
L(x,D)
:= Lpr(x, D) + ^bj(x)
—
+c(x)
satisfies with a suitable constant cp > 0 IMIw^.p < Cp (\\L(x,D)u VII
(2.53)
+\\U\\LP)
/
LP
forallueW2'P(G). Proof: For u £ C£°(G) (or u £ W 2'P(G)) we find with rj > 0 n
\\Lpr(x,D)u\\Lp
< \\L(X,D)U\\LP
+
„
T^b^-)-^-
+ \\C(-)U\\LP
<\\L(x,D)u\\LP+io\\u\\wi.r < \\L(x,D)u\\LP + r)\\u\\W2,P + ji(ri)\\u\\LP, where we used Corollary 2.1.8. But now the proposition follows immediately from (2.52). • Theorem 2.1.25. Let L(D)u =
^2 aaDau
be a uniformly elliptic operator
with constant coefficients aa 6 R. Then we have for all u £ W2,p(M.n)
( J2 \\D*u\\lpy/P < C\\L(D)U\\LP.
(2.54)
|a|=2
Proof: It is sufficient to prove (2.54) for all v £ C£°(R ra ). For ip £ C%°(Rn) we find
(L{D)
36
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
Since £<*<£(£) = ( D » A ( £ ) we find for |a| = 2
The polynomial £ i—• L(£) is homogeneous of degree 2 and by assumption uniformly elliptic. Thus for £ =£ 0 we find
*{m)\s<M-» •£«) for some CQ = co(n, Ao), Ao being the ellipticity constant and \a\ = 2. Next we define
thus |m(£)| < # for all £ € M". Since £(£) e Cg°(R n ), we find that | ( L ( D V ) A ( 0 | < ( l + l^|2)-("+1)/2 and the Fourier inversion formula gives D X * ) = (2vr)-"/ 2 / = (2TT)-"/
2
JR"
efa-«-il(L(I>)v)A(Ode Ms)
/
e»-« m(£) (L(D)v>)A ( 0 d£.
Thus by the Michlin-Hdrmander multiplier theorem, Theorem 1.3.12.3 (in the form of Corollary 1.3.12.4) we have ||D
= ||F- 1 (m(0(L(I>) V ) A )||Lp <
\\L(D)V\\Lr,
which implies (2.54). D Remark 2.1.26. A. Clearly (2.54) implies (2.44) in case of constant coefficients. We only have to take into account (2.17). B. Since Da and L(D) are local operators, we find with constants c p and c'p independent of G C R n , G open, the estimates ( £
p>||^) ^
< cp\\L(D)u\\LP
(2.55)
2.1 Second Order Elliptic Differential Operators
37
and \\
<4,(\\L(D)
(2.56)
for all
Lemma 2.1.27. Let L(x,D)
( c ^ s u p V |a Q (z)-a a (zo)| p < (\Xrf with some n < 2~N(p~1^p, where N = {a£NJi
n( n+
-
\a\ = 2}. Then we have for alluGW
\\u\\w2,p <
(2.57) ' is the cardinality of the set 2P
' (G) (2.58)
2 ^ , ( | | L ( X , D ) « | | L P + ||U||LP).
Proof: It is sufficient to prove that (2.58) holds for all u G C$°(G). Moreover, by (2.56) it follows that IMIw^.p < c'p(\\L(x0,D)u\\Lp < c'p(\\L(xo,D)u\\Lt,
+
\\U\\LP)
+ \\(L(x,D)-L(x0,D))u\\LP
+ \\u\\LP)
and it is suffices to prove that (2.57) implies (2.58) which will follow from cp\\(L(x,D)-L(x0,D))u\]LP
< \\\u\\W2,P.
(2.59)
But for u G C£°(G) we have (cp)p\\(L(x,D)-L(x0,D))u\\pLP = {c'p)p f I V {aa(x) JG '' l.",l = 2„
aa{x0))Dau(x)\Pdx '
< (c^f 2 i ^ - 1 > ( s u p ] T M * ) - ««(*o)| p ) Y J l6G |a|=2 l«l=2 ,/G 2/
\Dau(x)\pdx
n-iiw2.pi
where we used in the final step (2.57).
•
To proceed further, we need the following commutator estimate:
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
38
= ^2 o-a{x)Da be a differential operator with
Lemma 2.1.28. Let L(x,D)
|a|=2
coefficients aa G Cb(Rn).
For any
\\[
(2.60)
Proof: Observe that Up, ^2aa(x)Da
u=
[
^2
|a|=2
H=2
^aa{x)[
= \a\=2
and the lemma follows from Corollary 2.1.9. Theorem 2.1.29. Let L(x,D) = £) aa(x)Da
be a uniformly elliptic differ-
l«l=2
ential operator with coefficients aa G Cb(M.n, R). Further let G C Rn be a o
bounded open set. Then we have for all u £ W2'P(G) \\U\\W2,P<\J\\L(X,D)U
(2-61)
+\M\LP), LP
Ml
I
where Xp = Xp(G, aa). Proof: Once again, it is sufficient to prove (2.61) for all u G C(f(G). On G + B\ (0) the coefficients aa are uniformly continuous and therefore we may find a finite open covering I £/,- I of G such that J
V
(c'pr max
sup
/j
£
=
l,-,M
\aa(x)-aa(y)\" < ( i ) V ^ - 1 )
(2.62)
where c'p is as in (2.56) and as before we take N = n ' n 2 + '. We know by Lemma 1.2.1.3. B that there exists a partition of unity subordinated to (£7j)j=i,...,Af > i.e. there are functions tpj G Co°(R") such that 0 < tpj < 1, supptpj C t/,-, and M
^2
<Mp(\\Hx,D)(VjU)\\LP(Gr\Uj)
+
\\
which leads to IIVjw||wa.p(onUj) ^
2c
p(ll-C'(a;>-D)(¥'jw)IU>'(G)+lkju||Lp(G))-
(2.63)
2.1 Second Order Elliptic Differential Operators
39
Now it follows for u e C£°(G) by Lemma 2.1.28 that .
.
M
iMi^.P(G)= E / \D u\*>dx= E / ( j > ) V « i p < k <
a
\a\<2JG \oc\<2J° ~ M ^2M(P-X) / Y^\'PjDau\pdx G |a|<2 -' j=l
G
l«l<2
J'=1 .
< 2(M+2)(P-1) £
M
M
/• ( £ 1 0 - ^ tt)|P + £| [v? ., j ^ f )
|a|<2
i
j=l
•/G
J= l
dx
j=l
M
M
.
= 2(^»>o- )(x;iiw«H^ (Gni , i) +E E /
j=l |a|<2l/t?inG
J= l
Ife* 0>r«fe)
M
c
< (Ell^'ullwra-»(Gn^)+ Hl^co)(2.64) Further we find using (2.63) and Lemma 2.1.28 < 2c'p(\\L(x,D)(
+ ||«|| L , ( G ))
< 2 c ; ( | | ^ i ( a ; , £ » ) U | | L P ( G ) + ||[^,L(x,£>)]u|| L P ( G ) + | M U P ( G ) ) <2c'p\\L(x,D)u\\LP(G)+c\\u\\Wi,P(G)+2c'p\\u\\LHG) <
2cp\\L(x,D)u\\LHG)+r)\\u\\Wi,P{G)+c(r),p)\\u\\LP{G),
where we used Corollary 2.1.8 for 7? > 0 arbitrarily given. Thus we find M
EH^IvrMGrWi)
+
llttHwa-(G)
< 7(||Z(x, 2?)«||^(G) + 2 ^ | | « | | ^ . P ( G ) + HI£P(G)) • Taking 277jp = \ we finally arrive at I N I ^ , P ( G ) < 27(||£(x > Z7)«||* F(G) + | | « | | ^ ( G ) ) , which of course implies (2.61).
•
Chapter 2
40
Generators of Feller and Sub-Markovian Semigroups
It was crucial in the proof of Theorem 2.1.29 that we could work with a finite covering of G. If the coefficients aa are Lipshitz continuous, i.e. max.\aa(x)—aa(y)\
< n\x — y\
(2.65)
|a|=2
if follows that M in the proof of Theorem 2.1.29 depends only on diam(G) and K. Thus we have Corollary 2.1.30. Suppose that L(x,D)
]T} aa(x)Da
=
is a uniformly el-
l«l=2
liptic differential operator with coefficients aa € Cb(M.n, M) satisfying (2.65). Then there exists a constant fi = fi(p, K, R) such that \\u\\w'.p(G)
+ \\U\\PLP(G))
(2.66)
holds for all u € W2'P(G) with diam(G) < R. Now we will use the partition of unity considered in Lemma 2.1.23 and Corollary 2.1.30 to prove (2.44) for G = R n and L(x, D) = £ aa{x)Da with l"l=2
Lipschitz continuous coefficients. Theorem 2.1.31. Let L(x,D)
=
^ aa(x)Da
is a uniformly elliptic differ-
|a|=2
ential operator with coefficients aa £ Ct(R n , R) satisfying (2.65). Then we have for all u e W2
(2.67)
Proof: We may take the partition of unity from Lemma 2.1.23 with e = \ and we denote it by (tpk)keZn- Further we put Qk := suppV'fc. Note that diam(Qfc) < fn 1 / 2 . From Corollary 2.1.30 it follows now for all k £ Zn and u e W2>P{Rn) that HuV'fcll^.pCG) < ^(\\L(x,D)(uipl)\\LP
+ \\u\\Lp),
where \x depends only on p, L(x,D), and n. Now using Lemma 2.1.23 and intermediate estimates for the norm || • ||WI.P we find with rj > 0 sufficiently
2.1 Second Order Elliptic Differential Operators
41
small using the support properties of ipk IIUIIW2.P(G) — C Z ^ IIV'fculliV2.P
fceZ" < c'( Y, \\L(x,DMu)\\lP < c"( £
+ £
UU\PLP)
U V ^ K i?)tt||Jp + | | « | | ^ . , ( C ) + \\u\\lP)
fceZ"
which yields (2.67). D Combining Theorem 2.1.31 with Proposition 2.1.24 we finally arrive at Theorem 2.1.32. Let L(x, £>) = £ akl(x)^-
+£"
b^x)^
k,i=i
+ c(x) be 3
a uniformly elliptic operator with coefficients aki = aik, aki, bj, c e Cb(R n , R), such that max
\aki{x)-aki(y)\
l
Then (2.44) holds for all u €
W2'P(Rn).
Our next aim is to use (2.44) in order to prove that for some A > 0 the equation (\-L(x,D))u=f
(2.68)
has for all / € L p (R n ) a unique solution in W2
Clal(Rn, /
R). The operator L*(x, D) is defined by the relation (L(x,D)
f
)dx
(2.69)
42
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
for all ip, ip G C£°(R", E), i.e. (-l) | Q | d(a„(x)V(x))
L*(x,D)xP{x) = £ H£2
(2.70)
= £<(*)c^(z). I"l<2 Note that if L(x, D) is elliptic with principal part Lpr(x,D) = ^2
aa(x)da,
\a\=2
then L*(x, D) is also elliptic with the same principal part, in this case we have o„ = a,ki = aik for a = ek+£i by our definition of ellipticity. Definition 2.1.33. A. Let L(x, D) = J2 Oa(x)9 a be a second order (elliptic) H<2 differential operator with coefficients aa G Clba\Rn, R). Given / G Lp(Rn), 1 < p < oo. We call u G W2'p(Rn) an V-solution of (2.68) if this equality holds in the sence of the space L p (R n ). B. We call u G Lp(Rn) a weak V-solution of (2.68) if /
uL*(x,D)
J
f
holds for all 1).
(2.71) for ± + i = 1, p
p
Corollary 2.1.34. Every Lr'-solution of (2.68) is also a weak Lp'-solution of (2.68). To establish the existence of a weak L p -solution for A — L(x, D) we need a further estimate. Theorem 2.1.35. Let L(x,D) R>0 and c\, > 0 such that
be as in Theorem 2.1.32. Then there exists
M\u\\LP R
and u G
W2'p(Rn).
(2.72)
2.1 Second Order Elliptic Differential Operators
43
Proof: We consider on W2'P(Rn+1) the operator L(x, D) + Jf^. Since this operator as a differential operator acting on functions defined on R n + 1 fulfills the assumptions of Theorem 2.1.32 we have for all v G W2'p(Rn+1) \\v\\w*.r(M~+i)
LP(K7I+I)+IMILP(K"+I))-
(2-73)
We choose C S C°°(M) such that 0 < £ < 1, C|[_i i, = 1 and (L^ 1]c = 0. Further let r > 0 and u G W2'P(M.n). The function v(x,y) := ((y)eiryu(x) 2 n+1 belong to W 'P(M. ). First we observe that y) = u(x) (e
-^v(x,
- r 2 e ir »C(tf)),
which implies by our assumptions on £ f\ay)eiry\P\u(x)\Pdydx<2\
-i+i) = /
I£P(R"
LP(R")'
and IW2.P(R'
"/K • /_ T |a|<2 = / 7R»
/ ( ( l + r p + r 2 P ) | u | P + ( l + 2rP) J ^
2^ la,. i=i
+
_2-, la^aj J fc,(=i
>r 2p H|*L P ( R " ) + r l V
ILP(R")
+ ||V22„,IIP u Z-P(R")'
In addition we find d2v dy2
£eirL(x,
LP(R"+ 1 )
D)u - eir-u(r2{ - 2ir? - C")
< ||C(i(^,-D)w-^ 2 «)l|Lp(R"+i) + ||(2irC' + O l U p ( R " + i ) < 2||L(i, -D)u - r2u||z,p(K") + 2r||C'||LP(E") IMILP(R») + IIC"I|LP(R»)II U IILP(R")-
Chapter 2
44
Generators of Feller and Sub-Markovian Semigroups
Using (2.73) we get now ^IMUpdR") +r||Vw|| iP (K") + ||V 2 u||ip(H n ) < 3|H|VK2.P(R~+I) d2v dy
/ll VII
<3cp[\\L(x,D)v + —,l
LP(K"+1)
^
>J
2
< cp(\\L(x, D)u - r u\\LP(U") + (1 + r)||u||z,P(K")). With A = -y/r and r such large that c p (l + r) < ^- we find <MMILP(R™) +^ 1 / , 2 ||VM||LP(R«)
< 2c p ||Au-
+ HV^HiP^)
(2.74)
L(x,D)u\\LP(un~),
which gives in particular for A sufficiently large (2.72).
•
Corollary 2.1.36. Let L(x,D) be as in Theorem 2.1.32 and A sufficiently large. Then the equation (2.68) admits at most one LP -solution. Proof: Let u,v £ W2'P(Rn) be two solutions of (2.68) for some / e L P (M"). By Theorem 2.1.35 we find A||u - u||iy2,p(R„) < cp\\X(u - v) - L(x, D){u - I>)||LP(R") = cp\\Xu - L(x, D)u - (Xv - L(x, D)U)||LP(R") = CP||/-/||LP(R")=0.
D
Corollary 2.1.37. Let L(x, D) be as in Theorem 2.1.32 with coefficients aki = atk e C2(Rn,R), bj e C£(Rn,R) andc£ Cb(Rn,R). Then Theorem 2.1.32 and Theorem 2.1.35 also apply to L*(x,D). Next we give a necessary condition for the existence of a weak L p -solution to (2.68). Proposition 2.1.38. A necessary condition for (2.68) to have a weak IPsolution is that
J
fipdx
R"
JR
is valid for all ip 6 C£°(R n ) (or equivalently for all ip G W2>P'(Rn)).
(2.75)
2.1
Second Order Elliptic Differential Operators
45
Proof: By definition of a weak ZAsolution we have /
fipdx
—
/
u(L*(x, D) — \)
<\\u\\LP\\(L*(x,D)-X)
D
We prove now that (2.75) is a sufficient criterion for the existence of a weak L p -solution to (2.68). Theorem 2.1.39. Let L{x,D) = Si a |< 2 a c«(a;)9 a be a second order differential operator with coefficients aa £ Cj (R n ;R) and let L*(x,D) be defined by (2.69). Moreover, suppose that for some f € L P (R") we have (2.75). Then the equation Xu — L(x, D)u = f admits a weak Lp-solution. Proof: Consider the set Wx := {w e Lp (R n ); there exists
w -> Fxw
ffdx,
where
f(
In addition we have
\Fxw\
f
f
Hence FA is a continuous linear functional on Wx- By the^Hahn-Banach Theorem, Theorem 1.2.7.4, we may extend Fx to a continuous linear functional
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
46
on £ p '(R n ) which we denote once again by F\. For this reason we find some u G Lp{Rn) such that F\w = I
uw dx
holds for all w G L p '(R n ). For y> G C§°(R n ) we know that L*(x,D)
u(L*(x,D)ip — Xtp)dx=
/
fipdx,
which shows that u G L p (R n ) is a weak L p -solution of the equation L(x, D)u —
\u = f.
D
Corollary 2.1.40. Suppose that L(x,D) satisfies the assumptions of Theorem 2.1.32 with coefficients aki = aik e C%(R.n;R), bj G C j ( R n ; R ) and c £ C(,(R";R). In particular, L(x,D) is uniformly elliptic. Then there exists Ao > 0 such that for A > Ao equation (2.68) has for all f £ L p (R n ) a unique weak Lp -solution, 1 < p < oo. Proof: According to Corollary 2.1.37 we have for some Ao > 0 the estimate \MW*,P>
for all 9 e W2'P'(Rn) and A > A0. Given / G L p (M n ) we find for A > A0 /
f
and the corollary follows.
•
To proceed further we need the following regularity result which we will not prove, for a proof we refer to F.Browder [37]. Theorem 2.1.41. Let L(x, D) be a uniformly elliptic differential operator with coefficients akl = alk e C ^ + m ( R n ; R ) , bj G C ^ + 1 ( R " ; R ) and c e C ^ ( R n ; R ) for some m G No. Further let u G LP(R), 1 < p < oo, a (unique) weak Vsolution to the equation L(x,D)u-\u = / , A > A0. / / / G T^ m ' p (R";R), then u G Wm+2'P(Rn;M.) and the estimate ||«||wm+2,J> < C ( | | / | | w m , p + | | u | | j r , p )
holds.
(2-76)
2.1 Second Order Elliptic Differential Operators
47
Taking Theorem 2.1.41 for granted we have Theorem 2.1.42. Let L(x, D) be a uniformly elliptic differential operator with coefficients aki = alk 6 C£(R n ;R), bj G C£(R n ;R) and c G C b (R n ;R). Further let A > AQ sufficiently large. Then the operator L(x, D) — A and L*(x, D) — X are continuous and bijective operators from W 2 , p (R n ;R) to Z p (M n ;R) with continuous inverse. If in addition aki,bj, c G C™(Rn; R), m G No, then L(x, D) and L*(x,D) map Wm+2>p(Rn;R) continuously into Wm
< £
(a)||Sfird||oo||^+a-7«||Lp.
D
Next we will prove that L(x, D) extends to a generator of a Feller semigroup IT}00' ) as well as to a generator of an L p -sub-Markovian semigroup V /t>o ITf) for 1 < p < oo. First we turn our interest to the Feller semigroup. \ /t>o For this let L(x, D) be a uniformly elliptic differential operator of second order with appropriate coefficients. From Theorem 2.1.41 we know that for A > Ao the equation L(x,D)u — Xu = / has for all / G W 1 ' p (R n ;R), 1 < p < oo, a unique solution u G .W 3 > P (R";R). Further, by Theorem 2.1.42 we have for all u G W3>P(Rn; R) the estimate \\L(x,D)u\\Wi,P
Let us suppose that p > n. Sobolev's embedding theorem, Theorem 2.1.12, yields now that (L(x, D), W 3 ' P (R";R)), is a densely defined operator on Coo(Rn; R) and it is clear that L(x, D) satisfied on W 3 ' P (R"; R) C C£,(R n ; R) also the positive maximum principle provided that c(x) < 0. Thus we see that (L(x, D),W3'p(Rn;R)) satisfies all requirements of the Hille-Yosida-Ray theorem, Theorem 1.4.5.3. Theorem 2.1.43. Let L(x, D) be a uniformly elliptic differential operator with coefficientsakl = aik G C3(Rn;R), bj G C 2 (R";R) andc£ C£(Rn;R). Further assume that c(x) < 0 for all x £ Rn. Then the operator (L(x, D),W3 n, extends to a generator of a Feller semigroup.
48
Chapter 2 Generators of Feller and Sub-Markovian Semigroups In case of L p -sub-Markovian semigroups we find
Theorem 2.1.44. Let L(x,D) be a uniformly elliptic differential operator with coefficients akt = aik e C£(R";R), bj S cf(Rn;R) and c € Cb(M.n;R). In addition suppose that c(x) < 0 and
Then (L(x, D), W2'p(M.n; R)) is a generator of an Lp-sub-Markovian for 1 < p < oo.
semigroup
Proof: First note that Lemma 2.1.19 and Theorem 2.1.31 implies that (L(x,D),W2>p(M.n;R)) is a closed operator in L p (R n ;R), 1 < p < oo. Moreover, Theorem 2.1.20 yields that the operator L P (R"; R) is a Dirichlet operator, in particular, it is dissipative according to Proposition 1.4.6.12. Thus it remains to prove that for some A > 0 we have R(L(x, D) - A) = L P (R"; R), which is however the content of Theorem 2.1.44. •
2.2
Some Second Order Differential Operators with Non-Negative Characteristic Form as Generators of Sub-Markovian Semigroups
In this section we want briefly discuss some second order differential operators with non-negative characteristic form which are not elliptic but still have an extension to a generators of a sub-Markovian semigroup. We restrict our attention mainly to the Hilbert space case, i.e. to L 2 -sub-Markovian semigroups, and we will use the result from Section 1.4.7, i.e. the theory of Dirichlet forms, to construct these semigroups. However, since in this monograph we are mainly concerned with pseudodifferential operators which are not differential operators, we do not prove all results in this section. We start with symmetric operators in divergence form. Let
L D
^ ^ti:
(«<•>£)
own
2.2 Second Order Differential Operators
49
have symmetric coefficients aki = aik s C£(R n ;R) and assume n
^2
a
ki(x)^i > 0
(2.78)
k,i=i
for all i g R " . The operator is considered first as an operator on L 2 (M n ;R) with domain C £ ° ( ! n ; R ) . On C£°(M n ;R) we associate with L(x,D) the following form B{u,v):=[ ±akl(X)^l^ldx dx JR" QZX i
(2.79)
dx
*
which is symmetric and non-negative, i.e B(u,v) = B(v,u) for all u, v £
CQ°(R™;
and
B(u,u)
>0
R). Moreover we have the estimates
||L(x,D)u|| 0
u£f2'2(l";l),
(2.80)
and \B{u,v)\
u , u G W 1 , 2 (R n ;R),
(2.81)
implying that L(x,D) extends to a continuous operator from W2'2(M.n;M) to L 2 (R";R) and B extends to a continuous bilinear form B : W 1 , 2 (R n ;R) x W1'2(Rn;R) -> R. By Corollary 2.1.21 it follows that /
L(x,D)u{x)(u-l)+(x)dx<0
(2.82)
for all u e W2'2(Rn;R). Thus L(x,D) is a candidate for a Dirichlet opera2 n tor in L (R ;R), see Definition 1.4.6.7. In fact, Remark I.4.6.13.A yields that (L(x,D),C%°(Rn;R)) is closable and (2.82) still holds for the closure. Note that also the operator (L(x, D), W 2 ' 2 (R n ;R)) is closable and and its closure coincides with that of (L(x, D), C£°(R"; R)). It is convenient to denote this closure of L(x, D) by {—A, D{A)). Moreover, the symmetry of B implies estimate (1.4.324), i.e. \B(u,v)\ <
ciB^utf/^B^v))1/2,
(2.83)
50
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
where as usual Bi(u, v) stands for B(u, v)+(u, V)Q. In addition, since B{u, u) > 0, we find MlKB^u),
(2.84)
i.e. estimate (1.4.323) holds. Corollary 1.4.7.14 implies that (B,C§°(Rn;R)) 12 n and (B,W ' (W ;W)) are closable and have the same closure. Let us denote this closure again by B, its domain by D{B). Thus we have B(u,v) = (-Au,v)0
(2.85)
for all u G D(A) and v e D(B), and we have D{A) c D(B). Now combining Theorem 1.4.7.10 with Theorem 1.4.7.23 we obtain Theorem 2.2.1. Let L(x,D) = £ £ > l = 1 ^ ("fetO)^) be a second order differential operator with coefficients aki = aik S C\ (R™; R) such that Ylik l=i aki£k£i > 0 holds for all x £ R™ and £ € R". Consider L{x,D) as operator on L 2 (R n ;R) with domain C£°(R";R) (or W2'2(M.n;R)). Then it is closable, its closure is a selfadjoint Dirichlet operator which is the generator of a symmetric sub-Markovian semigroup (T t (2) ) t >o on L 2 (R";R) and therefore it generates a symmetric Dirichlet form (B, D{B)). Remark 2.2.2. In case of an elliptic operator we may characterize the domains D(A) and D(B), namely D(A) = W2<2{Rn;R) and D(B) = W^2(Rn;R). The first result follows from the estimate given in Section 2.1, especially from (2.39) combined with Theorem 2.1.32. Whereas the second result follows from (2.81) together with the observation that Y, [ akl(x)^l^-dx>X0[ \Vu(x)\2dx, j^i-V dxi dxk yR„' where Ao > 0 is the ellipticity constant, i.e. n
J2 °w(*)&& > A0|£|2. fc,i=i
However, for a general operator L(x, D) as in Theorem 2.2.1 we do not know explicitly D{A) or D(B). We only know the two inclusions W2'2(Rn;R)
c D(A) C L 2 (R n ;R)
(2.86)
2.2 Second Order Differential Operators
51
and Wh2(Rn;R)
c D(B) c L2
(2.87)
Later in this section we will discuss the question to improve (2.86) or (2.87) in order to obtain inclusions of the type D(A) C # 2 s ( R n ; M ) or D(B) C H3(Rn; R), respectively, for some s > 0. In a next step we want to add a drift term to the operator L(x, D) , i.e. a first order differential operator. Thus we consider the operator
*-»-t
£;(«*•)£)+£*•)£;•
(2.88)
where we assume that Y^k,i=i dx~ ( a f c '(')&r) satisfies the assumptions of Theorem 2.2.1. In order that L(x, D) extends to a Dirichlet operator, we assume according to Corollary 2.1.21 that bj G C£(R n ;R) and that
ydbj(x)>
(2.89)
holds for all x G R n . The linear form associated with L(x, D) is now given on C§°(Rn; by E(u,v) = Y]
/
aki(x)
du(x) dxi
dv(x) dx dxk
n
.
+J2 / hAx
du(x) dxj
v(x) dx.
(2.90) In order to apply results from Section 1.4.7 we need the sector condition to hold for E. Denoting by B the symmetric part of the form E, i.e. B(u,v):=E^(utv)=±
akl(X)d-^ld-^ldx,
f kj^i ^ K "
l
k
we are longing for the estimate
I E / bAx) ^Lv{x)dx dx
^ciBxi^uf^iB^v^Y'2
(2.91)
for all u,v G C7g°(Rn;R) (or H^ 1 ' 2 (R";R)). Clearly, for an elliptic operator (2.91) follows easily since I E /
bj(x)^-v(x)dx\<J2\Moo
du dx.
P o,
(2.92)
52
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
and ||^f:||o < ||w||wn,2, ||w||0 < ||v||nn,2. In case of non-elliptic operator we have to examine the non-symmetric part of E, i.e. the linear form E/
bj(x)^-v(x)dx,
(2.93)
more carefully. For this we recall Lemma 1.4.7.34 where we proved with tpj (£) =
u
j = 1
ax
JUL"-
j
/2
<EllML (lNlL +
dbj
Ox,
(2-94)
i/4
ll«ll^-,l/2||^IIVj,l/2-
Now let
^ - t : £(«<•>£ be a linear differential operator as in Theorem 2.2.1. In addition we assume for some A0 > 0 that
^•o^Elll^ 2
(2.95)
J£J
where J C { 1 , . . . , n}. We set (2.96) and find Bi(u,u)>(XoAi)\\u\\i,Jtl.
(2.97)
Since
we see immediately that l|w|lvi,i/2 < II«IIVJ,I-
Thus we obtain
(2.98)
2.2 Second Order Differential Operators
53
Theorem 2.2.3. Suppose that L(x,D) is as in Theorem 2.2.1 and satisfies (2.97). In addition let bj 6 Cl(Rn;R), j e J, be given functions such that £ € j Q^{X) ^ 0 for aM x £M.n. Then the bilinear form E(u,v)=±
f
^
a
A^JE"
)
^ M ^ dxi dxk
+
W f^jh-
3K
b3(x)^v(x)dx dxj
(2.99) originally defined on Co°(R ;R) is closable and its closure is a semi-Dirichlet form. Thus the operator L(x,D) + J2jeJ^j{x) 9% has a closure which is a Dirichlet operator and it generates a sub-Markovian semigroup on L 2 (R n ;R). n
Proof: It remains only to prove that E satisfies the sector condition. But from (2.94) we deduce using (2.97) b x
j( )^r-v(x)dx
S /
°xi
•j-ej./R"
j e J
< c'lMl^ilMl^.i <
ciBi^u^iB^v))1'2,
and the theorem is proved. D Example 2.2.4. Consider a matrix (aki)k,i=i,...,n with aki — aik 6 C j ( R n ; R ) having the property that for some 0 < m < n it follows m
m
k,l=l
j=l
for all i £ l " and all £ £ R m . Moreover we assume that aki = 0 if k > m and / < m, which implies by symmetry that aki = 0 for k < m and I > m, and further we require that n
y~]
aki(x)€k£t > 0
for all x € R n and all £ £ R " - m . a
x
£jb,i=i 5§T ( *i( )-£;)
In this case we have for the operator
estimate (2.95) with J = { 1 , . . . , m } .
In Theorem 2.2.3 we face the situation that the drift acts only in directions where L(x,D) is elliptic. In order to treat more general example we need
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
54
Definition 2.2.5. Let L(x,D) = J2k,i=i af^ \aki{x)-^\ be a second order differential operator with non-negative characteristic form, i.e. aki — aj/t and 12ki=iaki(x)£k£i > 0 for all x € Rn and all £ £ R n . Further assume akl e A. We say that L(x, D) satisfies a sub-elliptic estimate in the strict sense if for some s £ (0,1] and all compact sets K dW1 the estimate \\ufe < cK ((L(x, D)u, u)0 + \\u\\l)
(2.100)
holds for all u € C^(Rn), suppw C K. B. We call L(x, D) to satisfy a uniformly sub-elliptic estimate if for some e e (0,1] \\u\\2e
(2.101)
holds for all u e C£°(R n ). Remark 2.2.6. A. If L(x, D) is an operator satisfying a sub-elliptic estimate in the strict sense, then L(x,D) is hypo-elliptic provided that the coefficients belong to C°°(M.n). Moreover, in this case (2.100) is equivalent to ||«|| 2 e +. < cK(\\L(x,D)u\\3
+ \\u\\0)
(2.102)
for all u G C^°(K™), suppu C K, and s > 0. Operators satisfying (2.102) are called sub-elliptic operators and we will make some remarks to their analysis later on. B. The notion of uniformly sub-ellipticity seems not to be present in the literature, but see Example 2.2.15 which is related to Hormander-type operator. Since we have to consider certain anisotropic pseudo-differential operators q(x, D) satisfying with some 0 < e\ < e^ < 1 the estimates ||«||2 1 < (q(x,D)u,u)0+\\u\\20 for all u €
CQ°(]R")
< C l ||«||* 2
(2.103)
we are naturally led to this definition.
Proposition 2.2.7. Let L(x, D) given by (2.77) be an operator satisfying a uniformly sub-elliptic estimate for some e > \. Moreover let bj € C j ( R n ; R ) , 1 < j < n, be given functions such that Y^j=i Jx Dirichlet form is given by the closure of
originally defined on Co°(K n ;R).
— 0 holds. Then a semi-
2.2 Second Order Differential Operators
55
Proof: It remains only to prove that E fulfills the sector condition, i.e.
^ciBMu^iB^v))11/2 '
\J2 [ bj(x)^-v(x)dx 1
n
Clx
•_! JR
i
where n
Bi(u,u)
.
:= V
/
afeJ(a;)
9u(x) 9u(x) dx+ / 9a;; 9xfc
|u(a;)|s rfx.
Using Lemma 1.4.7.34 we obtain for u,v € Cg°(R n ;R)
I E / M * ) ^ ^ ) * *
ox
J=1JK"
j < \\u\\e\\v\\e
proving the proposition.
D
More important (see below) is the following result: Proposition 2.2.8. Let L(x,D) satisfy a sub-elliptic estimate in the strict sense with (2.100) to hold with some e > \. In addition let bj € C ^ R ^ R ) , 1 < 3< n, be given functions such that for a fixed compact set K C Rn we have supp bj C K for 1 < j < n. Then there exists at least one function c £ Ci(R n ;R) such that the operator d L(x,D)+J2bj(x)7r--c(x)
(2.104)
3=1
extends from C£°(R n ; R) to a Dirichlet operator. Proof: Denoting by B the linear form associating with L(x, D), i.e. nf
\
V^ /
<
,du{x)dv(x)
we prove first \T, f bj{x)^-v{x)dx i £-j j R n OXJ
(2.105)
Chapter 2
56
G e n e r a t o r s of Feller a n d S u b - M a r k o v i a n Semigroups
for all u,v G Cg°(R n ;R). For this let V G C£°(Rn;]R) such that tp\K = 1 and 0 < ip(x) < 1. Due to the support properties of the functions bj we find using Lemma 1.4.7.34
\±f 1
-=
1
bj(^v(X)dx\ ./R"
aa;
9
= \± f
i
-^lm{x)dx
bj{x)
' J = 1 ./R»
ax
j
< C 6 | | V m | | 1 / 2 | | V H I l / 2 < Cfc||V'u|| e ||V'«||e
d(ipu)(x) akl(x)^^
f k,i=ijRn
+ fI
°Xl
d(ipu)(x) •dx dxk
2 i 2 l tp%lj' {x)u {x)u {x) (x)dx t
f
E7K« fe(=1
, . $w(:r) 9u(x) OX;
dxk
•dx
k
Since
fe,(=i
for all x G R" we find with £ = (1 - V>2)1/2Vu that
dx
h-lfj^!
dx
l
k
implying that Bi(ipu,ipu) <
B(U,U)+CA
j
\u(x)\2dx,
2.2 Second Order Differential Operators
57
where CA
:= sup
"*>*>0£^>^?-*>2
Thus we obtain (2.105) which yields that for any function c e c(x) > 0, the bilinear form n
E{u,v) :=(-L{x,D)u~
p. b
J-rr- + c{-)u>v)o
J^
x
fc,(=i
= B(u,v) — 2~] I
i
bj(x)—
v(x)dx+
I
c(x)u(x)v(x)
dx
satisfies the sector condition. Moreover, from Y\
I
b x
i( )
%
u(x)dx+
/
c{x)\u{x)\2 dx
we deduce that for c(x) + | Y12 z=i al — ^> ^ follows that -B('U, u) > 0 for all u € C£°(R n ; R). A careful inspection of the proof of the Theorem 1.4.7.20 shows that the contraction property holds for E if we prove that #((0Vu)Al,u-(0Vu)Al)>0 holds for all (suitable) functions u. However, for the bilinear form B(u,v) + J*K„ c(x)u(x)v(x) dx this follows since L(x, D) — c(x) is a Dirichlet operator, while for 1 < j < n we have /
bj(x)-—((0Vu)Al)(u-(0Vu)Al)dx
which proves finally the proposition.
=0 •
We now want to discuss briefly a class of second order partial differential operator which satisfy certain sub-elliptic estimates. The class we are interested in is the class of the so-called Hormander-type operators, often also called the sum of squares of vector fields. Our aim is not a complete discussion of these
58
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
operators — this would be beyond the scope of our book, but we want to consider some aspects related to the fact that these operators occur as generators of semigroups. For 1 < j < n and 1 < I < m, m € N, let a\ : Rn -> R be given C°°functions and consider the vector fields
x :
(2-106)
< =E4(*)Jr-
As usual we obtain by [X;,Xfc] the commutator field, i.e.
[Xh Xk] = XtXk -XkXt = J2 c? &)•£-, 3=1
(2-107)
J
where
•^m^^^). The formal L 2 -adjoint of Xi considered as a differential operator is given by X
t = -Xi-E-£2-
(2-108)
3
j=i
The operators we are interested in are given by m
lP\x,D) = Y,X?
(2-109)
i=i
or m
LW(x,D) = -J2X?Xi-
(2-110)
i=i
A straightforward calculation shows that for the principal part of (2.109) and (2.110) we have L
n
m
i,j=l
1=1
P = £ ( E f l 5 W ° i W ) ^ 7 . " = 1.2,
(2-111)
2.2 Second Order Differential Operators
59
and since for x 6 R n and £ e R" n
m
rr, m
n
the operators l / 1 ^ ! , .D) and L^2>(x, D) have non-negative characteristic form. t n Moreover, in case of the operator L^(x, D) we find for u,v £ rCf° °°(R '° ;M) (L{2)(x,D)u,v)0
= - /
( E ^ f W O W ) ^
= E / I J^l JR" JR"
Xl{u){x)Xl{v){x)dx,
thus m
.
BW(u,v):=S2
(Xiu)(x)(XlV)(x)dx
(2.112)
1=1 • / R n
is a pre-Dirichlet form and by an analogous argument which led to Theorem 2.2.1 we conclude that the closure of B^ is a symmetric Dirichlet form on L2(Rn; R), hence the operator —L2(x, D) = J2i=i X*Xt extends to a generator of a symmetric sub-Markovian semigroup on L 2 (R n ;R). Since we may write
L^(x,D)
-±XtXl+±±d-^Xl,
= i=i
i
i=i j=i
the difference of L^(x,D) and L^-2\x,D) is a first order drift term. n a] € C£(R ; R), 1 < / < m, 1 < j < n, we find r
™"dalAx) dxj 3= '
n
a I
m
< max II E ^ £ll*m||o|M|o, ;• .Hoof--' — / = 1
whereas for A > 0 m
B^\u,u)
i2)
= B (u,u)+X(u,u)0
= E i=i
ll^ullo+AHlo-
For
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
60
Thus we find m
»
n
aoj(x)
| £ £ / 1
'~i^Xl{u){x)v{x)dx\
<9a:J
, _ , „•_, JR.*
<
c v—» „ „
„o
c^||x,H||oH|o < 2 £ U * H l o + (=1
ra2c,
;=i
implying that (u, v) : =
V)Q
= B(2)(u,t;) + ^ ^ / i = i j = i •'*"
-^i^H^Mx)^
(2.113)
X j
extends from C£°(R";R) to a semi-Dirichlet form on L 2 (R";M). Hence L^\x,D) extends from Co°(R n ;R) to a generator of a sub-Markovian semigroup on L 2 (R n ;R). It is not easy to characterize the domains D(B^) and D{L^-2\x,D)) or D(L^(x,D)), respectively, in terms of (classical) function spaces. Denoting by X the system X\,..., Xi, one may of course introduce Sobolev-type spaces Hx(M.n) as completion of Co°(R n ) with respect to the scalar product B\ . In order to get some reasonable scale of function spaces further assumptions on X are necessary. In fact, such assumptions are also necessary to get sub-elliptic estimates for L ( 1 )(x,D) or L^(x,D), respectively. Definition 2.2.9. Let X be a system of vector fields Xi,..., Xm on R™ with C°°-coemcients. A. We say that the system satisfies the Hormander condition if for any point x £ R n the vector fields Xi,..., Xm together with a finite number of iterated commutators of them do span the tangent space at x. B. We say that the system X satisfies a uniform Hormander condition if there exists an integer k independent of x € R n such that for all i e l " the fields X\,..., Xm together with all iterated commutators of length less or equal to k do span the tangent space TXR™. Clearly, by an iterated commutator of the system X we mean a vector field of the the type [Xh,[XiA---[Xh_1,Xlj}...]]],
he{l,...,m}
(2.114)
2.2 Second Order Differential Operators
61
and j — 1 is the length of the iterated commutator (2.114). The importance of the Hormander condition lies in Hormander 's theorem on the hypoellipticity of sums of squares of vector fields. Theorem 2.2.10 (L.Hormander). Let L(x,D) be given by (2.109) or (2.110) with vector fields Xi having coefficients in C°°(R";R). If in addition the Hormander condition is satisfied, then L(x,D) is hypoelliptic, i.e. sing supp L(a;, D)u = sing supp it, and L(x,D) satisfies a sub-elliptic estimates in the strict sense, i.e. for some e > 0 ||ti||? < cK((L(x, D)u, u ) 0 + HI?)
(2.115)
and \\U\\2B+S < cK{\\L{x, D)u\\s+\\u\\0), hold for all ueC§°(Rn),
s € R,
(2.116)
supp u C K, K C Rn compact.
Recall that sing supp u, the singular support of u, is the set of all points in R n having no open neighbourhood to which the restriction of u is a C°°function. A refinement of Theorem 2.2.10 is due to L.P.Rothschild and E.M.Stein: Theorem 2.2.11. Suppose that L(x, D) is as in Theorem 2.2.10 and assume in addition that for some open set G C R n we need at most iterated commutators of length k to span the tangent space at each x e G, i.e. in G we have a uniform Hormander condition. Then we have for all u € Co°(M n ), suppu C K C G, K compact, the estimates \\u\\\/k < cK ((L(x, D)u, u)0 + \\u\\l)
(2.117)
and N U + 2 / * < cK (\\L(x,D)u\\s
+ \\u\\0),
seR.
(2.118)
Remark 2.2.12. Theorem 2.2.10 is is proved in L.Hormander [139] and Theorem 2.2.11 was published by L.RRothschild and E.M.Stein in [238]. A proof of Theorem 2.2.10 using pseudo-differential operator techniques is due to J.J.Kohn [177]. This proof is included in the monograph [266] of F.Treves and it was given in great detail in the author's textbook [157].
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
62
Corollary 2.2.13. In the situation when Theorem 2.2.11 holds with K = E n and k = 2 we may apply Proposition 2.2.8. Then we perturb the operator L(x, D) by a drift with compact support and we will still get a semi-Dirichlet form and a sub-Markovian semigroup on L 2 (M n ;R). Let us consider some examples. Example 2.2.14. A. the Laplace operator satisfies the uniform Hormander condition since
B. The Gruschin operator, compare V.V.Gruschin [113], defined by v-
d2
v2
2
d2
,
satisfies the Hormander condition uniformly on E n with k = 2. We just have to take X\ = ^ - and X2 = xijfc-, and to observe that
[XuXa]= xid
£{ £)-xiwa(-£d f
d \
d f d \
d dx2
More generally, for I G N we find that _^_ dx\
2l&_ +Xl
dx\
satisfies the uniform Hormander condition with k = I + 1. C. On E 2 n + 1 we denote a typical point (xi,... ,x„,yi,...,yn,t) consider the vector fields Q
Pi
o
and we
Pi
and the operator
4 3= 1
d2
d2
.
d2
.
d2
\
„ „
,
l0x
d2
(2.120)
2.2 Second Order Differential Operators
63
The operator —AK is the Kohn-Laplacian (on the Heisenberg group), sometimes it is called the sub-Laplacian. The commutator relations for Xj and Yj are [Xj,Xi) = \Yj,Yl] = 0, [Xi,Yi]=0
l<j,l
forj^Z,
and
Thus Xi,..., Xn, Y\,... Yn satisfy on IR2™+1 a uniform Hormander condition with k = 2. Subelliptic estimates for the Kohn-Laplace operator are proved using pseudo-differential operators, by J.J.Kohn [178]. Note that a uniform Hormander condition leads in general not to uniform sub-elliptic estimates in the sense of Definition 2.2.5 since the operator L(x, D) may have unbounded coefficients. The following example gives an operator in R 2 satisfying a uniform subelliptic estimate. Example 2.2.15. Let o : R -» R be a C°°-function such that 0 < a(t) < 1, «![_!_!] = 0, a|[_2,2j<= = 1- Further let b : R —• R be a C°°-function such that fe|[ i,i] = I. &|[~.2,2]c = 0 a n d 0 < b(t) < 1. We consider the vector fields Xi = -z— oxi
a,ndX2 =
with p{x{) = a{x\) +xib(x{), L(x,D) = X 2 + X 2 2 =
p{xi)-—, 0x2
and the associated operator ^ V ( x i ) ^ .
Since
it follows that X\ and X2 satisfy a uniform Hormander condition with k = 2 provided either ^{x{) • &^{x{) ^ 0 for x G [1,2] U [-2, - 1 ] or b(:n) ^ 0 for zG[l,2]u[-2,-l].
Chapter 2
64
Moreover, for all u £
Generators of Feller and Sub-Markovian Semigroups
CQ°(R2;R),
\\u\\2/2
suppu C K, K CR2 compact we have
+ \\u\\2),
(2.121)
where
In addition, for u e C£°(R 2 ;R) and suppu C Q c , where Q = [-2,2] x [-2,2] we have since p2(x\) = 1 for |xi| > 2 that
< B(u,u) + \\u\\2. Now, take \ G C°°(R 2 ) such that X\Q = 1, 0 < x < 1, and suppx C [-4, 4] x [-4,4]. For any Cg°(R n ;R) we have
u = xu +
(l-x)u.
Since supp xw C [—4,4] x [—4, 4] and supp(l — x)u C Qc we find combining (2.121) and (2.122)
2NI?/2
+ \\xu\\l)
+ 5 ( ( 1 - X ) « , ( 1 - X ) « ) + ||(1-X)«HS. A straightforward computation using the properties of p and x leads now to \\u\\2/2
+ \\uf0)
for all u £ Co°(]R2), since x was arbitrary. Let X be a system of vector fields X\,..., Xm satisfying a uniform Hormander condition. In this case it is possible to introduce anisotropic Sobolev spaces W%P(R2;R) denned by W^'P(R2;R) := {u e LP{R2; R); Xau € LP{R2; R) for a = {au ..., a.), \a\ < k}, (2.123)
2.2
Second O r d e r Differential O p e r a t o r s
65
where the norm || • \\wk,P is given by
NI^.» = ( E l l * Q * ) 1 / P
(2-124)
|a|
and Xau = Xjai • • • XjaBu,
j
a i
e { 1 , . . . , m}.
These spaces where introduced by L.Hormander [142], see also the paper [47] of V.M.Chernikov and S.K.Vodop'yanov as well as C.-J. Xu [282], [283] who related these spaces using a type of Sobolev embedding theorem to spaces of Holder continuous functions when the Holder continuity is defined by an intrinsic metric obtained by the system. This metric was introduced in [220] by A.Nagel, E.M.Stein and S.Wainger, see also the paper [90] of C.Fefferman and D.H.Phong. In the paper [105] U.Gianazza and V.Vespri used these spaces to obtain analytic semigroups generated by Hormander type operators on [Wx'p(Rn)}*. Note that W^'2(Rn; R) is a Dirichlet space under the scalar product B[2), B[2) associated with L^(x,D). Hence also (B™, W£ 2 (R";R)) is a Dirichlet space. Furthermore, fix a system X of vector fields X\,... Xm satisfying a (uniform) Hormander condition and let (a,ki(-))k,i=i,...,m be a strictly positive defined matrix of C£°(R n ; R)-functions, i.e. aw = aik and m a
ki(x)£kZi > Ao|e|2,
Y,
A0 > 0.
fc,!=l
Consider the operator m
- Y, XliautfXi).
(2.125)
k,i=i
Using the paper [242] of L.Saloff-Coste and D.W.Stroock, it is also possible to construct symmetric sub-Markovian semigroups on L( 2 )(R n ;R) generated by an extension of the operator (2.125). Further we can find that m m
«
aki(x)Xi(u)(x)Xk(u)(x)dx
(2.126)
66
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
is a symmetric Dirichlet form with domain Wx' (R n ;R). In fact, if we are not interested in the hypoellipticity of the operator (2.125) we may reduce the regularity assumptions in (2.126) to L°°(R™;R)-coefficients and still obtain a symmetric Dirichlet form on w £ 2 ( R n ; R ) .
2.3
Some Properties of Pseudo-Differential Operators with Negative Definite Symbols
In some sense, starting with this section we come to the central theme of this monograph, namely to study these pseudo-differential operators which may extend to a generator of a Feller semigroup or a sub-Markovian semigroup. In this section we collect some first results. The class of pseudo-differential operators we are interested in is determined by Courrege's result, see Section 1.4.5. Thus we will consider operator q(x, D)u{x) =
(2TT)-"/2
/
eix-*q(x, OHO <*£,
(2.127)
where we assume that q : R™ x R™ —> C is a locally bounded function such that for every i £ l " the function q(x, •) : R™ —> C is a negative definite and continuous. For reference sake we give Definition 2.3.1. We call a function q : Rn xRn —> C a, (continuous) negative definite symbol if q is locally bounded (continuous) and for each x € R™ the function q(x, •) : R n —> C is negative definite and continuous. Note that for a negative definite symbol q for every compact set K C R™ there exists a constant cj< such that |g(*,OI < c / c ( l + |£| 2 )
(2.128)
holds for all x £ K and £ € R n . In fact we may take cK = 2 sup \q(x,r))\. xeK W
2.3 Pseudo-Differential Operators with Negative Definite Symbols
67
formula we see that a pseudo-differential operator with negative definite symbol has also the representation / ™ / N K - ^ / x d2u(x) q(x,D)u(x) = - 2^ akl(x)dxkdx[
(
^
(U{x-y)-u{x) + f , v
K"\{0} ^ •Wo
x
, . . du(x) . . .. +}^do(x)-g^-+c(x)u(x)
1 + lvl
V
\JI{X))
, t j i + \y\2 dxj J
2
^JrN{x,dy)
(2.129) with a,ki, dj, c and N(x, dy) as in Section 1.4.5. In particular — q(x, D) satisfies the positive maximum principle on Co°(R n ; W) and in addition, q(x, D) maps Co°(]Rn;R) into 5f,(R n ;E), i.e. it maps real-valued functions onto real-valued functions. The latter fact follows already from q(x, £) = q(x, — £) which holds for every negative definite symbol, see (1.3.122). We are interested in proving estimates for the operator q(x, D) in Sobolev spaces related to continuous negative definite functions, more precisely, in the Hilbert spaces H^'s(M.n) introduced in Section 1.3.10 for a fixed continuous negative definite function rp and s g R . Note that the two different representations of q(x,D), i.e. (2.127) and (2.129), give rise to different conditions on q(x, D) in order to prove estimates since different techniques have to be used. In this and the following sections we will not make use of (2.129), but later, in Section 2.8 we will combine both representations to get more general results. In some estimates we need a more precise control on constants. For this we note /
(l + \x\2rh/2dx=1T
7 r n/2 r (fc= 2 )
J,.2
)
=:~cn^
k>n,
(2.130)
and jm,n is the smallest constant such that
(i+iei 2 r / 2 <7 m ,n J2 i*ai
(2-131)
|a|<m
holds. An upper bound for ym,n is ( n + l ) m / / 2 . Finally, for a continuous negative definite function ip : R n —• M the constant c^ is defined as the smallest constant such that
V>(£)<^(1 + |£|2)
(2.132)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
68
holds. For a (negative definite) symbol q(x, £) we set q(V, 0 :=
(2TT)-^2
/
e-ix-"q(x, £) dx = Fx^n(q(x,
#)(»/),
(2. 133)
whenever this Fourier transform exists. Lemma 2.3.2. Let q : R™ x i —> C 6e a measurable function and further let ijj : R" —» R 6e a continuous negative definite function. In addition assume that C is m-times continuously differentiate. A. If for every multiindex a 6 NQ, |a| < m, the estimate Kq{x,i)\
(2.134)
holds we find for all p g Cg°(R n ) lfaz) A fo,OI < c v (l + h | 2 ) - m / 2 ( l + V ( 0 ) B. Suppose that there are functions <pa € L1(Wn),
(2-135) \a\ < m, such that
(2-136)
l^?(*,OI<Ma:)(l+tf(0) feoWs. Then we have for all k GNQ, \k\ <m, the estimate \q(v,0\ < %,n £
IMMl+M'r^a+V^))-
|a|
Proof: A. For /? G Nft, |/3| < m, we have if
f e-"-"
=
[
(dP(e-ix*>))
e-ix-*dP{
^ £ (!W+wo)ii0'3-nviui < wi+v(0), 7
V 7 /
which leads to
(1 + I»?|2)m/2|(W)A('/,0I < i.e. (2.135) is proved.
(2ir)-n/%(l+m),
(2-137)
2.3 Pseudo-Differential Operators with Negative Definite Symbols
69
B. We may use the calculation of part A to obtain e-ix\(x,Odx\
rf I
e-i*r>dPq{x,e)dx\
= I/
<
||^||LI(1+IK0)>
which gives for k < m
(l + W\2)k/2\q(V,0\<%n Y,
Wf/sh^l+m),
\0\
and (2.137) is proved.
•
To prove a first consequence of Lemma 2.3.2 we need L e m m a 2.3.3. Let k £ L ^ K " ) . Then we have for all u, v £ L2(M.n) \f 1
I
k(£-rt)u(r))v(t)dridZ
< ||fc|| L i|M|o|M| 0 .
(2-138)
JR" JR"-
Proof: Using the Cauchy-Schwarz inequality we find I/ 1
/
k(Z-T))u(rj)v(t)dTidz\ '
JWL" JR"
< JRn ((fRn
m-V)\dn)
<\\k\\%2\Mo(
[ v
I
JR"JR"
V 2
( ^ |fc«-tj)||«(r/)|2^)1/2KO|)
^
\KZ-v)\Wv)\2dr,dz)1/2 '
< ll*IMMIolMlo, where in the last step Young's inequality, Lemma 1.2.3.15, was used.
•
Now we may show P r o p o s i t i o n 2.3.4. Suppose that q is a function satisfying the assumptions of Lemma 2.3.2. A for m > n + 1. Then the operator q(x, D) maps H^'2(Rn) into the space Lfoc(M.n), and we have for every test function ip £ Co°(]Rn) the estimate \\
Note
70
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
that for u,v £ S(Rn) we have by (2.135) and Lemma 2.3.3 \(
&q)A(t-ri,,n)Hr))v(£)d'nd£,
[ (l + |e-r?| 2 )- (n+1)/2 (l + ^ ) ) K ^ I K 0 l ^ ^
< CV^n+iHl + ^(O^lloll^llo or \(
< c^llull^,^,
D
The proof of Proposition 2.3.4 shows already a principal problem when estimating a pseudo-differential operator with negative definite symbol in some of the spaces JJ*''(R n ). We need also a control on the Fourier transform of the function x — i > q(x, £), i.e. we need a control on q(rj, £). This problem is well known in the theory of pseudo-differential operators. For example in order to improve (2.139) to a global estimate , i.e. to ||g(x,D)u||o
(2-140)
Of course we have still to assume that q(x, •) : M.n —> C is for all x S R" a continuous negative definite function. In addition we assume that q\ : W1 —> C is also a continuous negative definite function. The operators q\(D) and 92(1, D) will be estimated separately. Note that decomposition (2.140) may arise when freezing the coefficients, i.e. q(x,£) = q(x0,Q + (q(x,£)
-q{xQ,£)).
Let us pose some general assumptions on q\ and 52-
2.3 Pseudo-Differential Operators with Negative Definite Symbols
71
Assumption 2.3.5. We assume that the function q : R n x R™ —> C is a continuous negative definite symbol having the decomposition (2.140) into a continuous negative definite function qi : R n —> C and a continuous function q2 : R" x R n -> C. Further let V : R n -> R be a fixed continuous negative definite (reference) function. A.l. The function q\ is continuous and negative definite and satisfies with 7o > 0 and 71,72 > 0 7 0 ^ ( 0 < R e g i ( 0 < iMO
for
all |£| > 1,
(2.141)
and |lmgi(0| < T U l W ( 0
for all $ € R".
(2.142)
(Note that (2.141) and (2.142) implies 1 + R e g i ( 0 < l + |«i(0l < n ( l + V ( 0 )
(2-143)
for all ( £ R n with some n > 0). A.2.m. For m 6 No the function x i-> 92(^1^) belongs to C""(R") and we have the estimate \dZq2(x,t)\
<<pa{x)(l+iKt))
(2-144)
for all a € NQ\ |a| < m, with function <pa € L ^ R " ) . Later we will add a smallness assumption for q2{x,Q- We start with estimates for the operator qi(D). Proposition 2.3.6. We assume A.l. For any s G R the operator q\(D) maps H*>a{Rn) continuously into the space H*'8~2(Rn). Further we have the estimates ||qri(D)u||^ iS _ 2 < ri||u||v.,«
(2.145)
||giCD)u||v>,»-2 >7o|Mlv.,»-Ao||u|U,*-2,
(2.146)
and
with a suitable constant XQ.
72
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
Proof: It is sufficient to prove (2.145) and (2.146) for all u e S(R"). For s € R we find using (2.143)
ll9i(^HI*,,-2 = / (i + ^(0)'_2l9i(0«(0l2^ a+V'(or 2 (i+V'(0) 2 i^)i 2 ^
i|2
proving (2.145). T o prove (2.146) observe that
(i + v(0)- 3 ki(0l 2 l«(0l 2 ^
hi{D)u\\la_2 = f
(i+^(0)'-2i?i(oi2i«(Oi2de
+ / -/Bi(O)
(l+V(0) s - 2 (Re 9l (0) 2 |^(0| 2 ^
> / >7o 2 /
(1 + ^(0)'" 2 W0) 2 |«(0I 2 ^
•/BJ(O)
>7 0 2 /
(i+iKomoi2^
-7o 2 / (i+^(0)' _2 i«(0i 2 de = 7o2H2i,s-7o / (l + V-(0)'"2|«(0|2de -702/
7Bf(0)
(i+^(0)'|fi(0i2de
>7oll«l| 2 i, s -7o 2 sup
(l+^))2HL-2,
«€Bi(0)
which implies (2.146).
•
Corollary 2.3.7. Suppose that lim ip(£) = oo. TTien we have in the situaK|-oo
izon o/ Proposition 2.3.6 /or even/ £ > 0 and s > 2 \\qi(D)u\\^iS-2 /or X£ > 0.
> (7o-£)||w|U,s-A £ ||u||o
(2.147)
73
2.3 Pseudo-Differential Operators with Negative Definite Symbols
Proof: Since lim tp(£) = oo we find for e > 0 a suitable constant pe > 0 K|-oo
such that
(i+^)r2<|(i+^)r holds for all £ G E™, |£| > /9£. Note that the case Ao = 0 is trivial. It follows that
%h\\l,.-* = >%[ (i + V(Or 2 KOI 2 ^ =A2/
(i+^or 2 i«(oi 2 ^ (i+iKoraifi(oi2#
+A^/ •/BP,(0)
2
<£ /
(1+^))*|«(0|2«
•'B»€(0)
+ A 2 ( sup (l+vco)'-2) / <e2\\u\\lfS
m)\2dt
+ X2e\\u\\20.
Hence (2.146) implies ||gi(£>)it||^)S_2 > 7o||w|]v,s-Ao||'u||v',3-2 > (o'o—e)||wjj^iS—A£||uj|0-
•
Remark 2.3.8. It is possible to get a smaller value for Ae, but we do not need it later on. Next we want to estimate the operator q2(x,D) assuming that q^{x,£) fulfills A.2.m with a suitable large m. We start with the following commutator estimate: Theorem 2.3.9. Let s > \ and assume that qi{%,£,) satisfies A.2.m with m > n + 2s. Then we have
J2 MLi|Mk2-+i
(2.148)
\ac\<m
for all u G H*'2s+1 (R n ), where Kn,m,s,-0 = (27r)" n / 2 2 2 s + 3 sc v ,7 m ,„c 7 l , m _ 2 s with c„, m _2s, 7m,n and c^, as in (2.130)-(2.132).
(2.149)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
74
Proof: For s > i we have using Lemma 1.3.6.21 and Peetre's inequality, Lemma 1.3.6.23,
es-idt
i(i+v(or-(i+#?)n=2S /
•Ai+v-w)172 < 2s ((1 + V ( 0 ) ^ + ( 1 + ^ ) ) ^ ) |(1 + V ( ^ ) ) 1 / 2 - (1 + ^ ( r / ) ) 1 ^ < 2 2 5 + 3 s ^ ( l + |^-77|2)s(l + V(r?))^i,
(2.150) where we use also estimate (2.132). Further, by Lemma 2.3.2.B we have
\q2^-V,v)\
(2.151) Now, for u,v e Cfi° (W1) we find using (2.150) and (2.151) \(([l+iP(D)y,q2(x,
D)]u, v)0\
= (27r)-"/ 2 | / 1
/
^-r/,r/)((l+V(0r
JR™ J R n s
v
- ( i + VM) )^)t>(£)^| < (27r)- n / 2 2 25 + 3 SCv ,7 m) n £
H^Hi1
|a|<m
x /
/
2
(l + | e - r ? | ) - t + ^ ( l
+
V;(T/))^
ti
W7?)||^)|rf^
J R " v/R" < «n,m,s,i/> 2 ^ HValU 1 l| u ||v>,2s+l|M|o, \ct\<m
which implies (2.148) and (2.149).
•
Corollary 2.3.10. Let s > 0, t > \ and assume that q2{x,£,) satisfies A.2.m with m > n + 2s + It. Then we have with a suitable constant c \\[(l+^(D)Y,q2(x,D)}u\\ij<2t
IIVa||Li||«|U,2.+2t+l. |a|<m
Proof: Since \\[(l+^(D)r,q2(x,D)}uU,2t=\\(l+^(D))t[(l+^(D)y,q2(x,D)}u\\0
(2-152)
2.3 Pseudo-Differential Operators with Negative Definite Symbols
75
and (l + iJ(D))t[(l
ij(D))s,q2(x,D)}u
+
= [(1 + V(£>)) t+S , q2(x, D)]u - [(1 + ^D)f
, q2(x, D)] (1 + ^(D))3 u
if follows from Theorem 2.3.9 that \\[(l+i>(D))\q2(x,D)}uU,2t <\\{(l+i>(D))t+s,q2(x,D)}u\\0 + \\l(l + WD))t,q2(x,D)}(l
+ i>(D))su\\o w
|a|<7n <
5
l|i/>,2t
|a|<m
X I IIVa||Li||w||v,2a+2t+l\a\<m
•
Now we want to estimate q2(x, D). Theorem 2.3.11. Let s > 1 and suppose that q2{x,£) satisfies A.2.m with m>n + [s]+l. Then we have for allt>0 and u 6 H^'t+2 (R n ) \\q2(x,D)u\\^tt
< Cn.m.t.vIIwllv-,*+2.
(2.153)
and for t = s we have Cn,m,a,V= (7n+l,nCn,n+l + K n , m , f , v )
HValli1)
2-/
(2.154)
|aj<m
where Kn<m>^
is defined as in (2.149).
Proof: First note that for u, v € S (R n ) we have /
q2{x,D)u(x)v(x)dx=
/
/
q2(£-rj,v)u(v)v(OdVd£
and applying Lemma 2.3.2.B yields l(q2(x,D)u,v)0\<%+hn
HValU1 /
Y, |«|
/R
• "
(l + l ^ - ^ l 2 ) - 1 ^
/ 7R
"
x{l + tl>(V))\u(ri)\\v(Q\dTidZ < 7n+l,nCn,n+l
^ llValli1 I M I I M I N O , |a|
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
76
which implies ||92(z, D)ll||o < 7n+l,nCn,n+l
^ \\fa ||l,i |M|i/>,2, |a|
(2.155)
i.e. (2.153) for t = 0. Next let s > 1 and observe that \\q2{x,D)uU,a
\\q2(x,D)(l+TP(D)y/2u\\0
<
+ \\[(l + Since (1 + tp(D))r' #lM-r( R n)
#^
2
w e
find
TP(D)y/\q2(x,D)}u\\0.
maps iJ^>* (R n ) bijectively and bicontinuously onto for
u
g
#V>,*+2(Rn)
t h a t
( X + ^(D))*'2
U =:
V €
(R") and therefore ||g2(a:,-D)(l + '!/'(-D))s/2w||o <7n+i,nC„,„+i
^
||^ a || L i||u||^ i S + 2 .
|a|
(2.156) On the other hand we may apply Theorem 2.3.9 with s replaced by s/2 to get \\[(l+1p(D))s/2
,q2(x,D)}u\\0
< Kn,m,f,V
Yl |a|<7n
W M M k H - l . (2.157)
which leads together with (2.156) to H ^ O E , D)U\\TP:S
< ( 7 „ + l , n C n , n + l + « n , m , f ,i/>) 2 - / H ^ a l U 1 ll U lllM+2 |a|<m
and the theorem proved for s > 1 and m > [s] +n + 2. For 0 < s < 1 the result follows by interpolation, see Theorem 1.3.10.16 and Theorem 1.2.8.7. • Remark 2.3.12. Note that our proof of Theorem 2.3.11 and (1.2.215) yields the estimate \\q2(x,D)u\\^it
Y,
lb«llLi|MU,t+2
(2.158)
[a|<m
for alH > 0 with m suitably large. In the following theorem the relation of the constants in (2.147) with s — 2 replaced by s and the constant cn>m)S)^, from (2.154) will become important. For simplicity let us restate (2.147) as ||gi(D)u|U,s >f77o||w||vv>+2-7i7,sNlo
(2.159)
2.3 Pseudo-Differential Operators with Negative Definite Symbols
77
which holds for all s and 77, 0 < 77 < 1. Further recall the estimate (2.153) for s > 1 and m > n + [s] + 1: ||52(a;,-D)'"||V),s < C,i,m,s,i/. X ] ||¥'a||L 1 ||w||v,3+2) |a|<m
(2.160)
where Cntin,s,^ = (7n+i, n c„,„ + i + Kn.m.f>)• Combining (2.159) and (2.160) we get ||g(ar,£>)ti||v,,5 > ||gi(D)u||v.,a - ||g2(a:,.D)u||^>J9 > Vlo\\u\\^,s+2
— {jno
- lr),A\U\\o
~ Cn,m,s,il)
— Cn,m,s,i>
] > J llVa IU 1 ll u IU,*+2 |a|<m
2J ll¥'«lll.1JlluIIV',H-2-7»J,slMlo. \a\<m
Thus we have proved Theorem 2.3.13. Suppose that q(x,£) = qi(£) + q2(x,£) satisfies A.l and A.2.m with m > n + [s] + 1 for some s > 1. Further assume that for some T? G (0,1) we have
Y, M i * <^>-
( 2 - 161 )
|a|<m
T/ien we have for all u £ H^'s (R n ) the lower estimate \\q(x, D)u\\^tS > 5o||w||v,a+2-7T),a||w||o
(2.162)
X ! lly<*IU 1>0 -
(2.163)
with &o:=rno-cn,m,s,il>
)a|<m
Remark 2.3.14. Later on, we see that often (q(x,D)u,u)o>0
(2.164)
holds for all u e fl^'2 (R n ;R). Taking in (2.162) s = 2 we find then for all A>0 \\q(x, D)u + Xu\\20 = \\q(x, D)u\\l + 2X(q(x, D)u, u)0 + X2\\u\\20 >\\q(x,D)u\\2+X2\\u\\2
Chapter 2
78
Generators of Feller and Sub-Markovian Semigroups
or
\\q(x, D)u + Xu\\0 > \\q(x, D)u\\0 + X\\u\\0 > ^o||w||v,2 - 7T(,2||W||O + M\u\\o-
Thus for A > 7^,2 we have under the assumption (2.164) \\q(x,D)u+Xu\\0>60\\uUi2.
(2.165)
In order to prove regularity results for solutions of the equation q(x, D)u = / we have to have a closer look to the Friedrichs mollifier. Proposition 2.3.15. Let Je be defined as in (1.2.78). For any s > 0 and u e H^'3 (Rn) we have JE{u)
e p | H** (R n )nC°° (R n )
(2.166)
t>o
and
\We(u)U,s < \\uU,s
(2.167)
os well as Um\\Jg(u)-u\\^a=0.
(2.168)
In addition, if for e € (0, p), p> 0, we have for some u e I? (R n ) \\J£(u)\\^s
< Cu,s
(2.169)
with a constant independent of e, it follows that u € H^'s
(Rn).
Proof: For t > 0 we find
\\Uu)\\%,t=
((i+^(0)%-e*«)A(oi2de
= (2TT)"/
(l+1>(t))%(t)\2\&(Q\2d£-
Since jE € CQ° (R n ), it follows that j e € 5 ( R " ) . Hence there is a constant cttS,e such that
(i+wo)*ij«(oi<*,M(i+w0r,
2.3
79
Pseudo-Differential Operators with Negative Definite Symbols
which implies ||J e (u)||$ t < ct,,, e ||u||$ , hence we have Je(u) £ fi H*'* (R n ), t>o and (2.166) follows with Proposition 1.2.2.17. Moreover, the Lemma of Riemann-Lebesgue, Theorem 1.3.2.1, implies
\ m \ = m)\ < w-n/2\\j\\v = (2vr)-"/2, which yields
iije(«)i|2 = (2^" /
(i+m)s\m)\2m)\2dt
< f (i+mynz)\2dt = \\u\\is. In order to show (2.168), observe that
\\M«)-«\\h=
I (i + ^(0)'IO-e*«)A(0-«(OI2« = (2*)" /
(l + ^ ( 0 ) ' l 5 « ( 0 - ( 2 7 r ) - n / 2 | 2 | T i ( 0 | 2 ^ .
Since j e ( 0 = j ( e O - 5'(0) = (27r)-"/ 2 and |j e (£)l < the dominated convergence theorem implies
(2TT)-"/2
for all £ £ I T ,
lim||Je(u)-«||^>t = 0 . e-»0
It remains to prove that (2.169) implies u £ H^s (R n ). Prom (2.169) it follows that (Ji/n(u))n>± converges weakly in H^'s (W1) to some element v £ H^'s (R"). By the linearity of the continuous embedding of H^<s (R n ) into L2 (R") it follows that (J 1 /„(it)) , converges also weakly in I? (R n ) to n
p
v. But by Proposition 1.2.3.17 we know that ( i converges strongly in L? (R n ) to u, hence u = v and u £ H^'3 (W1). D Theorem 2.3.16. Suppose that q2{%, £) satisfies A.2.m. Further lets>0 such that | s - l | + n + l < m. Then we have for alls £ (0,1] and allu£ H^'s+1 (R") \\\Je,q2(x,D)}u\\i,ta
< c\\u\\
with a constant c independent of e £ (0,1],
(2.170)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
80
Proof: First observe that
([J£,92(x, D)]u) A ( 0 = (2TT)-"/ 2 /
(2.171)
q2(d - r,, r,) (j(eQ - j(er,)) u(r,) dr,.
Further we claim |2\l/2
2\l/2
tf(eO-ito)l(i + Kla) '
(2.172)
with c* independent of e e (0,1]. In fact, since |j(f )l < (27r)-"/ 2 for all £ G R n , we have for |£ — 7?| > -§•
+ Ki2)l/2
to)-^)i(i
< 2(27r)-/ 2 (1 + |e| 2 ) V 2 < c' (1 + |£ - V\2)1/2 • On the other hand, for |£ - rj\ < ^ , since j £ S(Rn) 2
c ( l + |£| )
and therefore |Vj(£)| <
> the mean-value theorem yields for e < \/2 -1/2
2\l/2
\m)-feri)\{l
+ \Z\V
2 1/2
1+
2\ -1/2
(^
+
f)
_
2\l/2
(1 + KI2)
2
(1 + |£|2)1/2,
and (2.172) is proved. Now we get using (2.170), Lemma 2.3.2.B, (2.172), Peetre's inequality for continuous negative definite functions, and the estimate ((1 + ^ ) ) < < V ( 1 + |£| 2 )
lP(0)S/2(iJe,
\(l + =
(2TT)-"/ 2
/
&(£ - 7?, T?) (j(eO - j f o ) ) (1 + V(£)) s/2 6fa) ^
(1 + |f - ry| 2 )- m/2 (1 + Vrfa)) (1 + K - r?| 2 ) 1/2
< c'" / JR"
f
JR
, n
n-. -
(1 + ^ - r ? ! 2 )
(1 + IC-77I2)'
m
+
1
2
m+l +
(1 + ^ - r / ) ) |j-l|
(1 + Vfa))
l«-l|
'
2
(1 + Vfo))
|w(7?)|rfTJ,
«+l 2
Rr/)l*7
2.3 Pseudo-Differential Operators with Negative Definite Symbols
81
with c independent of £ e (0,1]. Now, we finally get by Young's inequality \\[Je,
< c||(l + | • |2)-=±i^=ii
+ (1 +
^))4l|fi()|((o
•
Remark 2.3.17. Theorem 2.3.16 was originally proved in [155], we followed here the more polished proof given by W.Hoh in [128]. Corollary 2.3.18. Suppose that q\{£) satisfies A.l and that 92(2,£) satisfies A.2.m. Further let s > 0 such that \s — 1| + n + 1 < m. Then for all u £
||[J e ,?(a:,D)]u||^ < c | | u | | ^ + 1
(2.173)
holds with a constant c independent of £ £ (0,1]. Proof: Recall that q(x,D) = qi(D) +q2(x,D). In order to see (2.173) we only have to note that [J£, q\{D)\ = 0 since both J e and q\{D) are translation invariant operators. The rest follows from Theorem 2.3.16. D Combining the estimates for q{x, D) we can derive the following regularity theorem. Theorem 2.3.19. Suppose that q(x,£) — qi(£) + q2(x,£) fulfills A.l, A.2.m with m > [s] + n + 2 for some s > 0 and (2.161). Further suppose that for some A e R and f £ H^'a (R n ) we have a solution u £ H*'8*1 (R n ) to the equation q\(x, D)u = q(x, D)u + Xu = / .
(2.174)
Then it follows that u £ H^*"*1 (R n ). Proof: Using Theorem 2.3.13 we find M Je(u)IU,.+2-7'Mll'>'e(")l|0 ~ HM(«)|U,. <\\q(x,D)JE{u)\\^s-\\\J£(u)\\^s < \\q(x,D)Je(u)
+
\Je(u)U,a
< \\J£(q(x, D)u + Xu)\\^3 + \\[Je, q2(x, = \\Je(f)U,s +
\\[Js,q2(x,D)}(u)U,s.
D)](u)\\^3
Chapter 2
82
Generators of Feller and Sub-Markovian Semigroups
Thus we get by Theorem 2.3.16 60\\Je(u)U,s+2
< ||Je(/)IU,. +
\\[JeMx,D)]{u)U,.
+ \X\\\J£(u)\\^iS + 'y11ts\\Je(u)h,s < | | / l k * +c||w||v,,s + c||u||,/,,s+i, with c independent of e € (0,1], and the theorem follows from Proposition 2.3.15. • Let us introduce the sesquilinear form B(u, v) := (q(x, D)u, v)0
(2.175)
which is associated to q(x, D) and defined on CQ° (M.n). Clearly we have the decomposition B(u,v) = Bqi(u,v) where Bqi(u,v)
+ Bq*(u,v),
:— (qi(D)u,v)0
(2.176)
and Bq2(u,v)
:=
(q2(x,D)u,v)Q.
Proposition 2.3.20. Suppose that qi satisfies A. 1. The sesquilinear form B91 has a continuous extension onto H^'1 (R n ) and the following estimates holds: |5 , 1 («,W)|
(2.177)
with T\ as in (2.143). Furthermore we have with some Ao > 0 \Bqi(u,u)\>ReBqi(u,u)>
(2.178)
Proof: It is sufficient to prove (2.177) and (2.178) for all u,v 6 Prom (2.143) we deduce
\B«(u,v)\ = \ f 1
qi(0u(0W)di
7K"
Re5(u,u)= f
>io!
Re^OKOI^
v(0|fi(0l a ^+/"
JBI(Q)
Re9i(0|fi(0l2#
JB1(O)
> 7 o | K , 1 - 7 o H ^ - sup |Re 9l (0-7o^(OI WAl l€l
S(Rn).
2.3 Pseudo-Differential Operators with Negative Definite Symbols and (2.178) is shown.
83
•
Remark 2.3.21. Since qi(D) maps real-valued functions onto real-valued functions, it follows that for u e H^'1 (R n ; R) we have B
9 l
M>7oH^i-AoHg.
(2.179)
Next we will estimate Bq2. Proposition 2.3.22. Suppose that A.2.m holds for m> n+2. u,v€ H^'1 (R n ; R) the estimate
Then for all
(2.180)
|a|
Proof: As in the proof of Theorem 2.3.11 we find for u,v e S (R n ) \Bq2(u,v)\
= I /
< 7n+2,n 2L,
q2(x,D)u(x)v(x)dx II^IU 1
|a|
x f [ (l + | ^ - r / | 2 ) - i ^ ( l + ^ ) ) | % ) | K 0 l ^ ^ JR™ JlR"
1/2
SWMEJI,.IU./J„„(1+I{-«-^(H||)
x (i+v>(7?))l/2 \m\ (i+^(0) 1/2 Moi <^+2,»(lV(V) 1/2 ^ HV-Hi1
«
|a|
/
/
(l + | ^ - 7 7 | 2 ) _ i ^ ( l + ^ ) ) 1 / 2 | ^ ) I K ^ ) l ^ ^
<«2 52 II^II^IMU.lNU,! |a|
with K2 = v / 2 ( l V c v , ) 1 / 2 7 n + 2 , n c n , n + 1 .
D
Combining Proposition 2.3.20 and Proposition 2.3.22 we obtain
(2.181)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
84
Theorem 2.3.23. Suppose that qi(£,) and q2(x,£) satisfy A.l and A.Z.m with m>n + 2. Then B is continuous on H^'1 (R n ), i.e. for all u,v e IT*'1 (Rn) we have |B(u,u)|
(2.182)
Further we have Theorem 2.3.24. Suppose that gi(£) and (72(2,0 satisfy A.l and A.2.m with m > n + 2. Assume further with «2 from (2.181) that
<5i:=7o-«2 Yl
IKIU 1 > 0 -
(2-183)
\a\
Then we have for all u 6 H^*1 (R n ) \B(u,u)\ > ReB(u,u)
> <$i|Hv,,i-Ao||u|| 0 \
(2.184)
where Ao is taken from (2.178). Proof: Using Proposition 2.3.20 and Proposition 2.3.22 we get for u € \B[u,u)\ >ReB(u,u)
> Bqi(u,u)
-
>7o\\u\\iil-\0\\u\\20-K2
\B"2(u,u)\
Yl \\<pa\w\n\l,i \a\
= 6i\\u\\l,i
- *o\\u\\o-
•
Corollary 2.3.25. In the situation of Theorem 2.3.24 we have for all A > Ao \B\(u,u)\
>ReB\(u,u)
= ReB(u,u)+X\\u\\ltl
> «i||u||^i.
(2.185)
Let us suppose that qi(£) and 92(2;, 0 satisfy A.l and A.2.m , m > n + 2. Further let f € L2 (R n ). Definition 2.3.26. We call u e H^,x (R n ) a variational solution to the equation qx(x,D)u
= q(x,D)u+Xu
= f,
A € R,
(2.186)
if Bx{u,
if) = B{u,
holds for all ip £ C%° (R n ), or equivalently for all
(2-187)
2.3 Pseudo-Differential Operators with Negative Definite Symbols
85
Note that a variational solution of (2.186) which belongs to H^'2 (Rn) satisfies (2.186) already in the strong sense, i.e. q\(x,D)u £ L2 (Rn) and (2.186) is an equality in I? (Rn). Using the Lax-Milgram theorem, Theorem 1.2.7.41, we deduce Theorem 2.3.27. Let n + 2, and take \0 from (2.179). Moreover assume (2.183), i.e. 6\ > 0. Then for every A > Ao there exists a unique variational solution u £ H^'1 (Rn) to equation (2.186) for all f £ L2 ( I P ) . Proof: First note that by
|(^/)o|<||/||o|M|o<||/|||Mki
(2.188)
every f £ L2 (Rn) defines a continuous linear functional on H^'1 (Rn). Moreover, Theorem 2.3.23 and Theorem 2.3.24 imply
\B\(u,v)\ < IMU.ilMki and \Bx(u,u)\
> ReBx(u,u)
> S^u^,
6t > 0.
Thus Theorem 1.2.7.41 gives the existence of a unique element u £ H^'1 such that B\{u,p)
(Rn)
= (
holds for tp £ CQ° (Rn), which proves the theorem.
•
We want to prove that the unique variational solution constructed in Theorem 2.3.27 has more regularity properties. Theorem 2.3.28. Let q{x,D) = qi(D) + q2(x,D), u £ H^'1 (Rn) and f £ L2 (Rn) be as in Theorem 2.3.27 and assume (2.141) with s = 0. Then u belongs to H^2(Rn). Proof: Denote by Je, e £ (0,1], the Friedrichs mollifier and let (ufc)fceN be a sequence in C§° (Rn) converging to u in H^'1 (Rn). It follows that B\(J£{uk),ip)
= (q(x, D)Je(uk)
+ XJE(uk),
= (Je((q(x, D) + X)uk),
86
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
Prom Corollary 2.3.18 we obtain (for k large) \\[Je,q2(x,D)]uk\\o
< c||ufc||v,,i < c||u||v-,i,
implying that [Je,q2(x, D)]uk —> w£ in L2 (R n ) for some we £ L2 (R n ) and ll^llo < c for e € (0,1] with c independent of e. Thus for fc —» oo we obtain B\(Je(u),
ip) = Bx (u, J£(), /)o - K ,
It follows that \\qx(x,D)Je(u)\\0
< ||J £ (/)||o + I M o < H/llo+c
or \\qx(x,D)JE(u))\\0<\X\\\u\\Q
+ \\f\\o+c,
which implies by Theorem 2.3.13 that ||^e(u)||v,2 < C for all e £ (0,1] with c independent of £ £ (0,1]. Thus we have u £ H*'2 (M.n) by Proposition 2.3.15. D Remark 2.3.29. Theorem 2.3.28 was proved in [155], but we borrowed the more polished version from W.Hoh [128]. Finally we have by Theorem 2.3.19: Theorem 2.3.30. Suppose that [t] + n + 2, t > 0. Moreover assume (2.161) as well as (2.183) ( with t instead of s). If\> A0, A0 taken from (2.178), and f £ H** (E n ), then there exists a unique variational solution u £ H^'t+2 (Rn) to (2.186). Note that Theorem 2.3.30 tells that every variational solution to (2.186) is already a strong solution. Finally in this section we will prove a commutator estimate for the operator qi(D) and elements of 5 ^ (R n ). Theorem 2.3.31. Suppose that qx fulfills A.l and let a £ Bs^ the commutator [a,qi(D)]u =
aq1(D)u-q1(D)(au)
(M n ). Then
(2.189)
2.4 Hoh's Symbolic Calculus
87
is defined on H^'a (K"), s > 1, and maps the space H^'s (R n ) into H^,s~1 (][£«). Moreover we have the estimate
continuously
l l h < ? i ( - P ) H k s - i < c Tl , s , v ,||o||v,, s+ i il ||w|[^, s .
(2.190) n
Proof: Of course it is sufficient to prove (2.190) for all u £ S (R ). note that ([a,qi(D)}u)A(0
= (aqi(D)u)A(0 =
(2TT)-"/ 2
/
First
(gi(D)(au))A(t)
-
a(£ - V)(qi(v) ~ qiiOMv)
dr,.
Further, we have by Lemma 1.3.6.21 and the generalization of Peetre's inequality to continuous negative definite function, Lemma 1.3.6.23, Mr?) - 9i(01 = kiV2(r7) - ql/2(0\\Ql/2(v) < l\'\
~ tl) ( 2
1/2
(l +
+
Q^iOl
l 3 9l
' « ~ r,)) + l ) ( l + « } % ) )
< ( l +
m
~ V)\ (1 + 1>{t - v))^
(1 + ^(r?)) V 2 \u(v)\dv-
Now, fi(-) (1 + V ( - ) ) ^ 1 € L1 (W1) by our assumptions and (1 + i>(-))3/2 \u(-)\ S L1 (M.n) since u € 5 ( E " ) (or H^s (R n )). Hence, Young's inequality implies l l ^ g i ^ l u l l ^ - ^ I K l + ^OJ^KkftCDJlurillo < c., n > ||a(-) (1 + VO))"* 1 M l (1 + ^ ( - ) ) V 2 |«(0lllo and the theorem is proved. D
2.4
Hoh's Symbolic Calculus for Pseudo-Differential Operators with Negative Definite Symbols
In the last section we solved the equation q\(x,D)u = / , A £ R sufficiently large, for a pseudo-differential operator with a negative definite symbol q{x, £)
88
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
having the decomposition q(x,£) = (?i(£) +<72(^,£) with q2(x,Q being a small perturbation of q\ (£). So far it was not possible to get rid of this smallness condition since the application of suitable partition of unity as in Section 2.1 fails due to the non-locality of the operators q\(D) and q2(x, D). In particular, for ifi, y>2 € CQ° (R n ) such that supp^j n supp R be a continuous negative definite function with LevyKhinchin representation 1>(t) = c+Q(t)+[
(l-cosx-$)i/(dx).
(2.191)
JR"\{O}
Suppose that for 2 < I < m all absolute moments of the measure v exist, i.e. /
|z|' v{dx) < 00,
2 < I < m.
(2.192)
./]R"\{0} *\{0}
Then 0 is of class Cm (R n ; R) and for a G NQ , \a\ < m, we have the estimate V>(£),
I W O I < cH ^ / 1,
2
(0,
oc = 0
M = l,
(2.193)
|a|>2
with co = 1, ci = (2M2) 1 / 2 + 2A1/2, c2 = M2 + 2A and ct = Mu 2 < I < m, where A is the maximal eigenvalue of the quadratic form given in (2.191). Remark 2.4.1. Note that for a measure v with compact support in R™\{0}, (2.192) holds for all k e N, k > 2. We will often need the subadditive function P • No - No k^p(k
):=jfcA2.
, 2.194
2.4 Hoh's Symbolic Calculus
89
Lemma 2.4.2. Suppose that the continuous negative definite function ip : R" -> R satisfies (2.192) for all k G N. For aM m G R and a// a e N J w /»aue l^(l+V(0)m/2|
(2-195)
Proof: Prom (2.193) it follows that
1^(1+^(0)1 < q«| ( 1 + ^ ( 0 ) 2 = £ ^
(2.196)
holds for all a G NQ which yields
Using formula (1.2.25) we get
cf(i+v(or/2 -(1+W0)
2^
<*.,-./W"ll
(1+,/,^ '
which gives
|0f(i+tf(O)m/2l
< cH (i+v(o)m/2 E n (!+^o) - "^ i} =C|a,(i+^)r/2
E
(l+^o)-^^"
m —p(|ot|)
,
where we used the subadditivity of p. O Definition 2.4.3. We say that a continuous negative definite function ip : R " - » R belongs to the class A if it satisfies (2.196) for all a G NJJ. Definition 2.4.4. A. Let m G R and ip G A. We call a C°°-function q : R " x l " ^ C a symbol in the class S™^ (R") if for all a ^ e N J there are constants ca$ > 0 such that |fl?fl£g(*,0l < c«/3 a + V K O ) 2 ' " ^
(2-198)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
90
holds for all x G R n and £ G R n . We call m G R the order of the symbol
q(x,£). B. Let tp G A and suppose that for a C°°- function 5 : Mn x Rn -> C the estimate \9£dgq(x, 0 | < c a/3 (1 + V ( 0 ) m / 2
(2-199)
holds for all a, /3 G N% and i G l ™ and £ G R n . In this case we call q a symbol of the c/ass fig1'* (R»). Clearly we have Sf^ (R n ) c S ^ (R n ). First of all we want to give examples for symbols in the class S™'^ (R n ). Example 2.4.5. From Lemma 2.4.2 it follows immediately that for ip G A the function £ 1-+ (1 + i>{£,))m/2 belongs to S™<* (1"). Example 2.4.6. Let v be a symmetric measure on R n \ { 0 } , i.e. ^ ( 5 ) = v(—B), such that the function ^ ( 0 := /
(1 - cos(y • £)) i/(dy)
(2.200)
./R»\{0}
is a continuous negative definite function. Clearly the function iKO ==
/
(1 - cos(y • £)) v(dy)
(2.201)
0<|J/|<1
is continuous and negative definite too and the measure xB (ou/oiO/M^J/) has all absolute moments of order greater or equal to 2. Further let / : R™ x R n —> K, /(a;, y) > 0, be a bounded measurable function such that for all a G NQ the functions (a;, y) H-> d"f(x, y) are bounded. We define the kernel /*(x, dy) := Xi7(o) U o } (»)/(*, v)"(dy)
(2.202)
and consider the symbol Q(x, £ ) • • =
(1 - cos(y • £)) //(a:, dy) /
Nf 1
• "" °
- / Joo<M
(2.203)
(l-cos(y£))/(x,y)z/(dy).
2.4 Hoh's Symbolic Calculus
91
First of all it follows that £ i—• q(x, £) is a continuous negative definite function. Furthermore, for (3 € NQ we find ! # « ( * . 01 <
/
(l-cos{y.e,)dPf{x,y)v{dy)
JO<\y\
< ll^/||oo /
(1 - cos(y • 0 ) v{dy) = c0 (1 +
^)).
J0<\y\
In addition, for a G NQ we get I 3 ? # ? ( * . 01 < %% f
(1 - cosfo • $)) f(x, y) v{dy)\
J0<\y\<\
,(/ V
'
isra-cosd/.ojiKdi/))
^0<|J/|<1
'
a
A
a
\y vo^ \y • i)\v{dy)
Jb<\y\<\
< C/3 c a (1 + V(0)
2-p(|a|)
2
,
where we argued as in the proof of Theorem 1.3.7.13. Thus we have proved q G S£>* (E n ) with V defined by (2.201). Example 2.4.7. Let / : (0, oo) —> M. be a Bernstein function satisfying f(s) < csr for s > 1 and some r G (0,1]. Further let ip G A and g G S™'^ (R n ), m > 2, be a symbol satisfying with 70 > 0 (2.204)
70 (l + V>(£))m/2'
Then we have / o q G S™-^ (R n ). Indeed , for any function g G C°° (R 2 n ) and any 7 G N2,™ we have, compare (1.2.25), M
3
^(/°s) = £/ ( i ) °S
£ 71H 71
H7j=7 73eNJ
^
v U^'a
(2.205)
1=1
Now, for a , / 3 g N J let 7 = (a, /?) G Ng" and put g = q(x, £)• In addition recall Lemma I.3.9.34.D which says that for the Bernstein function / the estimates \fU)(s)\
< ^jf(s),
j € No,
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
92
hold. Thus (2.205) yields i
I-YI
j' = l
QiH
\-ctj=a
l=\
Pi+---+Pj=P
-+Pj=P
h\
j
< c/(i+(,(!,o)E E n c1+^o)"ip(ia,i) /9i + -+/3 J =/3
< c a / 3 ( l + « ( a : , 0 ) r ( l + V'(0)'
-P(I«I)/2
where we used again the subadditivity of p and also the monotonicity of / . Hence we get \d?d?(f o q)(x,0|
< ^
(1 + tf(0)*-r (1 + V ( 0 ) _ £ i ^ m-r—p(|oi|)
= cQ,^ (1 + ^ ( 0 ) — * ^ , i.e. / o
9
e S p * ( r ) .
Example 2.4.8. Obviously the continuous negative definite function £ i-> |£| 2 , £ G R n , belongs to the class A. Now by definition a classical symbol q G S^0 (R"), m G R, is an arbitrarily often differentiable function g : R n x R n - » C such that I3?a£«(*,0l < c «,£ (1 + K l
2
) ^
(2.206)
holds for all a, ft G Nft and i £ l " , ( e R n . Since for all m G R TTI-IOI
m — p(|ai|)
2
(l + |e| )^<(l + i a ^ , it follows that S™0 (R n ) C S™'1''2 (R n ). Lemma 2.4.9. Let ip £ A. A. The sets S™<^ (R n ) and S ^ (R") are vector spaces. B. For ai G S^^(Rn) and q2 G S^ a -*(R n ) it follows that qx • q2 G m n Sp+ *^(R ). Thus \J S™>* (Rn) is a graded algebra (which respects the mem order of the symbols).
2.4 Hoh's Symbolic Calculus
93
Proof: Part A is trivial. In order to see Part B, we apply Leibniz' rule to find
1^(91 • <&)(*, 01 <
E
I0?'^i(*> mdfdfq2(x,
01
P'+P"=I3
E
"»i—c(l°'l)
(i + v(0)
*
">2-p(l°"l)
(i + tf(0)
2
.
a'-)-a"=a /3'+/3"=/3
and the subadditivity of p gives l ^ ( ? l - 9 2 ) ( s , 0 l < c'(l+V(0)m2+mrP
(2-207)
•
Note that Part B of Lemma 2.4.9 holds of course also for S™3''0 (R n ), j = 1,2. Next we introduce pseudo-differential operators associated with the symbol classes S™<* (R B ) and SJ*1'* (R n ), respectively. Definition 2.4.10. For 9 e SJ"-* (R") or q e Sj*'* (R") we define on S(M n ) the pseudo-differential operator q(x, D) by q(x,D)u(x)
:= ( 2 T T ) - " / 2 /" e <x *g(x, 0 ^ ( 0 # •
(2-208)
The classes of these operators are denoted by ^™<^ (R n ) and ^™,V' (R"), respectively. Obviously we have the inclusion tf™> (R n ) C tf™'^ (R n ). Since for fixed x € R" the function £ — i > q(x, 0 is polynomially bounded, it follows that q(x, D)u(x) is well defined for u £ ^ ( R " ) . Theorem 2.4.11. Let q & S^1^ (Rn). Then q(x,D) maps S (R n ) continuously into itself. In particular q(x, D) 6 \Er'n'^ (Rra) is continuous on 5 ( R " ) . Proof: For a, (3 e Nft take JV > |/3| + m + n. It follows that |^(^(27r)(-"/2) /
e f a -««(x,0fi(0de)|
< (2TT)(-"/ 2 ) ^ f
eix
Chapter 2
94
Generators of Feller and Sub-Markovian Semigroups
""""U^iX)'*'
e«-«
xdfD?q(x,Z)Dfu(Z)dt
£
|A?fi(Ol).
where we used the estimate (1 +•0(0) ^ ^ (l + l£l2)- With the notation of Section 1.2.6 we arrive at pa^{q{x,D)u)
< cpNt\a\(u).
(2.209)
Since the Fourier transform maps S (W1) continuously into itself, (2.209) implies the theorem. • To handle elements of VfJ, bols and oscillatory integrals.
(R")it is convenient to introduce double sym-
Definition 2.4.12. A. The class A consists of all functions a e C°° (R n x R") such that for all a, /? £ N£ and all 7? € R n , y £ R n the estimate < ^ ( l + M ^ ^ ^ a + bl2)^2
\9$tfa(r,,y)\
(2-210)
holds for suitable m G M, <5 6 [0,1) and r > 0. B. For a £ A and x G S ( R " x R n ) such that x(0,0) = 1 we define the oscillatory integral by Os~
/ JRn JUn
e-iyr>a(rl,y)dydi1:=
lim /
/
e-iy^X{£r],ey)a{r],y)dydr].
£-+0 ,/Rn ,/Rn
(2.211) n
Lemma 2.4.13. Let x G 5'(R ) suc/i t/ierf x(0) = 1. TTien we have uniformly on compact sets X{ex) -> 1
as £ -> 0,
(2.212)
and for all a £ NQ \{0} tue have uniformly on R n d£x(ex) - • 0
as £ -> 0.
(2.213)
2.4 Hoh's Symbolic Calculus
95
In addition, with a constant ca independent of e £ (0,1) we have for 0 < a < |a| \dZxi?x)\ < cae°{l + \x\2)-iM^.
(2.214)
Proof: Since a?x(ex) = e | a | (dyx(y)) \y=ex, (2.212) and (2.213) follow immediately. Now, write \dZx(ex)\ =
e°(eW-^\dy'x(y)\y=eX\),
which implies (2.214) for \x\ < 1. But for |a;| > 1 we find for 0 < a < \a\
e (|a, - ff) |a?x(y)|y= e x| =
fol(|a|-ff)|a?x(!/)lv=e*ll*r(|o|-
2
< ca (1 + |:r| ) and the lemma is proved.
2
,
•
T h e o r e m 2.4.14. For a £ A the oscillatory integral (2.211) exists and is independent of the choice of xProof: By our assumptions there exists m £ R, 6 € [0,1) and r > 0 such that
\d^a(V,y)\ < a
+ H ^ f l + MV"
holds for all a , ^ e N g and y, rj £ R". We choose 1,1'eN such that -2/(1-5)+m < - n
and-2/'+ r <-n.
(2.215)
It follows that (V,V) -
| (1 + I?/IT'' (1 - A , ) ' ' {(1 + h i 2 ) " ' (1 - Ay)1
is an element of L 1 (R n x 1"). Since (1 + M T * (1 - A„)' e~iy'n = e~iy'n and (l + M 2 ) - ' ' ( 1 - A / e ^ ' " = e ^ " ,
a(V,y)}\
96
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
we find by integrating by parts le ••=
/
e-tyr>x(£V,£y)a(v,y)dydv
=7R" [[JR* e^^l + H^'fl-A^Uh,^,!/))^^ = / / e-^(l + \y\Y' x (1 - A,)'' ((1 + \V\Tl
(1 - Ay)1 (x(sv,sy)a(r],y))) dydt}.
On the other hand, Lemma 2.4.13 implies with ca,p independent of 0 < e < 1 Wtf
(X(ev, ey)a(V, J/)) | < ca,p\a\\<*+0\ (l + M 2 )
2
(l + \y\2)T/ , (2.216)
where „
\a\k:=
max
sup {\d^a(V,y)\
|a+/3|
rn.+ 6\f>\
(l + \V\2)
2
T
(l + M 2 ) " ' } .
Since | ^ ( l + |r?| 2 ) s / 2 | < cs,a(l + \v\2)^
(2.217)
we obtain
I W + M 2 ) - ' (i - Avf (x(ev,eyHv,y))}\ m-2l(l-S)
*
,„
(i+M2r2,
which yields with lo = 2(Z + /') and cij> independent of 0 < e < 1 the estimate | (1 + \y\2)-1' (1 - A,)'' {(1 + M 2 )-' (1 - A,)' (x(eri,eyWri,y))}\
m-2Ul-S)
2
(i + M2)
-2l'+T
2
•
(2.218) Because (2.215) implies that the right-hand side of (2.218) is an element in L1 (E n x R"), Lemma 2.4.13 and the dominated convergence theorem leads to the existence of 03 - [
[ e - ^ a f a , V) dydr, = lim Ie
7R" 7R"
£->O
= f f e-'"-"(l + |y| 2 )-''(l-A,)''{(l + H2)-' JRn
JUL"
x (1-Ay)la(ri,y)}dydr). (2.219)
2.4 Hoh's Symbolic Calculus
97
The representation (2.219) does not involve the function Xi thus the value of Os-
!
e-iyia(rj,y)dydr)
I
is independent of x € S (R n x R n ) whenever x(0>0) = 1> and the theorem is proved. • Remark 2.4.15. Lemma 2.4.13 and Theorem 2.4.14 are taken from Chapter 1.6 of the book [189] of H.Kumano-go. Note that the proof of Theorem 2.4.14 gives also the following formula of integration by parts O.-f
I n
JR
n
e-iyr>T1aa(r),y)dydrl
= Os-[
JR
f
e'^D^a^^dydr,.
JRn JRn
(2.220) Now we may introduce double symbols. Definition 2.4.16. Let V £ A and m,m' € R. The class S™'™'^ (R n ) of double symbols of order m and m' consists of all C°°-functions q : R n x R™ x l n x l " ^ C satisfying
\d?dgd?:d£q(x,& x', Ol < ccficF (i + v(0) m / 2 (i + V-(0) m ' /2 (2.221) for all a, /3, a', /?' e N£. For q e S£ l,m ' ,V ' (R") we define on S (R n ) the operator q(x,Dx; x', Dx>)u(x) = (2*)=^
[
/ n
JR
/ n
JU
e«*-*
'>e+i*'*q(x,Z;x',Z')W)d£'dx'dS.
n
JR
(2.222)
Theorem 2.4.17. Let ip G A and q <E S^'m''^ (R n ). Then for u € S (R") the iterated integral (2.222) exists and defines a pseudo-differential operator in the c/asa*™ + m , , *(R n ). Moreover qL{x,Z):=0.-(2ir)-n is a symbol in S^+m q{x,Dx;x',Dx.)u
[
[
e-iy»q(x,Z+r,;x+y,0dyd7]
(2.223)
(R") and defines the same operator, i.e. = qL(x,D)u
(2.224)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
98
for all u £ S (R"). In particular, this implies that q(x, Dx; x', Dxi) maps continuously S (W1) into itself. Definition 2.4.18. The symbol <7L(X,£) in (2.223) is called the simplified symbol of the double symbol q(x, £; x', £'). Proof of Theorem 2.4.17: First observe that by Peetre's inequality we have (1+ /
5??]/U2N/2(1 + ^ - ^ ) ) M / 2 ,
-GH,
(2-225)
and since (1 + V(£)) < c/, (l + |£| 2 ) we get for all a, /3 £ N£ \&%tfq(x, t + T,;x + y,Z)\
+ M + V))m/2(l
+
tf(0)m'/2
+ M ) ) W / 2 < 4(! + I'/I2)|m|/2-
< 5(1 + rKQ^V
Therefore, for x, £ fixed, (77, y) H-> g(x, £ + 77; x + y, £) belongs to the class .4. and it follows that the oscillatory integral (2.223) exists. Next we claim that +m <7L(X, £) belongs to the class S™ '^ (R n ). For this we use the representation (2.219) to find
%dgqL(x,0 = ( 2 T T ) - " ^ f
f
e -'v"
x (1 - A , ) ' ' {(1 + \V\Tl(l
(l + |y|3)-''
- AyYqix^
+ ^x +
y^^dydr).
Taking I and I' sufficiently large, we may differentiate under the integral sign to get with (2.221) and (2.225) \3?tfqL{x,
0\ < caP f
(1 + M 2 ) - ' ( l + \y\2)-1'
/
x (1 + iP(£ + r,))m/2(l < c ^ ( l
+
+ V>(0) m ' /2 dydr,
^ ) ) ^ ,
i.e. qL{x,£) belongs to S?+m'''* (Rn). Now we will prove (2.224). For this let xeS{M.n x Rn) such that x(0,0) = 1. For 0 < e < 1 we put qe{x, £; x\ O = xMZ-?),
e(x-x'))q(x,
£; x', £')
and using Leibniz' rule and Lemma 2.4.13 we get for all a, /3, a', 0' £ N% with a constant capac'f)' independent of e £ (0,1] \d^dp'd^qe(x,£;x',a\
<
capa>0>(l+i>(O)m/2(l+4>(Or'/2(2.226)
2.4 Hoh's Symbolic Calculus
99
Define qu,e{x, £; x', $') := q£(x, £; x', £ > ( £ ' ) , su,e(x,Z;x'):=(2n)-n/2 ru,e := (27r)-"/ 2 /
eix'^qu,e(x,^x'^')d^,
f e-
ia:
'-« S u , e (x^;x')dx'
and fix I, n0 e N such that 2/ > n + m+, m+ = m V 0, and 2n 0 > n. Using once again the identity eix'<' = (1 + | x ' | 2 ) _ n ° (1 - A ^ ) n ° e i x '' £ ' we find for all /?' € NJJ, |/?'| < 2/, by integration by parts and Leibniz' rule
|#'< e (s, £; *') I < (27T)-/2 9f,' /
(i + | x ' | 2 ) - n 0 e - ' - « ' ( i - A ^ ) " 0 ^ ( ^ ^ ; x ' , O ^ '
<w n o (i+v^r / 2 (i+MT n o with Cq)U];ino independent of £. It follows that rUte{x,£) since (1 + ip(£))
<(27r)-"/ 2 (l + \tfyl
(2.227) is well defined and
e-ix'<(l-Ax,)lsu,e(x,tx')dx'
[
< Cg.u.J.no (1 + V ^ ) ) m / 2 (1 + K I T ' W " o , V > (1 + l e l 2 ) " 2 ^
(2.228) Thus the integral qc(x,Dx;x',Dx,)u(x)
= (2Tr)-n/2 [
eix*ru,e(x,Qdt
JR"
exists and the integral (2.222) is well-defined. Taking into account that the estimates (2.227) and (2.228) are uniform with respect to e e (0,1], we may successively apply the dominated convergence theorem to find with
?u(*,6*,,0=?(s,6*,.£W)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
100
that q(x,Dx;x',DX')u(x) = (2TT)-3"/2/
/
e 4 <*-*>« +fa '-« gtt (*, £; s ' . O <£'<*«£
/
JM" J R " J R "
= lim(27r)-3"/2 / £—>o
/" f
e^-^^'t'q^x^-x'^^dZ'dxdti
7R« 7R« 7R«
= lim^ £ (x, Dx\x'
,Dxi)u(x).
£->0
(2.229) On the other hand, for e > 0 we define QLAX,
0 =
(2TT)-"
/
/
e - ' " x N , ey)?(a;, £+77; x+y, £) dycfy,
7R" VR™
(2.230) implying by the definition of an oscillatory integral limgL,e(x,0=gL(a:,0,
(2-231)
E->0
and for ! i , ( ' £ N such that 2li > \m\ + n, 2l[ > n integration by parts yields
|giie(x,OI = c | / / e - ^ l + MV'Ml-A,)' 1 1
x
J l " JR™
{(1 + 'M 2 )"'' 1 (1 - A,) 1 ' 1 x(«7,«/)
y,0}dydV
x (1 + V>(£ + v))m/2 (1 + V>(0) m ' /2 dydr,
^ ( l +VKO)^ uniformly in e € (0,1]. Hence by (2.231) we have lim qL,E(x, D)u(x) = qL(x, D)u{x)
(2.232)
£-•0
for all u e 5 (R n ). The substitution x' = x(y) = x + y and £ = £(T/) = £' + 77
2.4 Hoh's Symbolic Calculus
101
shows however q£(x, Dx; x', DX')u(x) = (2TT)- 3 "/ 2 f
f
JRnJR"
= (2TT)-3"/2/
/
eil*-*'>e+i*'-?qe(x,t-,x',Z')u(?)dZ'dx'dZ
[ JRn
[
e^e-'yixi^ey)
JR"- JR"- JR»-
x q(x, £' + ri;x + y, £')&(£') dgdydr) = (2TT)-"/2/
eixt'qL,£(x,Z')u(e)de
JRn
=
qL>e(x,D)u(x). (2.233)
Combining (2.229), (2.232) and (2.233) finally gives q(x, Dx; x', Dx-)u{x) — qL(x, D)u(x), and the theorem is proved.
•
As a first consequence of Theorem 2.4.17 we have Corollary 2.4.19. Let ip G A. A. Ifqj G &?'•* (R n ), j = 1, 2, then qi(x, D) o q2(x, D) G tf™i+m>> (R«). 5. For any q G S ^ (R n ) there exists q* G S^^ (R n ), such that (q(x, D)u, v)0 = (u, q*(x, D)v)0 holds for all
(2.234)
u,veS(Rn).
Proof: A. We put q'(x,£;x',Z') := qi(x,£) • q2(x',£') and find q' G S^ l l ' m 2 ' , / '(R n ). Therefore the operator q'L(x,D) exists and belongs to the class ^t+m2,n> (Rnj a n d i o l u e S (R») it follows that gi(x,D)og 2 (a:,D)u(a;) = (2TT)- 3 "/ 2 /
e^qi(x,0
f
e-"'*
= g'(a;, D x ; x', Dx>)u(x) = q'L(x, D)u(x).
[
eix'<,q2{x',Ou{Odildx'd^
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
102
B. First note that for all no € N we have eix'-tq(x',£):J(x7)dx'
f n
JR
= (l + l £ l T " °
eix'<(l-Ax>r°q(x',0^)dx'
/ ./R"
q(x', D)u(x')v(x')
dx'
JRn
= (2TT)-"/2/
eix'iq{x\C)u(£)dt:^x>jdx'
/
JRn
JRn ix
= (2TT)-"/ [ e7K" JR™
= (27r)- n f
u(x) f
JR"
= (2n)-3n/2 =
[
^u(x){
^
e-i{x-x'Hq(x',£,)v(x')dx'dZdx
[
JRnJRn
u{x)j
eix'^q{x\^{^)dx'\dxdi >
f 7R™
f
[
ei(x-*')*+*li'-e'q(x',Z)v(?)d$'dx'd£dx
(u,q(x,Dx;x',D'x)v)0,
which proves the assertion with q*(x, D) = qi,(x,D).
•
So fare we have proved Theorem 2.4.20. The set * ~ ' * (R n ) :=
(J * " • * (Rn) is an algebra of meR (pseudo-differential) operators with the composition as multiplication and with involution *. Further, the graded structure of ty'jf'^ (M.n) is respected in the sense that we have (2.235) (2.236) and tf^'* (R^otf^' 1 * (E71) c $£ l+m '- ,/ ' /n» )•
(2.237)
2.4 Hoh's Symbolic Calculus
103
In every classical symbolic calculus for pseudo-differential operators asymptotic expansions of symbols play a fundamental role. In principle they allow us to reduce (up to lower order terms) operations on the level of operators to operations on the level of symbols. Such asymptotic expansions are in general not valid for the symbol class \J S^1 (M.n), ip e A. However for the class m€R
(J S™'^ (Rn) some asymptotic expansions up to order 2 can be shown. First mgR
we prove Lemma 2.4.21. Let i> e A and q G S™'m'^ (Rn) such that d%q(x, £; x', £') e s J ' - r f N W . * ( R »)
(2.238)
holds for all a € NQ . For all N € N the simplified symbol qj_, satisfies
|a|<JV
where qa(x,0
€ 5j l + m '-"(l"l).*( R n).
= D2.f%q(X,Z;x',t')\*>=*
(2.240)
Proof: By Taylor's formula we find
\a\
'
h\=N
~"
where
q^x,z;t,ri)=
f Jo
(l-tf^djqix^+t^x+z^'^dt.
Using Theorem 2.4.17, especially (2.223), we find (2ir)nqL(x,0 = E )c^N
+ E
-y°*~ a !
I
z,Z')\e=sdzdr,
*
I
I
e-iznrfqi{x,z^,r,)dzdr]
E ^ 7 «(^) + |-y|=JV E 3-*r(*.0-
|a|
+
J n
*"
—\°>-
e-iZVVad?q(x,Z;x
I
J
104
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
It remains to prove Ia = qa& 5 ^ + m ' - p ( | a | ) ' v ' (Rn)
(2.241)
for \a\ < N,
and J 7 e 5™ + m '- p ( i V ) ^ (Rn)
(2.242)
for | 7 | = N.
Let \a\ < N and take Xi,X2 6 ^0° (^ n ) such that xi and xi equal 1 in a neighbourhood of 0 e Rn. The definition of Ia and (2.220) yield (27r)"/ a = O. - f f e-"-"Tiad?q(x,£;x Jut"- 7R« = 0.-
f
+
e-izr>Dazd1q{x,£;x
[
+
z,Z%.=edzdTl z,?)\e=€dzdr]
e-i^Xi(ev)X2(ez)D^q(x,^x',a\x,=x+zdzdr] (I=J
= lim/ f e-,0 JR» „/K» = lim(2vr)"/ 2 f
X2{ez)e-nxi(^)Dax,d%q{x,Z-x',Z')\x,__x+zdz
= (27r)nD^q(x,e,x',e)\x'=x
(2rr)nqa(x,0,
=
because X2(ez)e~"xi(f) converges to (27r) - "/ 2 -times the unit mass at 0 e K" as e -» 0, and (2.238) gives qa e s™+™'-KI«l),V> ^ R „j_ Now let |7| = N. By (2.238), Peetre's inequality and the estimate l+ip(£) < c^ (1 + |£| 2 ) we find \%d;'dltfq^x,z;Z,T,)\ Jo ( l - t ) " < c ,a,a',/3,/3',7
1
^ ' ^ ' (dlq(x,t
/ (1 + ^
+ tV;x + z,OW=s) (1 + ^))m'/2
+ tv))^^
*t\
dt
JO
< Ca,a',p,P',-t\(i + ^ ) )
! : 1
^
1
(i + V ( 0 ) m 7 2 / ' ( i + V-to))
+m'_-p(JV) , j
(
|2
< c a , a W 7 ( l + ^ ^ ^ ^ ^ ^ ( l + M2)
\™~P(N)\
2
dt
|m-p(JV)|
*
Using (2.219) for /, n 0 € N such that 2/ > N + \m - p(N)\ + n and 2n0 > n it
2.4 Hoh's Symbolic Calculus
105
follows that |d^J7(x,OI = 1 / / e - " " ( l + |77|2)-'(l-A0); ' JR" J&n X { (1 + \A2) < <w/
/
n
° (1 - A , ) " 0
(i + N2)
W<^dPq^x,z^7i)})dzdn ^^(i
+
|2\-™o
N2)
m+mf — p(N) 2
x (1 + V»(0)
dzdr)
tn+m'-p(N)
<<W(1+V>(0) and (2.242) is proved.
•
Remark 2.4.22. The proof of Lemma 2.4.21 shows that qa and the remainder &* a r e i n t h e c l a s s 5 0 m + m ' - " ( | a | W (R») and C ' " ^ ' * (®n), respectively. Moreover they satisfy (2.199) with constants depending only on Ca,0,a',/3' a n d the double symbol q(x, £; x', £'). 9L-EJ«|
Corollary 2.4.23. Let ip £ A. A. For qi e Sp'^ (R n ) and q2 £ S™*^ (Mn) the symbol q of the operator q(x, D) := qi(x, D) o q2(x, D) is given by n
g ( x , 0 = qiM-tafaft+YtdetqifaftD^qifaO+qrrfaZ)
(2-243)
with qri(x,0 £ S^+m'-2^ (R»). 5 . For g G S™'^ (R n ) the symbol ofq*(x, D) is given by n d D
q*(x,D) = q(x,0+J2 ti *Mx'0+qr2(x,0
(2-244)
j=i M
'iAfe(i,Oe57"w(l").
Proof: We only have to apply Lemma 2.4.21 to the double symbols q'(x,£;x',£') := gi(x,£)g 2 (x',£) and g(x,£;x',£) := q(x',£), respectively, introduced in the proof of Corollary 2.4.19. • Next we want to introduce the Priedrichs symmetrization.
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
106
Definition 2.4.24. Let r £ CQ° (Rn) be a fixed non-negative function which is even, supported in the unit ball J5i(0) and satisfies / E „ r2(£)d£ = 1. For i/> £ A we define
F& o := (i+wor n / 8 r ((c - 6 ( i + m r 1 / 4 ) • For any q £ SJ, symbol
(2.245)
(K n ) we define its Friedrichs symmetrization to be the double
QF(t; x', O := /
F(£, CM*', C ) ^ ' , 0 dC-
(2.246)
(Note that q^ is independent of x.) Our aim is to prove T h e o r e m 2.4.25. For the Friedrichs symmetrization A, we /lave the estimates I
a
a'
of q £ S ^ (R n ), if> £
m-ipO)
|^^a?«He;^,OI
2
-p(P')
(i + WO)
2
•
(2.247) /n particular qp G S^1' (Mn) and i/ie simplified symbol qp,L belongs to S?'* (W1). Moreover, ifq£ S™'^ (IT), then 9-?P,i6Sr1,*(Rn)
(2.248)
holds. We need some auxiliary results. L e m m a 2.4.26. For all /3 £ N% we have with ^ , 7 ) 7 l £ s^pm)'i' flfj,«,o
(R n )
= (i+V'(0)-n/8 E ^ 7 , T . ( o ( « - c ) ( i + ^ ) ) - i ) 1 ' hl
x(^r-)((e-C)(l + V(0) _ i ). (2.249) Proof: For /3 = 0 we have (2.249) with
% (i + V>(£))m/2 = m (i + v(0) m/2 (i + Hi)T112 % (i + v>(0)1/2
2.4 Hoh's Symbolic Calculus
107
and differentiating in (2.249):
hl
_. Ii \ T i
x ((£ - 0 (1 + ^ ( 0 ) ) ' (S7r) ((£ - C) (1 + l K 0 ) _ i ) + x ^ l 7 1 (0 (tf - 0 ( 1 + ^(O)"*) 71 " 6 ' ( ^ ) ((£ -C)(i + V(0) _ i )
+ xJJU (0 ((£ -0(1 + ^(O)"*)71 (^^r) ((£ -0(1 + V>(0)_i) +xSJL (0 E ((* -0(1+v-tor*) 71 ^ fc=l
(^^r)^-0(1+^(0)"*) (2.250) with
x ^ U ( 0 = -^,7,71 (0 ( j + ^ ) (i + V'(O)-172 % (i + V(0) 1/2 , >S,7l(0=%^/3,7,7l(0>
*£ 7 , 71 (o=7i j (i+v
have to check that x j ^ e S~ip{m+1)',p
(Rn), I = 1 . . . . . 5 . Lemma 2.4.2
implies ( l + V ( - ) ) " 1 / 4 ' 6 W 1 / 2 ' * (IT) and ( 1 + ^ ( - ) ) " 1 / 2 ^ (1 + W ) 1 / 2 G SJ"1'* (R"). Since 0,o,o = 1, we see for /3 = 0 that xfn(-))"1/4,(1 + V ( " ) ) " 1 / % ( 1 + ^(')) 1 / 2 } e 55" 1 / 2 , *(R n ), which also gives that ^ , 7 , 7 1 G S" 1 7 2 '^ (R") for \/3\ = 1. Next note that dik (1 + ip(-))~1/4 & So~3/2,*(Rn) for |/?| = 1. Thus from Lemma 2.4.9.B we conclude for |/3| = 1 that xfnni £ SQ1^ (Rn). Since So1'* (IP) is stable under taking derivatives, we may apply Lemma 2.4.9 once again to get by induction X^, 7l7l
for all \p\ > 2. •
e
&o
(^")
108
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
L e m m a 2 . 4 . 2 7 . Let q G S£'* X/J.7,71
€
^
( R n ) , t G R and a G R n .
Then we have
with
(R")
H
Proof: Since for a G Nft we also have d«q(x,£) € S ^ ( R n ) , we m a y replace d%q(x, £) by
|7|
+ X/3,7,71(0(ar£ig)(x,|)(£<7)^ ft
*:=i
which proves the lemma, since % (1 + il>(£))l/i
G S g ^ (R«).
D
P r o o f o f T h e o r e m 2 . 4 . 2 5 : First we prove (2.247). Using L e m m a 2.4.26 and t h e support properties of r we find
\eg,d%,d%qF{£;x',Ol = I J a f n e , 0 # 9 ( * ' , C)af>(£', c) <*C < ( l + ^ ) ) - " / 8 ( l + V(£'))" l / 8 E
E
|v/3.7,-n(0V/»',y,7i«')|
M
/
7(<7'
( *~c V1 f ^ c Y
l«'-CI<(l+V-(C'))1/4 l£-CI<(l+
x (ffV)
1 4
(57V)
.(I+VCO) / /'" n/8
< <*,w a + m r
£'-C
1 4 ''Vu+v^o) / n/8
d2q(x',0dc
a+^(or
(2.252)
2.4 Hoh's Symbolic Calculus
109
where
//
/=
!#«(*', azq(x',oo d< 1/4
l«'-CI<(i+V'«')) l«-CI<(i+V>(0)1/4
Now, for a € W1, |<x| < 1, we have using the triangle inequality for ijj1/2
( i + ^ + (i+V'(C))1/M)
1/2 1/2
< (i+v(^))1/2 + (i+v«i+^(0) 1 / 4 ^) 1 / z
(2253)
< (i + v(0) 1/2 + c (I + (i + i&(0)1/4M) < c '(i + ^ ) ) 1 / 2 Thus, using the substitution ( = £+(1 4- V>(£)) Schwarz inequality that
i^v,oi 2 dc) 1/2 (/
i/i<(/ V
o is follows from the Cauchy-
-/|€-CI<(1+^(0)1/4
= (i+^(0)
n/8
( /
y
-/|«'-CI<(i+V-(€'))1/4
i^:^',e+(i+v(0)
x(l + ^ ' ) ) n / 8 ( /
ide)
V
1/4
2
^)i ^)
'
1/2
1^)1/2
< C (i+v(0) n/8 (i+v^or 78 ( i + m r / 2 , which together with (2.252) gives (2.247). Now we prove (2.248). We use expression (2.239) from Lemma 2.4.21, replacing p by ^p, and (2.247). It follows that qF,L-qFfi-
£
9F,a € S ^ " 1 ' * ( R n ) .
|a| = l
Thus it remains to show QF,a £ S ^ - 1 ' ^ (R B )
for |a| = 1
(2.254)
and ? F , o - g e ^ ( r ) .
(2.255)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
110
Let | a | = 1. It follows that
dtF{£, V) = d? ((i + V(0) _n/8 r (to - 0 (i + V-(0)~1/4)) = (1 + *(0)-" /8 ( - J r ) (fo - 0 ( 1 + ^(0)" 1/4 ) x(i + v(0) _ 1 / 2 ^(i + V(0)1/2 n fl r 1/4
1/4 +E( * ) (fo -oa+iKor ) (% &)flf (i+v-(0)~ fc=i
- (9Qr) ((, - o a+v-(or1/4) (i+^(0)_1/4. and consequently with a = {r\ — £) (1 + V'(O) 9F,a(s,0 = -DS'^(7F(«;a: , ) r)|x'= a ! = /
w e nn(
l
dfF(^,r])D^q(x,r])F^,r,)di1
= ~(i+^(or 1 / 2 ar a+^(0) r(a)^ g (x^+(l + V(0) 1 / 4 ^)^ + ^ ( i + V(0) 1 / 4 ^(i + V'(0)~1/4 fc=i
x /
^ 3 ^ ( ^ ) 2 ^ 9 ( 1 , g + (l + V ( 0 ) 1 / 4 ^ ) ^
-(i+^(0) _1/4 / (^o^M^zj-g^.e+a+^o) 1 ^)^ = / i + /a + J3We estimate each term separately. Since dxdf J
(x, £ + (1 + i/,®)1'* a) da
r\a)D«q
= 1 I r\a)dsxdlDaxq(x,i+{l+^))llAa) < £ \i\<\P\ 71 < 7
1^,7,7x1 / r2(a)\ (di^D-q) R
da (x,Z + (1 + ^))1/4c)
"
+
(l + ^)f4a))m/2
da
\\a^\da
2.4
Hoh's Symbolic Calculus
r2(a)da(l
111
+ m)m/2
= c(l +
m)m/\
where we used Lemma 2.4.27, suppr C .Bi(O), and (2.253), and further ( 1 + ^ ( 0 ) " V 2 ^ (1 + ^ ( 0 ) G 55" 1,,ft (R n ). Thus we get A 6 S ^ " 1 ' * (R n ). Analogously we find (1 + V ( 0 ) 1 / 4 9 | (* + ^ ( 0 ) ~ 1 / 4 | ^ akdkr{a)-r{o)Daxq
e
SJ"1'* ( M ")
and
((x,£ + (1 + V ( 0 ) 1 / 4 * ) da G S ^ * (R"),
implying that 1% G S^ 1 - '^ (R n ). In order to estimate 73 we use Taylor's formula to get 1 ^ (5 a r(a)) r{a)Daxq ( s , £ + (1 + V(£)) 1 / 4 a) = /
^
(5«r(a))r( < T)d ( T£)^(x,0 + ( l + V ' ( 6 ) 1 / 4 /
(aV(<7))r(a)
By the symmetry of r the first term vanishes and for the second term we get, using once again Lemma 2.4.27, dxd? J
• jf* (%£>*?) (x,£ + (1 + V ( £ ) ) 1 / 4 ^ ) dtda
(dfr(a))r(a)ak
\(dar(a))r(cr)ak\
< [
/ ' f e a f (a £jb ZJ««)(x,e+ (1 + V(O) 1/4 *<0
< c' /" |(9 a r(a)) r(a)ak\
Thus /
f
( l + V (f + (1 + V»(0) 1 / 4 i ( 7 )) ~
dtda
^
a
(aQr((0) V4 °)d° belongs to the symbol
class 5 ^ _ 1 / 2 ' V ' (H n ) which implies I3 G S™" 1 '^ (R"), and (2.254) is proved. In order to see (2.255), we use Taylor's formula to find
r(a)2q(x^+(l
9F,o(s,0= /
+
^))1/4a)da
= [ r(a)2(g(x,0 + S(l+V-(0) 1/4 ^^9(^0 •/»-
V
fc=i
112
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
+/( 1 -C^ 1 +^)) 1/2(r7 (^)M+ t ( 1 +^)) 1/ M^^ J
|7|=27-
°
The symmetry of r implies that the first order term vanishes and therefore
l7l=2
x /
7
'
/ (1 - t)r(a)2a^(djq)
(x,$ + t(l+ ^{Cif4 ^
7R" JO
A '
dtda.
(2.256) Now, from Lemma 2.4.27, suppr C -Bi(O) and (2.253), we may conclude that the right hand side of (2.256) defines a symbol in S0n~2'i' (R") and (2.255) is proved. • Now we can prove T h e o r e m 2.4.28. Let q e S™'^ (R") be real-valued. Then qF(Dx;x',Dx>) with domain S (R n ) is a symmetric operator in L2 (R n ). / / in addition q{x, £) is non-negative, then qF(Dx;x',Dx>) is a non-negative operator. Proof: For u, v G S (R n ) it follows that (qF(Dx;x',Dx<)u,v)Q = c f FfX ( I
= cfff
f e-ix'-^'<'qF{i-
e -«'-«+«'-€'
JR" JM." JRn
/
x', ?)W)
F&T,)q(x',r,)
JR"-
x F{£, rj) dr)u{i')d^dx'v{x)
d£
= cf f q(x',V)(f e »''^(r,#(r)^) x ( [
eix'
and the theorem is proved. D
dt'dx')
{x)vix)dx
2.5 Estimates for Pseudo-Differential Operators
2.5
113
Estimates for Pseudo-Differential Operators with Negative Definite Symbols Using the Symbolic Calculus
The aim of this section is to use the symbolic calculus developed in the last section to obtain estimates for the operator q(x,D) in related anisotropic Sobolev spaces. These estimates will enable us later on to construct Feller and sub-Markovian semigroups. We will need the theorem of A.P.Calderon and R.Vaillancourt which we will prove first following their paper [41]. We start with some auxiliary results. The first is a Cotlar-type lemma. Lemma 2.5.1. Let (Z, A, fi) be a measure space and Az : L2 (R n ) —> I? (K"), z G Z', be a weakly measurable family of uniformly bounded operators, i. e. the mapping z — i > Az is weakly measurable and for all z G Z we have \\AZ\\ < Mo.
(2.257)
Here, as usual, \\ • || denotes the operator norm. In addition suppose that k : Z x Z —> R+ U {0} is a non-negative function that give rise to a bounded integral operator Kop on L2 (Z; /z), i.e. Kop : L2 (Z; /J,) —> L2 (Z; n), Kopu(z) = / k(z, z')u(z')
Jz
dz',
with operator norm \\Kop\\ = M.
(2.258)
If the inequalities \\AZA*Z,|| < k2{z, z')
and \\A*ZAZ, \\ < k2(z, z')
(2.259)
holds, then the operator A:=
f Azdz
(2.260)
Jz is bounded in L2 (Rn) with norm \\A\\ < M. Proof: For z\,...,
z^m € Z, m G N, define
Tm :— \\AZlAZ2AZ3..
.AZ2m\\.
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
114
Multiplying the two inequalities A*
Tm<\\AZlA*Z2\\...,\\Az and Tm<\\AZl\\\\A;2AZ3\
\A*z2m-2AZ2m-l\\\\~Z2r,
we obtain Tl < ||A z l ||||A I l A; a ||||A; a A„||.....||A, a m _ 1 ^ a m ||||A: a r a ||. Using (2.259) we get further Tm < M01/2k(zi,z2)k(z2,
k{z2m-i,z2m)Ml/2
z3) •... •
= M0k(z1,z2)k(z2,z3)
• ...•
k(z2m-i,z2m)-
For a measurable set N C Z such that n(N) < oo and for u G I? (R n ) such that ||u||o < 1 we find
(//•^(//•^r-ir = A izi A dz
\{jN " )UN « ')"--
A
( /
S
z2m-i
dz2rn^!J ( / AZ2m dz2mJ
II1l/m ' AZ2m_xA*z dzi... dz2mu\\ "° 1/n A A* dzx... dz2m\
f f / . . . / AZ1A*Z JN
JN
JN
./A
(/"7 x
A
u
A*
< Mo [ f { [ . . . [
k( ,z )...
Zl 2 <• Jz Jz k(z2m-i,Z2m) dz2 ... dz2m-i j dz^dz JN JN
Mo [ JN
[
k^m-1\z1,z2m)dz1dz2ri
JN
where k® denotes the i-times iterated kernel function. Thus we find
Azdz
Azdz u
UN )(L )T
l/m
2.5 Estimates for Pseudo-Differential Operators XN(zi)^2m-1Hz1,z2m)X(z2m)dz2mdziy
< (Mo J J /
\ 1/ro
< [Moii{N)M2m-1} which gives
A dz
115
A dz u
, l/m
[(L * )(L * )T I/-
(M 0 /i(AT)M 2 m - 1 )
l/m
and as m —> oo it follows that <M2,
A,dz
J.
A,dz
<M,
and the lemma is proved.
•
Our next auxiliary result is L e m m a 2.5.2. The function E(x) := \x2e-xX{x>o}(x)
(2.261)
defined on M. is a fundamental solution of the differential operator (l + 4^) , i.e. l +
Tx
E
(2.262)
= £°
in the sense of distributions. Proof: From the definition of E it follows for all tp e C§° (R) and all e > 0 that
l-^H
V{x)dx
+ V{e)e-* ~ 2ee-* {
116
Chapter 2 Generators of Feller and Sub-Markovian Semigroups (\x2e~x)
Since (l + -^f
= 0 for x > 0, it follows as e -» 0 that
<(l + ^)V V >=^(0), which is of course equivalent to (2.262).
•
Now we can prove the Calderon-Vaillancourt theorem: Theorem 2.5.3. Let q : W1 x R" -> C be a function such that for all a, (3 e N%, |Q;|, \/3\ < 3, the partial derivatives d@d?q(x, £) exist, are continuous and satisfy the estimates \d^q(x,O\
(2.263)
Then the pseudo-differential operator q(x,D) which is defined on S {W1) by q{x,D)u{x)
= {2Tv)-n'2 [
e^q{x^)u^)di
JR"
extends to a bounded operator from L2 (Rn)to itself. Proof: We prove the result for the case n = 1. For this define
Using the function E from Lemma 2.5.2 we find « ( a : , 0 = / / 9(s,t)E(x-s)E(£-t)dsdt.
(2.265)
JRJR
Thus on S (R) the operator q(x, D) has the representation q(x,D)u(x) = (27T)-1/2 I eix-( f f g(s, t)E(x - s)E{£ -1) dsdtu{() d£ JR 1 2
= (27T)- / f
JR JR
f g(s,t)
JRJR
f eix^E{x
- s)E{£, - t)u{£) d£dtds.
JR
(2.266) Defining the operators Astu(x)
:= (27T)-1/2 / eixiE{x
- s)E(£ - t)u(£) d£
JR
= E(x -
S)(2TT)- 1 / 2
/ eix*E(Z JR
-t)u(£)d£,
2.5 Estimates for Pseudo-Differential Operators
117
we find q(x, D)u{x) =11
g(s, t)Astu(x)
dsdt.
JUJR
Put z — (s,t) € R 2 and observe that z = (s,t) i-> g(s,t) is measurable and bounded. Hence it is sufficient to prove that the operator Az = Ast satisfies the assumptions of Lemma 2.5.1 with (Z,A,n) = (R 2 ,B< 2 \ A<2)). Since H^Hoo < 2 e - 2 it follows that \\Azu\\2o<\\E\\l\\F-1(E(--t)u(.))\\20
=
\\E\\l0\\E{--t)u(.)\\l
< ||J5|£,NM|£|li>lft or ||AZ|| < 4e~ 4 =: M 0 .
(2.267)
Next we define the function
k\s,t):=4(i+\s\r3(i+\t\r3 with a constant CQ which will be determined later. Note that k £ L1 (R 2 ) and we put M : = c o ( f ( l + |t|)-3/2dt)2.
(2.268)
Now, let (s,t), (s',t') S R 2 and note that H3t,s't'(x,y) :=
(2TT)-1/2
f ei{x-yHE(x
- s)E{£ - t)E{y - s')E{Z - t') d£
= E{x - s)E{y - s')(27r)- 1 / 2 / e^^^E^
- t)E{£ - t')d£,
JR
is the kernel of the operator A„tA*,t,. With the transformation u>:=x—y,
r):=£ — t
and 6t := t — t' > 0
(2.269)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
118
we find for the integral in (2.269) (2 7 r)- 1 / 2 e i " t f e^EWEin
+ 0t) dt]
JR
= {2ir)-l'2-eiu't-6t 4
e ^ V (»7 + ^ V " 2 " dV
/ Jo
Analogously we may handle the case 9t < 0 and we obtain a similar expression with —8t. Thus it follows that \H,t,Mx>v)\
^ ce-W/2(l
\x-y\)-3E(x-s)E(y-s'),
+
implying that /
/ \Hst,s>t>{x,y)\2 dxdy
JR JR
< c ' e - | 0 t | / f(l
\x-y\)-6E2(x-s)E2(y-s')dxdy.
+
The substitution u + v = x — s and u — v = y — s' for &s := s — s' > 0 yields / / (1 + \x - y\)-6E2(x
- s)E2(y - s') dxdy
JRJR
= 2 [ f (l + \2v +es\)~6E2(u
+
v)E2{u-v)dudv
JRJR
= i / (1 + \2v + es\)~6 ° JR
f
(u2 - v2fe~iu
dudv
JU>\V\
\v\ = c f (l + \2v + 6s\)'6e-^dv. Ju
(2.270)
Since we have the Peetre-type estimate (1 + \2v + 9S\)-6 < c ( l + | ^ s | ) - 6 (1 + \2v\)6 , it follows further from (2.270) that /
/ ( l + | s - y\)-6E2(x
- s)E2(y - a') dxdy
JRJR
6
/ (1 + \2v\)6 e-l»l dv < c" (1 + |0 S |)- 6 • JR
2.5 Estimates for Pseudo-Differential Operators
119
Now we arrive at \\AstA:,t,\\2
< [ [ \Hat,s
= which gives
\\AstA*s,tl\\2,t-t').
(2.271)
Since the kernel of A£tAai# is given by (2Tr)-^2E^~t)E(r]-tf)
[ Ju
e-ix^~^E{x-s)E{x-s')dx
which is nothing but # t s , t v (£,??) , it follows further that
II^A't'H 2 < k2(s-s',t-t').
(2.272)
Thus the operators (A z ) z= ( Sit )eE 2 satisfy the assumptions of Lemma 2.5.1 with Mo = 2e~ 2 and M given by (2.268). Hence we have proved the theorem for n = 1. But the general case follows now by observing that we may handle each pair of variables (XJ,£J), 1 < j < n, separately. • Now we may return to the class of pseudo-differential operators considered in Section 2.4, and we will follow once again closely the work [127] and [128] of W.Hoh. First we prove Theorem 2.5.4. Let q £ S^ (M.n) and let q(x,D) be the corresponding pseudo-differential operator. For all s £ R the operator q(x,D) maps the space H^'m+S (IP) continuously into the space H^'3 (W1) and for all u e H*<m+S (R n ) we have the estimate \\q(x,D)u\\^,s
< c\\u\\,ptm+s.
(2.273)
Proof: First note that it is sufficient to prove (2.273) for all u e 5 (R n ). For s = m = 0, i.e. q £ SQ (R"), the assumptions of the Calderon-Vaillancourt theorem, Theorem 2.5.3, are fulfilled which implies that lk(*,.D)u||o
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
120
holds for all u G L2 (R n ) and c depends on the constants ca,p, \a\, |/3| < 3, from (2.199). Next let s — 0 and m arbitrary. It follows that q(x, D)u(x)
= (2TT)-"/2
eixtq(x,o(i+i>(orm/2(i+mr/2HOdt;
/
and q(x, £) (1 + ip(£))~
belongs to <SQ (R™) which yields the estimate
\\q(x, D)u\\0 < c\\ (1 + V(£>)) m/2 «||o =
c\\u\\^m.
Now, by Corollary 2.4.19 we know that (1 + ip{D))s/2 oq(x, D) G %+m^ which leads to \\q{x, D)u\\^tS = || (1 + ip(D))3'2 q(x, D)u\\0 < and the theorem is proved.
(R")
c\\u\\^t3+m,
•
Next we give a sharp Garding inequality. Theorem 2.5.5. Let q G S™''* (R n ) be non-negative, i.e. q(x,£) > 0 for all x G R™, £ G R n . Then there exists a constant K > 0 such that Re (q(x, D)u, u)0 > -Jr||u||J > S ! -i
(2.274)
holds for all u G S^R"). Proof: Denote as before by qF(Dx; x', Dx>) the Friedrichs symmetrization of q(x, D). From Theorem 2.4.25 we know that q - qF,L G S^'1^ (R™), hence m 1+S by Theorem 2.5.4 the operator q(x, D) -qF{Dx\ x', Dx>) maps H^ (R n ) continuously into H^'* (R n ), s G R. Moreover, by Theorem 2.4.28 the operator QF(DX; X', DXI) is non-negative. This implies now Re (q(x, D)u, u)0 = Re (qF(Dx; x', Dx>)u, u)0 + Re ((q(x, D) - qF(Dx; x', Dx>)) u, u)Q >Re((l+ij(D))-—(q(x,D) - qF(Dx; x', Dx,))u, (1 +
^(D))I^u)0
> -\\(q(x, D) - gF(Dx; x', Dxl))u\\^_^
>-K\\u\\l^.
U
H ^ ^
2.5 Estimates for Pseudo-Differential Operators
121
We want to examine the sesquilinear form B(u,v):=(q(x,D)u,v)0
(2.275)
more closely. Theorem 2.5.6. Let q e S'™''0 (R n ) be real-valued and m > 0. It follows that \B(u,v)\
(2.276)
holds for all u, v G 5(IR n ). Hence the sesquilinear form B has a continuous extension onto H^'^2 (W1). If in addition for all x e l n 9(a:,0>To(l + ^ ) ) for some 70 > 0 and R>0
m / 2
for\£\>R
(2.277)
and
lim V(£) = 00 hold, then we have for all u G H^,m^ ReB(u,u)
(2.278) (W1) the Garding inequality
> ^rWuWln-XoWuWl.
(2.279)
Proof: Using Theorem 2.5.4 we find \B(u,v)\^\(q(x,D)u,v)0\ = ((l + TP(D))-m/4q(x,D)u,(l+iP(D))m/4v) \
I /ol
<4uU,?\\vU,%> which proves (2.276). Now if (2.277) holds, for A sufficiently large we have 7 o ( 1 + V ( £ ) ) m / 2 for all x G R n and £ G R n . Hence the symbol r ( s , 0 = - K I M f t ™-i
Chapter 2
122
Generators of Feller and Sub-Markovian Semigroups
ReB(u, u) > 7o||«||v,, f - # H l ^ s = i -A||u||S.
(2.280)
For m — 1 < 0 we have \\u\\, m-i < ||u||o or — IML ™-i > — |Mlo which yields II
ReB(u,u)
nip,
2
II
H
n
nifi,
II
2
II
>y0\\u\\l^-(K+\)\\u\\l
(2.281)
But forTO— 1 > 0 it follows from (2.278) that for any e > 0 we have
(l + iKO) 2 * 1 < e 2 ( i + V(0) m / 2 +c 2 ( £ ), which leads to ll«lljlfflfi<e||«|||l¥+c(e)||u||g. Taking £ = ^
we arrive once again at (2.279).
(2.282) •
Rerricirk 2.5.7. Note that the proof of (2.279) also yields the estimate ReB(u,u)
> ^H*,,
f
-Ai||u||$)2!-i.
(2.283)
In fact, forTO— 1 > 0 we have ||u||o < ||u||^, m-i. or —|Mlo > —||u|L ™=ii and (2.280) gives (2.283) for m - 1 > 0. In case TO-l<0butm>0 condition (2.278) leads to
i < £ 2 ( i + v ( o r / 2 + i ^ (1+^))"^, which implies also (2.283). It is also possible to prove a lower bound for the operator q(x, D). T h e o r e m 2.5.8. Let q e S™>^ (R n ) be real-valued and assume (2.277) as well as (2.278). Then for s > —m we have y l M k m + . < \\q{x,D)u\\%tS+c\\u\\lm+s_i_ forallueHxi''a+m{Wl).
(2.284)
2.5
Estimates for Pseudo-Differential Operators
123
Proof: We set rs(x, £) := q(x, £) 2 (1 + ip(£))s and observe that r . ( x , 0 > 7 g ( l + V'(0) m + a
for |^| > il.
From Corollary 2.4.23.B we know that the leading term in the expansion of the symbol q*(x, £) is given by q(x, £) = q(x, £). Thus we get \\q(x, D)u\\%tS = ( ( 1 + i>{D))sl2 q(x, D)u, (1 + i>(D))s/2 q{x, D)u)Q =
(q*(x,D)(l+i>(D))sq(x,D)u,u)0
= Re (rs(x, D)u, u)0 + Re (r(x, D)u, u)0 with f(x, D) e $2(*+m)-i,^ ^ R „ ^
NQW w e m a y a p p l y T h e o r e m
2.5.6 in form
of Remark 2.5.7 to obtain
and the theorem is proved.
D
In order to use Theorem 2.5.8 to obtain regularity result for certain solutions of the equation q\(x, D)u = f, we have to examine the Priedrichs moUifier more closely in the context of the symbolic calculus introduced in Section 2.4. For this let j be as in (1.2.77) and js(x) := e""j'(f), £ > 0. The Priedrichs moUifier is given by Mu)(x)
= (je*u)(x) = (27r)'n/2
f
e^j&mt)
d£.
(2.285)
Hence J£ is a pseudo-differential operator with symbol (j £ )(£) = j(e£)L e m m a 2.5.9. For any continuous negative definite function i\> £ A, see Definition 2.4.3, and any e > 0 the function £ — t > J(e£) belongs to the symbol class n S°p^{M. ). Proof: Since d£j(e£) = e|a|9"5'('7)U=ef \dfj(eO\
we
find
for
l£l < 1 and a e Ng
< e | a | sup (1 + V ( £ ) ) P ( H ) / 2 Halloo (1 + V ( 0 ) " " ( | a | ) / 2 • l€l
124
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
But for |f | > 1 it follows since j G S (1") that
i#-<*)i * £w(i Yiy /a(i+ ^^)~ p(iai)/2
•
For |f | > 1 and 0 < e < 1 we find for |a| = 1 e^(l
+ m)PiM)/2 (1 + K I 2 ) 1 / 2
^ e(l + |f|2)1/2 - ^ ( 1 + 1^2)1/2 (i + K | 2 ) l / 2 C
2 C
- \l
+
H??'
~ *-
But for |a| > 2 it follows that e' a l (1 + ^ ( 0 ) p ( | a | ) / 2 (1 + 1^12)1*1/2 - ^ -^
(
£l Q i+eW|f| 2 1 + K|2)H/2
( £
2
+ £
2
m
N / 2 -C^(1
+ m
W/2
- * V
Thus we have Corollary 2.5.10. For any ip G A the Friedrichs mollifier is a pseudodifferential operator with symbol j(e£) G S£'^ (E n ) and for 0 < e < 1 the constants in the estimates (2.198) are independent of e. Now take q € S™-* (W1) and consider the commutator [q(x, D), J £ ] = q(x, D)JE-J£q(x,
D)
(2.286)
which has the double symbol 9foOS"(eO-J'(eO0(s',O-
( 2 - 287 )
From the asymptotic expansion (2.239) it follows that [q(x,D), Je\ has order m - 1, i.e. it maps #V-,m-i+« (jjn) continuously into i ^ ' 8 (R") and the estimates for the remainder terms are uniform for 0 < e < 1. For these reasons we have proved
2.5 Estimates for Pseudo-Differential Operators
125
Theorem 2.5.11. For q G S£*'* (R n ) and s g l there is a constant c independent of e, 0 < £ < 1, such that
|| [q(x, D), J£]w|U,s < C I M I ^ - H , - !
(2.288)
holds for all u G # * " » - i + * (£«). For solving the equation q\{x, D)u = / we may now proceed as in Section 2.3 and set Bx{u,v)
= (q(x,D)u,v)Q+X(u,v)0.
(2.289)
From Theorem 2.5.6 we know that B\ extends to a continuous sesquilinear form on H^'m/2 (R n ) which we denote once again by B\. As in Definition 2.3.26 we call u G H^,m/2 (Rn) a variational solution to the equation qx(x, D)u = q(x, D)u + Xu = f
(2.290)
for all A G R and / € L2 (Mn) if Bx(u,
(2-291)
holds for all 0, is real -valued and fulfills (2.277). Then for all A > Ao, Ao taken from (2.279), there exists for all f G L2 (R n ) a unique variational solution to (2.290). Proof: First we note that for / G L2 (R™) a linear continuous functional on jji/>,m/2 ^ n ^ j g g j v e n by y, ,_> ( ^ y) 0 - This follows from the obvious estimate |(V,/)o|<|No||/||o<||/||oNkm/2-
The rest of the proof is completely analogous to the proof of the Theorem 2.3.27. • Theorem 2.5.13. Let q G Sm^ (R n ), m > 1, be as in Theorem 2.5.8. For f G H^'3 (R n ), s > 0, any variational solution u G H^-ml2 (R n ) to (2.290) belongs to H^m+S (R n ).
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
126
Proof: For 0 < e < 1 set ue := J£(u). Then we have by Proposition 2.3.15 that ue G # ^ - t + m (R") for alU G R and for t < s we find (q\(x, D)ue,
= (V, Mf))o
+ ([q\(x, D), JE]u,
or \\qX(x,D)ueU,t
< \\Wf)U,t
+ \\Mx,D),
Je]u\\^t.
From Theorem 2.5.8 applied to q\(x, D) we deduce that ||Ue|U,t+m < C\\qx(x, D)ue\\^,tt
+ Ci\\ue\\
< C2 (\\fU,t+m + ||u e || Now, for t = t0 := ^ Theorem 2.5.4 that
+ \\[qx(x,D),Je]u\\xptty
we find by Theorem 2.5.11, Corollary 2.5.10 and
KIU,?+i < C2 (ll/IU,. + H k ? + HU.f - i ) i which yields by Proposition 2.3.15 that u G H^+s (R n ). But now we may repeat our argument with ti — ^^ < t0 + \ to obtain u G H^'^+1 (R n ), and further recursive applications yield the theorem. •
2.6
Feller Semigroups and Sub-Markovian Semigroups Generated by PseudoDifferential Operators
In this section we will see how it is possible to use the estimates for the operators considered in Section 2.3-2.5 to extend these operators as generators of a Feller semigroup and certain L p -sub-Markovian semigroups, respectively. Our aim is to use the Hille-Yosida-Ray Theorem, Theorem 1.4.5.3, to get the Feller semigroup. Then we will apply Theorem 1.4.6.20 in order to get the L2sub-Markovian semigroup. Finally we will use some interpolation techniques to construct the L p -sub-Markovian semigroups for 2 < p < oo. In case of symmetric operators we get also the L p -sub-Markovian semigroups for 1 < p < 2.
2.6 Some Feller and Sub-Markovian Semigroups
127
For this purpose let q(x, £) be a negative definite symbol and consider on C%° (R";R) the operator -q{x,D). Since C£° (R n ;R) is dense in C^ (M n ;R) and — q(x, D)satisfies on C£° (R n ; R) the positive maximum principle, see Theorem 1.4.5.6, an application of Theorem 1.4.5.3 "only" requires to solve for some A > 0 the equation qx(x, D)u = q(x, D)u + \u = f
(2.292)
for a dense set (in C^ (R"; R)) of right-hand sides / in the space C$° (R n ; R). This problem is too hard to attack. Our strategy is to consider q(x, D) on a large domain where (2.292) is easier to handle. For this it is necessary to prove that — q(x, D) also satisfies the positive maximum principle on a larger domain. Theorem 2.6.1. Let D{A) C Coo(R n ;R) and suppose that A : D(A) -> Coo(R";R) is a linear operator. In addition assume that Co°(R n ;R) C D(A) is an operator core of A in the sense that to every u £ D(A) there exists a sequence (
lim
\\A
k—»oo
If A\c°° satisfies the positive maximum principle on CQ° (R";R), then it satisfies the positive maximum principle also on D(A). Proof: We have to prove that for any u € D{A) such that u(x0) = sup u(x) > 0 it follows that AU{XQ) < 0. For u and x0 as above choose
x£Rn
¥> £ C£° (R n ; R) such that 0 we have now sup (u + r]tp)(x) = U(XQ)+T] > 0.
Since C£° (R n ; R) is a core of the operator A, there exists a sequence (pk)k&N, fk S C£° (R n ; R), such that lim ||Vfe-(u+w)||oo - 0 k—*oo
and lim \\Aipl-A(u
+ r)=0-
k—*oo
(2.293) For each fc € N take a point xk £ R n where the function
(2.294)
128
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
Suppose that there is a neighbourhood U{XQ) of XQ such that Xk € U(XQ) for all k. Then there is an e > 0 such that (u + rj xQ and that ^(xk) > 0 by (2.294). Since A satisfies the positive maximum principle on CQ° (Mn; K) we get Au(x0) = A(u + r)(f)(x0) — T]A 0 was arbitrary, thus we have maximum principle on D(A). D
AU(XQ)
< 0, i.e. A satisfies the positive
Remark 2.6.2. Theorem 2.6.1 was proved in concrete situation in [154] and [162]. The proof presented here is taken from W.Hoh [128] who, however, modeled it after these papers. Now let us come back to equation (2.292). Note that for solving (2.292) we will not use the positive maximum principle, hence we may consider complexvalued functions, i.e. we may work in spaces of complex-valued functions. The following strategy for solving (2.292) was suggested in [153]. Let us suppose that we may extend q\(x, D) to some space H^'3 (R n ) where if> is a fixed real-valued continuous negative definite function. In addition suppose that ip satisfies
tf(0 > coltr
(2.295)
for some CQ > 0, r 0 > 0 and all £, |£| > R. Then it follows from Theorem 1.3.10.12 that for s > -2ro n
H+* (R ) ^ Coo (K n )
(2.296)
and ^||oo
^
Cs,ro,n\\'U'\\TP,S
(2.297)
2.6 Some Feller and Sub-Markovian Semigroups
129
holds. Thus assuming (2.295) and having in mind some properties of the operators considered in Section 2.3 to Section 2.5 we may consider operators q\(x, D) satisfying q\{x, D) : ff*.*°+« (R n ) -» H*'*0 (R n )
(2.298)
\\qx(x, D)u\\^t0
(2.299)
and < c||«||^,t 0+2)
where to > ^ (and CQ° (R n ) being an operator core). Since now we have Hi>,to+2
( R « ) °c Hi,,to
(Rnj
to+2
c
Q^ (fl^n) j
n
t
foIlows
t h a t
qA
(X] £ ) ,
Le_
-qx(X,
D)
n
with domain H*< (R ) is a densely defined operator on Cx (R ). In virtue of Theorem 2.6.1 the operator -q\(x, D) satisfies on H^to+2 (R n ) also the positive maximum principle. So far we have reduced an application of the Hille— Yosida-Ray Theorem, Theorem 1.4.5.3, to solving the equation q\(x, D)u = / in the space H^>to+2 (R n ) for (all) / G H^'to (R n ). The advantage is that we may now work in the scale of Hilbert spaces H^'s (R n ) and we may consider first variational solutions. Suppose that for some A > 0 there exists for all / e L 2 (R n ) a variational solution u G H^'1 (R n ) to (2.292), i.e. we may extend the sesquilinear form Bx(u,
(2.300)
to H^1 (W1) and that for any / € L 2 (R n ) there exists u G H*'1 (R n ) such that Bx(u,
(2.301)
holds for all ip G H^'1 (R n ). Then we may try to get some "regularity" theory of elliptic type, but now in the scale H^'3 (R n ): Prove that / G H^'s° (R n ) always implies for a variational solution u G H^<1 (R n ) that u G H^'ao+2 (M n ). But a sufficient condition for the existence of variational solutions is beside the continuity of B\ the validity of a Garding inequality, and the regularity result might be deduced from certain lower bounds for q(x,D). It turns out that we may apply this strategy described above in several situations. First we consider the operators handled in Section 2.3. Let t/j : R n —» R be a fixed continuous negative definite function satisfying
tf(0 > <*KI'
(2.302)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
130
for some Co > 0, ro > 0 and all |£| > 1. Further recall the definition of the constants cn,fc, 7 m , n and cy, given in (2.130)-(2.132). We want to apply Theorem 2.3.30 with tQ such that H^to (R") <-> C^ (R") holds. Thus we have to take t0 > f-, say ro '
to:=
n + 1. ro
(2.303)
Assumption 2.6.3. We assume that the function q : W1 X W1 —> C is a continuous negative definite symbol having the decomposition 9(*,0=<7i(0+
(2-304)
such that with ip as in (2.302) the following conditions hold: A.l. The function qi is assumed to be continuous and negative definite and to satisfy with 70 > 0 and 71,72 > 0 7oV-(0 < R e 9 l ( 0 < 7 i ^ ( 0
for all |^| > l ,
(2.305)
and I l m g i ( 0 | < 72 R e g i ( 0
for all £ G R n .
(2.306)
A.2.mo. Set rao = to + n + 2 = \ — \ + n + 3, note to > 1. We assume that x i-» q2(x, £) belongs to Cm° (Rn) for all ^ e R " and that we have the estimate \d:q2(x, 01 < P*(x) (1 + 1>(Q)
(2-307)
for all a & NQ, |a| < m 0 , with functions
llValUi < T 7 o ( — A -
).
(2.308)
n,m0,-£,i/>
Note that A.3.m 0 implies that (2.162) holds with 60 = ;j7o and (2.184) holds with i = ^70, too. Theorem 2.6.4. Suppose that Assumption 2.6.3 holds with to Then —q(x, D) extends to a generator of a Feller semigroup.
ro
+ 1.
2.6
Some Feller and Sub-Markovian Semigroups
131
Proof: Consider the operator (A, D{A)) on Coo(R";R) with domain D(A) = if^>' 0 + 2 (R";M) and A = -q{x,D). It follows by Proposition 2.3.6 and Theorem 2.3.11 that Au G tf^'*0 (R";R) if u G D(A), and therefore Au e C c o t R " ; ! ) and (A,D(A)) is a well defined operator on ( ^ ( R ^ R ) with a dense domain. Moreover, by Theorem 2.6.1 it satisfies on D(A) the positive maximum principle. Further we may apply Theorem 2.3.30, thus for A > A0, A0 taken from (2.170) we find for every / G H^to+2 (R";R) satisfying (A - X)u = -q\(x,
D)u = f.
The theorem follows now by the Hille-Yosida-Ray Theorem, Theorem 1.4.5.3. Remark 2.6.5. For real-valued symbol Theorem 2.6.4. was proved in [155], an extension to complex-valued symbol was given in [156]. The next step is to use our estimates in order to prove that —q\(x, D), A sufficiently large, extends to a generator of an L 2 -sub-Markovian semigroup. The idea is to prove that — q\(x, D) extends to a generator of an L 2 -contraction semigroup and then to apply Theorem 1.4.6.20. Theorem 2.6.6. Under the assumption of Theorem 2.3.27 and (2.162) with s = 2 the operator —q\(x,D) defined on .H^' 2 (R";R) is for A > Ao, A0 from (2.179), a generator of a strongly continuous contraction semigroup on i2(Rn;R). Proof: We will apply the Hille-Yosida theorem, Theorem 1.4.1.33. From Section 2.3 we know that (-q\(x, D), H^'2 (R™; R)) is a densely denned closed operator on L2 (R"; R) and for A > A0 the equation q\(x, D)u = f is uniquely solvable for all / G L 2 ( R n ; R ) with a solution u G # ^ 2 ( R n ; R ) by Theorem 2.3.28. It remains to prove that —q\(x,D), A > Ao is dissipative. But for all peCg3 (R n ; R) we have (qx(x,D)
>0
which extends to all u G H^'2 (R";R). Now the dissipativity of A > Ao, follows from \\TU+qx(x, D)u\\2 = r 2 || W || 2 +||g A (^, and the theorem is proved.
•
D)\\2+2T(U,
qx(x, D)u)
-qx{x,D),
Chapter 2
132
Generators of Feller and Sub-Markovian Semigroups
Corollary 2.6.7. Under the assumptions of Theorem 2.6.6 and Assumption 2.6.3 with t0 = [^1 + 3 the operator (-q\(x, D), H^-2(Rn;R))
is for
all A > Ao a Dirichlet operator and generates an I? -sub-Markovian semigroup. Proof: We only have to apply Theorem 1.4.6.20 for p = 2 with U = 4>,t and to note that [qx(x, D)]'1 (H^to+2 (Rn; R)) = H 0+2 ( R « . R ) c D{A^ t0+4 n ^• (R ;R). • Remark 2.6.8. We have seen that —q\(x,D) extends to a generator of a Feller semigroup (Tt(o° )t>o and —q\(x,D), A > Ao, extends to a generator of an L 2 -sub-Markovian semigroup (T} '' )t>o as well as to a Feller semigroup (T t (oo),A )t>o. Note that T t (oo) ' A = e- A 'T t ( o o ) . Clearly (e At T t (2) ' A ) t > 0 is a strongly continuous L 2 -semigroup and its generator is (—q(x, D),H^'to+2 (M";R)). However, in general we are not able to prove that this semigroup consists of L 2 -contractions. We only know the estimate ||e At T t ( '' ||n < e At ||u||o, i.e. following Definition 1.4.1.8, it has growth bound A. In case that we have constructed a symmetric Feller semigroup we may improve this estimate by Theorem 1.4.6.25. In fact, this theorem yields that (Tj |L 2 nc oo )t>0 extends to an L 2 -sub-Markovian semigroup (T{ )t>o with generator being an extension of —q(x, D) and it is clear that the semigroups (Tt' )t>o and (eXtTj '' )t>o must coincide. Thus we find that eXtT^ = T} '' is an L 2 -contraction. Note that it follows further that (q(x, D)u, v)o extends to a symmetric Dirichlet form with domain i T ^ R ^ R ) . Therefore we have B(u,u) > 0 for all u € H^'1 (Rn;R), B being the extension of (q(x,D)u,v)0, which yields the improved Garding inequality J B(u,u)+e||«||g
> cw||«||^ fl ,
£>0.
(2.309)
Next we turn our attention to the class of pseudo-differential operators considered by W.Hoh in [127]-[130]. The estimates provided in Section 2.5 allows us to follow the same strategy as before. First we get Theorem 2.6.9. Let tp : R n —> R be a continuous negative definite function in the class A, see Definition 2.4.3, and suppose in addition (2.295) to hold. If q(x, £) is a negative definite symbol belonging to S^ (R n ) and satisfies q{x,Q>6(l
+ il>{Q)
(2-310) n
for some 6 > 0 and all £ S R , |£| sufficiently large, then —q(x, D) defined on CQ° (Rra; R) is closable in Coo (Rra; R) and its closure is a generator of a Feller semigroup.
2.6 Some Feller and Sub-Markovian Semigroups
133
Proof: We may argue as in the proof of Theorem 2.6.4. We choose to > £ and find by Theorem 2.5.4 that -q(x,D) : # V ' t o + 2 ( R n ; l ) -» n ff*.*° (R ;R) c CooO&T;!) is a densely defined operator on Coo(R n ;R) with domain H^'to+2 (Mn;M), and further by Theorem 2.6.1 it satisfies on #iMo+2 (jjn. ^ t n e p 0 S i t i v e maximum principle. Moreover, by Theorem 2.5.12 for A > A0, A0 taken from (2.279), there exists for all / e L 2 ( R n ; R ) a unique variational solution u € H^'1 (R n ;R) to the equation q\{x,D)u = f and Theorem 2.5.13 tells us that for / e H^'to (R n ;R) this solution belongs to i I v ' ' t o + 2 ( R n ; R ) . Hence all conditions of Hille-Yosida-Ray Theorem are satisfied and the theorem is proved. • As in the case before we obtain also L 2 -sub-Markovian semigroups. Theorem 2.6.10. Under the assumption of Theorem 2.6.9 the operator (-qx(x,D),H^'2(Rn;R)), A > A0, A0 as in (2.279), is the generator of an L2-contraction semigroup which is in virtue of Theorem 1.4-6.20 an L2-subMarkovian semigroup. Hence (—qx(x,D), H^>2 (Rn;R)), is a Dirichlet operator and (B A ,fl' v '' 1 (R n ;R)) is a semi-Dirichlet form. Remark 2.6.11. In case of a symmetric operator —q(x,D) all considerations in Remark 2.6.8 apply also to the operators handled in Theorem 2.6.9 and 2.6.10. So far we construct for two classes of pseudo-differential operators Feller semigroups and L 2 -sub-Markovian semigroups. In both cases certain (disturbing) restriction show up. Either the symbol has "coefficients" x >—• q{x,£) which have only a small and controlled oscillation, or the function £ — i » q(x, £) must have certain smoothness properties which in general a continuous negative definite function does not share. However starting with one of these two classes and applying perturbation theory we may enlarge the class of pseudodifferential operators generating Feller and (or) L 2 -sub-Markovian semigroups. We will come back to this question in Section 2.8. In the remaining of this section we want to discuss the question how (if possible) to get L p -sub-Markovian semigroups for p S [1, oo) starting with the Feller and i 2 -sub-Markovian semigroups just constructed. The first part of these considerations is of more abstract nature and it will be applied later on also to the semigroups obtained via a perturbation of the semigroups just constructed. Although we may use Theorem 1.4.6.20 to get some L p -intermediate semigroups, we prefer to use some interpolation result developed in [86].
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
134
Let — q\(x,D) be first defined on CQ° (R";R) and suppose that it extends to a generator A^°' of a Feller semigroup (TJ°°'' )t>o and to a generator i j , po > 1, of an L Po -sub-Markovian semigroup (T^po>' )t>0, respectively. In order to interpolate the operators T}°°'' and T^'' we have to overcome a difficulty, namely the fact that T^'' u is pointwise defined whereas T^0'' u is only almost everywhere defined. The following considerations are taken from our joint paper [86] with W.Farkas and R.L. Schilling. A function u G Bb (R n ; R) is uniquely determined on R™ whereas an element v £ Lp (R n ; R) is an equivalence class of functions which may differ on a set of Lebesgue measure zero. By writing u £ Bb (R™; R) n LP (R™; R) we mean the uniquely determined element in Bb (R n ; R). Let S^ : LP (R n ; R) -> LP (R n ; R) and S(°°> : Bb(Rn;R) -» 5 6 ( R n ; R ) be two linear operators and let u £ B 6 ( R n ; R ) n L p ( R n ; R ) . Then S^u £ £ b ( R n ; R ) and S^u e L p ( R n ; R ) . For the latter we have S^u = S^v almost everywhere whenever v = u a.e., i.e. although u £ Bj,(R";R) n L p ( R n ; R ) can be constructed as a uniquely determined function, S^u is still an equivalence class of functions and all v £ IP (R n ;R) such that u = v almost everywhere are mapped via S^ into the equivalence class of S^u. Definition 2.6.12. We write S<*)\LPnBb=s£),Bt
(2.311) n
n
if and only if for u € V (R ; R) n Bb (R ; R) we have S(P)U
=
S(OO)U
a e
(2.312)
Since S(°°) u is uniquely determined, this allows the interpretation S^u £ Bb (R n ; R) n IP (R n ; R), i.e. we may choose S^u as representative for S^u. Theorem 2.6.13. Let S ^ be a linear contraction on LP0 (W1) and S(°°) a linear contraction on Bb (R n ) such that S^\L?onBb
= S$nBb.
Then there exists for every p, l<po
(2.313) a linear contraction S^
on
(2.314)
and therefore 5^UponLp=^P°0)nLP.
(2.315)
2.6 Some Feller and Sub-Markovian Semigroups
135
Remark 2.6.14. Theorem 2.6.13 is a type of Riesz-Thorin interpolation theorem, compare Theorem 1.2.8.1, with the space L°° (R™) of the interpolation couple being substituted by Bb (M.n). We follow the proof from [86]; we do not know another precise reference for this situation. The proof, however, follows the standard proof, see for example the monograph [24] of C.Bennett and R.Sharpley, Theorem 2.2 on page 196. Proof of Theorem 2.6.13: We denote by S the operator 0
o(po)|
„
_
c(°°)
^ • - J v r '\LPonBb — &LronBb-
Since for | + i = 1 ||5||LP_LP
= supdlSuHip; = sup{
u € Lp{W)
f
and
||U|| L P
= 1}
(Su)(xHx)dx\l
where the supremum is taken over all u £ V (R n ) with ||w||z,p = 1 and all v G IP (R™) with ||w|| iP ' = 1, it is sufficient to show that (Su)(x)v(x)dx
/
< 1
(2.316)
for all simple function u,v satisfying ||u[|ip = IMIxy = 1. Such functions u and v have a representation J
K
U=
a XA
Y1
i i
and
v
j=i
= YlbkXB^
(2.317)
fe=i
where the sets A,, j = 1 , . . . , J, and Bk, k = 1,...,K, are two families of pairwise disjoint Borel sets with finite Lebesgue measure, and the coefficients Oj and bk satisfy J
K in
52\a,rx HAj) j=i
= £ N P ' A ( " > ( 5 * ) = 1. fc=i
For each z G C let a(z) := ^ and let 6 e (0,1) be such that a(6) = i , i.e. i=£ = ±. Further, for z G C define Po
P
'
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
136 and set F{z):=
f
(Suz)(x)vz(x)dx.
F{9)=
I (Su9)(x)v0(x)
dx = /
(Su)(x)v(x)
dx
the desired estimate (2.316) will follow from Hadamard three lines theorem, Theorem I.2.5.1.B, if we show that F is analytic in ft := {x + iy; 0 < x < 1 and y £ M.} and continuous and bounded in f2 with \F(iy)\ < 1 and |F(1 + iy)\ < 1 for all t / £ R . First note that F(z) JJ
K K
= EElflilaW/°Wl6*l^ /
(^»XB fc (s)e <(ai * a ' +BI * 6 *>dx
which shows that z i-> F(;z) is an entire function. It also reveals that F is bounded in ft since the real part of a(z) is bounded there. Applying Holder's inequality and using the fact that 5 is an Lpo -contraction we have \F(iy)\<
f
\(Suiy)(x)viy(x)\dx
< II^Mii/llLPollVij,!!^ <
(2.318)
||wi„||LPo||u<»||LpJ.
Moreover, since a{6) = A and Rea(iy) = —, we have IIPO
.!«(*)/<*(«)
PO
j |l j E|i° u
^iy II LPO — /
(2.319) and similarly, since 1 — a(#) = —, K
(2.320)
A"
p A(n)
= Ei^i ' fc=i
(5 fc ) = i.
2.6 Some Feller and Sub-Markovian Semigroups
137
Prom (2.318)-(2.320) we conclude that \F(iy)\ < 1. In order to estimate |.F(1 + iy)| observe that
itii+iy(x)i = | W (x)r( i+ ^/«w
= i
(2.321)
since a(9) = - and R e a ( l + iy) = 0, and \v1+iy(x)\
=
| w ( s )(l-a(l+*v))/(l-a(«))
=
(2.322)
J^JP'
because of 1 - a(6) = 1 - ± = ^ and Re(l - a ( l + iy)) = 1. Using (2.321), (2.322) and the fact that S is also a contraction on Bb (R™) we arrive at \F(l+iy)\<
/
\(Sui+iy)(x)v1+iy(x)\dx
< \\Su1+iy\\Loo and the theorem is proved.
\v{x)\p' dx = 1,
\(vi+iy(x)\dx<
•
Remark 2.6.15. A. Clearly the consideration of Theorem 2.6.13 holds for Bb (R n ) substituted by L°° (R n ). B. Let (T t (po) ) t >o be an Z,p-sub-Markovian semigroup on V° (R n ;R). For u e LPo (R n ;R) n L°° (R n ;R), p 0 > 1, we find using the decomposition u = u+ — u~ for t > 0 that |T t (po) u| = T t (po) u+ - T t ( p o ) u - < T t (po) u+ = T^o)(u+
+Tt{po)u-
+ u-) = Tt{po)\u\
implying for u G LPo (R n ; R) n L°° (R n ; R) further 1
|T(P0)u| =
T (PO)
/
«
\ I < T(P0) / J « L N
x
which yields ||T t (po) u|| L ~ < ||u||L=o.
(2.323)
Since every u G L°° (R n ;R), u > 0 a.e., is a monotone limit of a sequence (u„)„GN, uv e LP0 (R™; R ) n i ° ° (R n ; R) and uv > 0 a.e., we may extend T t (po) to L°° (R n ; R) and it is easily seen that this extension is independent of the special
138
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
choice of approximating sequences for u+ and u~. This extension (T}°°')t>o is a contraction semigroup of sub-Markovian operator on L°° (Rn; R), but it is in general not strongly continuous. Hence according to part A of this remark all intermediate semigroups (Tt )t>o, po < P < oo, do exist as L p -sub-Markovian semigroups. Now let —q\(x, D) be a pseudo-differential operator with negative definite symbol which extends to a generator A£°' of a Feller semigroup (T^' )t>o 2 ( as well as to a generator .4^ ' of Z, -sub-Markovian semigroup (Tt '' )t>o with D(A^') = H^'2 (R n ;R) for some continuous negative definite function ip : W1 -> E. Moreover denote by (T t (oo) ' A ) t > 0 the extension of (T t (oo) ' A ) t > 0 to the space Bb (R n ;R) which according to Theorem 1.4.8.1 and Remark 1.4.8.2 always exists. Corollary 2.6.16. Let —q\(x,D), A > 0, be as above and suppose that for all t>0 the operator Tl \L2nBb andT{ \L2nBh coincide. ThenT} '' \L*nBb extends for all 2 < p < oo to a contraction T t on V (R"; R). The family (Tt )t>o is an LP-sub-Markovian semigroup and —q\{x,D) extends to an Lp-Dirichlet operator A£ . Remark 2.6.17. A. Note that for the operators considered in Theorem 2.6.6. and Theorem 2.6.10 the assumptions of Corollary 2.6.16 are fulfilled which follows for example from the proof of Theorem 1.4.6.20. B. It remains to determine the domain of A£ in terms of function spaces, see however Section 3.2 and 3.3.
2.7
Further Analytic Approaches for Constructing Feller and Sub-Markovian Semigroups
In Section 2.3-Section 2.6 we considered pseudo-differential operators with negative definite symbols. In particular our intention was to work with rather general continuous negative definite functions ip ( or £ i-> q(x,£)) and to develop tools to handle the associated operators. However, another way is also possible. One may use existing "classical" theorems and calculi for the pseudodifferential operators and restrict oneself to symbols q(x, £) having the additional property that x H-+ q(x, £) is a continuous negative definite function. In
2.7 Further Analytic Approaches
139
this section we briefly want to discuss some results in this direction, but we will omit often the proofs and refer to the literature. We will handle three approaches. First we apply the strategy of Section 2.6 to symbols in the class &p,6 0^™)- Then we indicate how to construct a parametrix (by using the method of E.E.Levi [198]) to the "parabolic" problem (§i + q{x,D))u = 0, u(0, x) = f(x), and how this parametrix can be used to get a transition function and hence a (Feller) semigroup. Finally we shortly mention results related to a Weyl calculus for pseudo-differential operators. Note that generators which are fractional powers of given generators will be examined in Section 2.9 within the general context of subordination. Our main reference for handling operators with symbols in the class S™6(Rn) is the monograph [189] of H.Kumano-go. Definition 2.7.1. Let m e R, 0 < 6 < p < 1, 6 < 1, and q : Rn x R n -> C an arbitrarily often differentiable function. We say that q(x, £) belongs to the symbols class S™g (Rn) if for any pair of multiindices a, [3 £ NQ there exists a constant ca^ such that \dlI%q{x,Z)\
< c a / 3 (l + |£| 2 f +4 " 2 '~ PM
(2-324)
holds. E x a m p l e 2.7.2. A. For m G N 0 and aa e C%° (R"; R) the function q(x,£) = T,\a\<ma'x(x)£a belongs to the class S%0 (Rn). B. For any r € R the mapping £ i—• (l + |£| 2 ) r is a symbol in the class S[, 0 (R"). C. The function q(£) = i£i + 5Z™=2 £j being given, we find that for
eix
Definition 2.7.3. Let q <E S%s(Rn), Condition H.m' if |g(x,OI > 70 (1 + l ^ l 2 ) " 7 2
6 < p. The symbol q(x,£)
(2.325) satisfies
for some 70 > 0, m' < m, \£\ > R, (2.326)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
140 and
dgD?q(x,£) q
M
< T W (1 + \$\2)
atpi-fi°i 2
(2.327)
holds for all a,/3 GNQ and |£| > i? with some 7o,a,/3 > 0. Using the symbolic calculus explained in [189] we obtain the following estimates for operators with symbols in the class 5"^ (R"). Theorem 2.7.4. A. Let q e S%6 (Rn), 6 < p, and s e R. Then q(x, D) is a continuous operator from Hs+m (R™) into Hs (R n ) and we have the estimate ||g(x)D)ti||.
(2.328)
B. Suppose in addition that q(x,£) satisfies Condition H.m. (sharp) Garding inequality \\q(x,D)u\\s > jo\\u\\s+m-\\\u\\0 holds for u£
Then the (2.329)
Hs+m(Rn).
Theorem 2.7.5. Let q € S™6 (W1), 8 < p, be real-valued and suppose that Heq(x,£) satisfies Condition H.m. Then we have for all u S Hm (R™) \(q(x, D)u, v)0\ < c||ii|| m/2 ||t>|| m/2
(2.330)
> 7i||w||m/2- A iH u llo-
(2.331)
and ReB(u,u)
Remark 2.7.6. Although we give no proofs of these results and refer to H.Kumano-go [189], it should be clear to the reader that in principle we may use the considerations of Section 2.4 and Section 2.5 to get these estimates. Definition 2.7.7. We say that q(x, £) belongs to the class CNS™6 (Rn) if q(x, £) belongs to S™6 (R n ) and if £ H-> q(x,£) is for all x G R™ a continuous negative definite function. Clearly, for q £ CNS™S (R") we must have 0 < m < 2. Example 2.7.8. Suppose that Y^jk=iadk{x)Z,jt,k, for all x £ R" positive definite, i.e.
akj = a,jk € Cg° (R n ;R) is
n
Yl aJk(x)Zjtk > 0
(2.332)
2.7 Further Analytic Approaches for all x € R n and ^ a
x
141
£ R.
r
For all 0 < a < 1 the function (x,£) H-> T
(Y^,k=i jk( )£j£k + o ) ~ o is f° r any To > 0 an element in the class n CNSfy (R ), compare Corollary 1.3.6.9. Now, with the same arguments as in Section 2.6. we have T h e o r e m 2.7.9. Let q € CNS™6 (W1), 6 < p, be a real-valued symbol satisfying Condition H.m. Then the operator —q(x,D) extends to the generator of a Feller semigroup (T}°°')t>0. Further, for A > Ai, Ai taken from (2.331), —q(x,D) extends to the generator of an Lp-sub-Markovian semigroup
(Tlp)'x)t>0,2 0 = c-« ( x > € ) t +o(l)
(2.333)
uniformly for x £ K, K C R" compact, and £ € R n . Another way of constructing Feller or sub-Markovian semigroups starting with a pseudo-differential operator is to construct a fundamental solution to the problem U
^1]X' +q(x, D)u{t,x) = 0
&ndu(0,x) = f{x).
(2.334)
We will first explain how to use such a fundamental solution to construct the semigroup (s), then we will briefly discuss the construction of fundamental solutions for some classes of pseudo-differential operators generating Feller and/or sub-Markovian semigroups. In the papers we will refer to, mostly the case of time-dependent coefficients is handled. However, due to the scope of this treatise we state results only for stationary coefficients. Definition 2.7.12. Let q : R" x R" —> C be a continuous negative definite symbol in the sense of Definition 2.3.1. We call a function T : (0, oo) x P x R n —> R a fundamental solution to the problem (2.334) if for fixed y £ R" the function (t, x) \-» T(t, x, y) satisfies the equation dT(t,x,y)
dt
- +q(x, Dx)T(t, x,y)=0
(2.335)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
142 and lim f
T(t,x, y)f(y) dy = f(x),
x e R".
(2.336)
t \ 0 7K"
Remark 2.7.13. Since q(x, £) is independent of t it follows for all s > 0 that lim /
T(t-s,x,y)f(y)
x e Rn.
dy = f(x),
(2.337)
t \ s JR™
Example 2.7.14. Let q(x,£) = ip(£) be x-independent and suppose that for alH > 0 the function £ — i » e - * ^ ^ belongs to the Wiener algebra, see Definition 1.3.2.8. Then a fundamental solution to the problem (2.334) with ip(D) instead of q(x, D) is given by T(t,x,y)
= (2ir)-n/2
f
e *(*-vK e -**<
n
€)
d£.
(2.338)
JR
Indeed, let / € S(R") (for simplicity). It follows that 9 r (
y,j/) = at
(2TT)-"/2
= thus
dr v)
^'
/
/
e^-y^i-^^e-**^
d£
7E"
-ip(Dx)T(t,x,y),
+ il>(Dx)r(t, x, y) = 0. Further we know that r(t,x,y)f{y)dy
= (2ir)-n'2
f
e - ^ / ( £ ) e ^
which leads for all x £ K™ to
lim f T(t,x,y)f(y)dy = f(x). Now let T(t,x,y) be a fundamental solution to (2.334). Suppose in addition F(t,x,y) > 0 for a l H > 6 and x,y e R n . Moreover we assume that (x, y) t-> r ( t , x, y) is measurable, J*Rn T(t, x, y) dy < 1, and that the ChapmanKolmogorov equations f r{t,x,y)T(s,y,z)dy
= r(t+s,x,z)
(2.339)
JR"-
do hold. In this case pt(x, A) := J^ T(t, x, y) dy gives a transition function, in particular pt(x,dy) = T(t,x,y)X(-n\dy) is a sub-probability measure. Hence we may define the operator Ttf{x)=
J JR"
f(y)Pt(x,dy)=
f JRn
f(y)T(t,x,y)dy.
(2.340)
2.7
Further Analytic Approaches
143
It is clear that (T t ) t >o has on Bb (R n ) the semigroup property and each of the operators Tt is a positivity preserving contraction on (Bb (R n ; M), j| • ||oo)Whether (Tt)t>o gives a Feller semigroup or an L p -sub-Markovian semigroup dependents of course on properties of T(t, x, y). In case that we may identify (Tt)t>o with a certain strongly continuous contraction semigroup, it is reasonable to expect that its generator is some extension of —q(x,D) with an appropriate domain. Thus we are concerned with the problem of constructing a fundamental solution to problem (2.334). When working with a symbolic calculus in Section 2.4 (or in the beginning of this section), the idea was that L A is the symbol of a first approximation of the operator [q(x, -D)] - 1 . This idea led us to the concept of a parametrix for q(x, D). The construction of a fundamental solution to (2.334) uses a different idea of a parametrix, namely to consider a fundamental solution of an operator with constant coefficients as first approximation of the fundamental solution of the original operator. This idea is due to E.E.Levi [198] for handling parabolic differential operators and it is explained in detail in A.Friedman's monograph [94] or that of S.Ito [147]. For classical pseudo-differential operators such a construction was done first by C.Tsutsumi [273], see also her paper [148] published under the name C.Iwasaki, and and is presented in the monograph [189] of H.Kumanogo. In connection with Markov processes and related semigroups it seems that A.N.Kochubei [176] was the first using this method of handle certain pseudodifferential operators. Let q(x,D) be a pseudo-differential operator with a continuous negative definite symbol and for x0 £ R" denote by q(x0, D) the operator obtained from —q(x,D) by "freezing" the coefficients at XQ. As indicated in Example 2.7.14 a fundamental solution to the problem Oil
— +q{xo,D)u = 0,
(2.341)
u(0,x)=f(x),
is given by r X 0 (t,a;,y) = (27r)-"/ 2 f
i{x y e
- ^e-tq{xo^
d£.
(2.342)
Let us define Z(t,x,x0):=Tx°{t,x,y).
(2.343)
Levi's method consists in the Ansatz to find a fundamental solution T(t, x, y)
Chapter 2
144
Generators of Feller and Sub-Markovian Semigroups
to the problem (JlL
— +q(x,D)u
= 0,
u(0,x) = f(x),
in the representation r(t,x,y) = Z(t,x,y)+
J
I
(2.344)
Z(t,x,z)$(z,r,y)dzdr
(2.345)
and $ : M.n x (0, oo) x E™ —> C (or M.) is a function to be determined. Suppose that
(— +q(x,Dx)y(t,x,y)
= 0.
Then (2.345) gives an equation for the function <J>, namely (—+q(x, Dx)) (Z(t, x, y)+f
/
Z(t, x, z)$(z, r, y) dzdr) = 0,
and by a formal calculation we find $(x, t, y) = (— + q(x, Dx)^j Z(t, x, y) +
[(q(x,Dx)-q(x0,Dx))Z(t,x,y)]$(z,r,y)dzdr. JO 7K"
(2.346) Thus the task is to make for concrete operators (or symbols) these calculations correct and most of all to solve equation (2.346) in order to determine with the help of $ a fundamental solution T(t, x, y) to problem (2.344). In the following we discuss two classes of operators where this method was successfully applied to, first we have a look to A.N.Kochubei's results obtained in [176], then we introduce the result of V.N.Kolokoltsov [180]. In both cases, as already mentioned in the beginning of this section, we do not present the proofs. In [176] A.N.Kochubei used the parametrix method to construct a fundamental solution to the problem du — +q{x,D)u
= Q, u(0,x) = f(x),
(2.347)
where the symbol of q(x, D) is given by m
9(*>0 = 5]gi(z,0, 3=0
(2.348)
2.7 Further Analytic Approaches
145
and each qj(x, £) is homogeneous of some degree 7,-, 70 > 1 and 0 < 7, < 70 for j = 1 , . . . , n. Of course the symbol qj(x, £) have to fulfill additional hypotheses, in particular hypotheses which guarantee that the theory of hypersingular integral operators developed by S.G.Samkois applicable, compare [243] and the references given therein. The following theorem shall give the flavour of the type of results obtained by A.N.Kochubei. For more general results and details we refer to his paper [176]. In particular it should be noted that he treated in his paper the case of time-dependent coefficients, i.e. symbols qj(x, t, £). Assumption 2.7.15. A. The symbol qo(x,£) is assumed to be homogeneous with respect to £ of degree 70 G (1,2). For N G N, N > 2n + 3, suppose that q0(x, •) G CN ( R n \ { 0 » and assume for |a| < N \Dtqo(x,Z)\
eeK"\{0},
(2.349)
as well as < C„\x-y\x\t\-»-M
\D?(qo(x,Z)-qo(y,t))\
(2.350)
for all x,y G Mn, £ 6 M"\{0} and \a\ < N with some A G (0,1). In addition we suppose the ellipticity condition Req0(x, 0 > Co > 0 for all x G W1, £ G 5 n _ 1 .
(2.351)
B. For the symbol qj(x,^) 1 < j < m, it is assumed that they are with respect to £ homogeneous of degree jj, 0 < 7^ < 70, and 7, ^ 1. Further for |a|
(2.352)
\DI(qi{x,i)~qi(y,m
(2.353)
and
are required. The homogeneity assumption on qj (x, £), j = 0 , 1 , . . . , m allows to represent each of the pseudo-differential operators qj(x,D) as a hypersingular integral operator. Thus we have qs(x, D)u{x) = cn^
j
Qj(x, A ) ^j0&
dh,
(2.354)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
146
where (Alhu)(x) = £fc=o( -1 )* : ( l k )u(x - kh) with / € N, and it follows that
*(*, 0 = <W,7i jT_ ^ = ^ 0 , (x, A ) ^ .
(2.355)
Following S.G.Samko, we will call Vtj ix, -rK 1 the characteristic of the (hypersingular integral) operator qj(x, D). Theorem 2.7.16 (A.N.Kochubei). Suppose that Assumption 2.7.15 is fulfilled and that the operator q(x,D) = ^2T=0qj(x,D) has a representation as sum of hypersingular integral operators each with a characteristic being even and non-negative. Then a fundamental solution to the problem (2.347) exists and gives a transition function of a Feller semigroup. Remark 2.7.17. Note that the assumption of Theorem 2.7.16 imply that £ H-> q(x, £) is a continuous negative definite function. In his paper [180] V.N.Kolokoltsov constructed Markov processes, hence semigroups, by starting with pseudo-differential operators q(x, £) with symbol given by
(2.356) wherea € (0,2), y= \y\rj and fl(x,drj) is akernelon RnxB(Sn~1). Clearly£i-» q(x, £) is a continuous negative definite function and it has the representation q(x, 0 = K|° J
| ( | | | , v) f M(a, drj).
(2.357)
Assumption 2.7.18. We suppose that for the symbol q(x,£) we have the representation (2.357)and that for the kernel n(x, drj) C i < / U-^- r,)\afx(x,dr]) 2, a real-valued measure is given for each fixed x e Rn by d"n(x, drj) which is of uniform bounded total variation with respect to a;. Theorem 2.7.19 (V.N.Kolokoltsov). Under Assumption 2.7.18 there exists a fundamental solution T(t, x, y) to the problem ^f + q(x, D)u = 0, u(0, x) — f{x), which determines a Feller semigroup by setting Tt{00)u(x)=
(
T(t,x,y)y(y)dy.
(2.359)
2.7 Further Analytic Approaches
147
For details of the proof and estimates for F(t, x, y) we refer to the paper [180] of V.N.Kolokoltsov as well as his lecture notes volume [181] and the most recent related paper [58] of V.G.Danilov and S.M.Frolovitchev. Finally in this section we will briefly discuss results due to F.Baldus [13], [14], who used the Weyl calculus of pseudo-differential operators to construct Feller semigroups. Besides an appropriate formulation of the Weyl calculus he used the theory of inverse-closed Frechet operator algebras as introduced by B.Gramsch in [109]. Unfortunately we need quite a lot of special notions to state the results. In some sense the following considerations are more or less an extended summary, for a detailed discussion we refer to [13] and [14], but also to L.Hormander's original paper [140] and his monograph [141]. Denote by the standard symplectic form on R™ x R™, i.e.
(2.360)
and for a positive definite quadratic form 7 on 1 " x R n we set
7°((*.0.(*.0) (2.361) In the following a metric on R m does mean a family 7 = (7 x )xeR m ! of positive definite quadratic forms on R m , which we may interpret as Riemannian metric and denote it sometimes by 7(cfx, dx). Given a metric 7 on R" x R™. We say that it splits if we have for each (y, 77) € R" x R n
7<,,„)((*,0, (*,0) = ill)(^x)+^v)&0
(2-362)
with two metrics 7W and 7^) on R n . Definition 2.7.20. A. A metric 7 on R m is called a slowly varying metric if there exists a constant Oy such that for x,y £ R m satisfying jx(x—y, x—y) < •£it follows that —7x(z, z) < 7y(z, z) < Cy-yx(z, z)
(2.363)
holds for all z € Rm. B. Let 7 be a slowly varying metric on R m . A function M : R m —> R + is called ^-slowly varying if there is a constant CM such that for all x,y G Rm
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
148
with 7X (x — y, x — y) < — we have —M(x)
< M(y) < cMM(x).
(2.364)
CM
Next we may introduce the notion of a Hormander metric and that of (sub-) admissible weight functions. Definition 2.7.21. A. A slowly varying metric 7 on R" x R™ is called a Hormander metric if there exists constants c 7 > 0 and 7V7 e N such that for all (x, 5 ) 6 E " x R" we have 7(x,o(-,-)<7( r x,£)(-.-)
(2-365)
and for all (x, £), (y, rf) € R" x R" f r ^ ^ H )
(2.366)
B. Let 7 be a slowly varying metric on R" x M". We call M : R™ x R™ -> R + a "/-admissible weight function if M is 7-slowly varying and satisfies with CM > 0 and NM £ N
Q±1
x R".
Denoting for a metric 7 on R n x R n the function /i 7 by
M(*,0) :=sup
(0^M) ;
MGR
XR
' ^>*(°'0>}' (2.368)
we give further Definition 2.7.22. Given a Hormander metric 7 on R™ x R". A function M : R™ x R™ —> R + is called sub-j-admissible weight function if there exists a 7-admissible weight function M0 such that M < M0 and for some m € N and c > 0 it follows that K?M0 < cM.
(2.369)
If M and JJ are both sub-7-admissible we call M an invertible sub-^-admissible weight function.
2.7 Further Analytic Approaches For (y, rj) £ l
n
149
X Rn and u : R™ X R" -> C we set
d
(v,r,)u(x> f) = ((y, rj), V2„u(x, £)),
where V2n is the gradient in Rn x R n and (•, •) the scalar product. Further we set for a metric 7 on Rn x Rra and k G N,
|«|?»->(x,0
(2.370) Definition 2.7.23. Given a metric 7 on R" x R n and a weight function M : Rn x R n -*• R+. The symioZ class S(M,-y) consists of all functions q € C°° (Rn x R") which satisfy for all k e N 0 sup J ^
;
(x, 0 e R" x R" and 0 < / < k J < 00.
(2.371)
Now we can introduce the operator associated with S(M, 7). Definition 2.7.24. Given q G S(M,7). We define the associated Weylpseudo-differential operator qw(x, D) : S (Rn) -> 5 ' (R n ) by qw(x,D)u(x)
= (2ir)-n
f
f
e^'yXq(^-,Au(y)dyd^.
(2.372)
The set of all operators qw(x, D) with symbol q € S(Af, 7)is denoted by *(M)7). Example 2.7.25. For 0 <6 < p
Taking in addition the weight function M(x, £) = ( l + |£| 2 ) S(M,7)=S%s(Rn).
we find (2.373)
Note that the Weyl-pseudo-differential operator qw (x, D) we are interested in can be transformed into the "usual" form q(x, D)(u) = (27T)""/2 /
eix
(2.374)
Chapter 2
150
Generators of Feller and Sub-Markovian Semigroups
see [141], Theorem 18.5.10, or [13], Theorem 4.7. Let us denote by B (L2 (R n )) the set of all bounded linear operators A : 2 n 2 n L (R ) -+ L (R ) which have a bounded inverse, and denote by <]>(M,7)_1 the set of all qw(x, D) G \I>(M, 7) with inverse in \I>(M, 7). Now we can state the result of F.Baldus [13]: Theorem 2.7.26. Let 7 be a Hormander metric on R™ x R™ which splits and assume # ( l , 7 ) n £ (L2 ( R " ) ) _ 1 = ^ ( l ^ ) - 1 .
(2.375)
Further let M be an invertible sub-j-admissible weight function satisfying with some k G N and CM > 0 (mhy)k<^,
(2.376)
where /i 7 is given by (2.368). If q £ S(M,f)
sup •
|gl7l'i°(^0.-;
and satisfies in addition
(x,£) eM.n x W1 and 0 < I < k\ < 00
(2.377)
for all k e No, |A+g(x,0+c,| > c,(A+M(x,0)
(2-378)
/or a// ( 1 , 0 € R n x R n , A > Xq > 0 and cq, cq > 0, and
£»-M(s,0
(2.379)
is a negative definite function, then the operator —q(x,D) : CQ° (R™; R) —> Coo (R n ; R) is a densely defined operator on C^ (R"; R) which extends to a generator of a Feller semigroup. Remark 2.7.27. A. As we will see by the following examples, Theorem 2.7.26 is the most general result so far for constructing Feller semigroups generated by a pseudo-differential operator with a symbol q e C°° (R n x R") where £ •-> q(x, £) is negative definite. B. Condition (2.375) is often called the spectral invariance of ^(1,7) in B(L2(Rn)).
2.8 Some Perturbation Results
151
Example 2.7.28. A. We know already that the class 5 ^ . (E n ) is included in Baldus' calculus, thus Theorem 2.7.26 induces Theorem 2.7.9. B. Symbols of the mixed homogeneity are partly included. C. The class S™^ (R n ) considered by W.Hoh in [127], see Definition 2.4.4, is included when working with the metric
7(^(
2.8
Some Perturbation Results
In this section we want to extend the class of pseudo-differential operators generating a Feller or an L p -sub-Markovian semigroup by perturbing a generator —q(x,D). Let us denote the perturbation by —p(x,D). In order to assure that — q(x, D) — p(x, D) is a generator we want to apply (variants of) Theorem 1.4.4.3. For this we have to assume that —p(x,D) is dissipative and (—q(x, D))-bounded. However, to get a Feller or an £ p -sub-Markovian semigroup we need more, namely that — q(x, D) — p(x, D) satisfies also the positive maximum principle or is an L p -Dirichlet operator, respectively. Thus we may immediately formulate the following result: Theorem 2.8.1. A. Let —q(x, D) be a pseudo-differential operator extending to a generator of a Feller semigroup. Denote the domain of this extension
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
152
by D^ (q(x, D)) C Coo(]R";]R). Let -p(x,D) be a pseudo-differential operator defined on D^°°\q{x,D)) which maps D^°°\q(x,D)) into Coo(]Rn;lR). n If —p(x,D) is C00(M ;'R.)-dissipative, (—q(x, D))-bounded, and if in addition —q(x, D) —p(x, D) satisfies on D(-°°^(q(x, D)) the positive maximum principle, then {—q{x, D)—p(x, D), £)(°°) (q(x, D))) is the generator of a Feller semigroup. B. Let —q(x, D) be a pseudo-differential operator extending to a generator of an Lp-sub-Markovian semigroup, 1 < p < oo, and denote the domain of this extension by D^(q(x,D)) C L P ( 1 " ; E ) . If -p(x,D) is V-dissipative, (—q(x, D))-bounded, and if in addition (—q(x, D)—p(x, D), D^ (q(x, D))) is an LP-Dirichlet operator, then (—q(x, D) —p(x, D), D^ (q(x, D))) is the generator of an Lp -sub-Markovian semigroup. Using Proposition 1.4.3.25 we get the following corollary: Corollary 2.8.2. Let 0 < a < 1 and fa : [0, oo) —» R the Bernstein Ja
=
^
function
•
A. If (—q(x, D), D(°°) (q(x, D))) is a pseudo-differential operator generating a Feller semigroup, then the operator (—q(x, D) — qf" (x, D), D^-°°^(q(x, D))) is also a generator of Feller semigroup. B. Suppose that the pseudo-differential operator (—q(x,D),D^p\q(x,D))) is a generator of an IP -sub-Markovian semigroup. It follows that the operator —q(x,D) — qf<*(x,D) with domain D^p\q{x,D)) is also the generator of an Lp-sub-Markovian semigroup. Remark 2.8.3. A. In contrary to classical pseudo-differential operators, and in particular to partial differential operators, we do not have the notion of a principal part for the operators under consideration, i.e. there is no obvious way to define the principal symbol of q(x, £) if q(x, •) : Rn —> C is just a continuous negative definite function. Hence there is no obvious notion of "lower order terms". The content of Corollary 2.8.2 is that using fractional powers we may construct to a given pseudo-differential operator certain operators which behave as lower order perturbations. B. Note further that in Theorem 2.8.1 it is clearly not necessary to assume that —p(x, D) satisfies the positive maximum principle or is an L p -Dirichlet operator, respectively. C. It is possible to prove Corollary 2.8.2 for every Bernstein function / satisfying lim *-^- < 1. x—»oo
In order to apply Theorem 2.8.1 in concrete situations we are faced with several problems
2.8 Some Perturbation Results
153
— we need to find appropriate (—q(x, D))-bounds for —p(x, D); — we have to prove that —p(x, D) is dissipative in the space C"cx)(™n! ^ ) or L p ( E n ; E ) , respectively; — we have to show that — q(x, D) —p(x, D) satisfies the positive maximum principle or is an ZADirichlet operator, respectively. The last two problems may be overcome by the assumption that —p(x, D) itself satisfies on D^°°\q{x,D)) the positive maximum principle or is an LpDirichlet operator on D^p\q(x,D)), respectively. Indeed, it is obvious that if —q(x, D) and —p(x, D) are operators satisfying on D^°°\q{x, D)) the positive maximum principle or are both Dirichlet operators on D^p\q(x, D)), then their sum — q(x, D) — p(x, D) has the same property. Further, as an operator satisfying the positive maximum principle the operator — p(x, D) is dissipative on C o o ^ R ) , compare Lemma 1.4.5.2, and as an L p -Dirichlet operator —p(x,D) is dissipative on Z*P(R;R), see Proposition 1.4.6.11 together with Proposition 1.4.6.12. We know already the structure of the operators satisfying the positive maximum principle. In order to construct interesting perturbations we will now use a very recent structure result for Lp-Dirichlet operators which is due to R.Schilling [250] and is interesting in itself. Theorem 2.8.4. Let (T}p')t>o be for some p > 1 an V-sub-Markovian semip n n group on L (R ;R) with generator {A^, D{A^)). J/C£°(R ;R) C D(A^) and if A : C£°(R n ;R) —> C(R n ;R), then —^4|c°°(K";K) W a pseudo-differential operator with symbol p : R n x R™ —> C such that £ H-+ p{x, £) is for all i £ l " a continuous negative definite function. Furthermore, if A defined on Co°(R") is symmetric, satisfies the positive maximum principle and maps C^'(Wl) into L2(M.n) then A satisfies on C£°(R";R) / (Au)(x){(u(x)~l)+)p-1dx<0,
(2.380)
i.e. (A, C Q ° ( R " ; R ) is a Dirichlet operator. In particular every symmetric pseudo-differential operator —p{x, D) with a symbol p : R n x R™ —> C such that £ i—• p(x,£) is a continuous negative definite function which maps Co°(R";R) into L 2 (R n ;R) n C(R n ;R), is a Dirichlet operator. In [250] R.Schilling proved a more general version of this theorem. The importance of this theorem in our context of perturbation results is that it makes the assumption on the perturbation —p(x, D) to be a pseudo-differential
154
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
operator with symbol —p(x,£) where £ H-> p(x,£) is a continuous negative definite function also in case of Dirichlet operators very natural. Now, once we restrict ourselves to perturbations —p(x, D) as described above, it remains to prove their —q(x, D)-boundedness. According to Definition 1.4.4.1 an operator (Q, D(Q)) on a Banach space (X, \\ • \\x) is A-bounded where (A, D(A)) is a further linear operator on X if D(A) C D(Q) and \\Qu\\x
ueD(A),
(2.381)
holds for some a € [0,1) and j3 > 0. For concrete operators we will have to work in some spaces H$'s(Rn;R), for p = 2 see Section 1.3.10 and for general p £ (1, oo) see Section 3.3 for the definition of these spaces. In particular, we usually have a result stating that the graph norm of — q(x, D) considered as an operator on LP(E™;E) is equivalent to the norm || • || ff *,2. If the interpolation estimate \\u\\Hi,,. < e\\u\\Hi,.2+c(e)\\u\\Lr,
0 < s < 2,
(2.382)
holds for any e > 0, it is clear that any pseudo-differential operator p(x, D) mapping Hps{Rn; E), 0 < s < 2, continuously into i p ( E n ; E) is automatically —q(x, ,D)-bounded provided ||g(a;,I>H|LP-|-||u||z,p ~ ||it||^,2. But we know that (2.382) holds from Proposition 1.4.3.25, especially (1.4.180). Indeed, since ||u|| H «,j = ||(id+V,(-D))S//2'"||z,p we have to apply Proposition 1.4.3.25 with a = s/2. As already mentioned, so far no general results on the L p -boundedness of the class of pseudo-differential operators we are interested in are known. However the situation is dramatically different in the case p = 2. In fact the operators considered in Section 2.3-2.5 and 2.7 are in general "L 2 continuous", i.e. map some spaces H%'2(E";E) into L 2 (E n ;M). Let us state a typical result in this direction. Theorem 2.8.5. Letq\(x,$) be as in Theorem 2.6.10. In particular q e S^. In addition let p 6 S^~v<^ for some 0 < rj < 2 be a symbol such that £ i-> p(x, £) is continuous negative definite function and suppose that —p(x, D) maps C£°(R"; E) into L 2 (R"; E) n C(R n ; R). Then the operator -q(x, D) - p(x, D) extends from Co°(E n ; E) to a generator of a I? -sub-Markovian semigroup. So far our discussion was straightforward: just adapting the case we are interested in, i.e. generators of Feller or Lp-sub-Markovian semigroups, to general perturbation results. The following discussion is a bit more sophisticated.
2.8
Some Perturbation Results
155
Its aim is to enlarge the class of Hoh's symbolic calculus. We follow closely W.Hoh [128], see also [130]. The underlying idea is that a continuous negative definite function with a Levy measure having a bounded support is arbitrarily often differentiable and admissible for constructing a symbolic calculus in the sense of Section 2.4. Now, given any continuous negative definite function ip with representation
1>(t)= [
(l-cosj/.fMdO
(2-383)
JRn\{0}
then the decomposition
VK0=V>«(£)+<M0 = f
(1 - cos y • Z)v(dQ + f
JBR(0)\{0}
(1 - cos y • Z)v{d£)
JB°R(0)
yields a smooth continuous negative definite function and a bounded one. Thus, if ip satisfies a certain minimal growth condition so will tpR, implying that ^R{D) will be a Vfl(-D)-bounded perturbation. Now let p € S?'^ with representation p(x,0= /
(l-cosy£)N(x,dy).
(2.384)
JR«\{0}
A decomposition as P(x^)=
(1 - cos y • £)N(x, dy) 7B H (O)\{O}
+ /
{\-cosy£)N{x,dy)
JB%(O)
promises a decomposition of — p(x, D) into an operator PR(X, D) with symbol in some class S^ and a bounded perturbation PR(X, D) such that under certain conditions —PR(X, D) generates a Feller semigroup as well as the original operator —p(x, D) = ~PR(X, D) — PR(X, D) by a perturbation argument. We are going to make this argument precise. As in Section 2.4 let ijj : W1 —> R be a continuous negative definite function such that ^ ( 0 > c|£r / 2 ,
|£| l a r g e and 0 < r < 2,
(2.385)
156
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
holds. Recall the definition of the symbol class S™'^, see Definition 2.4.4. Let p G S2'^ and assume in addition that £ H-> p{x, £) is a continuous negative definite function. Moreover we suppose that p(x,£) has the Levy-Khinchin representation p(x,0=/"
JRn\{0}
(l-ooay£)N(x,dy)
(2.386)
and satisfies the uniform estimate
(2-387)
p(x,0
In order to decompose p(ar,£) let 9 G C£°(R n ) be such that 0 < 0 < 1 and 8 = 1 in a neighbourhood of 0 and define N!(x,dy)
:= 6(y)N{x,dy)
and
JVafocfy) :=
(l-6(y))N(x,dy).
(2.388) Clearly, Ni(x,dy) and N2(x,dy) are both Levy kernels each being associated for a fixed a; to a real-valued continuous negative definite function. We decompose p(x, £) now according Pi(x,0=
[
(l-cosy-QNifady)
(2.389)
[
(l-ooByt)N2(x,dy).
(2.390)
JRn\{0}
and p2(x,Z)=
JK"\{0}
i.e. p(#,£) = Pi(£,£) +P2(^,£)- For i £ R " fixed the support of the measure Ni(x, •) is bounded, hence £ h-> p(a;, £) is a smooth function. Thus we are close to the situation discussed in Theorem 2.6.9. Theorem 2.8.6. Assume (2.385) and that pi G S%'^. In addition suppose that for some 6 > 0 and T sufficiently large Pi(x,0+r>6(l+m) holds for all x G R™. Then —pi(x,D) semigroup.
(2-391) extends to a generator of a Feller
Our aim is to prove that in the case of Theorem 2.8.6 also —p(x, D) = —pi(x,D) —p2{x,D) generates a Feller semigroup. We need some auxiliary results.
2.8 Some Perturbation Results
157
L e m m a 2.8.7. There is a bounded measure fi on M"\{0} such that J 12 (l-cos ( i - c o s!y/-OM^) -0Md0 = = T T TrM 2
[
<2'392)
•\{0}
and
(l + | £ | 2 ) / « < o o
f
(2.393)
JR"\{0}
holds. Proof: According to Corollary 1.3.6.13 the function y >—* m{y) = 1 + [ ia is positive definite, and since m(0) = 1 by Bochner's theorem, Theorem 1.3.5.7 there is a probability measure jx on R" such that
irw = Lcosy-^^)-
< 2 - 394 >
In particular (2.394) implies that Ji has no atom at 0 and therefore we have
/
7R"\{O}
(li--ccoossiy/ - 0 / i ( d eO) = l1- -r r7r 3- i^?2 = T 7 7^^ 7 2 i + \y\
1+
|j/r
*\{0}
and (2.392) follows with \x = /x|Rn^o}- Since y, is bounded, in order to prove (2.393) we need to show that
/
|£| 2 /i(^)<°o.
H\>\ For 0 < a < 2 we find with c ^ a ^ - r ^ ) / ^ ! - ! )
(2.395)
the Levy-Khinchin representation |£| a = < * , « /
(l-cosyO|»|-a_nrfj/
(2.396)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
158 which yields
I
\t\"tidt)
JRn\{0)
= cnta [
t)\v\-a-ndyn(d£)
[ (1 - cosy •
\y\2-a-ndy +
<^a{ J\y\
I
\y\-a-ndy
J\y\>i
_a-2^T(^yn W
Cn,a
2
! 1 -pf^zcA \2 - a
^ a)
with wn being surface area of Sn~1. Now, taking into account that the Vfunction has a pole of order one at 0 we obtain by a monotone convergence arguments that oo > lim /
lei" KdO > lim /
and the lemma is proved.
|^|>(«) = /
If | V K ) ,
D
This proof is taken from W.Hoh [128]. In fact in [165] together with R. Schilling it was proved that
\y?
f
i + M2
(l-e-^MQdt
with 1
f°°
9(0 = 7; /
(27rA)-"/ 2 e-l«l 2 /2A e -A/2 dA
2 Jo
Proposition 2.8.8. If (2.387) holds then N2(x, dy) has with respect to x e R " a uniformly bounded mass. Proof: As in Lemma 2.8.7 denote by fi the measure satisfying (2.392) and
2.8 Some Perturbation Results
159
(2.393). It follows that N2(x, K"\{0}) = /
(1 - 6{y))N{x, dy)
./R"\{0}
= c f f (1 - cos y • 0 ti<%)N(x, dy) JRn\{0} JUn = c /
p(x, £) /i(d£)
proving the proposition.
D
P r o p o s i t i o n 2.8.9. The operator p2(x,D) extends to a continuous operator from C(,(M") into itself as well as from J3f,(Mn) into itself. In each case it has the operator norm 2 sup A^(x,]R n \{0}). Proof: We use the finiteness of AT2(x, •) and the representation (2.390) of p2(x, £) to find for u e C^(Rn) -p(x,D)u(x)
=
JRn\{0}
{u(x+y)-u(x))N2(x,dy).
Further, for tp £ C^(Rn\{0}), /
(2.397)
y?(0) = 0, it follows that
JK"\{0}
= -(2TT)-"/2 f
f
(1-cosy
£)0(fl d£N2(x, dy)
•/R™\{0} JR"-
= -(2TT)-"/2/ JR"
P2(x,0mdt
Since x t-* p2(x,£) is continuous we deduce now that N2(x,dy) depends continuously on x with respect to the vague topology. This implies already the extension properties claimed, whereas the representation (2.397) yields the bound 2 sup N2(x,Rn\{0}). D a:eK"
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
160
Since p2(x, D) is a nonlocal operator we may not deduce that u e Coo(Rn) implies that p2{x,D)u € Coo(R n ). Suppose that N2(x,dy) has the property that for every e > 0 there exists R > 0 such that N2(x,BcR(0))<s
(2.398)
uniformly in a; £ R n . For u G Co°(M") clearly p2(x,D)u further, for x ft supp it it follows that p2(x, D)u(x) = /
(u(x + y) - u(x))N2(x,
is continuous and
dy)
•/R»\{0}
= /
u(x +
y)N2(x,dy)
JR"\{0\
and further \p2(x,D)u(x)\=
/
u(x +
y)N2(x,dy)
JM"\{0}
< ||u||00JV2(x,supp'u(a; + -))But if (2.398) holds it follows that JV 2 (x,suppu(x + •))—» 0 as |x| —> oo. Proposition 2.8.10. Letp2(x,^) sup p2(x, £) -> 0
and N2(x,dy)
be as before. Then
as £ -> 0
(2.399)
implies (2.398). Proof: Assume that (2.399) holds true and take tp e Cg°(Rn), 0 < (p < 1, <^(0) = 1, supp 1. Then we have N2(x,BR(0))
< f
(pR(0) -
f
(l-cosy.Z)0R(Z)N2(x,dy)
VR"\{0}
= (27T)-"/2 / JRn\{0}
< <
JRn
(27T)-"/2 f P2(X,0I^(0I^ n (2TT)-"/ 2 / P2(x,oR mm\dz+c (27T)-"/2/
p2(x,^)^|v(^)l^
[
M*,t)\
2.8 Some Perturbation Results
(l + \$\2)Rn\Rt;\-(n+3)dZ
+ c' f
161
P2 (x,o)IMlLi+c"ir
where we used that (p 6 S(Rn). N2(x, 5^(0)) -^ 0
3/2
,
Thus we proved that
as R -> oo
implying of course (2.398).
•
Corollary 2.8.11. If (2.399) holds thenp2(x,D)
maps C ^ R " ) into itself.
Proof: We know already that p2(x, D) maps Cf,(Rn) continuously into itself. Since Co°(]R™) is dense in Coo(]Rn) the corollary follows from Proposition 2.8.10. • Remark 2.8.12. A. Note that (2.399) is equivalent to sup (p(x, £) -p(x, 0)) -> 0 as |£| -> 0
(2.400)
x€R n
since from (2.389) it always follows by the uniform boundedness of the Levy measure Ni(x,-) that sup(pi(x,^)-p1(x,0)) -»0
as|f|->0.
x€K n
B. As proved in W.Hoh [128], condition (2.399) is in fact equivalent to (2.398). An independent proof is given in R.Schilling [248]. Summing up we arrive at the following result due to W.Hoh [128]. Theorem 2.8.13. Let p € S2'^ satisfy (2.387) and (2.400). Assume further that £ H-> p(x,£) is negative definite and has the decomposition (2.389) and (2.390). If —pi(x, D) extends to a generator of a Feller semigroup, so will -p(x,D). For discussing an analogous perturbation result in L 2 (R n ;]R) for a symbol P(x,Q=
/ Jm.n\{o}
(l-cosy-£)N(x,dy)
162
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
we make an assumption on the kernel N(x, dy) in (2.384), namely that N(x,dy)
= n(x,y)fi(dy)
(2.401)
where /i is a fixed Levy measure on R"\{0}. Now we decompose according to Ni(x,dy)
= 6(y)n(x,y)fj,(dy)
N2(x,dy)
= (l-0(y))n(x,y)(i(dy)
= nx{x,y)^{dy)
= n(x,y)m(dy)
N(x,dy)
(2.402)
and n(x,y)n2(dy) (2.403) where 6 is as in (2.388). The corresponding decomposition of p(x,£) is given by p(x,£)=p1(x,£)+p2(x,£)=
== n2(x,y)fi(dy)
/
=
(l-cosy-^)n1(x,y)fj.(dy)
•/R"\{0}
+ /
(1 - cosy • £)n 2 (x, y)ix(dy)
JR»\{0}
(2.404) Proposition 2.8.14. Assume that n(x,y) is bounded, i.e. n(x,y) < M for some M > 0. Then p2(x,D) is a bounded operator from L p (R";R n ) to LP(Rn; R) for 1 < p < oo, i.e. \\p2(x,D)u\\LP
(2.405)
holds for all u e Lp{Rn; R). Proof: It is sufficient to prove (2.405) for all u G C£°(Mn; R). Observe that for u e C£°(R";R) -p2(x, D)u(x) = / (u(x + y) - u(x))n2(x, 7K"\{0} = / JRn\{0}
u(x + y)n2(x,y)fj,(dy)
y)^{dy)
-
n2(x,y)fi{dy)u(x). JR"\{0}
Since /K„v r0i n2(x, y)fi(dy) < c, it follows that we only need to discuss the first
2.8 Some Perturbation Results
163
term. For v € V' (R n ; R), p' = -2-^ we find that / 1
JR"
/
u(x +
y)n2{x,y)iJ,(dy)v(x)dx
7 K " \{0} \{0}
<M
I
/
|u(x + y)||r;(x)|«ia:/i3(dj/)
•/K™\{0} JRn
<M
t
\u(x+y)\Pdx)1/P(
(f
JR"\{0}
y
n
JR
'
f
\V(X)\P'dx)1/P
WR™
Mdy)
'
= M/i 2 (R»\{0})||«|| L p||T;|| Ll ,. Dividing by \\v\\LP> and taking the supremum over all v e L P '(R";R) with ||U|| L P ' = 1, estimate (2.405) follows. O Since we have good L 2 -bounds within Hoh's calculus we can derive the following perturbation result which is again due to W.Hoh [130]. Theorem 2.8.15. Letp(x,£) =Pi(x,£)+P2(x,£) be as in (2.404) and assume that pi € Sp'^(Rn) is such that for some 6 > 0 and r large
(2-406)
P I ( S , 0 + T > « ( 1 + V(0) holds for all i f f " . \\p(x,D)u\\L2
Then we have on Co°(Rn;R)
the estimate
< c\\u\\H4,,2
and \\u\\Hi,,2 < c(\\p(x,D)u\\L2 + \\u\\L2), and —p(x,D) extends to H^,2(Rn;M.) semigroup.
as a generator of an L2' -sub-Markovian
Proof: For pi(x,D) we know all estimates stated in Section 2.5, especially we know that the graph norm of pi(x, D) is equivalent to || • ||#«,2. Now the boundedness of p2(x, D) in L 2 (R n ; R) implies the theorem. • Let us give a sufficient condition on n(x, y) in (2.401) implying that pi(x, £) satisfies the assumptions of Theorem 2.8.15. Proposition 2.8.16. Suppose that for n(x,y) NQ the estimate \d^n(x,y)\<Mp
in (2.401) we have for all /? €
(2.407)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
164
with Mp independent ofx,y
for all x,yeRn,
n(x, y)>r)>0 thenPl
6 R n , and assume (2.408)
E Sf>2(M.n;R)and (2.406) holds.
Proof: For (3 G N£ and \a\ > 1 we have \d?dPpi(x,0\
= Sf ^
/
(1 - cosy • 0"(^,y)Mi(dy)
./R"\{0}
\ya
^M^/
cos^(y)\^(dy),
and further by the Cauchy-Schwarz inequality for \a\ = 1
\d?dgPl(Xlt)\ < MpWjfh^J
f v
mn2(y dnidy))1'2
./tt"\{0}
f (1 - coB(y • Mmidy)1 VK"\{0} '
v
' 2
and for \a\ > 2
\y\lalm(dy) = caP.
\d?dZPl(x,Z)\ <MP( JK«\{0}
For Q = 0 w e have always
\d^Pl(x,0\<MQ(l + ^)), thus pi G 5^' 2 (R n ;lR). Now, in order to prove (2.406) we use (2.408) to find Pi(x,0=
JRn\{0}
>ri f
(l-cosy-£)n(x,y)iii(dy) (1 - cosy • OMi(dy) > # ( £ ) - 2 W 2 ( E " \ { 0 } )
•/K"\{0}
which gives for r sufficiently large the the derived estimate (2.406).
•
Remark 2.8.17. It is clear that we can combine the conditions of Theorem 2.8.13 and 2.8.15 in such a way that we will obtain ZAsub-Markovian semigroup for 2 < p < oo with generator — p(x, D).
2.8 Some Perturbation Results
165
The following example is taken from W.Hoh [128]. Example 2.8.18. Consider N
where N € N, bj : R n —> R+ are bounded functions, and ipj : R n —> R are continuous negative defined function having the Levy-Khinchin representation ^•(0=/
(1 - cosy •£)/!,• (dy). "\{0}
Then /x := X^,=i Aj is a Levy measure with corresponding continuous negative definite function ip(£) = X)- = 1 i'jiO- We may use the Radon-Nikodym theorem to find functions hj : R n —» R + , \hj\ < 1, such that \ij = hj/j,. This yields a representation of p(x, £) as N
P(x,€)=
/
(l-cosy-Q'S2bj(x)hj(y)[j,(dy),
i.e. we have (2.401) with n(x,y) = J2j=i bj{x)hj(y). Therefore, if bj G C6°°(Rn;M), 1 < j < N, it follows that P l e 5^' 2 (R) and (2.407) holds. In addition bj (x) > TJ > 0 then N
Pi(x,£)
= Y] / >T]
JRn\{0}
(1 - cosy •
^)bj(x)9(y)hj(y)fi(dy)
(1 - cosy • £)9(y)n(dy) n
V^(O-2r,((l-6)^(R
\{0}),
hence we have also (2.406). Remark 2.8.19. Some of the pseudo-differential operators q(x, D) generating Markov processes play a fundamental role in quantum physics, for example the relativistic Hamiltonian (—A + m 2 ) 1 / 2 — m. Therefore it is natural to look at perturbations through potentials, i.e. at operators of the form q(x, D) + V(x) with a suitable potential V(x). A central question is first of all whether for a symmetric q(x, D) the operator q(x, D) + V(x) is selfadjoint. This gives a
166
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
restriction for the potential class, for example V(x) should belong to some Kato-type potential class defined with the help of q(x, D) (or the corresponding Markov process), compare for example M.Demuth's and J.van Casteren's monograph [68] or K.-Th.Sturm [260] and [261], and the references therein. Using a related Feynman-Kac formula a stochastic spectral analysis for the operator q(x, D) + V{x) can be developed. We will come back to this question in volume III, the major reference is of course the monograph [68] of M.Demuth and J.van Casteren. Finally let us discuss non-local perturbations of second order elliptic differential operators in L P (R";R). Assume that
L(x,D) = J2 akl^^a7,+T,b^^r+<x) fei/=1
OXkOXl
,=i
(2- 41 °) OXj
is a second order differential operator satisfying a^i = aik € C^(R™;R), bj g C£(Rn;R) and c € C t (R";R) such that c(x) < 0 for all x € R n , and for all i € R " , ( £ R " we require with some A0 > 0 that J2k,i=i aki(x)£k£i > A 0 |£| 2 . In addition let us suppose £ " = 1 -^{b^x) - £Li $£*h > 0. In this case 2 n it follows that that (-L(x, D), W >P(M. ; R)) is a closed operator on £ P ( R " ; R) hence it is also an L p -Dirichlet operator. Moreover we have the estimates \\L{x, D)u\\LP < c||u||w2,P
(2.411)
and \\u\\W2,P
+ \\U\\LP)
(2.412)
for all u € W 2 ' p (M n ;R), compare Section 2.1 Now let p(x, D) be a pseudo-differential operator with negative definite symbol. In order to handle -p(x, D) as an L p -perturbation of L(x, D) it is more convenient to use its Levy-type-operator representation. We are interested in operators p(x,D)u(x)=
(u(x)-u(x-y))u(x,dy)
(2.413)
JRn\{0}
where for some 0 < a < 1 c
v,a < 0°
JR"\{0}
(2.414)
2.8 Some Perturbation Results
167
holds with c„>a independent of x G Rn, or in operators
p(x,D)u(x)
= f
(u{x)-u{x-y)-J2
. *
|2
^ K ^ d y )
(2.415)
with /
(lA\y\a+1)v(x,dy)
< c„,i + a < oo
(2.416)
JM"\{0}
for some 0 < a < 1 and c„,i + a independent of x. It turns out that L p -estimates for the operators (2'.413) and (2.415) can be obtained in Sobolev-Slobodeckij spaces which are special Besov spaces. For s S R+\No and 1 < p < oo we set Ws'p(R.n; R) := B",,(R n ; R)
(2.417)
where the Besov spaces B* ? (R") had been defined in Section 4.11. For s £ N the spaces are the spaces defined in Section 2.1. In W S ' P (R";R), s G R+\No, we have the equivalent norm ||«||w".i> = ||w||ivw.p ,f^,W
R
„V
\x-y\(—\»X)P
\x-y\n
y
J
(2.418) From the interpolation properties of Besov spaces we can deduce that for s > so > 0 and £ > 0 the estimate ||u||w*o.p <
£||U||VK S 'P+C(£)||U||Z,P
(2.419)
holds, compare H.Triebel [267], p. 186, Remark 2.4.2.6. Proposition 2.8.20. Suppose that p(x, D) is given by (2.413) with a kernel u(x, dy) = g(x, y)\^ (dy) where g : R™ x W1 —» R + is a measurable function satisfying with some 0 < K < fi < 1 the estimate \g(x,y)\ < , ~)f+II. Then we have for all u £ W > p ( R n ; R) the estimate \\p(x, D)u\\LP < c\\u\\Wn,p.
(2.420)
168
Chapter 2
G e n e r a t o r s of Feller a n d S u b - M a r k o v i a n Semigroups
Proof: For u 6 W' P (IR"; R), 0 < K < y. < 1, we get |p(x, D)u(x)\p =
/ 1
<2
(u(z - y) - u(x))g(x, y) dy
7K»\{O} P
(u(x
I
- j / ) - u(x))g(x,
y) dy
JO<\y\
/
(u(x - y) - u(x))g(x, y) dy
J\y\>i
••2P(h(u)(x) + I2(u)(x)). For I\u{x) it follows by Holder's inequality, ^ + \ = 1, P^
/
p'
(u(x - y) - u(x))g(x, y) dy
J0<\y\
/"
<
|(u(x-y)-u(g))|
~ Jo<\v\
|j/| + lt -"
W w |l j«/ l| w
~ ^0<|W|<1 X
V
dy
)
\(u(x-y)-u(x))\P\y\»-K
f
-
|11/1"+*-" y|'
W O < M < I |j/|"+ K -^
C m
M n
Jo<M
< cn
M^
bF~
|(u(a; — y) - u(x))\p
dy
\y\W
|j,|n'
dy
which leads t o
V
1
;
,M
' S
c
''PVAn
Hw
In order to estimate l2(u)(x) consider for v € If'(Rn;R) / JR"
/
|y|"
(2.421)
n ) /x,K,p||u||iy>.P.
the form
(u(x -V) ~ u(x))g(x, y) dy v(x) dx
J\y\>l
(u(x-y)-u(x))g(x,y)x\y\>i{y)v(x)dydx
= JR" ,/R"
<M
[
f
JR" JR<*
\u(x-y)-u(x)\\v(x)\X^0-dy n+K \y\
2.8 Some Perturbation Results
<M
I
169 \u(x - y) - u(x)\p dx) '
( I
f
p+1
1
<2.MMr,Miyl
r^dy
= Cn,K||«l|Lp||«|| L p
which yields /
(u(—y)-u(-))9(;y)dy
]
J\y\>i
<
(2.422)
CnJ\u\\LP,
LP
and combining (2.421) with (2.422) we arrive at (2.420). D Proposition 2.8.21. Suppose that p(x,D) is given by (2.415) with a kernel v(x,dy) = g(x,y)\(n\dy) where 5 : 1 " x R" -» R+ satisfies for some K € (0,1) the estimate g(x,y) < 1 1 J+n., • For 0 n u e W' ^ (R ;R) the estimate \\p(x, D)u\\LP <
(2.423)
CIIUIIIVI+M.P.
Proof: For u G W 1 + ^ P ( R n ; R ) (or Cg°(R n ;R)) it follows that p(x,D)u(x)=
/ JM\{0}
=
(u(x) -u{x-y)-J2,
J
V
~{
(u(x) - u(x - y) - 2
+
! + \y\
l\
(u(x)-u(x-y)-J2
= (Jiu)(x) + (J2u)(x) +
(J3u)(x).
Now, for Jiu we find with v € I p ' ( R n ; R), \ + ± = 1, / 1
JK"
(Jiu)(ar)v(.x)dx
)"(x>dy^
. 2 n~
|2
ax
3
'
- | - ^ - ) f f ( x , j/) dy
Vi " S T " ) flfo y) d »
170
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
y
\°l{x))
f f Ux)-u(x-y)-±^ w(a;)5(a:, 2/)X{|y|>i} ^
<M
/
dy
(|u(x)-u(x-y)||'?;(x)|
I du(x) . . . ,\ X{|»|>1}(») rfx n + l + K dy ^ »
E iauix) 1
<M
J
E
\u(x)-u(x-y)\pdx)1/P\\v\\LP,
{(( ll # w II II ax,- IILP
< 2Mw„( «
L
His))
i ^ n + i ^ «*»
P+ —
) v\\LP> /
r~2-Kdr
2cunM =
1 +
K
\\u\\wi.r\\v\\LP'
,
implying (2.424)
||Jiu||z> < C I | | U | | W I , P .
For Jiu it follows by the mean-value theorem that \J2u{x)\p=
I J0<\y\
HE/7 H 1
= 1 J 0
J0<\y\
^
L{x)-u{x-y)-y\yj~^-)g(x,y)dy fr{ OXj 1 du(x) dxi
du(x — ty) dxj
jg{x,y)dy dt\
2.8
Some P e r t u r b a t i o n R e s u l t s
n V
I du(x) I dxi
r
j =l
171 du(x—z) I dxi I
"—r~i
JO<\z\
du(x)
f
1
dz)
du{x—z} ip dxj
^
\n+K—\i
\Z\P»
WO<|*|<1
Wo<|,|
i \"
—r-r—,
J
/ I
fltt(x)
<
du(x-z)
ip
dz,
\z\r
\z\w
0<|2|<1
which yields that I
\J2u(x)\p dx < cJ2 [
f
L*2l
j=l-
||^2W||LP <
dXj
Izlw
I
1
Yz\r
dzdx,
(2.425)
C2||tt||vyi+^, P .
Finally, to estimate J3 observe that
Saij './o
<
bj||y| 2 M ^ 1 arj lyo< M
cte,
Mw„ ^
7o<|y|
,|2—n —/c dy
1du(a;)
j'=l
9x9
implying II^IILP < C3||U||WI,P.
(2.426)
Combining (2.424), (2.425) and (2.426) yields (2.423), i.e. the proposition is proved. D
172
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
Thus in each of the cases treated in Proposition 2.8.20 and 2.8.21 the operator —p(x,D) is according to (2.419) an L(x, Z))-bounded operator on W 2 ' p (R n ; R) and as before it is clear that —p(x, D) is also an L p -Dirichlet operator. In particular it is dissipative and —L(x, D) —p(x, D) is an L p -Dirichlet operator, too. Therefore we arrive at Theorem 2.8.22. Let L(x,D) be a second order elliptic differential operator as in (2.410) and suppose that p(x, D) is either as in Proposition 2.8.20 or Proposition 2.8.21 then the operator (-L(x,D) - p(x, D), W2>P(Rn;R)) is an Lp -Dirichlet operator generating an LP -sub-Markovian semigroup.
2.9
On Semigroups Obtained by Subordination
Let (Tj )t>o be an L p -sub-Markovian semigroup, 1 < p < oo, or a Feller semigroup for p = oo. In Section 1.4.3 we studied the general procedure of subordinating in the sense of Bochner. It was shown that for any Bernstein function / the subordinate semigroup (Tt(p' )t>o is again L p -sub-Markovian or Feller, respectively, and on the level of generators an operational or functional calculus was developed. The essential result was that if A^ is the generator of (Tt )t>o then the generator A^'f of (T t ( p ) J ) t > 0 is given by A^^ = —f(-A^). Clearly we may apply subordination to the semigroups constructed in the last sections and we will obtain automatically new examples. The purpose of this section is to combine the general theory of subordination and its underlying functional calculus with a symbolic calculus for pseudo-differential operators being a generator of an L p -sub-Markovian or Feller semigroup. Let us outline the situation in the case of translation invariant semigroups, i.e. semigroups characterized by a fixed continuous negative definite function V> : R n -> C according to (T t u) A (£) = c-**«>u(0
(2.427)
for u 6 S(Rn; R). (Since in this case all the operators T^p', t>0, coincides on 5(R"; R) for 1 < p < oo, we omit p and write only Tt, and A for the generator etc.) If / is a Bernstein function with corresponding convolution semigroup (Vt)t>o, supp^t C [0, oo), we find for the subordinate semigroup ( T / « ) A ( 0 = e-*«>«(0.
(2.428)
2.9 On Semigroups Obtained by Subordination
173
Further, the generator of (Tt)t>o is given on 5(R n ;R) by -tl>(D)u{x) = - ( 2 7 r ) - n / 2 f
e™*il>(Z)u(Z)dt
(2.429)
and that of (T/)t>o by -tf(D)u{x)
= -(foi>)(D)u(x)
=
-(2TT)-"/2
ete*/MO)«(Ode-
/
(2.430) Thus the symbol of A is O and (Rx)\>o, respectively, *W(0 = T TH AT+^ V(£)
and
" (V* 'A) A( 0W
A + */ W 0 ) '
as well as for the semigroups (as stated before) a(Tt)(0
= e~** (0
and
CT(T/)(£)
=
e-*
/W€))
.
Now, suppose that the pseudo-differential operator — q>(x, D) with symbol q : R" x R" -» C such that £ i—• g(x, £) is continuous negative definite function, extends from S(Rn; R) to a generator of an Z p -sub-Markovian semigroup, 1 < p < oo, and a Feller semigroup for p = oo. We denote all these semigroups again simply by (T t ) t >o and the generator(s) by A. Then it is clear that for the subordinate semigroup (T/)t>o, its generator A? and its resolvent CRJJA>O we will have in general a{Ttf){xti)^e-tfl9lx'()),
(2.431)
(2.432)
g
(2 433)
and
(*0('»Q*A+/(W
-
However the question arises whether the operators with symbols as on the right-hand side of (2.431)-(2.433) will give some good approximations of the operators we are interested in. There are two cases of particular interest. First suppose we know for a given semigroup (Tt)t>o that Tt has a kernel representation Ttu(x) =
JRn
Yt(x,y)u(y)dy
174
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
with upper and lower estimates for the kernel Tt(x,y). semigroup (Tt )t>o will also have a kernel given by
Then the subordinate
/»oo
r/(x,y) = / Jo
r3(x,y)r)t(ds),
and we will have immediately bounds for T{(x, y). The second interesting case follows from the observation that by Lemma 1.3.9.9 the function £->/(<7(*,0)
(2-434)
is a continuous negative definite function too. Hence the pseudo-differential operator —(/ o q)(x,D) with the symbol —f(q(x,£)) is a candidate for an operator having an extension generating an L p -sub-Markovian semigroup and a Feller semigroup, respectively. Denote this semigroup (if it exists) for a moment by (St)t>o- Then we should expect that St has "approximately" the same properties as Tt. But we will have complete knowledge of the symbol of its generator which (as we will see later in volume III of this treatise) give us a lot of information on St as well as on the corresponding stochastic process. As an example for the first case we will discuss the subordination of certain diffusion semigroups, i.e. semigroups generated by certain second order elliptic differential operators. Then we will turn to the general case where so far only a few first results are known. Finally we will pay more attention to fractional powers where much more results are available. We need some additional background material on second order elliptic differential operators, some results we will just summarise, often results will be provided with some (technical) proofs. The selection is determined by preparing some parts for the discussion in Chapter 3 and volume III. Our first aim is to understand in principle subordinate diffusion semigroups. More precisely we will restrict our attention to semigroups generated by second order elliptic differential operators in divergence form L(X,DH,),±±-(ak,(^)
( 2 .435)
where aki = atk € C% (R.n; R) such that n 2
Ao|£| < Y, °«(*)&G < Ao W k,l=l
(2-436)
2.9 On Semigroups Obtained by Subordination
175
holds for all x <E R n and £ e R" with some A0 > 0. Since
k,i=i
K
'
k,i=i
it follows with bi := £ * = i 2 ^ that
;=i
fc=i
K
;=i
fc=i
K
fe=i
K
and according to Theorem 2.1.44 the operator (L(x, D), W2'p(Wl; R)) is a generator of an L p -sub-Markovian semigroup. In fact it also generates a Feller semigroup. Moreover, the associated parabolic operator TJT — L(x, D) has a fundamental solution, (compare Definition 2.7.12). Definition 2.9.1. A function r : (0, oo) x R n x R n —> R is called fundamental solution of the operator j ^ — L(x,D) if (t,x,y) H-> T(t,x,y) ( = r t ( x , y ) ) is continuous, T(t, x, y) > 0, x — i » T(t, x, y) is a C2-function and 1i—> T(t, x, y) is 1 a C -function such that dT{t
^'y)~L(x,Dx)T(t,x,y)=0, or and for all / € C b (R";R)
(2.437)
lim / T(t,x,y)f(y)dy = f(x) t-»o J i '
(2.438)
uniformly on compact subsets of R". Theorem 2.9.2. For aki = alk 6 C^(R";R) satisfying (2.436) the operator TjT — L(x,D) has a unique fundamental solution T(t,x,y) = Tt(x,y) which satisfies the Chapman-Kolmogorov equations T(t + s,x,y)=
(
T(t,x,z)T(s,z,y)dz,
and in addition for f e C 6 (R"; R) n £ P ( R " ; R) we have Tt(p)f(x)
= Ttf(x) = [ T(t,x,y)f(y)dy,
which implies that T t
t>0
/ias a kernel representation for t > 0.
(2.439)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
176
Remark 2.9.3. The general version of Theorem 2.9.2, i.e. for operators with ^-depending coefficients and lower regularity, was obtained by A.M.Il'in, A.S.Kalashnikov and O.A.Oleinik [144]. We used a bit E.B.Dynkin [77], [78] for our formulation. A standard reference is the monograph [94] of A.Friedman, see also that of S.Ito [147]. Of course the heat operator ^ — A n is included in Theorem 2.9.2 and the corresponding fundamental solution is given by T0(t, x, y) := (47rt)-" / 2 e - | *-*' | 2 / 4 t .
(2.440)
Moreover, if a^ = aik G 1R satisfy (2.436) and Q = (a^) denotes the corresponding matrix a fundamental solution of the constant coefficient operator
dt
^
dxndxi
is given by
compare (1.3.130). If Q = 7 id we will write V instead of T 7 i d . Now, accordingly to our philosophy stated above we should try to compare T(t,x,y) from Theorem 2.9.2 with TQ^°^(t,x,y), i.e. TQ^x°\t,x,y) should be a first order approximation of T(t, x,y). However, the Aronson estimates we are going to prove, i.e. the estimates Klr
71
(t, x, y) < T(t, x, y) < K2T^ (t, x, y)
(2.442)
are even better since they allow us to compare T(t, x, y) and therefore (T t ' p )t>o with the densities of two convolution semigroups and the corresponding operator semigroups, respectively. Note that (2.442) yields immediately results for subordinate semigroups (which we will discuss later in this section), just recall the formula TOO
Ttfu(x)=
Jo
r
/»oo
Tsu{x)r)t(ds) = / n / JR JO
T(s,x,y)r,t(ds)u{y)dy.
We are now going to prove (2.442) following the work [84] of the E.Fabes and D.Stroock, see also D.Stroock [258]. In doing so we will encounter techniques which will become useful later on.
2.9
On Semigroups Obtained by Subordination
177
P r o p o s i t i o n 2 . 9 . 4 . Forp < n, p > 1, the Sobolev inequality (n -
i»-p
\)p
(2.443)
< y/n{n — p):I|V«|| LP
holds for all u e W1'p(Rn)
=
H£(Rn).
Proof: For p = 1, u £ C ^ ° ( R n ) and x = ( x i , . . . , x „ ) G 1 " it follows t h a t \u(x)\<
[Xj / J-oo
d |^—u(xi,...,yj,...,xn)\dyj a Vj
f II du{x) < / Ju ' dxj
with yj in the j t h position. T h u s we have
i^)r/"-1<(n/R^)i^)1/n"1. Integrating with respect t o x\ and applying Holder's inequality yield /|u(x)|n/n-1dxi l/n-l
dyi
-u(yi,x2,...,xn)\dyi
,dx« )
dx\
=
(XI^ (yi '" 2 '---'"" ) W 1/ "" 1 X(nXI^I^0 1/ " _Wi
-
( /
\-^—u(yi,x2,---,Xn)\dyi)
n(/./.l^h-0
l/n-l
l/n-l
Now, continuing with integration with respect to x 2 , . . . , xn and using t h e same arguments we arrive at
and using t h e geometric-arithmetic mean inequality Fill,-/—i
*(jW*r*U£
10"O»O i dx. 1 5x,- '
Chapter 2
178
Generators of Feller and Sub-Markovian Semigroups
Finally, since Y^j=i 1 g . I — V ^ V i ^ x ) ! we obtain Hullin/n-i < -J=||Vu|| L i,
(2.444)
i.e. (2.443) for p = 1. But from (2.444) we can easily derive (2.443). For this let q > 1 and observe that MIL^-D
1_
< 4= / |V(M*)|dx •/HI"
= -?= /
lul'-^Vulda:
(2.445)
< -^iiitir^i^nvtiiu,. Now, take g := ^ L ^ P and estimate (2.443) follows first for u 6 C£°(R n ) but then by density for all u € W 1 , p (R n ). D Remark 2.9.5. We have taken the proof of Proposition 2.9.4 from W.Ziemer [290], Theorem 2.4.1, the original proof is due to L.Nirenberg [224]. Note that the constant in (2.443) is not optimal. For p = 1 the optimal constant is ——} .-,/„, compare W.Ziemer [290], Theorem 2.7.4. Using the generalized Holder inequality ||«||L-
< IMl£»IMIi7\ «€i*(R n ),
which holds for s < r < t and ^ = j + Corollary 2.9.6. ForuG W1'P(En),
J
(2.446) we obtain
p < n, the estimate
IMU'
(2.447)
holds for i = A + (1 - A)(^=£). In particular for r = p = 2 we find A = which yields
ra
?2
«liL.
«llit 4 / n < ^,211 Vu||ia||u||^in.
(2.448)
2.9 On Semigroups Obtained by Subordination
179
Let us return to the operator L(x, D) given by (2.435), (2.436). For ( e R n denote by L^(x, D) the operator Lfc,
D)u := e-*-xL{x, D)(ex^u{x)).
(2.449)
Thus we have for u G S(Rn) daki{x)
Lt(x,DMx)= J2
{^r1^u^+£u^
fe,j=i
, * (+ >. / N ,. du(x) cram
. du(x) au\x)
d2u(x)> \ i o'u(x)
(2.450) With Q(x) = {aki{x))k,i=i,...,n we find further for u € 5(E"), u(x) > 0, that (Li(x,D))u2p-1dx
/
= /" (£,Q(x))u2pdx-2{p-l)
fci^i-'R"
dxkdxi
2pdxkoxi
= f (£,Q(x)Z)u2pdx-2(p-l) yR" -(2p-l)
[
(£,Q(x)Vu)u2p-1
f
f
dx
)
{£,Q(x)Vu)u2p-ldx
JR"
(Vu,<2(x)Vu>u 2p_1 da\
Using the relation u p _ 1 V u = | V ( u p ) we arrive at / JR"
{Li{x,D)u)u2p~1dx=
f
{£,Q(x)0u2pdx
JR" 2{P l)
f ~ P JR< 2(P - -^1-) [
(t,Q(x)V(up))updx (\7{up),Q(x)V(up))updx. (2.451)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
180
Now we estimate each term on the right-hand side of (2.451). By (2.436) we find first (Z,Q(x)0u2pdx<±-
/ JR"
\tf J u2vdx, o Jm.
A
and next for r) > 0 using the fact that (£, Q(x)Q is a scalar product and that for a, b > 0 we have a-b < j]c? + -£-b2 2{p l)
~
P
(£,Q(x)V(up))updx\
f jRi
' 2
<^LzA(f ^Q(x)V(u")) dx)1/2([ u^dx)1'2 V P W R » ' 7K" ' ^ 2 ^ 1 ) / 1 {^Q{xmy{uP)jQ{xW{uP))dx\i/2n P WR" ' WR" < 2 ( p - - l ) / J|l_ J ( V ( u P) | Q( a .) V ( u P)) d a ; + J . I P V A 0 ./ R » 4r7 ,/ R „ < 2 ^ 1 ) / j ^ ?
| V K ) 2
^
1 /• ^
1/2
u2Pdx\
'
u2pdx\
/
N
and finally, by (2.434), _2(p- L 2) /• P
( V («") > g(x)VK))dx
,/K"
<_ A o (fcil)/ |VK)|^,. Adding up we find / (Lt(x,D)u)u2*-ldx<(l-\Z\2 + ?f±) f u2pdx JK" VAO 2p?7 / J K ~ +
(2fezi)^,{|._A0^'))/|VWfifc.
and with 77 := 2 ^"|2 we obtain /
(Le{x,D)u)u2*-ldx<-^
[ |V(0| 2 cfa+-£-|£| 2 /
2 W
"dx
|2
= — / |VK)|2dx+f|^ hll|. P Ju"
Ao
(2.452)
2.9 On Semigroups Obtained by Subordination
181
Now, by (2.448) we have
I J!
f ^<^2||VK)|||2l
or C
u2Pdx)1+Vn
(f
1 llull2p(1+2/™)
Thus we have proved L e m m a 2.9.7. For a// u G 5 ( E n ) and a/Zp > 1 the estimate -
/
M ii2p(l+2/n)
(Xe^^uju^icte < - g " 7 II
L
^
/ n
i£|2
+ ^ M %
(2-453)
ll£/*^
holds with Co = co(n, Ao). Our next aim is to relate (2.453) to (Tt)t>o, the semigroup generated by L(x, D). We use the representation (2.439) for {Tt)t>o and we want to estimate Tlu{x)
:=e- x *Zt(e<-'°u(-))(a:) * /• , = e-x< / r(tfXly)e»My)dy-
(2.454)
Since we have no a priori knowledge on the growth of y — i > r ( t , x, j/) we have n to require that u G Cfi°(M. ). For these u > 0 it follows by the fact that Tt is positivity preserving that Jt\\Th\\%
= j j ^
\e-*
= 2p [ = 2p [
L £ (x,D){Tlu){Tfu) 2 r- 1
dx.
Since ^lltf «||& =
2p\\Tfu\\lV^\\Ttu\\L>P
we obtain from Lemma 2.9.7 that | P M | L * < -|l|lft.||^-||I<«||£F^-+^||7<«||i>
(2.455)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
182
holds for t > 0 and p > 1. Taking p = 1 and looking to the less sharp inequality
jt\\Tfu\\L,<^\\Tt^u\\L,, an application of Gronwall's lemma, compare [120], p. 169, yields the estimate \\Tfu)\L2<eM2t/x°\\u\\L,.
(2.456)
Lemma 2.9.8. Let w € C([0, oo)) be a non-negative, non-decreasing function and suppose that u € C 1 ([0, oo)) is a non-negative function satisfying for p 6 [2,oo)
"'<" S -g(4w-)
«'+*'"(')+^«e>
(2-457)
for all t > 0. Then for each 6 > 0 f/iere is a constant K = K(co, 6) < oo suc/i that u(t) < {Kp2)n'APu(t)e-m2t,XoP holds for all
t(1-p)n/4p
(2.458)
t>0.
Proof: A straightforward calculation for the function v(t) :=
e-M2pt/Xou(t)
yields first v'{t)
= _fe e -|fl 2 P*/Ao u W + e -|«| 2 Pt/Ao u , (i)
and therefore using (2.458)
4p V K I 2 '
A0
V
V
. . -4,-n f\f\2p
,
N
(K, / i ( p - 2 ) n / 4 p
w(f)4P/"
n
Thus we find d /e ^ S ^ ^ N
2co «^m2« i(P~2)
(ft V u ( i ) 4 p / n J -
~n~e
°"
w(f)4p/n
4 p/n
„+4p . .
2.9 On Semigroups Obtained by Subordination
183
or c4|€|Vt/nAo
U(t)4p/"
/"* ^ I f l 2 -
2co ~
w(s)4P/n
Tl J0
2co >
Sp_2
1
4 7-7T7-T7 n w(t) P/"
r /
*fts£e ^ °
p_2
SP
Z
dS,
where we used that u(0) > 0 and that u is non-decreasing. Since
*P
X
( 1 _ (1 _ ^ / p 2 ) p - l ) e ( 4 | C | 2 p 2 / « - 4 « | « | 2 / " ) t / A o )
we arrive with co
tf:=^infMl-(l-«Vp2r1)}>0 at e4|e|
2
p2t/nA0
u (t)4p/n
+p-l
-
which implies (2.458).
Kp2bj{tyP/n •
Now we set pk := 2fe, Uk{t) := \\Tfu\\LPk and wfc(t) := max s{pk-2)^uk(s).
(2.459)
0<s
Supposing
||M||L 2
= 1 it follows from (2.456) that
and (2.455) together with Lemma 2.9.8 yield a;fc
+ 1 W < (AkK)n/2k+2es\tft/2'<\o ; w t (t) - V
But (2.460) implies the existence of ci — ci(n,X0,6) supu;fc(t)
/ 2 4 6 o) < oo such that (2.461)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
184
Combining all these auxiliary results we are in a position to prove an upper estimate for T(t, x, y). Theorem 2.9.9. There exists a constant A = A(n, Ao) > 0 and a = a(X0) > 0 such that T(t,x,y)
< jLe-l»-*lV««
(2.462)
holds for all t 6 (0, oo) and i , ! / 6 R n . Proof: Taking into account the definition of Wk(t), (2.459) and the fact that lim uk(t) = \\Tf ul|oo) we arrive first at k—too
Halloo < ^ e
2
l ^ ° N |
(2.463)
L 2
for u G Co°(M"). By our assumption (Ti)t>o is a symmetric semigroup implying that (T/)*u = Tfu and by duality (2.463) yields ||r«U|U2<^e2l«l2'/*°H|Ll.
(2.464)
Using the semigroup property T^tu = (if o Tf)u we get
||4«IU~ = \\T?{Tlu)\\L~
i.e.
sup /
e"xiT(2t,x,y)u(y)evidy< ^ i V A o /
for all « e C J ° ( I " ) , hence (e'^r(2t,x,.)e^\u(.))o implying e-*tT(2t,x,y)eyt<^eW2t/*°,
< J^ e 4l«l a «Mo| H U
Mdx
2.9 On Semigroups Obtained by Subordination
185
or o2
r ( 2 t , x , y ) < -Ij^lCI'VAoge-C—v)^ Now we may take £ := ^(y 2
r(2i,x,y) < A -
— x) to arrive finally at
4xSt|x-„|i
*n|„-*|2
64('ix0 — " - K —
e
(2.4 6 5)
„2 =
_^ e -Ao|!/-.|Vi6t
which implies (2.462). D We will now derive a lower bounds for T(t,x,y). some lemmata.
For this we prove first
L e m m a 2.9.10. For all if G C£(R";R) the estimate f e - ^ 1 2 (
e-7TM2
f
e'^^
\Vy{y)\2dy (2.466)
holds. Proof: With ut{dy) = fit(y)X^{dy) = (47rt)-n/2e-^^4tX^(dy) where X( \dy) is the Lebesgue measure it is clear that (2.466) is equivalent to n
/
i/2(dy) < I
\V
(2.467)
for all tp G C£(R";R) such that / R „ i/2(dy) = 0. The density of C£°(R n ;R) in L2(E.n,i/i/2(dy)) implies that it is sufficient to prove (2.467) for all i/2{dy) = 0. Consider St
(2.468)
i.e. (S't)t>o is the Ornstein-Uhlenbeck semigroup. From properties of the heat semigroup (Brownian semigroup) it follows immediately for
(x) — 0
and lim St
(Ax-x-Vx)St(p{x),
t-*oo
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
186
which means that Ax — x • V x is the (pre-)generator of (St)t>o- It follows that -r. I
\St
at ,/flJn
St
,/jJn
and an integration by parts yields \St
-| /
\VySMy)\2vl/2(dy),
= -2 /
(2.469)
which says that t H-> \\Stip\\L^(U",vl/2(dy)) is a non-increasing function for (p e CS°(Rn;]R) or even
= /
-2* (y - e~lx) dy
ip(y)Vxfit.
JR"
5
= e-< /
(V y
(2-470)
- e-'5 t (V^)(x) with the obvious interpretation of St(Vtp). Thus using (2.468) and (2.470) we arrive at /
\
jRn
[°°e-2t( JO
[ V
\St\VV\{y)\2vl/2{dy))dt
JK"
'
< I \V HStVlli^pi"^ 1/2(dy)) and the lemma is proved. D Lemma 2.9.11. There is a constant 71 = 71 (Ao) such that for all \x\ < 1 /
e-^2logr(l,y,x)dj/>-7l
(2.471)
JR"
holds. Proof: For simplicity let us set v(s,y) := T(s,y,x) G(s):=
f Jnn
e-«M2logv(s,y)dy.
and
2.9 On Semigroups Obtained by Subordination The fact that (Tt)t>o generated by L(x,D) semigroup implies by (2.439) that f v{s,y)dy=
f
187 is a sub-Markovian (and Feller)
T(s,y,x)ldy=(Tsl){x)
Indeed, in our case it is also clear that T s l = 1, i.e. L„ v(s,y)dy = 1. Thus G(s) < oo and we seek for a lower bound for G(l). For this we observe that
G
'^ = i L e~*M° logw(s'y)dy = L e^v{2^y)iv^y)
dy
= e Mv L{yMv y)dy
L ~ ^) ^
= _
r
/e—*\y\
2
/ Wv{Tt77z>Q(y)Vv<s,y))
V{S, y) J e - l v l (2n-^— + Y^lA, JRn v{s,y) v2(s,y) 7R"
a
=
= 2n[ JR« +
f
y)) dy
e-^(y,Q(y)^f)dy V(S, y)
e -^ ( Y^) )Q{y) Yi±A )dy V(S,
7K«
= 2TT f
Q(y)Vyv(s,
y)
v{s, y)
2
e~*M {y, Q(y)Vy
log v(s, y)) dy
e-*M2(Vylogv(s,y),Q(y)Vy\ogv(s,y))dy.
+ JRf
n
Since \2ir(y,Q(y)Vy\ogv(s,y))\ < 2n2(y,Q{y)y)
+
-(Vylogv{s,y),Q(y)Vylogv(s,y))
we find 2TT /
e-*M2(y,Q(y)Vylogv(s,y))dy>-2ir2
f JRn
7R"
_ I f
2
e-*\v\
(yy
l o g t , ( S ) y)>
e-*M'(y,Q{y)y)dy
Q ( y ) V j / l 0 gt;(5, y)) dy
implying G ' ( S ) > - 7 + T /
e-^2|Vylog^(S,y)|2dy.
(2.472)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
188
In particular, (2.472) implies that G(s) + 7S is non-decreasing on [5,1]. But Lemma 2.9.10 we know that /
e-*M\logv(s,y)-G(s))2dy
2
f
e-*M'\Vylogv(s,y)\
dy
which gives with (2.472) e-*M2(logv(s,y)-G(s))2dy
G"(s)>-7+c' I
(2.473)
with 7 and c' depending only on A0. The function r H-> ( ° s r ^ \3)) j s a n o n . increasing function on [e2+G(s\ 00) and further, by Theorem 2.9.9, we know that sup v(s, y) < K with K being an absolute constant. This implies by 7<«<1
(2.473) for 5 £ [±, 1] that (?{,) > - 7 + c ' ( 1 ° g ^ - G ( 7 ) ) 2 /
e-"M\(s,
y) dy.
(2.474)
The integral in the right-hand side of (2.474) we can estimate according to / e'7rM\(s,y)dy> f e-'M'v(s,y)dy - e 2+G(s > 2 a Jv(s,y)>e + M JR" >e~*R2
[ J\y\
v(s,y)dy-e2+G^
= e-«R2(l-
[ v(s,y)dy)-e2+G^, J\y\>R ' where we used that J*R„ v(s, y) dy = 1. Prom Theorem 2.9.9 we may also derive that there exists R\0 > 0 depending only on A0 such that v
sup
/
v(s,y)dy
<-. l
h>s>lJ\y\>Rx0
Now, since s H-> G(S) + 7s is non-increasing on [A, 1] we may deduce from (2.473) that there are 6\0 and M\0 depending only on A0 such that G'(s) > SXoG(s)2
for
provided G(l) < -M\0. we have proved that
*e[i,l]
Now, (2.475) implies that G(l) > --£-
_2_ G(l)>-max(^-,MAo) 5A0
(2.475) and therefore
2.9 On Semigroups Obtained by Subordination
189
or by the definition of G(l) /
_,r|j/|2 e
log v(s, y) dy > - 7 1
and the lemma is proved.
•
Lemma 2.9.12. For all x,y € -Bi(O) we have r(l,x,i/)>e-271.
(2.476)
Proof: Observe that the Chapman-Kolmogorov equations yield logr(l,x,y)
= log( J
J(±,x,z)T(±,z,y)dz)
>log/* /" =
lo
g/
1
1
r(- ) ar,z)r(-,2,j/)i/ 1 / 2 (dz)
> f
\og{T{\,x,z)T{\,z,y))vl/2{dz) e-*l*l2logr(i
= / +
e-^T(\,x,z)Y{\,z,y)dz
x,z)dz
e-*l«lalogr(i
f
y,z)vy2(dz)>-2lu
where we used Jensen's inequality, the symmetry of T, i.e. T(t, x, y) — T(t, y, x), and Lemma 2.9.11. But the estimate logr(l,x,2/)>-27i is equivalent to (2.476).
•
Before we will finally prove a lower estimate for T(t, x, y) we need a scaling result. Lemma 2.9.13. Denote by QrA{x), r > 0 , ( G R n , the matrix Q{rx + 0 = (afc/(rx + £))k,i=i,...,n and by T® and T^ ' the fundamental solutions corresponding to ^ — V(QV-) and to & — V(Qr'^V-)> respectively. Then we have YQ(r\
r x + £ , ry+£) = r-nTQr,t
(t, x, y).
(2.477)
190
Chapter 2
Generators of Feller and Sub-Markovian Semigroups
Proof: For if e C^°(Rn) set u(t,x):=
f
and u)(t, x) := u(r2t,rx
+ £).
It follows that J U ( t , x) = (V, Q'-«Vw(i, •))(*)
and
lim w(t, •) = ¥>(•)•
But now we may apply the uniqueness result for solutions of the initial value problem for second order parabolic differential equations to find /
for all ip G C£°(R n ) which yields (2.477).
y{y)TQr'\t,x,y) dy
= / •
We may restate (2.477) as
for all | ^ | , ^ | G -Bi(O). In particular, since (2.476) holds also for TQ obtain for \x — y\ < It1/2 the estimate r^(t,x,y)>^e-2^
' we
(2.478)
which is of course a lower bound for T^(t, x, y) close to the diagonal x = y. Now let x, y G R™ be arbitrary points with \x — y\ > 1. DeterminefcG N such thatfc< \x — y\2 < k + 1 and set 771
xm := x+——-(y-x) K+ 1
Bm:=Bi(xm),
m =
*Vk
0,...,k+l.
For £ m G Bm and £ m _i G -B m -i it follows that |sro
sm—11 ^ |sm
^
1
4\/fc
^ m | r \Xm
I
V&TT
Xm_i| + |Xm_i
?m-l |
'
1
4V3k
l
2>/3fc
l
2
v ^ T T - v/FTT
2,9 On Semigroups Obtained by Subordination
191
implying by (2.478) that rQ
GfcTr e " , - 1 ' u ) - ( fc+1 ) n/2e ~ 271 ^ * n/2 e- 271 -
But now the Chapman-Kolmogorov equations yield
r0(1
"'» ) -/ a -7 a r0 (^T- fi ) re (^i'^ 6 )-" ••' r "(fcTT & ' ! ') d?1 "-' i? '
+i
^•- »' (-(^)"'T =-271
= jfc"/ 2 e - 2 ^(^i
fc
— 2*Vi
Thus taking /3 > 0 such that ^ ^ — k < \x - y\2 < k + I, TQ(1, x, y) > e-^e-pk
> e _ / 3 we find, remember that
>e'^e-^-^.
(2.479)
,1/2 j
As before we may restate (2.479) first for TQ to arrive finally at
' and then rescale as before
Theorem 2.9.14. There exist constants B = B(n, Ao) > 0 and(3 = /3(Ao, n) > 0 such that TQ(*,:r,y)>^e-l*-*'l2//3t
(2.480)
holds for t > 0 and all x, y € W1. Clearly, Theorem 2.9.9 and Theorem 2.9.14 yields Aronson's theorem, i.e. the estimates (2.442) with «i, «2 and 71,72 depending only on Ao and n. It is important to note that both T 71 and T 72 are kernels representing convolution semigroups, more precisely we have that T t , 7i u(x) = J V* (t, x, y)u{y) dy is the semigroup generated by (an extension of)
jjAn.
Chapter 2
192
Generators of Feller and Sub-Markovian Semigroups
Remark 2.9.15. As stated before when deriving D.Aronson's estimates we followed the paper [84] and [258] and not Aronson's original paper [7]. This choice is motivated by the fact that we later on in Chapter 3, we rely much more on Nash-type inequalities and [84] as well as [258] have taken up the ideas of J.Nash [221] in their proofs. In order to calculate the symbol of the generator of (T/) t >o we need to have a closer look at (Ti) t >o. Define ff(rt)(x,0:=e-fa*rt(e*'«)(x)=
T{t,x,y)e-i^-^dy.
f JRn
Since for u e <7£°(R")
JRn
= (2ir)-n/2
eix^e-ixiTt(e^^)(x)u(^d^
f JR"
= (27r)-"/ 2 / I = f
T(t,x,y)Jv*iH£)dydt
r(t,x,y)u(y)dy=(Ttu)(x)
JRn
holds, we find that Tt is a pseudo-differential operator with symbol ^dy
(2.481)
JR*>
and by the Aronson estimates, (2.481) is well defined. We want to use (2.481) to re-calculate the symbol of L(x, D). We find using the symmetry of L(x, D) and the Aronson estimates ±a(Tt)(x,0
= j
I
T(t,x,y)e-^-y^dy
= /
^-T(t,x,y)e-^-yy^dy
= J
{L{x,D)T(t,x,y))e-**-vKdy
JRn
T(t,x,y)eiy<(L(x,D)e-ix^dy
= / n
JR
= [ JR"-
T(t,x,y)eiy<e-ixML(^D))(x,Ody
2.9 On Semigroups Obtained by Subordination =
193 T(t,x,y)e-i^~y^dy.
I
Since T(t,x,y)e-ilx-yHdy=l
lim / t->o Jw t we arrive at
a(L(x, D))(x, 0 = jt°(Tt)(x,
OI*=o-
(2.482)
Now we may turn to subordinate diffusion semigroups. Let / be a Bernstein function with associated convolution semigroup (r]t)t>o, supp^t C [0, oo), and suppose for simplicity that / has the representation /»00
(l-e-sr)Mdr),
/(s)= / Jo+ 10+
compare (1.3.229) in Theorem 1.3.9.4. From the Aronson estimates it follows that the symmetric semigroup (Tt)t>o is an L p -sub-Markovian semigroup for 1 < p < oo as well as a Feller semigroup on Coo(lRn;R) which extends to CbFeller semigroup in the sense of Definition 1.4.8.6. These properties will also hold for the subordinate semigroup (T/) t >o, / being a Bernstein function. In addition we find for u G C b (K n ; R) '
T3u(x)r]t(ds) = / J0
0
/
T(t,x,y)u(y)dyr,t(ds).
JUn
Now, in order to handle /
/ jRn
J0
T(t,x,y)u(y)dyqt(ds)
note that Tlj(t, x,y) = -
•, n /2 e ~^~ 3/ ^ 47: ' t ) i-e- ** *s t n e density of a Gaus-
sian convolution semigroup (^)t>o- Therefore OO
Jo
is a further convolution semigroup as was proved in Proposition 1.3.9.10. In particular f / u{y)Vt' {dy) Jw
< oo
194
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
for all u e Cb(M.n; R) and {y\ )t>o is just the convolution semigroup associated with the continuous negative definite function £ i-> f(fj |£| 2 ). Therefore we find /*oo
Ttfu{x) =
r
Tsu{x)r)t{ds) = J0
/*oo
/ JwinJO
T(s,x,y)nt(ds)u(y)dy
and we have proved P r o p o s i t i o n 2.9.16. Suppose the L{x, D) and T(t, x, y) are as in the Theorem 2.9.2 and denote by (Tt)t>o the semigroup(s) generated by (extensions of) L(x,D). For any Bernstein function f the subordinate semigroup (T/) f >o has a kernel representation T{u{x)=
rf(t,x,y)u(y)dy
[
(2.483)
n
JR
with i»00
Tf(t,x,y)
=
T(s,x,y)nt(ds)
(2.484)
Jo and for T* we have the estimates /•OO
Kl
Jo
i»00
T^(s,x,y)r}t{ds)
where r»(t,x,y)
=
(4
f ) n/2
e
|a
y|/ Tjt
~ '~ *
Jo
^(a.x.yMtfa) (2.485)
-
Further, since the convolution semigroup (u{ )t>o give rise to Feller semigroups we may extend (T/) t > 0 by taking monotone limits to C 6 (R n ; R). Thus it makes sense to consider a(T/)(x>0=e-»«T/(ei(-'«))(x)= / Jun
rt(t,x,y)e-«x-y> dy.
Assuming for convenience that (T/) t >o is even a CfFeller semigroup then we may easily argue as before to get P r o p o s i t i o n 2.9.17. / / (T/) ( > 0 is a Cb-Feller semigroup then the operator T t is a pseudo-differential operator with symbol
(2.486)
2.9 On Semigroups Obtained by Subordination
195
Next let us have a look at the generator Af of {T/)t>o, in particular Af\ c~(R";R)- Using the representation (1.4.132) from Theorem 1.4.3.5 we find /•OO
Afu(x)=
/ Jo
(T3u-u)ti{ds).
Since we may apply Ts to function in C(,(R";M) we find further /•OO
e-ix
= / Jo
=
e- ix '«((T,e i (-- J) )(x) - e' 1 *) n(ds)
r {L
e_ix$r(s y)eivi dy 1
(ds)
'*- - ) ^ '
thus we find /•OO
/ (a(Ts)(x,0~lMds). Jo
Note that if log(a(Ts)(x,£))
(2.487)
is well defined then (2.487) reads as
/•OO
a(Af)(x,£)
= -
(l-e-^^°s(-(^)(-.«)))) M (rf s ).
( 2 .488)
Jo In particular in the translation invariant case, i.e. o-(Ts)(x,0
= e-t«<W
we find
a(Af)(x,£) = -m,QO) as we should expect from the general theory. In general different "approximations" of a(Ts)(x,^), i.e. of T(s, x,y), will lead now to different "approximation" of a(Af)(x,£). For example one may try to approximate F(s,x, y) as done in E.E.Levi's [198] parametrix method for constructing T(s, x,y), see also A.Friedman [94] for details. Let us turn to the general situation, i.e. let A = —p(x, D) be a pseudo differential operator generating a Feller or an L p -sub-Markovian semigroup and assume for simplicity that Co°(M n ;R) C D(A). However, for / let us suppose that it is a complete Bernstein function, compare Definition 1.3.9.27 with representation
/w
= f IT?***
(2.489)
196
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
where p is a measure such that JQ < J_rp(dr) < oo, compare Theorem 1.3.9.29. According to Theorem 1.4.3.10 and its Corollary 1.4.3.11 in this case we have /•oo
Afu=
/ Jo
ARxup(dX).
(2.490)
Introducing the operator rx{x, D)u{x) := (2vr)-"/ 2 /
ete*
*
u($) d£
(2.491)
it follows that /•OO
Afu=
/ Jo
{(A-X)+X)Rxup(dX)
/•OO
= / ( - id Jo
+XRx)up(d\)
/•oo
/»oo
= / (Xrx(x, D) - id)up(dA) + / Jo Jo
A(i?A - r A (z, D))up{dX).
Since for u e C § ° ( R n ; R ) /•OO
/ Jo
(ArA(a;,Z?)-id)u/!)(dA)
- ( ^ j ^ f x^*wa« = -[(/op)(i,£>)]u, we find with pA(:r, -D) = p(rr, D) + X = X — A and Kx(x,D)
= id-Px(x,D)orx(x,D)
(2.492)
that /•OO
A ' u = -[(fop)(x,
D)]u+ / A-RAKA(:r, D)up(dX). Vo
(2.493)
Thus in order to claim that the symbol of Af is given by
+ "lower order terms"
(2.494)
2.9 On Semigroups Obtained by Subordination
197
we need to estimate OO
\RxKx{x,D)up(d\) i
more carefully. Note that for p(x,£) = p(£) we have K\(x,D) = 0! First of all let us remark that the problem lies already in defining "lower order terms". Note that, in general, p(x, £) has no expansion into homogeneous symbols, i.e. in general we cannot expect that p(x, £) ~ YHjLo Pi (x> £) w ^ n Pi being with respect to £ homogeneous of some degree rrij, (ra_j)j>o being strictly decreasing (to - c o if {Tfij)j>o is an infinite sequence). But even if such an expansion would hold, po(x,£) would in general not be our candidate for a "principal symbol" as seen by symbols of mixed homogeneity such as (£1,62) >-• | £ i | a i + l&l" 2 , 0 < ax < a2 < 2. The definition of K\(x,D) makes it obvious where our problem lies. We have to control the composition p\(x, D) or\(x, D), r\(x, D) given by (2.491). It is clear that in case where we are "closest" to a classical symbolic calculus we should expect the best result. The calculus we will look at first is of course F.Baldus' Weyl calculus, see [13], which we discussed briefly at the end of Section 2.7. Taking into account some observations from [163] he could prove the following result, where we use the notation of Section 2.7. Let 7 be a Hormander metric on W1 x R n which splits and assume that #(1,7) U 5 ( L 2 ( K " ) ) _ 1 = ^ ( l ^ ) - 1 . Further let M be an invertible sub-7admissible weight function such that there exist constant a > 0 and CM > 1 with ^ < M < CMh~a, /i 7 given by (2.368). In addition let p e S(M,j) be such that £ 1—> p(x, £) is a continuous negative definite function satisfying \p(x,£)| > cM(x,£), c > 0 . Forp(x,D) : C§°(]Rn;R) -> C ^ I R " ; R ) we assume (p(x, D)u, v)0 = (u, p(x, D)v)0
(2.495)
(p(i,D)u,u)o>Ao||u||g
(2.496)
and
for some Ao > 0. For the complete Bernstein function / , f(s) = J0°° s + we assume in addition f°°
1
J A ^r/'( d A ) < 0 0 where rj := 1 A i , 0 < a < ^, compare [163], Theorem 6.1.
rp{dr),
(2-497)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
198
Theorem 2.9.18 (F.Baldus). Under all the assumptions stated above on p(x, £) and f, —p{x, D) extends to a generator A of a Feller semigroup and if we denote the generator of the subordinate semigroup by A* we have with Mf -.= 1+fop that fop e S(Mt,i), -{fop)(x,D): Cg°(R n ;R) -» Coo(Rn;R) and A* = -(fop)(x,D)+p1(x,D) withpi{x,D)
(2.498)
E *(/ii- 2 < r ,7) C #(1,7).
Corollary 2.9.19. / / the operator L(x, D) in (2.435) has coefficients a,ki G C£°(R™;R), then Theorem 2.9.18 is applicable and in particular we have that a(Af)(x,0
= -f(a(L(x,D))(x,0+qi(x,0
(2.499)
where lim _ _ £ £ ^ - _ = 0
(2.500)
provided that lim f(s) = oo. For the proof of Theorem 2.9.18 and its corollary we refer to F.Baldus [13], Section 6.5. The symbolic calculus of W.Hoh, compare Section 2.4, is closer to our philosophy since combined with his perturbation results, see Section 2.8, it allows to start with some quite general continuous negative definite symbol and then to consider an analysis in function spaces related to their symbol (or a modification of it). We are now going to prove a result analogous to Theorem 2.9.18 within Hoh's calculus, and now we will use the notation of Section 2.4. As a first result recall that if p £ S™'^(Rn) and / is any Bernstein function, then / o p is also a symbol within Hoh's calculus. More precisely, we have by Example 2.4.7. Example 2.9.20 (W.Hoh). Let / be a Bernstein function satisfying f(s) < c\sr for all s > 1 and some 0 < r < 1, note that r = 1 is always possible. Let p e S™^(Rn), m > 2, where ip is in the class A, i.e. satisfying (2.196) for all a € No, and suppose that p(x,£) is real-valued and elliptic, i.e. ?(*,£)> co(l+V(0)m/2
(2.501) n
n
for some CQ > 0. Then / o p e S^-*"(R ) = S^(R ). In particular, if in addition £ H-> p(x,£) is a continuous negative definite function then / op is an elliptic negative definite symbol in S™>^r (Rn).
2.9 On Semigroups Obtained by Subordination
199
Remark 2.9.21. According to the results in Sections 2.4-2.5 we know in case that p{x, £) is an elliptic continuous negative definite symbol in S" l '^(K n ) and if f(s) = sr, 0 < r < 1, then -pf{x,£) = -(p(x,£))r is the symbol of an operator extending from Co°(]Rn;E) to a generator of a Feller semigroup which we denote for a moment by (St)t>o- Thus it is of interest to compare (St)t>o and (T/) t > 0 , (T t ) t > 0 being the semigroup generated by —p(x,D). We will come back to this problem later. In order to estimate K\(x, D) for p G S^'^ffi71) it is convenient to extend Hoh's calculus to a parameter dependent calculus. It turns out that for our special purpose a very simple version of such a calculus is sufficient and we state the results needed without proofs, a rigorous parameter dependent symbolic calculus including all proofs can be found in a forthcoming paper [134] with W.Hoh and A.G.Tokarev. Note that once the symbolic calculus of Section 2.4 is established a parameter dependent calculus can easily be developed along the lines of the considerations of G.Grubb [111], see also her highly readable survey [112]. Using Corollary 2.4.23.A we find that for an elliptic continuous negative definite symbol p e S^(Rn) the symbol of K\(x, D) is given by
where p € S2p^(Rn)
implies that ^'f^'Jl?'^'®
is a
symbol in S ^ ( R n )
and pn(x, £, A) belongs to the class ^ ^ ( I " ) , but it is important to note that \2pn(x,£, A) is bounded. Now, in order to estimate the L 2 -norm of K\(x, D) we handle in detail the term ^/dtjP(x,QDx.p(x,Qs
For j — 1 , . . . , n it follows that
\\({dflD^p%DH = II \ ( p + A ) / Ho ((P + A) m+A+ (z?))2(i+ ))_3/2 - oi( l(+ y+y)fo ^« ^ A + ^ ( D ) ) - ( l + V(I>)) 2
2
3/2
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
200
-IK(I+A+V(-))2J1
j
where we used the fact that
,
«
%,p(*.Q^PCX,0(1 + A + HQ)2
is a symbol of order zero uniformly bounded in A, and thus we may apply Theorem 2.5.4. Now
a+*«»•?_ s i _
(2.503)
implying the estimate
Note that (2.503) can be changed to (1 +1/>(£))3/2
(1 + A + V(0)2
1
1
~ (1 + A)1' (1 + V(0) 1/2 -"
for any 0 < r) < A which yields
IK (P + xy ){x>D)u
o-
(TTAFI|U|I^-1/2-
(2 506)
"
The structure of the symbol pr(x, £, A) implies in particular that (2.504) and (2.506), respectively, do hold for the corresponding pseudo-differential operator too. Thus we have established Lemma 2.9.22. Let p(x,£) definite symbol and define Kx (x, D) = id -px withp\(x,
£ Sp',l'(Mn) be an elliptic continuous negative
(x, D) orx (x, D)
D) = p(x,D) + Aid and
rx(x,D)u(x)
= (2n)-n/2
[ e™< i ^ t t K JRn p(x, 4) + A
Then for all 0 < r\ < A we have the estimate \\Kx(x, D)u\\0 < ^ ^ H i M - i / 2 -
(2-507)
2.9 On Semigroups Obtained by Subordination
201
Note that K\(x, D) is an operator of order — A, i.e. we have || K\(x, D)U\\Q < 'll'"l!^',—1/2, but we need some of its "regularity improving order" in order to get an estimate with some decay in A. Using Lemma 2.9.22 we may prove an estimate for Af - (—f°p){x, D) which is given by (2.493). c
Theorem 2.9.23. Let p G Sp'^(M.n) be an elliptic continuous negative definite symbol generating an L? -sub-Markovian semigroup. Suppose in addition that f(s) = f0 sj_rp(dr) is complete Bernstein function with representing measure p satisfying in addition f£° -jW/^dr) < oo. Then the estimate \\Afu-(-fop)(x,
D)u\\0 < c\\u\\0
(2.508)
holds. Proof: Prom (2.493) it follows by Lemma 2.9.22 that /•OO
\\AU + (/ o p)(x, D)u\\0 < /
\\XRxKx{x,
D)u\\0P(dX)
Jo f°°
1
where we used also the fact that ||Ai?,\|| < 1.
•
Remark 2.9.24. A. Recall that sa = ^ ^ 1 fi° ^^-r" dr, 0 < a < 1, is a complete Bernstein function and for 0 < a < A, s H-* sa satisfies the additional condition of Theorem 2.9.23. Intuitively, and trivially justified for p(x,£) = ip(£), for f(s) = sa, 0 < a < ^, we should expect A* to have order 2a, i.e. ||^w||o < cH^H^a, note that 0 < 2a < 1 under our assumptions. The same holds clearly true for (—/ op)(x, D). Thus Theorem 2.9.23 says that A? — (—(/ o p))(x, D)u is of order lower than A? and —(/ op)(x, D), i.e. it is a lower order perturbation and —(/ op)(x,D) should be considered as a first approximation of A?. B. Assuming instead of J^° -^j2P{dr) < oo the condition f£° \p{dr) < oo for 0 < rj < A, we find by (2.507) the estimate ||A'«-(-/op)(x,I>)u||o < H l ^ - x .
(2.509)
But now we may only allow f(s) = sa for 0 < a < rj < \. C. Our considerations leading to Theorem 2.9.23 are very close to those given in [163] and [164] with R.Schilling. The only difference is that we used now Hoh's symbolic calculus in order to derive estimates in the scale i7^' a (K n ).
202
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
As we will discuss in Chapter 3 and in particular in volume III of this treatise we can deduce many properties of the semigroup and also of the corresponding stochastic process generated by an extension of — p(x, D) when knowing the symbol p(x, £). Therefore approximating a Feller or sub-Markovian semigroup and a corresponding stochastic process by a semigroup or stochastic process, respectively, of which we know explicitly the symbol of the generator will give a chance to get approximate properties of the original semigroup or stochastic process, respectively. This programme should in particular apply to subordinate semigroups and processes. Suppose that A = —p(x,D) generates (Tt)t>o and that / is a Bernstein function. From a functional calculus we may deduce some "abstract" properties of A? and (T/)t>o but in general we can not determine explicitly a(Af )(x, £) the symbol of Af. On the other hand, a symbolic calculus may lead to a first approximation to a(A^)(x, £) for example by considering —(f°p)(x, £). Note that if £ — i > p(x, £) is a continuous negative definite function so is £ — i » (f op)(x, £), thus we may raise the question whether —(fop)(x, D) extends to a generator of a Feller or a sub-Markovian semigroup too. This is for example true if p € S ^ ( R n ) is an elliptic continuous negative definite symbol which is due to the monotonicity of the Bernstein function / By the observations made above we may sometimes consider the operator A* — (—/ o p)(x, D) as a "lower order" operator compared to A? and (fop)(x, D). In the following we will use such a control on A* — (—/ op)(x, D) to indicate how it is possible to estimate the difference of the corresponding semigroups. We restrict ourselves to one example of X,2-sub-Markovian semigroups using Hoh's symbol classes and combine it with the considerations made in [163] with R.Schilling. Thus let p(x,£) G S^(M.n) be an elliptic continuous negative definite symbol such that A := —p(x, D) generates an Z/2-sub-Markovian semigroup (Tt)t>o- Further let / be a complete Bernstein function such as in Theorem 2.9.23 and suppose that —(/ op)(x,D) extends also to a generator of an L 2 -sub-Markovian semigroup which we denote by (St)t>o- The aim is to estimate T / - St. By Theorem 2.9.23 we know that /•OO
Qu •= Afu-{-(fop)(x,D))u
= / Jo
XRxKx(x,D)up{d\)
is a bounded operator on L2(Rn). Therefore, (T/) t >o can be considered as a perturbation of (5 t ) t >o. Denoting the contraction semigroup generated by Q
2.9 On Semigroups Obtained by Subordination
203
by Rt := e~c^t, t > 0, note that the power series converges in i 2 ( E n ) since Q is bounded, we may apply Trotter's product formula, Corollary 1.4.4.15, to obtain (St/nRt/n)nu.
T/u := lim n—»oo
Furthermore, since T/u-Stu=
f T{_s{Af-(-(fop)(x,D))Ssuds Jo
(2.510)
we get Theorem 2.9.25. In the situation described above the estimate \\T{-St\\
(2.511)
holds. Proof: From (2.510) it follows using (2.508) that ||T/« - Stu||„ < / | | r / _ , | | | | ^ - ( - / o p ) ( x > D ) | | | | 5 , | | | H | o d a Jo < ct||u||o implying (2.511)
•
Remark 2.9.26. Suppose that p{(x,B) :=T/XB(X) a,ndpt(x,B) := StXB(x), B S /£?("•) with \(n\B) < oo, have the interpretation of a Markov transition function. Then (2.511) gives the estimate /
\p{(x,B)-Pt(x,B)\2dx
< ct\<-n\B)
(2.512)
/K"
i.e. we have a comparison for integrated transition probabilities. Remark 2.9.27. Using embedding theorems for the spaces H^'s(M.n) it is also possible to obtain pointwise estimate for p\(x, B) —pt(x, B). For this and many other related estimates for subordinate elliptic diffusion semigroups we refer to the paper [163] and [164] with R.Schilling.
204
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
2.10
Pseudo-Differential Operators with Variable Order of Differentiation as Generators of Feller Semigroups
In the last section we explored some one-parameter semigroups and their generators obtained by subordination. We will extend these considerations to the case where the Bernstein function s \—• f(s) is substituted by (x, s) — i > sr^ n with r : M. —> R being a continuous function such that 0 < r(x) < 1 holds. Let p : R n x R " - > C (for simplicity we will later on take R instead of C) be a continuous function such that £ — i > p(x, £) is a continuous negative definite function. Then it follows by Lemma 1.3.9.9 that £->P(*,0r(x)
(2-513)
is once again a continuous negative definite function implying that the pseudodifferential operator Au(x) =-(2TT)-^2
f
eix-tp(x,£,)rix)u(£.)d(,
(2.514)
is a candidate for a generator of a Feller semigroup. For p(x,£) = |£| 2 we obtain (—A)r(x\ but a bit more convenient is to look at (1 — A)r(x\ For \
( =FjjJ J
r x
()
, thus the
name "operator of variable order of differentiation" is justified. For some "classical" symbols p(x, £) operators of variable order of differentiation had been studied by A.Unterberger and J.Bokobza [274], and in particular by H.G.Leopold [195], [196]. Studies with the symbol (1 + \£\2)r^ leading to Feller semigroups had been done jointly with H.-G.Leopold [162], and further investigations are due to A.Negoro [222], and in particular to K.Kikuchi and A.Negoro [173], [174]. Note that F.Baldus [13] can also handle some operator of variable order of differentiation within his Weyl-Hormander calculus. In this section we follow once again W.Hoh and use his symbolic calculus for pseudo-differential operators with negative definite symbols — but now we consider the case of operators of variable order of differentiation, see [128] and in particular [129]. Using the notation of Section 2.4 we may state the main result of this section. T h e o r e m 2.10.1 ( W . H o h ) . Let ip : M.n —» R be a fixed continuous negative definite function such that its Levy measure has a compact support and that ip(0 > co|£r,
|£| large and r > 0,
(2.515)
2.10 Pseudo-Differential Operators with Variable Order of Differentiation
205
holds. Let q £ Sp'^(M.n) be a real-valued negative definite symbol which is elliptic, i.e. we have q(x,t)>So{l+iKt))-
(2-516)
Further let m : E n —> (0,1] be an element in C£°(E n ) satisfying M-n<-
(2.517)
where M := supm(:r) and 0 < fi := inf m{x). Consider the symbol (x,Z)»p(x,0:=q(x,Om{x)
(2-518)
which has the property that £ — i > p(x, £) is a continuous negative definite function. The operator -p{x,D)u(x):=-(2n)'n/2
f
eixtp(x,£)u(£)dZ
(2.519)
maps C£°(E n ;E) into Coo(E n ;E), is closable in Coo(E";E) and its closure is a generator of a Feller semigroup. R e m a r k 2.10.2. In volume III of this treatise we will use probabilistic methods to relax some condition in Theorem 2.10.1 The proof of Theorem 2.10.1 requires some preparations. P r o p o s i t i o n 2.10.3. Let p(x,£) be as in Theorem 2.10.1. Then we have for all e > 0 the estimates \d?dPp(x,Z)\ < Caa.pMO.+M))-'*^ implying thatp(x,£)
belongs to
(2.520)
S2pM+£'^(Mn).
Proof: We have to estimate d?d£P(x,0
= d?dZ(q(x,0mix))
= d^d^(expm(x)logq(x,0).
(2-521)
Prom (1.2.28) it follows that with / = |a| + |/?| |^p(s,OI <
exp(m(x)}ogq(x,t))x V
5Z px+-+pl'=p i'=o,i,...,i
c
{<*s,p*}*[[
(2.522)
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
206
where QaifiJ {x, 0 = df df (m(x) log q(x, £)) = £
Qdr'Jrn(X)dfd?logq{x,0.
(2-523)
Now, from (1.2.26) with A; = \aj\ + |/f | > 0 we may derive
dfd?iogqM=
c { a i J i > n \x{x%
^ 1
j
a +---+a''=a
•
%
~1
By our assumptions q(x,£) is an elliptic symbol in S%'^(M.n) which yields further that p(i°'l)
5 1 +---+5 f e =a : ' i = l
where we used the subadditivity of p. In addition we have always | log q(x,£)\ < clog(l + V(0) < c £ (l + i>(0)£/21 with / as in (2.522). For m G C6°°(Rn) it follows therefore from (2.523) that
Combining finally (2.524) with (2.522) we arrive at
\d^p(x,0\
i+...+c/=a
i=i,...,r
ii (i+vcor/20 i=i,...,r
r=o (
^^K^a+v^))""11^, and (2.520) is proved. The second statement follows now from the estimate P(x, £) < c(l + ip(£))M proving the proposition. •
2.10 Pseudo-Differential Operators with Variable Order of Differentiation
207
We want to prove Theorem 2.10.1 with the same strategy as we proved for example Theorem 2.6.4, i.e. we will first introduce the notion of weak solutions for p(x,D)u = f and prove that for A sufficiently large the existence of a unique weak solution to p\(x, D)u = p(x, D)u + Xu = f in some Hilbert space. Then we will prove the regularity of weak solutions, and an application of Theorem 2.6.1 will also give the positive maximum principle for —p{x,D) on a suitable domain. First we need to construct a suitable Hilbert space for dealing with p(x, D). The idea is taken from the theory of ^-coercive operators as considered by Chr.Simader and I.Louhivaara. For A > 0 we set p\(x,£) = p(x,£) + A and further Bx(u,v)
:= (px(x,D)u,v)0
for u,v G C^(Rn).
(2.525)
Since the proof of Theorem 2.5.6 yields the estimate \(q(x,D)u,v)0\ for all u,v€
< c||u||,/,,t|M|i/>,t
C£°(R") and q e S^iW1)
we find immediately
Lemma 2.10.4. The bilinear form B\ has for alle > 0 a continuous extension to H^'M+i(Rn) provided M - fi + e <\. In particular \B\(u,v)\
< c||u||v,,M+f IM|I/>,M+§
holds for all u,v <E
(2.526)
H^M+i(M.n).
In the following we will denote the continuous extension of the bilinear form Bx to H*'M+t (R") again by Bx. We get also a lower bound for Bx for A large, but not in the space H^'M+i(Rn). Since by our assumptions q(x,£) is elliptic, there exists 6 > 0 such that P(x,0>6(l+4>(Or
(2-527)
for all £ € R" where // is the largest lower bound for m(x). Proposition 2.10.5. For A > Ao, Ao sufficiently large, Bx(u,u)>6-\\u\\l^ holds for all u € i T ^ R " ) .
(2.528)
Chapter 2
208
Generators of Feller and Sub-Markovian Semigroups
Proof: From (2.527) we know that Q(x,£) := p(x,£) - 6(1 + ip(£))» 6 S2M+E'^(M.n) and Q(x,£) > 0. We may now argue as in the proofs of Theorem 2.5.5 and Theorem 2.5.6 to find (p(x,D)u,u)0-6\\u\\l^
> -c\\u\\lM+i_i
>
S
-\\u\\%tlt-c(6)\\u\\l
where for the last step we used the fact that M + § — \ < H-
•
To overcome the difficulty that the upper and lower bounds for B\ do hold in different norms we work now as Chr.Simader and I.Louhivaara, [200], and [201], with the symmetric part Bx(u,v)
= -(Bx(u,v)+Bx(v,u))
of B\ on H^'M+i(En), \B\{u,v)\
(2.529)
i.e. the smaller space. Obviously we have
(2.530)
and c
Bx(u,u)
>-\\u\\ltli
(2.531)
for A > Ao. In particular B\ is a scalar product on H^'M+i (R™) and therefore we may consider the closure of # ^ M + § ( ] R n ) c i T ^ R " ) with respect to this scalar product. Denoting this Hilbert space by ^ ( R " ) with corresponding norm || • ||PA = Bx' (•, •) we have the continuous embeddings H^,,M+i
( R ») ^ Hpx(Rn)
ff*"(ln),
*->
(2.532)
and C£° (R") is dense in H^ (R n ). Lemma 2.10.6. The bilinear form B\ is continuous on HPx(M.n). Proof: We may apply Corollary 2.4.23 to find that ^(px(x,D)+p*x(x,D))
= =
where r\ G S% ~1+£'^(Wl) \Bx(u,v)\
=
^(px(x,0+Px(x,D))+r1(x,D) px(x,D)+n(x,D),
and we used that p(x,£) is real-valued. Now, since
\(px(x,D)u,v)0
< \\{(px(x,D)+; g K ^ ^ ' ^ + P A C ^ - D K ^ o l + KnCar.DKwJol |5A(«,i;)| + |(ri(ilI>)u>i;)o|,
2.10 Pseudo-Differential Operators with Variable Order of Differentiation
209
the continuity of B\ on Hpx (Rn) and the estimate |(ri(x,D)u,u) 0 | <
c\\u\\i>tM+,_i\\v\\iJ:M+i_i
which follows from Theorem 2.5.6, imply the result if we note that M+§ — \ < fj, which yields the continuous embeddings HPx(Rn)
--> i T ^ ^ R " ) ^
^'M+t-i(Rn),
i.e. the estimate ||u|| v , ) M + f _i
D
Now we are in a position to prove an existence result. Theorem 2.10.7. Let p(x,£) be as in Theorem 2.10.1 and A > Ao sufficiently large. Then for every f £ i / ^ - ^ R " ) = ( # ^ ( R n ) ) * there is a unique H^^(Rn) such that px{x,D)u
= p(x,D)u + Xu = f
(2.533)
holds. Proof: Since by Lemma 2.10.6 and Proposition 2.10.5 B\ is a continuous, coercive bilinear form on Hpx, by the Lax-Milgram theorem, Theorem 1.2.7.41, there exists to any / £ H^-^(Rn) a unique u £ Hpx(Rn) such that Bx(u,v) = (f,v)
(2.534)
holds for all v £ HPx(Rn). Take a sequence ( U „ ) „ € N in C£°(R") converging in Hpx(Rn) to u. Since for v £ C£°(R n ) the mapping u >->• (u,v)0 has a continuous extension to ff*M-2^-£(R>>) and since p\{x,D) : H^^(M.n) -> # ^ - 2 M - 2 ( R n ) is continuous, it follows from (Px(x, D)uk, v)0 = Bx{uk, v),
v £ C0°°(R"),
that for k —> oo (px(x, D)u, v) = Bx(u, v) = (/, v) holds for all v £ C£°(R n ). Thus px(x,D)u HPx(Rn) we find Bx{u1-u2,v)
=0
(2.535) = f. For two solutions « i , u 2 £
Chapter 2 Generators of Feller and Sub-Markovian Semigroups
210
for all v G C^(Rn) and by the continuity of Bx on JP*(R n ) we have B\{ur u2, v) = 0 for all v G # P * ( R " ) , hence
-
C
- | | u i - u 2 | | ^
= 0
•
Our next step is to prove that if u is a solution to (2.533) and / is more regular, i.e. belongs to some space H^'k(M.n) with A; > —/x, then u is also more regular, i.e. belongs to some space # W ( R n ) such that # ^ ( R n ) g HPx(Rn). We need first Lemma 2.10.8. The function p'^1(x,^)
= —,—A
. belongs to the class
Proof: We use (1.2.27) to find with / := \a\ + \0\ that
'*< ** ^ ) i < ^ ^
E f n I
PA(g>0
Using (2.520) we get for any e > 0
\df a?Px(x,o\ <
saiM^+^))=ei^ML
and the ellipticity assumption (2.516) finally yields by the subadditivity of p
•
Theorem 2.10.9. Letp{x,£) be as in Theorem 2.10.7 and letue Hpx(M.n) C H^fW1) be the solution to (2.533) for f e H^'k(Rn), k > 0. Tfcew for all e>0 it follows that u e i l ^ + ^ - ^ R ™ ) . Proof: We may apply Corollary 2.4.23 and find p^(x,
D)oPx(x,
D) = id+r(x, D)
(2.536)
with r e 55" t,v, (R") where - t = ( - 2 M + £ ) + ( 2 M + £ ) - 1 = 2(M-n+e~)
< 0.
(2.537)
2.10 Pseudo-Differential Operators with Variable Order of Differentiation
211
Since p\(x, D)u = / we deduce from (2.536)
u = pj 1 (x, D)opx(x, D)-r(x, D)u = p^ix, D)f-r(x,
D)u.
Since further p~1(x,D)f e #*.*+2/'-*(R«) and r(x,D)u e #*•"+* (R n ) we find that u £ H*^+t^k+^-e\Rn). A finite number of applications of this argument finally gives u G i T ^ + ^ - ^ R " ) . D Now we come to Proof of Theorem 2.10.1: From (2.515) it follows that for s > n/2r the space H^,s(M.n) is continuously embedded into C 0O (R n ). For e > 0 satisfying M - / x + e < l / 2 w e have that p £ S2M+£^(M.n). If we choose k > n/2r it follows that #,fc+2M+e(Rn) a n d #V>,fc(K") c a n both be considered as dense subspaces of C 00 (M"). Hence -P(X,D)
: Jf*-.*+2^+*(R») _^ ff*,*(R»)
is a densely definite operator on Coo(R™) with a continuous negative definite symbol. Theorem 2.6.1 yields that —p(x, D) satisfies also the positive maximum principle on H^>k+2M+E^ln). Finally, combining Theorem 2.10.7 and Lemma 2.10.8, for A > A0 sufficiently large, for each / £ #V>,fc+2(M-^-£)(Rn) there exists an element u € H^'k+2M+e{Wl) such that px(x,D)u = / , i.e. k 2M £ n n (px(x, D), H*- + + (M. )) considered as an operator on Coo(M ) has a dense range. Now the Hille-Yosida-Ray theorem, Theorem 1.4.5.3, gives Theorem 2.10.1. As in Section 2.6 we may argue further (recalling that B\{u,u) > 0) to derive Corollary 2.10.10. The operator —p\(x,D) extends to a generator of an L2-sub-Markovian semigroup and (B\,Hp*(E.n)) is a semi-Dirichlet form. Our interpolation results now yield Corollary 2.10.11. For A > Ao and 2 < p < oo the operator extends to a generator of an Lp -sub-Markovian semigroup.
-p\(x,D)
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Chapter 3
Potential Theory of Semigroups and Generators Chapter 3 discusses aspects of potential theory. We start with a treatment of capacities and continue with abstract Bessel potential spaces J-r,p associated with an L p -sub-Markovian semigroup, see Section 3.2. In order to get applications to concrete operators we have to characterize the spaces J-r,p in terms of function spaces which is done in Section 3.3. In Section 3.4 Stein's LittlewoodPaley theory for sub-Markovian semigroups is discussed. Section 3.5 is devoted to global properties of sub-Markovian semigroups. Topics such as invariant sets, transience, recurrence or abstract potential operators are on the agenda. The final section indicates how Nash-type and Sobolev-type inequalities can be used also for non-local operators.
3.1
Capacities and Abstract Bessel Potential Spaces
In this section we first discuss the general theory of Choquet capacities and then we will discuss a lot of concrete capacities associated with sub-Markovian semigroups, Dirichlet forms and kernels. Since we will apply the concept of capacities in several distinct situations we prefer now to work on a general Polish space G, i.e. a complete metrizable Hausdorff space with countable base. Of course, as long as it is not stated otherwise, measurability will refer to the Borel-cr field B(G). We start with introducing analytic subsets of G.
214
Chapter 3
Potential Theory of Semigroups and Generators
Definition 3.1.1. Let G be a Polish space. A. A set E C G is called a Ka -set if it is a countable union of compact sets. B. A set E C G is called a Kas-set if it is the countable intersection of Ka-sets. C. We call a subset E C G an analytic set if there exists a Ka$-set F in a compact Hausdorff space H and a continuous mapping / : F —> G such that E is the image of F under / , i.e. E = f(F). We write Ana(G) for the family of all analytic subsets of G. Example 3.1.2. A. If K is a Kas-set and C C K is a closed subset, then C is also .K^-set since
C = CnK = Cn( f| U Kkn) = f) U (CnKkn) fceNneN fc€Nn£N
and each set C l~l iffcn is compact. B. Every compact subset K of G is an analytic set. Indeed, every compact set K is a .ftT^-set and therefore, taking as compact Hausdorff space the set K itself and as mapping the identity we see that K is the image of Kas-set of a compact Hausdorff space. Remark 3.1.3. A. Our definition of analytic sets is taken from G.Choquet's paper [48]. In more recent treatises such as R.F.Bass [17], J.Bliedtner and W.Hansen [30], or C.Dellacherie and P.-A.Meyer [64] the following definition of analytic sets is given: A set E C G in a Polish space is called analytic if there exists a compact Hausdorff space H and a K^g-set C C H x G such that E is the projection of C onto G. Since we may embed G into a compact space, it is obvious that analytic sets in the sense of this definition belong to the class Ana(G). It requires some topological efforts to show that in the cases we are interested in the converse holds too. The essential step is to show that for any Polish space G the projections of closed sets F c NN x G onto G coincides with Ana(G). We refer to the monograph [49] of G.Choquet or the lecture notes of R.Burckel [39]. B. For further definitions and concepts of analytic sets we refer to the work [49] of G.Choquet, Definition 8.12, L.Carleson [45], or the lecture notes [39] of R.Burckel. C. For a discussion of the relation of Souslin sets and analytic sets we refer to R.Burckel, G.Choquet, and especially to K.Jacobs [167], Chapter XIII, and
3.1 Capacities and Abstract Bessel Potential Spaces
215
C.A.Rogers [236]. Lemma 3.1.4. The family Ana(G) is closed under countable unions. Proof: Let (-E„)neN, En C G, be a sequence of analytic sets. For each of these sets exists a compact Hausdorff space Hn, a Kag-set Fn C Hn and a continuous mapping / „ : Fn —• G such that fn(Fn) = En. Without loss of generality we may assume Hn n Hm = 0 for all n, m G N. Now, on UneN ^ n we introduce a topology by taking as it base the system of all sets which are open in one of the spaces Hn. Denote by H the one-point-compactification of the topological space UneN Hn- Thus H is a compact Hausdorff space. We define the mapping / : UneN^ n ~* @ by f(x) = fn(x) for x G Fn which is obviously continuous and maps UneN Fn o n *° UneN Fn- It remains to prove that F:=Un S N -^« ^s a K^s-set in H. For any n G N there exist K^-sets Fnk such that
F={jFn={J(f)Fkn). neN
ngN fc€N
Since by our assumption the spaces Hn are mutually disjoint, we find that F = flfceN(UneN Fnk)- Since UneN Fnk is a -f^-set in H it follows that F is a Kag-set in H implying the lemma. • Remark 3.1.5. Since every open set is the union of countable many compact sets, Lemma 3.1.4 and Example 3.1.2.B imply that all open sets are analytic. Lemma 3.1.6. The family Ana(G) is closed under countable intersections. Proof: Let (.En)neN, En C G, be a sequence of analytic sets and suppose that this sequence is decreasing (otherwise consider the new sequence (E„) n eN defined by E[ := Ei and E'n := £ „ _ ! n En). We have to show that E := f] En G Ana(G).
(3.1)
neN
By assumption each set En is the continuous image of a .fC^-set Fn which belongs to a compact Hausdorff space Hn, i.e. En = fn{Fn) with / „ : Hn —> G and fn : Fn —> G is continuous. We define the space H := \[ Hn and set F := Y[Fn c H. By Tychonov's n€N
neN
theorem the space H is compact. Let us introduce the sets F* := FJ Yj where neN
Chapter 3 Potential Theory of Semigroups and Generators
216
Yj = Hj for j ^ n and Yn — Fn. It follows that F* is a J ^ - s e t in H since Hj is compact and Fn C Hn is a K^s-set. But F = f]nen F*, thus F is a Kas-set in H. Let us denote by n : H —> i7„ the family of (continuous) projections and define the continuous mappings / „ by fn°nn : H —f G. Note that fn(F) = En. By continuity of / „ the sets Dki := {x G F;fk(x)
=
fi(x)}
(3.2)
are for all k, I £ N closed subsets of F, hence again .ft^-sets in H implying that D := f l Dv
(3.3)
is a K^s-set in H. Further we find for all k, I € N
(3.4)
MD = II\D.
Thus the function f : D —> G, f = fk\o for some fc G N, is a continuous function from the Kag-set D C H into G and the image of D under / is for all k € N given by / ( D ) = fk(D)
C fk(F) = fk(Fk)
= Ek,
which yields f(D) cf]En
= E.
(3.5)
n€N
On the other hand for y € E we find for all k e N some xk G Fk such that fk(xk) = y which leads to x := ( # i , . . . ,xk,...) G D and f(x) = y, i.e. E = f(D) implying the lemma. • Theorem 3.1.7. The Borel sets in a Polish space G are analytic sets. Proof: By definition the Borel sets form the smallest family of subsets of G containing all compact sets and which is closed with respect to complements and countable unions. Denote by C the system of all analytic subsets of G having an analytic complement. Since compact and open sets are analytic, it follows that C contains all compact sets. Clearly, C is closed with respect to countable union. Hence C contains all Borel sets and the theorem is proved. •
3.1 Capacities and Abstract Bessel Potential Spaces
217
Definition 3.1.8. Let G be a Polish space. A function cap* : V{G) —> R+ U {oo} is called an outer Choquet capacity on G if the following properties are fulfilled if Ai, A2 £ V{G) and Ax C A2, then cap*(Ai) < cap*(A 2 ), if
(A,)„€N,
(3.6)
Av G V(G), is an increasing sequence
then cap*( 1 1 ^ ) = sup cap*(A„);
(3-7)
for any decreasing sequence (-A„)„eN of compact sets Av £ V{G) it follows that cap*( f] A„) = inf cap*(A„).
(3'8)
For a given outer Choquet capacity we define the inner Choquet capacity on G, cap, : V(G) -> R+ U {oo}, by cap„(A) := sup{cap*(B); 5 C A
and 5 compact}.
(3.9)
Note that (3.7) implies cap, (A) < cap*(A)
for all A C G.
(3.10)
Definition 3.1.9. Let G be a Polish space and cap* an outer Choquet capacity on G. A set Ac G is called capacitable with respect to cap* if cap*(A)=cap,(A)
(3.11)
holds. In this case we write cap(A) := cap*(A) and call cap(A) the Choquet capacity of the set A C G (with respect to a given outer capacity). Remark 3.1.10. A. Instead of outer Choquet capacity, Choquet capacity, etc, we will often write just outer capacity, capacity etc. B. Clearly, all definitions make sense for a general Hausdorff space and we will use this fact sometimes, i.e. we will study sometimes outer capacities on Hausdorff spaces which are not Polish. C. Note that there is some redundancy in Definition 3.1.8. Using (3.7) we may deduce (3.6): Consider the increasing sequence Ai,A2, A2,..., Ai C A2. It follows that cap*(^2) = cap*(M Av) = sup cap* (Av) = max{cap*(Ai),cap*(A2)} > cap*(Ai).
218
Chapter 3
Potential Theory of Semigroups and Generators
We decided to put (3.6) into the definition since this is most common in the literature. Our next aim is to prove that all analytic subsets of G are capacitable, thus in particular all Borel sets are capacitable. For this we need some preparations which we take from R.F.Bass [17]. L e m m a 3.1.11. In a Hausdorff space G all Ka&-sets are for all outer capacities capacitable. Proof: Let A be a .ft^-set, i.e. for compact sets Aki we have
A=f]Ak=f][JAkl, and without loss of generality we may assume that for each k the family (Aki)i<sN is increasing. Now let cap* be an outer capacity on G and suppose that a < cap*(A). We will construct a compact set B C A such that cap* (B) > a which will prove the lemma. Indeed, suppose that in this situation cap„ (A) < cap* (A) holds. Then there exists a such that cap, (A)
cap* (A),
but in addition we have according to (3.9) a < cap*(5) < sup{cap*(5'); B' c A
compact} = cap„(A) < a,
which gives a contradiction. By induction we construct first compact sets Bn C An, n S N, such that for Dn := AnBi n . . . n B n we have cap*(Z)n) > a. From (3.7) we deduce cap* (A) = c a p ^ A n A i ) = sup cap*(AD An). l&N
We choose now Zi large enough in order to have that cap*(A n Aut) > a and set Bi := Aid a s w e u as Di := A n B\. Assume that B\,..., B n _ i and Dn-i are already constructed. Since A C An it follows that a < cap*(D n _i) = cap*(Dn..in
An)
and therefore, once again by (3.7) cap*(D„_i) = sup cap* l€N
(Dn-tnAni).
3.1 Capacities and Abstract Bessel Potential Spaces
219
Now we take ln large enough to get cap*(.D n _i C\Anin) > a, and define Bn := Anin and Dn := D„_i n Bn. Thus we find Dn c An and cap*(D n ) > a. Now we put En : = B 1 n . . . n B „ and 5 := f ) n e N B n = f ) n g N £ „ . Since each set i? n is compact and the family (En)n&^ decreases, it follows that B is compact. Further we have Dn C En implying cap*(E n ) > cap*(D n ) > a. Now we use (3.8) to get cap* (B) > a. But in addition we have Bn C An which yields B C A and the lemma is proved. • In proving the next result we use the definition of analyticity given in Remark 3.1.3A. Lemma 3.1.12. Let A C G be an analytic set and let C be a Kas-set in the space H x G where H is a compact Hausdorff space having the property that 7r(C) = A, here ir : H x G —• G is the natural projection. On V(H x G) we define the function c*(B) := cap*(7r(5)),
B £ V(Hx G).
(3.12)
Then c* is an outer capacity on H x G. Proof: It is clear that c* satisfies (3.6) and (3.7). We are going to prove (3.8). First note that for any decreasing sequence (JBn)neN of compact sets Bn C H x G it follows that 7r(p|„ 6N Bn) — flneN n(Bn). Since we always have 7r(P|neNJB„) C n«eN 7 r (^n) ^ remains to prove the opposite inclusion. Let x e f]ne^^(Bn)We are looking for y E r\n€^Bn such that 7r(y) = x. Since for any x £ DneN ^(^n) the family ((7r _1 (x)n J B n ) rae N has the finite intersection property, i.e. for any finite sequence we have that (~|m=1((7r_ (x)) n BUm) is non-empty, it follows that n^Li( 7 r _ 1 ( a ; ) n -Bn) i s non-empty, and any y e n ^ L ^ 7 1 " - 1 ^ ) n Bn) leads to ir\y) = x. Thus it follows c*( f ) Bn) = cap* (TT( f | Bn)) - cap* ( f ) 7r(B n )). nGN
ngN
n€N
The continuity of 7r implies that for compact sets Bn the sets Tr(Bn) are compact too. Thus we have cap* ( p i n(Bn)) proving the lemma.
= inf cap*(7r(Bri)) = inf c*(Bn), •
Now we are in a position to prove
220
Chapter 3 Potential Theory of Semigroups and Generators
Theorem 3.1.13. Let G be a Polish space and cap* any outer Choquet capacity, then every analytic set A is capacitable with respect to cap*. Proof: Since A c G i s analytic there exists a .ftf^-set C in the space H xG where H is a compact Hausdorff space having the property that 7r(C) = A and IT : H x G —> G is the natural projection. Prom Lemma 3.1.11 and Lemma 3.1.12 it follows that C is capacitable with respect to the outer capacity c* defined by (3.12). Assume that cap*(^4) > a. If we can show that there exists a compact set E C A such that cap*(.E) > a we may conclude as in the proof of Lemma 3.1.11 that cap„(A) > cap* (.A), implying that cap,, (A) = cap*(A), i.e. the capacitability of A. Now, since cap*(A) — c*(C) > a and C is capacitable, there exists some compact set J C C such that c*(J) > a, but then E := TT(J) satisfy cap*(i?) = c*(J) > a and E C A is compact. • Corollary 3.1.14. The Borel sets in a Polish space are for all outer capacities capacitable. In our definition of an outer capacity we started with a set function defined on V(G). Often it happen that we can define easily a function c* on a subfamily of V{G). In this case the question arises whether we may extend c* to the whole family V(G) as an outer capacity. The following result is taken from the monograph [102] of M.Fukushima et. al., see also C.Dellacherie and P.A.Meyer [64]. Theorem 3.1.15. Let G be a Hausdorff space and denote by QQ the family of its open sets. Assume for a function c* : OQ —+ R+ U {oo} ifA,Be
OG and Ac
for all A,B£
B then c*(A) < c*{B);
(3.13)
OQ we have
(3.14)
c*(AuB)-rc*(AnB)
(3.15)
c*((J An) =supc*(,4 n ). Then the function cap*(5) := inf{c*(A); A e OG
and
Be
A},
B G V(G),
(3.16)
is an outer Choquet capacity extending c* to V(G). Furthermore cap* is countable subadditive.
3.1 Capacities and Abstract Bessel Potential Spaces
221
Note that (3.15) implies (3.13), but we consider it to be convenient to state it separately. We prove first an auxiliary result. Lemma 3.1.16. Let c* be as in Theorem 3.1.15 and denote the family {A G OG;C*(A) < oo} by 0GjiniteFor any pair of sequences (An)nen, (Bn)„eN such that An C Bn and An, Bn G OG,finite we have n
n
c*( | J Bk)-c*(
n
( J Ak) < YJ{c*{Bk)-c\Ak)).
k=\
fe=l
(3.17)
fc=l
Proof: We prove (3.17) by induction. For n = 1 this is just a triviality. Now suppose that (3.17) holds for n € N. We set n
n
An := | J An
and
Bn := | J
Bn.
fc=i fc=i
Since Bn U Bn+i = Bn U {Bn+1 U An) and i n c f i „ n (B„+i U An) it follows from (3.13) and (3.14) that c*{Bnl)Bn+1)
+ c*(An)
< c*(Bn U {Bn+l U An)) + c*(Bn n (Bn+1 U A n ))
(3.18)
< c*(Bn+1)+c*(AnUAn+1).
(3.19)
Adding (3.18) and (3.19) yields c*(BnUBn+1)
+ c*(An) + c*(Bn+1 U A) + c*(An+1)
< c*(Bn) + c*(Bn+i U An) + c*(Bn+1) + c*(An U ^ + 1 ) , or, since BnL)Bn+i implies
= Bn+1 and AnUAn+i
= An+i,
the induction assumption
c*(B n + 1 ) - c*(An+x) < c*(Bn) - c*(An) + c*(Bn+1) -
c*(An+i)
n
< ^ ( c * ( 5 f c ) - c*(Ak)) + c*{Bn+l) k=\
=
J2(c*(Bk)-c*(Ak)), k=i
-
c*(An+1)
Chapter 3 Potential Theory of Semigroups and Generators
222
and the lemma is proved.
•
P r o o f of T h e o r e m 3.1.15: Clearly cap* is an extension of c* to V{G) which is monotone in the sense that cap* {Bx) < cap* (B2)
for Bx c B2,
and which is subadditive, i.e. cap*(B!U5 2 ) < cap*(Si)+cap*(5 2 ). Next we prove that for any increasing sequence (A„)ne^, that cap* ([\An)=
supcap*(A„)
An C G, it follows
(3.20)
holds. Since cap* is monotone and subadditive it is sufficient to prove under the assumption that sup ca,p*(An) < oo it follows that 7l£N
cap* ( I J An) < sup cap* (.A„). By the definition of cap* for any e > 0 there exists Un G OQ,finite such that An C Un and cap*(.4 n ) < c*(Un) < cap*(A n ) + ^ ,
(3.21)
which yields lim c*(Un) = lim cap*(,4 n ). n—too
n—»oo
Furthermore, for m < n it follows that cap*(A m ) < c*(UmnUn)
< c*(Um) < cap*( J 4 m ) + ^ r ,
and consequently we have n
Yl(c%Uk)-c*(UknUn))
<e.
(3.22)
3.1 Capacities and Abstract Bessel Potential Spaces
223
Now we apply Lemma 3.1.16, i.e. (3.17), to get n
n
c* (\JVk)-
n
c*(Un) = c* ( ( J Uk) - c* ( [ J (Uk n Unj)
• ' fc=l
k=i *;=!
fe=i fe=l
n
<^(c*(t/ f e )-c*(^nt/ n ))< £ , fc=l
which leads to it
lim c*( M % ) < lim c*(U"fc)+e. n—»oo
^ ,=
'
1
(3.23)
fc—>oo
For the set C/ := \J£=l C4 we have C/ e 0 G and A C J7. Thus (3.15) implies n
cap* (A) < c*(t/) = lim c*( I ) t / A n—>oo
V
, fc=l
(3.24)
'
Combining (3.22) and (3.23) we obtain cap* (A) < lim cap*(j4„)+e, which gives (3.20). Finally we prove for a decreasing sequence of a compact sets An C G, n £ N, that oo
cap* (C]An) V
„=i
= inf cap*(A n ) '
(3.25)
"£N
holds. Again it is sufficient to assume that cap*(n^Li -^n) < °° an inequality, namely,
an<
^ *° P r o v e
oo
inf cap*(yl n ) < cap* ( f] An). V "e N n=i '
(3.26)
Now, for e > 0 there exists U £ Oojinite such that H^Li ^ n C !7 and c*(£/) < cap*(n^Li-^n) + £• But v4n C f/ from some n and (3.26) follows. Finally, let us remark that the subadditivity of cap* and (3.20) imply the countable subadditivity of cap* and the theorem is proved. • The first class of capacities we are going to consider are capacities which are associated to an L p -sub-Markovian semigroup (Tj)t>o- In particular we will use these capacities to control exceptional sets. We start with
Chapter 3 Potential Theory of Semigroups and Generators
224
Definition 3.1.17. Let (T}p')t>o be an L p -sub-Markovian semigroup on £ P (R™;R). We define its gamma-transform (V}p')r>o by 1
V^u:=—-
f°°
t^^e-'T^udt,
u€lp(R";R).
(3.27)
Remark 3.1.18. Clearly, by Vr(ds) = x ( o,oo)(s) ! YFyS r/2 - 1 e- s A( 1 >(d S )
(3.28)
a convolution semigroup (r/ r ) r >o with support in [0, 00) is given, namely the (modified) T-semigroup, see Example 1.3.9.15. It is associated with the complete Bernstein function / ( S ) = i l o g ( l + S ).
(3.29)
Thus the gamma-transform ( K )r>o of any L p -sub-Markovian semigroup (Tj )t>o is once again an L p -sub-Markovian semigroup which is obtained from the original one by subordination. Therefore we have V^oVrf=V^\r2-
(3.30)
||V; ( P ) U|| L P < ||«|| L p;
(3.31)
YrniV^u
=u
inLp(Rn;R);
(3.32)
r-»0
and 0 < u < 1 a.e. implies 0 < V r (p) u < 1 a.e.
(3.33)
The following result is taken from our joint paper [86] with W.Farkas and R.Schilling. Theorem 3.1.19. Let (T^)t>o be an Lp-sub-Markovian semigroup with generator {A^\D{A^)). For all r > 0 and all u e Lp(M.n; K) we have V^u
= (id -A^)-T'2u.
In particular, each of the operators V}
(3.34) is injective.
3.1
Capacities and Abstract Bessel Potential Spaces
225
Proof: Let / be as in Remark 3.1.18 the Bernstein function f(s) — \ log(l+s). We know already that Vrip) = T r ( p ) , / where (T^p)J)r>0 denotes the semigroup obtained by subordinating (T r ) r >o with respect to / (or (r)r)r>o), see Section 1.4.3. On D(A^) the resolvent of (A^\D(A^)) satisfies for A > 0 /•OO
Jo Using the Dunford calculus for unbounded operators, see Section 1.2.7, applied to the operator id — A^ we find (id-A^)~a
= ^ - /V*((£+!)id-A^)-1^.
(3.35)
Here the path 7 C p(id—A^) surrounds the spectrum a(id—A^), and is negative oriented, i.e. clockwise. Since the integrand is analytic, we can deform it to get 1*00
(id-A^)-s
X~S{(X+1) id-A^^udX.
= ^ ^ -
(3.36)
Jo
T
By the resolvent formula we have /•OO
(id-AW)-°
=
S
00 r°°
sin sir 7T
/»0O
A"s(/
^
Jo
r/»oo J0
e-(x+V%ip)udt)d\
\-°e-(*+i)tT(p)udtd\.
Observe that the term A s e ( A + 1 )' is positive. Thus Tonnelli's theorem yields for s G (0,1) /•OO
/ Jo
/*00
nOO
/ Jo
\-°e-(x+Vtdtd\=
/ Jo
/•OO
= / Jo
rOO
/ Jo
X-'e-^+^dXdt
/»00
/i-Vd/i/ Jo
t*_1e_tdt
Chapter 3 Potential Theory of Semigroups and Generators
226
< A- s e -( A + 1 ) t ||u|| L P ) we find for 0 < s < 1 that
Since WX-'e'^+^T^uW^
/•OO
(id-A&)-u = ^ ^ 7T
/*00
/
/
Jo
JO
_ sins7r f oo r/»oo
vr J0 J0 i
A-^e-^1)'dXT^udt X-3e-Xtd\e-tTtip)udt
f°°
where we used the formula T(s)r(l — s) = ^~=, So far we have proved for 0 < r < 2 that (id-A(p))-r/2u
= V}p)u,
0 < s < 1.
ueD(A(p)).
(3.37)
Using the functional calculus, see Section 1.2.7, on the left hand side, and the semigroup property on the right-hand side, we see that (3.37) extends to all r > 0. Finally, the strong continuity of the semigroup (K- ) r >o yields that (3.37) holds also for r = 0 and the theorem is proved. • Remark 3.1.20. From Example 1.4.3.27 it follows that for every strongly continuous contraction semigroup (Tt)t>o its gamma-transform is an analytic semigroup. Using the injectivity of V} type spaces: •F r , p (R n ; R)
:=
we may define the following Bessel potential
V}p)(Lp(Rn;
R))
(3.38)
with norm
IMI*,P := N L P for u = VrMt/.
(3.39)
Clearly, the spaces (J> | P (R";R); || • ||^-r ) are separable Banach spaces. Moreover we have Corollary 3.1.21. In the situation of Theorem 3.1.19 it holds f r , p ( l " ; R) = £>((id -A^Y'2).
(3.40)
3.1 Capacities and Abstract Bessel Potential Spaces
227
Proof: Obviously we have u <E J>,p(]Rn; R) if and only if u = Vr(p) v for some v 6 L p (R n ;R). Since Vr{p) = (id-A^)-r/2 we have u 6 :F r , p (R n ;R) if and p) r/2 only if u = (id -A<- )- v which implies D((id - A ( p ) ) r / 2 ) = k-, P (K n ; R)- • Our next aim is to prove the equivalence of the norms || • ||^ r and Wi-A^Y'2 • ||iP + || • ||LP. For this we show first Lemma 3.1.22. Let (A^P\D(A^)) semigroup (T t (p) ) t > 0 . For allr>0 we have
be the generator of an V-sub-Markovian and all u € £>((id ~A^)r>2) = JF r]P (R n ; R)
\\U\\LP <\\{)A-A^)r'2u\\Lr.
(3.41)
Moreover, if s,r > 0 then there is a continuous embedding •7>+s,p(R"; R) ^ ^>, P (R n ; R).
(3.42)
Proof: Since id = {id-A^)~rl2 (id-A^)r lows by the contraction property of V}p' that IMILP
= \\vr^ (id-A^y/2u\\LP
< ||(id
I2 = Vrip)(id-A^)r/2
it fol-
-AV>y'2u\\Lr,
whereas (3.42) follows from
||(id-A(p))r/2u||Lp = I K i d - A ^ r ^ i d - A ^ ^ I U p = WV^iid-A^^uWLP
< \\(id-A^)^u\\LP.
D
Lemma 3.1.23. Let A, B be two generators of Lp-sub-Markovian semigroups such that A — B is a bounded operator which extends to L p (R n ;R). Then we have \\(-A)a-(-B)a\\<\\A-B\\.
(3.43)
Proof: For A > 0 and u £ Lp(E.n; R) we find (-A)R$u-(-B)R%u
= u- \R$u -U + XRfu
= \R%u - \R$u = \R${B -
A)Rfu,
which implies \\(-A)Ri-(-B)R?\\
< ||AJtf || + ||Afl?|| < 2
228
Chapter 3 Potential Theory of Semigroups and Generators
as well as \\{-A)Ri-(-B)Rf\\
< ||AJtf ||||B-A||||i2?|| < M Z J ? I .
Now, for any s > 0 we have X<£-\\A-B\\
implies
^
•
2 ^ 2 ^ , - M A
and £„.
n„
,-
\\A- B\\
2 +e
\\A-B\\
which yields ||(_^)<«_.(_S)a||
\"-m-A)Ri-(-B)R*\\d\ Xa-1\\(-A)Rt-(-B)Rf\\d\
Jo
+ ^rLT *
r-'M-VRi-i-WfWdx J$\\A-B\\
V ; * Jo A+||A-5|| sinmr y~ /2 + eN p - 311 TT 7 f ||i4 _ s|l V 2 J A + | | A - B | | A
~
< ^ ( 2 + V 2 J
£
x
)I(|)Q||A-Sr 7r
y0
A + ||i4-B||
and for e —• 0 we find ||(-A)a-(-£)"|| <||A-5||a and the lemma is proved.
•
Corollary 3.1.24. Let (A^p\D(A^p))) be the generator of an W-sub-Markovian semigroup. Then (A^p' — id, D(A^P>)) is also a generator of an Lp-subMarkovian semigroup and for 0 < a < 1 we have ||(-i4)a-(id-A)a||
(3.44)
3.1 Capacities and Abstract Bessel Potential Spaces
229
Theorem 3.1.25. Let (A^^iA^)) be the generator of an L?-sub-Markovian semigroup (T£p')t>o. Further let 0 < r < 1. Then for all u £ D(A^) we have
i(ii(-^)f U |i LP + \\U\\LP) < n(id-^>rU||LP
In particular we find D((~A^) )
-A^)r).
= D((id
Proof: Using (3.44) we find ||(-J4<*>) P U|| L P - \\Cid-A^)Tu\\LP
< \\{-A^fu<
(id-A^)ru\\LP
\HLP
which gives using (3.41) ||(-j4<*>) r u|| L , +
||«||LP
< \\(id-A^Yu\\LP
+ 2\\u\\LP
r
<3\\(id-A^) u\\LP. On the other hand we have \\(id-A^)ru\\LP
- \\(-A^)ru\\LP
< \\(id-A^)ru-
{-A^)ru\\LP
< ll«||z>, or \\{id-A^Yu\\LP
<
| | ( _ A W ) - - U | U P + ||U|| L P
and (3.45) follows. Since D{A^) = D(id-A^) and since D((-A^)r) and D((id—A^)r) are obtained by completion of D(A^) with respect to the equivalent norms \\(-A^Yu\\LP
+ \\u\\LP
and
||(id-A<*>) r u|| L p,
respectively, we have proved the theorem noting that D(A(-P^) = D(id— is an operator core for both (—A^Y a n d (id— A^YD
A^)
We want to extend Theorem 3.1.25 to k + r, k G No, and 0 < r < 1. Theorem 3.1.26. Let (A( p ) ,D(A ( p ) )) be as in Theorem 3.1.25 and let k € N0 and 0 < r < 1. Then there exists constants 71,72 such that for any u 6 D((-A^)k+1) 7!(||«|UP
+ \\(-AM)k-*u\\LP)
<
\\(id-AW)k+ru\\LP
<72(||"||Lp + | | ( - ^ ) f e + ^ | | L P ) .
230
Chapter 3 Potential Theory of Semigroups and Generators
Furthermore, we have D((-A^)k+r)
-A^)k+r).
= £>((id
Proof: We will use a result due to L.Hormander namely Theorem 1.2.7.10. For this assume for a moment that D((-A{p))s)
= D{{id -A(p))s)
(3.47)
holds for all s > 0. It follows that the operators ((-Atoy,D((-AV>y)
and
((id-A^)s,D((id-A^)s)),
s > 0,
are closed and we obtain from (1.2.154) ||(-^))SU||LP <
Cl(||(id-^))
S
U||L?
+ || U || L I ) ),
3 > 0,
as well as ||(id-j4<*>)'u|| L p < c2(\\(id-A^yu\\L„
+ \\u\\LP),
s > 0,
implying the theorem. Thus it remains to prove (3.47). For any generator (^4, D{A)) of a strongly continuous contraction semigroup we have for s, t > 0 D((-Ay+t)
= {ue D((-Ayy,
(~Ayu
e
= {ue DU-A)');
(-Ayu
G
D((-A)')} D((-A)*)}.
Thus, assuming that (3.47) holds for s — k + r we find D((-A(p))k+1+r)
= {u£ D(-A^);
-Awu £
= {ue D(id -AW);
-A^u
= {u£ D((-A^)k+r); k+r
= {u£ D((id~A^) ); {p) k+r
= { t i £ D((id -A ) ); =
D((-A{p))k+r)} e D((id
~A^u
-A{p))k+r)}
e D(-A<*>)}
-A^u
£ {p)
D(id-A^)}
u - A u e D(id
-A{p))}
D{{id-A^)k+r+1).
This is the induction step A; to A; + 1. The beginning of the induction k = 0, 0 < r < 1, was given in Theorem 3.1.25, hence the assertion follows. • Remark 3.1.27. As it will become clear in the following section, (3.45) (and (3.46)) has the interpretation that the norms induced by "Bessel potentials" are equivalent to the norms induced by "Riesz potentials".
3.1 Capacities and Abstract Bessel Potential Spaces
231
Now let us return to the spaces ^> )P (R n ; R) and discuss a first example. Example 3.1.28. Let tp : R™ —> R be a fixed continuous negative definite function with associated convolution semigroup (/it)t>o and sub-Markovian semigroup (T^')t>o, 1 < p < oo. The generator of (Tj )t>o is the operator {A)) and we know that S(R") c D(A&>). On 5(R n ) the generator has the representation A<*>u(x) = -iP(D)u{x)
= -(27r) ( - n / 2 ) /
e f a , t y ( O u ( 0 d£.
(3.48)
This follows from Example 1.4.1.13 and the fact that on 5(R n ) the operator A (p) coincide with the operator A^2\ Further, for u e S(R n ) we find V^u
= (id-A^)-r/2u
= (id-A^)~r/2u
=
V^u,
and VrWu(x) = (27T)-"/2 / efa-«(l+^(0)-r/3fi(0de. JR" which implies on S(R n ) that IMI*.., = | | ( 1 + ^ ( £ > ) ) P / 2 « | | L - = WF-W+MyVmUp.
(3-49)
In particular, for p = 2 we have by Plancherel's theorem I M k , 2 = ||(l+V'( J D)) r/2 «l|0 = | | ( l + V ( ' ) ) r / 2 H l 0 = ||«||^r showing that •F r , 2 (R n ; R) = ff*,r(R"; R).
(3.50)
Moreover, for V>(£) = |£| 2 we find IMI*.,, = IMILP for « = V^v
= (1 - A ) - r / 2 u ,
i.e. in the case ^* riP (R n ;R) is just the Bessel potential space H£(Rn;R), Definition 1.3.11.9.
(3.51) see
Remark 3.1.29. Let ip : M.n —» R be a continuous negative definite function. In Section 3.3 we will study in detail Bessel potential spaces associated with
Chapter 3 Potential Theory of Semigroups and Generators
232
Now we are going to introduce a one-parameter family of capacities associated with a sub-Markovian semigroup (T t (p) ) t >o on L"(R";R), 1 < p < oo. For an open set G C R n we define cap r p (G) := inf{||u||^. p ; u e JF r , p (R n ; R)
and u > 1 a.e. on G},
(3.52) where _F riP (R n ;R) is as in (3.38). Furthermore, for an arbitrary set i c R " we set cap r>p (A) := inf{cap r p (G);
A C G and G C R n open}.
(3.53)
Definition 3.1.30. Let (Tj )t>o be an L p -sub-Markovian semigroup, 1 < p < oo, and r > 0. W,e call c a p r p the (r,p)-capacity associated with the semigroup (Tt(p))t>o. We will need some preparations in order to prove that cap Choquet capacity.
is an outer
Theorem 3.1.31. Let 1 < p < oo. For any open setG C R™ uratficap (G) < oo there exists a unique element UQ € J> i P (R";R) such that UQ > 1 a.e. on G and cap riP (G) = || U G ||£ r p
(3.54)
holds. Moreover, there exists f £ Lp(M.n; R), / > 0 a.e. with the property that UG = Vr(p)f. Proof: Recall that a Banach space (X, \\ • \\x) is called uniformly convex if for any e > 0 there exists a 6 > 0 such that ||a;||x < M, ||t/||x < Af and Ik - y\\x > £ implies ||x + y\\x <2M -6, see K.Yosida [288], p.126-127. By a result of J.A.Clarkson [53] the spaces L p (R n ;R), 1 < p < oo, are uniformly convex. Since J>, P (R";R) is the isometric image of L p (R n ;R), the spaces ^v,p(K";R), 1 < p < oo, are uniformly convex. Now let uG' ^ uG' be two elements in J> ) P (R n ;R) satisfying the assumptions of the theorem. Clearly, liu^ + u{G]) G J>,j, and ^(u^ + u^) > 1 on G. Now we use (3.52) and Clarkson's Theorem with M = (cap r p (G)) 1 /P to find
caPr,p(G)1/P < | | ^ 4 ^ I L . , P * V2 ™Pr,P(G)1/P-*) which is a contradiction thus UG must be unique. To prove the existence of UG we use the IP-version of the Banach-Saks theorem, see W.Rudin [240],
3.1 Capacities and Abstract Bessel Potential Spaces
233
Theorem 3.13, for a proof of this theorem, the I?-version is our Theorem 1.2.7.2. Thus we use that for any sequence {uv)v^,
uv G L p (M n ; R) converging
weakly to some u G L P (R";R) there exists a sub-sequence {uVl)i€^ such that converges strongly in L P (R"; R) to u G Lp(M.n; R). Since we
(w E i l i O
assume that ca,prp(G) is finite, there exists a minimizing sequence {UQ )„ew, G.fr,p(R™;R) such that lim llu^HJ-
=cap r j P (G).
The sequence ( / ^ e N defined by u £ } = Vrip)f%] is bounded in L P (R";R), hence it has a weakly convergent subsequence and this subsequence has a further subsequence with the property that its arithmetic means converge strongly to the weak limit fa € L p (R n ;R). We denote this subsequence by (fa'^)leNThus we have
sdiht^-to The linearity of V}
P
= 0. LP
yields
N
lim
I^E^-K^/G"
=0.
1=1
Note that for each N G N we have 4r Yli=i ua — 1 a - e - o n ^ an< ^ further « G := K / G G ^ > , P ( E " ; 1 R ) , which proves the existence of UQ. It remains to prove that fG > 0 a.e. Since K (p) is a positivity preserving operator in the sense that / > 0 a.e. implies Vr(p)f > 0 a.e., it follows that Vr(p) fG < Vr{p) f+ but H/^IILP < II/GIILP. Therefore K ( P ) / G " > 1 a.e. on G and | | / + | | i P < cap PiP (G), implying that fG = fa by the uniqueness result, and the theorem is proved. • Definition 3.1.32. Let uG = Vr(p) fG be as in Theorem 3.1.31. We call uG the (r,p)-equilibrium potential of the open set G with respect to the semigroup (Tj )t>o- The function fa is called the (r,p)-capacity function of the open set G with respect to the semigroup (T^ )t>o-
Chapter 3 Potential Theory of Semigroups and Generators
234
Remark 3.1.33. The proof of Theorem 3.1.31 yields that for an open set G c l " w e have
capr,p(G) = H/GIILP = inf{||/||£ p ; / € Lp(Rn;R)
s.th. / > 0 a.e. and Vr^f
> 1 a.e. on G}.
Theorem 3.1.34. Let (T^p')t>o be an Lp-sub-Markovian semigroup, 1 < p < oo, on L P (R";E). Its (r,p)-capacities, r > 0, have the following properties \^(A)
< cap r)P (,4)
for all A G B(E.n); < cap r 2 p (yl) for all A C W1;
r i < r2
implies ca,prip(A)
pi < p2
implies cap r p i (A) < caprp2(A)
Ac
B
(3.55)
for all A C E ;
implies cap r p (A) < c a p r p ( 5 ) ;
ca P r , p ( [ J A„) < J2 cap r , p (A l/ ),
(3.56) (3.57) (3.58)
A, c R n
(3.59)
and cap
(f]Ku)=
lim cap
(#„) = inf cap
(AT,,)
(3.60)
/or anj/ decreasing sequence of compact sets Kv C M.n. Proof: For u = Vr f we find using the contraction property of (V}p ) r >o and the definition of || • \\rriP that
IMI*,, = WPfWr^
= H/IU- > IIK ( P ) /HL, = ||u|| L ,.
i.e.
\H\LP < \\u\\rr>p, which implies (3.55). The semigroup property of (VsP')r>0 and its contraction property yields for n < r2 that .F r2iP (]R";R) C ^ r r i , p (R";R) and IMI^,,, < llull^2.p w m c n l e a d to (3.56). To prove (3.57) suppose that p\ < pi and consider the (r,p2)-capacity function / G > 0 a.e., UG = Vr/G, for an open set G. Using Holder's inequality we find 1 < (K (pa) f G ) P 2 / p i < Vr{p2\fG)P2/pi 2/pi Pl a.e. on G. Since fG G L (R";R) we arrive at ca P r , P l (G) < ||K (P2) (/G 2/P1 )II^, P1 = | | / e | | £ , = cap r , p2 (G)
3.1 Capacities and Abstract Bessel Potential Spaces
235
implying (3.57). Obviously (3.58) is trivial. For proving (3.59) we note first that for open sets G„ C E™, v € N, the inequality cap r p ((J„ 6 N G„) < E ^ c a p , . ^ , , ) n o I d s - N o w I e t A.v c R™ b e arbitrary. For £ > 0 and i / £ N there exists G„ c E™ open such that <»Pr,p(4,0 < capr>J,(G„) < cap r p (A I / ) + —. Thus we find ca
Pr, P ( U
A
")
ca
-
Vr,P ( U
^ I]
ca
Gv
)
Pr,p(G„) < J ] (ca P r i P (A„) + | ; )
<e + ^caprp(^), i/GN
which yields (3.59). Denote by UA„ the (r, p)-equilibrium potential for Av and set UA„ = ip) Vr fA„, IA„ G £ P ( E " ; E ) , / A „ > 0 a.e. and / := s u p / A „ . Clearly we have i/gN
/ P ^ E „ e N /£„>
thus
/
€
LP(Rn;E) and in addition
IIK-(p)/ll^,p<EcaP^(^)i/GN
Since V} f > 1 a.e. on Ui/eN^"' ^ follows further that
caPr)P [J A, < ||VW/||^p which proves (3.59). Finally we prove (3.60). For this let G C R™ be open and K := f\v€N Ku c G. Thus A*^ C G for some i>0 = vQ(G) and it follows caPr.pW < hm cap r p (A' l / ) < inf {cap r p (G);K C G
and G C M" open}
= cap r p (A'), and (3.60) is proved.
•
Note that it remains only to prove (3.7) in order to identify c a p r p as an outer Choquet capacity; this will be done in Theorem 3.1.53. For this we need an additional assumption on Jv,p(E n ; E).
236
Chapter 3 Potential Theory of Semigroups and Generators
Definition 3.1.35. Let (T^p')t>o be an Z,p-sub-Markovian semigroup, 1 < p < oo, and for r > 0 let J>, p (R n ; R) be defined as in (3.38). A. We call Fr>p(Rn;R) weakly regular if .F r , p (R n ;R) n C 0 (R n ;R) is dense in.F r , p (R n ;R). B. The space .7> jP (R n ;R) is called regular if it is weakly regular and if •Fr,p(R"; R) n C 0 (R"; R) is dense in (C 0 (R"; R); || • ||oo). C. The space J> i P (R";R) is contraction regular if there exists a subspace W C .F r , p (R n ;M) n Co(R n ;R) such that W is dense in (frtP(Rn;R), || • ||jFr,p) and (Co(R"; R), || • ||oo)j and in addition W has the property that u € W implies |u| G W. We give some first results concerning the question whether J>)P(R™;R) is (weakly) regular. Lemma 3.1.36. For k G N the space J"fc+2,P(Mn;R) is dense in
Fk,p(Rn;R).
Proof: We know from Corollary 3.1.21 that (id-A^yk/2Lp(Rn;R),
JFfc>p(R";R) =
where (A^, D(A^)) is the generator of the semigroup (Tt)t>o defining the scale ^"fe,p(Rn; R). Let u = (id -A^yk/2f G ^ fe , P (R"; R) with / G L p (R n ; R). p n Since D(A^) is dense in L (R ; R) we find for every e > 0 some ipE G D(A&>) such that \\f - (pe\\Lp < s. Set he := (id-A^) ) ^ / i £ G JT fc+2]P (R";R) and furthermore \\u-(id-AM)=^he\\rkiP = ||(id-AW)-fc/2(/ l
= \\f-(id-AM)- h£\\Lr, which proves the lemma.
(id-A^y^KW^ =
\\f-
•
Proposition 3.1.37. Let k G N and suppose that the set Co(R n ;R) n D((-A^)k) is an operator core for (-A^)k. Then J" 2A:;P (R n ;R) is weakly regular.
3.1 Capacities and Abstract Bessel Potential Spaces Proof: First note that D((id -A^f) Co(K";R)n^ 2 fe,p(lR n ;R) =
= D((-A^)k),
237 hence we have
C0(Rn;R)nD((-A^)k).
Noting that C0(M"; R ^ J ^ ^ R " ; R) is an operator core for (~A^p))k we choose for every u € J"2fc,p(Kn; R) a sequence (uv)v&i, uv € 6*0(R"; R) D JSfc.pCR"; R), which converges in the graph norm of (—A^) k to u. This implies
\K-uy2k:P = \\(v^)-\Uv-u)\\LP = as v —> oo.
\\(id-AM)k(u„-u)\\LP^0
•
Thus it is possible to reduce the regularity problem for J> ] P (R n ; R) to find a good operator core for A^ or (—A&))k, and we will come back to this fact later on. Definition 3.1.38. Let (Tj )t>o be an L p -sub-Markovian semigroup, 1 < p < oo, and for r > 0 let c a p r p be its (r,p)-capacity. A. A set N C R n is called an (r,p)-exceptional set with respect to (T{p')t>o if cap rp (iV) = 0. B. A statement is said to hold (r,p)-quasi-everywhere with respect to (Tt )t>o if there exists an (r, p)-exceptional set N such that the statement holds on R n \iV. We will use the abbreviation (r,p)-q.e. for (r, p)-quasieverywhere. C. A real-valued function u defined (r, p)-quasi-everywhere on R™ is called (r,p)-quasi-continuous with respect to (Tt )t>o if for any e > 0 there exists an open set G c R " such that cap (G) < e and U|G<= is continuous. D. Let u be as in Part C. We call u (r,p)-quasi-continuous in the restricted sense with respect to (T}p')t>o if for every e > 0 there exists an open set G c l " C R£o such that u|Kn \G is continuous and cap r p (G) < e. E. A function u € J> i P (R n ;R) is called an (r,p)-quasi-continuous modification of u € J^v,p(Rn; R) if u is quasi-continuous and u = u almost everywhere. Proposition 3.1.39. Ifu e J> ) P (R";R) is an (r,p)- quasi-continuous and u > 0 a.e. on an open set G C R n , then u > 0 (r,p)-q.e. on G.
function
Proof: Let e > 0 and take B£ open such that c a p r p ( 5 e ) < e and that u\ga is continuous. For N — {u < 0} we find by the continuity of u\Bc that N n B% = 0 , recall that u > 0 a.e. Thus it follows that N C Be implying
Chapter 3 Potential Theory of Semigroups and Generators
238
0 < cap rp (JV) < c a p r p ( 5 £ ) < e. Taking the limit e - » 0 w e find that N is an (r, p)-exceptional set, i.e. u > 0 (r,p)-q.e. on G. To proceed further we first derive a Chebyshev-type inequality for capacities. Lemma 3.1.40. Let u £ .Tv^R"; R)nC(R"; R). For every 77 > 0 the estimate cS,Pr.tP({\u\>r]})<^\\u\\^p
(3.61)
holds. Proof: Since {|u| > 77} C {u > 77} U {—u > 77}, i.e. ca
Pr, P ({M > V}) < cap r , p ({u > 77})+ca Pt%p ({-u > 77})
it is sufficient to prove (3.61) with right-hand side ^Hull^r for all u > 0. Thus let u £ . ^ ( R ^ R ) n C(R n ;R), u > 0, be given. For 77 > 0 we put Gr),u = {u > r)}. Since u is assumed to be continuous, GntU is an open set. Further we have GV!U = { U > T 7 } = { ^ > 1 } and we find with vn = ^ that vv £ {v £ J-r,p]vv > 1 a.e. on GVtU}, hence Ca
Pr,p(Gr?,«) =
which implies (3.61).
Ca
„. p
Pr,p{" > V} <
?r,P
1
V"
•
A reader with a good background in the theory of Dirichlet forms may already have noticed before that some of our proofs for results related to (r, p)-capacities and the spaces ,7>,p(]R™;R) differ a little from the usual proofs as they are given for Dirichlet forms, compare for example the monograph [102] of M.Fukushima, Y.Oshima and M.Takeda. The reason is that in general in ^> ] P (R n ;R) the truncation property does not hold, in particular u £ ^" r ,p(R";R) does in general not imply that \u\ £ .F r ,p(R";R). In Lemma 3.1.40 this fact also caused another constant in estimate (3.61), namely 2 instead of 1, which is the correct constant whenever u £ J> ] P (R n ; R) implies |u| £ ^ r ,p(R n ;R). We refer also to P.Malliavin [207], Lemma IV.1.2.5 and Theorem IV.2.2. In Section 3.3 we will discuss more detailed the truncation property. Theorem 3.1.41. LetJrr:P(M.n;R) be weakly regular, then eachu £ JrrtP(Mn-R) admits an (r,p)-quasi continuous modification u which is unique up to (r,p)quasi everywhere equality. Moreover we have for every 77 > 0 capr,p(N>77)<-||ii||p,T.p.
(3.62)
3.1 Capacities and Abstract Bessel Potential Spaces
239
Proof: Given u 6 .7> )P (R n ;R) we take a sequence ( 2-/'}) <
2
"^
+ 1 2
" ^ " ^ = 2--1.
For Am := \Jv>m{\ 1~v/p} we find by (3.59) in Theorem 3.1.34 that c a p r p ( A m ) < 2 • 2 _ m + 1 . Thus on A1^ the sequence (
J lim
: = < V—KX)
\
(f]m>iAm) -
/
Finally, let u S J> ) P (R n ; R) with (r, p)-quasi-continuous modification it as constructed before. In particular let (tpv)v&i, fu € .7>,P(R™;R) n C 0 ( l " ; l ) , the approximating sequence as before. For e > 0 there exists an open set Ge such that cap r p (G £ ) < e and u uniformly on GCE. Now take T] > £\ > 0 and observe that {|«(X)| >T)} C {| 7 7 - f f i } U G £
for ^ sufficiently large. Using (3.61) we find
cap r > p {H>r ? }<2 ( 7 ? _ £ J p + £ . For v —> oo, E\ —+ 0 and e —> 0 we finally arrive at (3.62).
•
It is convenient to introduce the notion of a nest. Definition 3.1.42. Let (Tt )t>o be an L p -sub-Markovian semigroup. A. A sequence (Fk)k€N of closed sets Fk C R n is called an (r,p)-nest Fk C Fk+i for all k € N and lim cap r ) P (i^) = 0.
if
k—too
B. An (r,p)-nest is called regular if for each k € N it holds supp(xFfeA(")) = Fk.
240
Chapter 3 Potential Theory of Semigroups and Generators
R e m a r k 3.1.43. Let (jFfc)fc€N be an (r,p)-nest and set G := (Ufcli-Ffe)C (XLi Fk- S i n c e Fk C Fk+l, i.e. Ffcc D Ffcc+1 it follows that
=
OO
ca P r i P (G) = cap r , p ( f | Ffcc) < capP,p(F,c) fc=i
for any I £ N, hence caprp(G)
= 0.
L e m m a 3.1.44. Let (Fk)keN be an (r,p)-nest and set F'k := suppxF k - Then (F{.)keH is a regular (r,p)-nest and Ff, C Fk for all k G N. Proof: The last statement is trivial. Further we find A (n) ( i ? ,c T O =
J
{x)xpk {x) dx =
f ^
{x) dx = 0
JF'kc
VR"
since F'k is the smallest closed set whose complement has with respect to the measure XFk^n^ measure zero. From the definition of c a p r p , see (3.52), we deduce now that caPr,p(*D = cap r , p (i^ c ), implying the lemma.
•
For a given (r, p)-nest we define C({Fk})
:= {u : R™ -> R;
u\Fk is for each k € N continuous},
(3.63)
hence u £ C({Fk}) is quasi everywhere continuous. Further, extending each function u : R n —• R to R£, by setting u(oo) = 0, we put Coo({Fk}) •= {u : R ^ -» R; u|Ffcu{oo}
is for each k e N continuous}. (3.64)
It follows that Coc({Ffe}) C C({Ffc}) and C U R " ; R) C Coo({Ffc}),
C(R n ; R) C C({F fc }).
The next result is essentially adapted from [102], Theorem 2.1.2.
3.1 Capacities and Abstract Bessel Potential Spaces
241
Theorem 3.1.45. Let (T t (p) ) t>o be an Lp-sub-Markovian semigroup. A. For a countable family S of(r,p)-quasi continuous functions ((r,p)-quasi continuous functions in the restricted sense) there exists a regular (r,p)-nest {Fk)k€H such that S C C{{Fk}) (S C C^iFk})). B. Let (Ffc)fcgN be a regular (r,p)-nest and let u £ C{{Fk})- lfu>0 a.e. then u(x) > 0 for all x £ \JkxL1 Fk. Proof: A. For an (r, p)-quasi-continuous u there exists a sequence (Fk)keN of closed sets Fk C K" such that cap r p (F j f) < £ and u\p is continuous. It follows that (Ffc)^^^, Ffc := Ui=i -^'> ^s a n (r",p)-nest and u G C({Fk}). Now, for S = {ui; I s N } , a countable family of (r, p)-quasi-continuous functions ui, we choose for each Z G N an (r,p)-nest (Fj; )ksn such that ui G C({FJ^'}), ca Pr , p (F fc (0 ' c ) < ^ \ . We set Fk := f ) S i *k°- It follows that oo
c a p ^ F ^ ^ T c a p ^ F i ^ r , thus (Ffc)fceN is again an (r,p)-nest and S c C({Fit}). Finally, we may regularize (Ffc)fcgN as in Lemma 3.1.44. B. Suppose that u(x) < 0 for some x G Fk. Then there is an open set U{x) such that x € U(x) and u{y) < 0 for all y G U(x) f~l Ft, recall that u|.Ft is continuous. Since (Fk)keN is regular, it follows that \(n\U(x) D Ffc) > 0, which is an contradiction to u > 0 a.e. D Remark 3.1.46. As it is explained in [102], Lemma 2.1.4, the result of Theorem 3.1.45.B can be localized: Let G C R n be an open set and u (r, p)-quasi continuous on G. If u > 0 almost everywhere on G, then u > 0 (r,p)-quasieverywhere on G. Extending Lemma 3.1.40 we prove Theorem 3.1.47. Let (Tt)t>o be an Lp-sub-Markovian semigroup and, asn sume that J> i P (R ; R) is weakly regular. Then each u G ^> )P (R n ; M.) admits an (r,p)-quasi continuous modification in the restricted sense. Proof: We may follow the proof of Theorem 3.1.41 to find that (Fk)k^, Fk := Al is an (r,p)-nest and that for any k the sequence (VI/|F1SU{OO})J/€N converges uniformly. If we define u as in the proof of Theorem 3.1.41, we deduce that u G Coo({Fc}) and u = u a.e. •
242
Chapter 3
Potential Theory of Semigroups and Generators
Definition 3.1.48. A sequence (u„),/eN> uv € .7>, p (R n ;R), is said to converge (r,p)-quasi-uniformly to u £ .7>]P(R™; R) if for any e > 0 there exists an open set Ge C R n with cap r p (G £ ) < e and (u„)„gN converges uniformly on G% to u. Proposition 3.1.49. Let .7> ]P (R n ;R) be weakly regular and let {uu)u£^, uv € •7>,P(R™; R), be a sequence converging with respect to ||• ||jr tou G J> )P (R"; R). TTien there exists a subsequence {uvi)i^ such that (uVt)leti converges tou (r,p)quasi-uniformly. Proof: There exists a subsequence (uVl )zgjj such that \\uvi — u]\^ which implies by (3.61) that cap r , p ( { | u „ - f i | > I } ) < l\\uVl-u\\^p
<
-^j
<2-2-(.
Let Am := Uz> m {K< - «| > j } , thus c a p r p ( A m ) < 2 £ , > m 2 - ( =: £ m and lim em = 0. Now take an open set such that Am C Gm and c a p r p ( G m ) < 771—*00
2em. On Gcm we have for / > m > 1 that \uvi — u\ < \ which proves the proposition. • Corollary 3.1.50. A. If{uv)u^n, uv € C F n p ( R " ; R ) , is a sequence converging to u € J> ) P (R";R) in the norm || • \\^rp, then a subsequence (UUI)I<=N converges (r,p)-quasi-every where to u. B. If (uI/)I/epj is a sequence in J> ]P (R n ; R) converging in J> )P (R"; R) to u € .F ri p(R n ;R), and if (uv)V£N converges (r,p)-quasi-everywhere to some function v, then v = u (r,p)-q.e. The next result extends Theorem 3.1.31 to arbitrary sets. Theorem 3.1.51. For any set A C R n with finite (r,p)-capacity there exists a unique element UA in the set {u € J> i P (M n ;R);u > 1 (r,p)-q.e. on A} minimizing the norm || • \\j?r . This element UA is non-negative and satisfies cap r]P 04) = | M | £ r i j F . Definition 3.1.52. The function UA is called the (r,p)-equilibrium of the set A.
(3.65) potential
Proof of Theorem 3.1.51: The uniqueness, existence and positivity of UA can be proved as the corresponding assertions of Theorem 3.1.31. Thus the theorem follows once (3.65) is proved. For £ > 0 there exists an open
3.1 Capacities and Abstract Bessel Potential Spaces
243
set G C R n such that A c G and cap r p (A) > cap r p (G) - e. Since by Proposition 3.1.39 the (r, p)-equilibrium potential UG of the open set G belongs to the set {u € J>, P (R"; R); u > 1 (r,p)-q.e. on A} it follows that
capr,p(G) = ||«||5rriii > IMI?^ p Taking the infimum in this inequality with respect to G D A we get cap r p (A) > ||w>i||^ . Now we are going to prove the converse inequality. For this let UA be an (r, p)-quasi-continuous version of UA and for e > 0 choose an open set G£ such that cap r p (G e ) < e, UA\G% is continuous and UA> I on AH G%. We denote by UQC the (r, p)-equilibrium potential of the open set GE. Further we set Be:={x
€GC£;
uA{x) > l - e } U G e .
The set Be is open and Ad Be. In addition we have UA + UGS > 1 — £ a.e. on Be. Therefore it follows that capPJ>(i4) < cap PiP (B e ) < (1 - e)^\\uA
+«G.||^>J,
<(l-£)-*(|M|^,p+£)*> which yields (3.65) since e > 0 was arbitrary.
D
Now we are in a position to prove that cap r p is an outer Choquet capacity. Theorem 3.1.53. Let J> ) P (R n ;R), 1 < p < oo, r > 0, be weakly regular. Then for any increasing sequence ( A , ) ^ ^ , Av C R n , it follows that cap r , p ( M Av) = sup cap y \7N "eN
(vl).
(3.66)
Corollary 3.1.54. For any weakly regular space ^>iP(R™;R), 1 < p < oo and r > 0, the capacity cap r is an outer Choquet capacity. Proof of Theorem 3.1.53: Obviously we may assume that the right hand side of (3.66) is finite. Set A := \Jv^A,y and denote by uv = UAU the (r,p)equilibrium potential of the set Av. Since ||w»/||^- = c a P r , p ( ^ ) < c < oo with a constant c independent of v, we may use the Banach-Saks theorem and argue as in the proof of Theorem 3.1.31 to find a subsequence (uUl)teN such that (jST E I I I U " ) N € N converges strongly in ,F r , p (R n ;R) to some u € ^>, p (R n ;R). Let us denote for a moment by VN the function JJ J2i=i u"i > anc ^ take an
Chapter 3 Potential Theory of Semigroups and Generators
244
(r, p)-quasi-continuous modification SJV ofujv- It follows that liminfujv(:r) > 1 JV-+00
(r,p)-q.e. on Av for each v € N, hence on A. From Corollary 3.1.50. A we may deduce that a subsequence of (VN)NGN converges (r,p)-q.e. to an (r,p)-quasicontinuous modification u of u. This implies that u e {w £ J>iP(]Rn; R); w > 1 (r,p)-q.e. on A} and further cap r p (A) < \\u\\t
= lim \\vNl || p Z—•oo
<
sup
||u„J£;
<sup||u„||^
*; = l,...,Ar i
= sup c a p r J A v ) .
D
i/GN
3.2
First Results on ZASub-Markovian Semigroups in their Associated Bessel Potential Spaces
In Section 1.4.8 we extended a Feller semigroup (Tt )t>o to the space •Bt(]R™;R) by constructing the sub-Markovian kernels pt(x,dy). For the extended semigroup (Tt )t>o we had the representation (1.408), i.e. f^)u{x)
= [
u(y)Pt(x,dy).
(3.67)
From (3.67) there is a straightforward way (as we will see in volume III) to construct an associated Feller process ((Xt)t>o, Px)xeWLn by defining the transition function Px{Xt
€A}=
Pt(x,
A) = ftioo)XA(x)
(3.68)
for a (bounded) Borel set 4 C 1 " . A major theme in the modern theory of stochastic processes is to construct and to study a process starting with an L p -sub-Markovian semigroup. In particular, for p = 2 this is the subject of the theory of Dirichlet forms, see the fundamental paper [97] of M.Fukushima as well as the monographs [99], [102], [251], [35], [204]. Given an Lp subMarkovian semigroup (Tt )t>o on L P (R"; R) one is tempted to define a transition function by pt(x,A)
= Ttip)XA(x),
\(n\A)
(3.69)
3.2 Lp-Sub-Markovian Semigroups in their Bessel Potential Spaces
245
However, the left hand side in (3.69) is only defined as an Z p -function, i.e. it is only unique up to a set N(A,t) of Lebesgue measure zero \(n\N(A, t)) = 0. Since an application of the Kolmogorov theorem requires a simultaneous control of all the sets N(A,t), we cannot construct the process along this line. The idea of M.Fukushima was (for p = 2 and (Tj ) t >o symmetric) to use quasi-continuous modifications of Tj 'u for sufficiently many u in order to overcome this problem. In this section we will discuss first in the case of analytic sub-Markovian semigroups how it is possible to obtain suitable refinements oiT^'u which are eventually good enough for an application of the Kolmogorov theorem. In this first part of the section we will assume the semigroup (Tt )t>o to be analytic, often we will also require that (Tt(p) ) t >„ is symmetric (on L P (R"; R)nZ, 2 (R; R)). One should recall that symmetric sub-Markovian semigroups are always analytic, compare E.M.Stein's result [255], or Theorem 1.4.2.12. Further, by the result of A.Carasso and T.Kato [46], or Theorem 1.4.3.26, many subordinate semigroups are analytic even if the original semigroup was not. One of the fundamental properties of an analytic semigroup (Tt )t>o with generator {AW,D(AW))iB Tt{p)u G f ) D{{-A(p))k),
u G Z7(R"; R),
(3.70)
fc€N
compare Proposition 1.4.2.5, in particular (1.4.92). Combining (3.70) with Corollary 3.1.21 and Theorem 3.1.26 yields Theorem 3.2.1. Let (Tt with generator (A^,D(A^)).
)t>o be an analytic Lp-sub-Markovian semigroup Then we have for all u G Lp(Rn;R)
Ttip)u € p | ^ r , P (M n ; R) = P | D((-A^)k). r>l
(3.71)
fcSN
Note that Theorem 3.2.1 has the important Corollary 3.2.2. Let {T^p')t>o be an analytic Lp-sub-Markovian semigroup with generator (A^P\D(A^)) and assume that for some ko £ N we have D((-A^)ko) C C(R n ; R) D L p (R n ; R). Then it follows for allt>0 that Tt(p) : Lp(Rn;R)
-» C(R";R)nZ7(R";R).
(3.72)
In particular, for any bounded Borel set A C R n the function x t-> T{ XA{%) is continuous in the sense that the equivalence class of T^'XA admits a unique continuous representative. p
Chapter 3 Potential Theory of Semigroups and Generators
246
Definition 3.2.3. Let (Tt )t>o be an L p -sub-Markovian semigroup. We call (Tj )t>o a strong Lp-sub-Markovian semigroup if (3.72) holds for t > 0. R e m a r k 3.2.4. A. Note that Definition 3.2.3 has a close analogy to the definition of a strong Feller semigroup, compare Definition 1.4.8.6. B. From Corollary 3.2.2 it is obvious that one of the most important problems in the theory is to characterize the spaces ^>iP(R™; R) in terms of function spaces. We will turn to this problem in the Section 3.3 An analytic L p -sub-Markovian semigroup is in general not a strongly Lpsub-Markovian semigroup and in order to get a "nice" representative for TtXA we will use the capacities cap r introduced in the previous section. We start with a few auxiliary results. Lemma 3.2.5. Let (Tt )t>o be an analytic V-sub-Markovian semigroup. p n Further let (uu)v€n, uv € L (R ;R), be a sequence converging in Lp(Rn;R) to u E £ P (R";R). For all 0 < r < 2 it follows that (T t (p) u 1/ ) l/6N converges to T t (p) u in .F r ,p(R n ;R). Moreover T t (p) u - • u as t -+ 0 in \\ • \\^p, provided Proof: The analyticity of T t ( p ) ) t > 0 yields that T t (p) u„, T t (p) u € J" r , p (R n ;IR). Further we find for s[p) := e-'T t ( p ) S[p\uv
- u)||*.. p = ||(id -A^y'2sf\uv
- u)
I
LP
p
= ||(id-AW) ^(id-^ ))5t(p)K-w)||LP < \\(id~A^)sip)(uv-u)\\LP
<
V
%u-u\\LP,
where we used the fact that the semigroup [e~tS[ )t>o is also analytic, (id-A^)-*/2, s > 0, is an L p -contraction, see Theorem 3.1.19 and (3.31) and for the last line we used (1.4.101). Thus we arrive at \\T£p\u„-u)yrp
<
CJ\\UV-U\\LP,
implying that Tt(p)tt„ -> T^p)u in J> i P (R n ;R) as v - • oo. The last statement follows from \\TJp)u - u\\Tr
-
(id-A^)r'2u\\Lv
3.2
L p -Sub-Markovian Semigroups in their Bessel Potential Spaces
and the continuity of (I* )t>o-
247
•
Lemma 3.2.6. Let u e C 0 (R";R), u > 0, and let (u fc )* eN , uk G C 0 (R n ;R) D X,p(R™; M), be a sequence such that uk —> u in L P (R"; R) and uk —» u uniformly onW1. Then there exists a sequence (vk)ke^, vk G C 0 (M n ;K)nL p (R"';]R), such that 0 < vk < vk+i < u, vk —> u uniformly and in L p (R n ; R). Proof: Define K :— suppu and wk := uk\K that
|u-Sfc| < \u-wk\ <
and sk := wk V 0. It follows
\u-uk\,
hence sk —• u in L p (R n ; R) and uniformly. Now set ^fc := (si V...Vs f c )Au, thus ffc > 0 and t^ < ffc+i < u. In addition we have 0 < u — vk = u — (si Au)V.. .V(sfe Au) and further |u — ufc| = u — Ufc < u — (sfc A u) = |(sfc A u) — u| 1 (sk + u - \sk — u\ — 2u) = -\(sk -u)2
\sk - u\
< Isjfc - U
which implies the lemma.
•
Further we need a variant of the monotone class argument. Lemma 3.2.7. Let H be a family of non-negative functions on R n such that MI, «2 G H and \\Ui + X2U2 > 0, then A\ui + A2U2 € H; if (uy)j/eN is
a
(3.73)
sequence in H increasing to u G L p (R n ; R),
then u belongs to H; for every open set G C R" there is an increasing sequence (u„)„eN of functions uv G H which converges pointwise to XGThen H contains all non-negative Borel functions in L P (R"; R), 1 < p < 00.
Chapter 3
248
Potential Theory of Semigroups and Generators
Proof: For a bounded open set F C R™ we set DG := {Ac
G; A is a, Borel set and XA G H}.
It follows that DG is a Dynkin system that contains because of (3.74) and (3.75) all open subsets of G, thus Do = B^n\G) for a bounded open set. Now let u G LP(R™;R) be any non-negative Borel function and take a sequence of bounded open set G„, v 6 N, increasing to R™. It follows that UXGV G H for all v G N and further that U\GV increases to u. Hence u G H. • The following lemma is taken from [102]. Lemma 3.2.8. Let F be a set and G C F be a countable subset. Further let S be a countable family of mappings s : F x F —> F. Then there exists a countable set H such that G c H C F and s{H x H) C H for all s G S. Proof: Let (SZ); € N be a sequence of mappings s; : F x F —> F such that each s £ S appears infinitely often in this sequence. Further set GQ = G and for k G N set Gk+1 = Gk U sk+i{Gk x Gk). The set H := \J£=1 Gk has the desired properties. In fact, for x,y G H and s G S we find k G N such that x,y G Gk and s = Sfc+i- • Proposition 3.2.9. Le£ (T^ )(>o 6e an analytic LP-sub-Markovian semigroup on L p (R n ;R) and for some 0 < r < 2 suppose that J> ) P (R n ;R) is contraction regular, see Definition 3.1.35. C. Then for the countable set Q + ttere exists a regular nest (F°)ve^ and sub-Markovian kernel (pt)teQ+ o n (K n ; 5") «uc/i ttai T t (p) «(-) := /
«(y)p t (-, dy) G Cocd^ 0 })
« G Coo(RB; R).
Moreover, Tt u is an (r,p)- quasi- continuous modification ofT^u LP(R";R)n£(Rn;R).
(3.76) for all u G
Proof: Let W C J> l P (R n ;R) D C 0 (R n ;R) be the subspace required in the definition of contraction regularity, see Definition 3.1.35.C. In particular, u G W implies |u| G W. First we prove the existence of a countable set V C W such that V is dense in (Co(R n ; R); || • ||oo) and further, for u,v eV and a G Q it follows that |u|, u + v, av € V. To construct V let (ufc)fcgN be a dense sequence in (C 0 (R n ;R); || - ||oo)- Since W c C70(R";R) is dense, we can choose for all k, v G N some uk,v G W such that ||ufc — Ufc)V||oo < -p- The set (ufc,i/)i/eN i s
3.2 Lp-Sub-Markovian Semigroups in their Bessel Potential Spaces
249
dense in (Cb(R™; R); || • Hoc). Now we define the mapping s'0, s[ and sa, a G Q, from W x W into W by s'0(u,v) = \u\,
s'^UtV) =u+v,
sa(u,v) = au.
Applying Lemma 3.2.8 with F = W,G = {uky, k, v £ N} and S = {s'0, s[} U {sa; a £ Q} we obtain a countable set with the desired properties. Set
H0:= ( J Tt(p)(V).
(3.77)
t€Q+
Since V C W C J"r)P(]Rn;M) and JrrtP(Wn;M) is invariant under T t (p) , recall that the semigroup is analytic, it follows that Ho C J>, p (R n ;]R). According to Theorem 3.1.47 we may choose for each u G Ho an (r,p)-quasi continuous modification u in the restricted sense. We put HQ := {u;u G Ho}- From Theorem 3.1.45. A we deduce the existence of a regular nest (F^)keN such that # o C Coo({F^}). Let Fo := \J?=1 Ffc° and fix t G Q+. By Theorem 3.1.45.B it follows that (Tt(p)(u+v))~(x) (Tlp\au))~{x)
= (Tt(p))~(x)+(Tt(p)v)~(x) = a(T t (p) )~(*)
for all x G YQ, u, v e V,
for all x G Y0,
and 0
u£V,
implies 0 < (T t (p) u)~(a;) < 1 for all x0
£Y0.
Thus we find |(T t (p) u)~(a;)| < Hwlloo for all x G YQ and u £ V.
(3.78)
We claim now that for all a; G YQ there exists a linear positive functional lx on CooOR^R) satisfying lx(u) = (T}p)u)~{x)
for all u £ V, x £ Y0,
(3.79)
and \lx(u)\ < \\u\loo for all u G Coo(R n ;R), x £ Y0. For u £ Coo(R";R) choose (uu)u^,
(3.80)
uv £ V, such that lim \\uu — u\\oo = 0. v—»oo
It follows that
{TIP)UU)~{X)
converges uniformly on
YQ
to a limit lx{u). Since
Chapter 3
250
Potential Theory of Semigroups and Generators
Zx(|w|) = lim fcdu,/!) > 0, it follows that lx is positivity preserving and linear, v—*oo
as well as (3.79) and (3.80). Observe that l.(u) : Y0 —» R, x ^ lx(u) belongs to Coo({i^}) since (T t (p) u„)~ G C^Ff}). Thus by a variant of the Riesz representation theorem, lx admits a representing measure pt(x, •) such that lx(u)=
/
u(y)pt(x,dy),
x € To,
(3.81)
and pt(x, R n ) < 1 for x G Y0.
(3.82)
We extend p t (x, •) by defining pt(x,A) = 0 for all x G Y§ and A G £<"). It follows that this extension, which we denote once again by pt(x,A), is n (p) a sub-Markovian kernel on (R ; B^) satisfying (3.76). We put f t u := /„„ u(y)pt(; dy) for u € L p (R n ; R). Now, let w G C 0 (R"; R), u > 0. We choose a sequence (ufc)keN, Uk € V, converging to u uniformly. For any w G V, u < w, it follows that Vk := (OVu^) A to belongs to V and converges to u uniformly and in L p (R n ;R), recall that uVv = ^(u+v+\u — v\) and uAv = ^(u+v— \u — v\). We know that lx(vk) is an (r,p)-quasi-continuous modification of T} Vk and that lx(vk) —* lx(u) uniformly on To, i.e. in particular /
uk(y)pt(x,dy)^
u(y)pt(x,dy)
(r,p)-q.e.
On the other hand it follows that TJp)vk - • T t (p) u in .F r ,p(R n ;R) due to Lemma 3.2.5. Thus Corollary 3.1.50 implies that x H-> fRnu(y)pt(x,dy) is an (r, p)-quasi-continuous modification of u G Co(R n ; R), u > 0. Now, set H := {w G £ P (R"; R)nB(R n ; R);
u > 0 and f t (p) w = T^u
(r,p)-q.e.}.
We know that {u € C 0 (R n ;R);u > 0} c H, implying that H fulfills (3.73) and (3.75) in Lemma 3.2.7. Thus it remains to prove (3.74) in order to see that Tf u is an (r,p)-quasi continuous modification of T^'u for all u G L P (R"; R) n £ ( R n ; R), u > 0. For this let (u„)„ eN be a sequence in H increasing to some u G L P (R";R). In particular we have uv, u > 0. It follows by the dominated convergence theorem that u„ —> u in L p (R n ; R) which yields by Lemma 3.2.5 that T t (p) u„ -> T t (p) u in || • ||^ r p . Therefore, by Proposition 3.1.49 a subsequence {Tj u„fc)fcgN has a modification (Tt
uUk)k£N that converges
3.2 i/p-Sub-Markovian Semigroups in their Bessel Potential Spaces
251
(r, p)-quasi-uniformly, hence (r, p)-quasi-everywhere, to an (r, p)-quasi continuous modification of T^'u. But f$p)uVk(x)=
I
uVk{y)pt{x,dy)
^
I
u(y)pt(x,dy)
on Yo- Hence f, u is by Corollary 3.1.50.B an (r, p)-quasi-continuous modification of Tt'u. Given u € Lp(Rn; R)n.B(R n ; R) and decompose u according to u = u+—u~. Each of the functions u+ and u~ belongs to L»(Rn; R) n B(Rn; R). It follows that Tj u+ and T t u ~ are (r, p)-quasi-continuous modification of T t ' u + and Tt'u~, respectively. Hence, ft(p)u+ - fip)uis an (r,p)-quasi continuous modification of T^'u, and clearly we have 3 , a = T^'u+ — T}p'u~. Hence the proposition is proved. • By Proposition 3.2.9 we have constructed for each t 6 Q + a kernel pt{x, dy) on ( l n ; £") such that for every u E X p (R n ; R) n B(Rn; R) an (r,p)-quasi continuous modification of T t u is given by T\u. Our next aim is to construct kernels pt(x,dy) on (R B ;B") such that (pt(x,di/)) te Q+ form a family of subMarkovian kernels satisfying the Chapman-Kolmogorov equations and leading to a quasi-continuous representation of Tt u. We set •F r , p (R n ; R) := {u; u = u
q.e. and Jv, p (R n ; R)}-
Note that there is a small abuse of notation. .Fr,p(Rn; R) consists of equivalence classes of functions which coincide in the complement of a set of (r, p)-capacity zero. Lemma 3.2.10. Let H C ^" r , p (R";R) n B b (R n ;R) being the smallest algebra over Q such that V C H,V taken from the proof of Proposition 3.2.9, and for all t e Q + and u £ H it follows that T^'u G H. Then H is countable. Proof: We consider the family of mappings from (J> iP (R n ; R) nBb(Rn; into ^>, P (R n ; R) n Bb(Rn; R) consisting of
si(u,v) = u + v, s
2,a(UiV)
—
S3(tl,u) = Si,t(u,v)
aU
'
a
e
Q>
U-V,
= f}p)u
t£Q+,
R)) 2
Chapter 3 Potential Theory of Semigroups and Generators
252
and apply Lemma 3.2.8 with G = V and S = {si, S3}U{s2)0; a £ Q}U{s4 )t ;t £ Q+}. • Lemma 3.2.11. There exists a regular nest (Fk)keN with the following properties H C Coo({Fk}),
Fk CF%, fc £ N,
where F% is taken
from the proof of Proposition 3.2.9 ;
(3.83)
oo
for all x € Y\ := \ \ Fk and for all u £ H there exists k=i
(3.84)
a sequence (tk)k€N, tk £ tQ+, decreasing to 0 such that T^'u(x)
—> u(x) for all x £ Y\;
for all x £ Yj, t,s £ Q+, and u £ H we have (3.85)
(fWofW)u(x)=f%lu(x).
Proof: For each u £ H there exists a sequence (tk)keN, tk £ Q+, decreasing to zero such that (3.84) holds (r, p)-quasi everywhere, compare Corollary 3.1.50 and recall Lemma 3.2.5. Since H is countable, we can using a diagonal argument choose (tfc)fceN independent of u £ H, i.e. there is a set N C R™, cap rp (iV) = 0, such that (3.84) holds for all x £ Nc. Next we apply Theorem 3.1.45 to find a nest (Ff.)ksN such that (3.83) holds for (Ff,)ksN and N C R " \ (Jfct=i Fk- Taking the regularization (Fk)keN of (F^keN in the sense of Lemma 3.1.44, we obtain (3.83) and (3.84), whereas (3.85) follows from Theorem 3.1.45 B. In the sense of a.e. equality we have TtoTsu
= TtoTsu
= Tt+Su =
ftofsu
= ts+tu
a.e.,
ft+su,
i.e.
which implies that ft o fsu = fs+tu
q.e. on Y\ := LCLi F„.
d
Now we may prove Theorem 3.2.12. Let [Tt )t>o be an analytic V -sub-Markovian semigroup and suppose that Frp(Rn;lRL), 0 < r < 2, is contraction regular. Then there
3.2 Lp-Sub-Markovian Semigroups in their Bessel Potential Spaces
253
exists a family of sub-Markovian kernels pt(x,dy), t 6 Q + , on ( R n , S n ) and a Borel set Y2 C Y\, Y± as in Lemma 3.2.11, with the following properties (pt(x, dy) from Proposition 3.2.9 J cap rj ,(y 2 c ) = 0
and pt(x,Y%) = 0 for all x£Y2,t£
pt+s(x,A)=
pt(y,A)ps(x,dy),
/
igl",
Q+,
A&Bn,
(3.86)
s,t G (3.87)
and f u(y)pt(x,dy)
teQ+,
u € Lp(Rn;R)
nB(Rn;R) (3.88)
JUL"
is a quasi-continuous modification
ofT^u.
Proof: Since \W(Y{) = 0 it follows that T t (p) xyc = 0 a.e. But f t ( p ) xrf = pt(x,Yi) is an (r,p)-quasi continuous modification of T^XY? by Proposil) (1) c tion 3.2.9. Thus there exists a Borel set Y^ C Yu cap r p ((y i ) ) = 0 and pt(x,Yf) = 0 for all i e F , and t G Q + . By induction we find a sequence (Y{k))ken of Borel sets such that Y{k+1) C Yfk) C Ylt ca,Prp(Y1{k+1)) = 0 and pt(x, (y/ fc) ) c ) = 0 for all x G Y^k+1) and t G Q + . The set Y2 = f]T=iYik) fulfills (3.86). We define now
Jpt(s,A) ,x€F 2 ,
Pt(x,A) := <
AeB™ , .
(3.89)
In order to prove (3.86) it is sufficient to show Pt{Psu)(x)
= Pt+Su{x)
for all i £ l n , t , s € Q + , u€V,
(3.90)
where Ptu(x) = /
u(y)pt(x,dy).
Note that both sides in (3.90) (or (3.87)) do vanish for x G Y2C. For x G Y2 it follows from (3.86) that Pt(P3u)(x)
= ft{p)(Psu)(x)
= f t ( p ) ( f W«)(x),
but from (3.85) it follows for these x and t, s that T t ( p ) (fWn)(x) = {f$u)(x)
= P t+S u(rr),
Chapter 3 Potential Theory of Semigroups and Generators
254
implying (3.90), i.e. (3.87). Finally, (3.88) follows from Proposition 3.2.9.
•
Note that still some work is required to pass in Theorem 3.2.12 from t 6 oOur aim is to find some additional characterization of the equilibrium potential. For this some duality theory is needed. Thus we assume that 1 < p < oo and that the dual semigroup (Tt(p )t>o of (Tt )t>o which is a contraction semigroup on LP (R n ;R), p' = ^ j , is itself LP -sub-Markovian. This is indeed a serious restriction. There are L p -sub-Markovian semigroups with a dual semigroup not being sub-Markovian, compare E.M.Ouhabaz [229], or [203]. First we need a characterization of the dual space ^ (R n ;R) of .7>,p(Rn; R). For (T t (p) ) t > 0 and (T t (p) *) t > 0 as above we know that ((id -A^)~r'2)*
= (id -A^*)-r'2
(3.91)
holds. Writing LP instead of L P (R"; R), Tr,v instead of Fr,p(Rn; R) etc. we set as before !FT,P = (id—A^)~r/2(Lp) and in addition we write FriP.:=(id-A<*>)-r'2{l/),
P' = ^
-
(3-92)
Clearly, J>,j>* > r > 0, is the scale of abstract Bessel potential spaces associated with (Tj )t>o- In order to determine J>,p» we use the method of negative norms due to P.Lax [191], see also K.Yosida [288]. Thus on Lp', p' = -^j, we introduce the family of norms
f gudx \\9]\H_ry := sup
(3.93) I
11 (4 11 > II II-' r,p
Clearly, || • ||w_^ , is a norm for r > 0 and 1 < p < oo. Define the space H-r,P' as completion of LP with respect to the norm || • \\n_r , > i-eH-r,p, •.= 17Hn-^'.
(3.94)
Let g G H-r,p' and {gu)ueN, 9v G LP , be a sequence converging in || • ||- r , P to g. We define the linear mapping Tg : J> )P -> R by Tg{
gu
(3.95)
3.2 Lp-Sub-Markovian Semigroups in their Bessel Potential Spaces
255
Lemma 3.2.13. The mapping Tg is independent of the choice of the approximating sequence (gu)veNProof: Suppose that (gv)v^ and (/i„)i/eN, gv, hv £ Lp , are two sequences converging in H-r,P' to g. For all tp e Tr%v it follows that /
{gv-hv)ipdx
<
\\gv-hv\\n_rp\\ip\\^rp,
JUL"
implying that lim /
(gv — hv)y>dx = 0.
Now we find Tg(
u—>oo 7lR n
= lim ( / = lim /
gv(fidx hvipdx+
I (gu — hu)(pdx)
hvipdx — Th(
i/—>oo 7]R«
for all ip € J-r,pLemma 3.2.14. The mapping Tg is a continuous linear functional on Tr^v, i.e.TgeJ*tP. Proof: Let (gv)v^n, 9v £ Lp , be a sequence converging in H-r,P' to g. Given if = (id -A^)-r'2ijj € TT,P, ip e W, ip ^ 0, we find / R „ g„
JRn (id -AM')-r'2gv(id
-A^y/\
dx
IMI*V,,
M\Fr,T r 2
fRn(id-AW*)- / gi/-^dx LP
Since ||<7i/||ft
, < c we find ^n(xd-A{p)*)-r/2g^dx
\\{id-A^)-/'gv\\Lrf
= sup ipeL"
\\n
(3.96)
Chapter 3 Potential Theory of Semigroups and Generators
256
Furthermore it follows that
v—»oo
,/Rti
I
(id-A^*)-r/2gi/(id-A{p))r/2
= lim I / V—>OC> ' J R ™
'
r 2
< lim
2
snp\\(id-A^*)- / gu\\LP,\\(id-A^y/ v\\LP
< c|M|*.,„, which proves the lemma.
•
Theorem 3.2.15. With !Fr,P and H-r,P
as before we have
F*r>p = H-r,p.
(3.97)
Proof: We know already that H-r,P> C F*p- Let g G !F*tp. It follows that \(9,v)\
= c\\iP\\LP
for all ip G J> )P ,
with some ip G L p . Thus we have
\(g,(id-A^)-r/^),\
(3.98)
for all V G Z p . Define # 5 : U - • R by ffflW0 == <5, (id - ^ W ) - r / V ) ,
(3.99)
i.e. Eg G (L p )* by (3.98). Hence, there exists hg G V', p' = ^zrj, such that Hg(lp) = /
hglpdx
for all ip e Lp, or, (g^id-A^)-^)
= f
hgijdx.
(3.100)
Now, take a sequence (/i^gN, ^ G J>, P «, such that /ij, —> /i in Z/p , recall that •7viP. C L p is a dense subspace. With gu G £ p defined by (id — A&>*)~rl2gv =
3.2 Lp-Sub-Markovian Semigroups in their Bessel Potential Spaces
257
hv we arrive at ( 5 , ( i d - A ( p ) ) - r / V ) = lim /
Kipdx
= lim J
{iA-A{-p)*)-r''2gv-^dx
= lim f
g„(id-A{p))-r/2ipdx
lim (&,,<£), i.e.
(3.101)
lim (gv,f) = (g,f). u—*oo
On the other hand it follows that /R„(fl* - gjvdx
/„„(&„ ~
hp)(id-AW)-r'2
m\rr,p SL II " I /
"7* 111,*''
implying that (#„)„eN is a Cauchy sequence in H-r,P'• Thus there exists g € T-L-rp such that gv —> <j in W- r ,p which yields for 9? e J> iP (3,¥>) = lim (&/,¥>) =
{g,f),
i.e. 3 = 5 implying J7* C H-r,P' and the theorem is proved. D We are looking for a different characterization of T*v. llvll^r+t v ^ or a n V e Fr+t,p, r, t > 0, it follows that
Since ||y||.Fr.,p <
IMIw_ (l . +0iP , < \\g\\n_ripl for all g £ H-r,p' • For h £ Lp we define ( i d - i ( p ) ' ) r / 2 f t = 9 £ W-r.p'
(3.102)
by (g,
h{id-A^)r/2ipdx
(3.103)
Chapter 3
258
Potential Theory of Semigroups and Generators
for all ip G Tr,P. Thus given h G Lp', it follows that (\d-A(-P>)r/2h G H-r,P'On the other hand, given g G H-r,p' there exists a sequence {gv)v£N, gv € Lp , such that gv —» g in H-r,P'- Defining hv := (id— A^*)~r^2gu we find gvipdx=
hu(id-A^)r/2(pdx.
[
As in the proof of Theorem 3.2.15 it follows that (/i„)„eN is a Cauchy sequence in V and for its limit h G Lp we find h(id-AW)r/2ipdx.
(g,
(3.104)
Thus we have proved Theorem 3.2.16. For the dual space T*v of !Fr,p it holds T*r%v = (id -A^*yl2Lp'
(3.105)
with norm \\9\\r;,v = \\9\\H_r,tf = \\h\\Lr,, forg = (id-A^*)r/2h,
(3.106)
h G Lp .
Prom (3.106) it is clear that (id-A(-p>)r/2 n.—r,p'
=
is an isometry from Lp' onto
J~r,pm
We will also use a straightforward extension of the definition of the support of u, supp u, for u G f*p = H-r>p'. For G c R " and u G -F*^ we say u\c = 0 if (u, y>) = 0 for all ip G ^>, p D C;, such that supp o and (T( )t>o be as before with corresponding generators P (A^ \D(A^)) and ( . A ^ D ^ O O * ) ) , respectively. We want to study the (r, p)-equilibrium potential UQ for an open set G more closely. In order to be conform with the theory of convex minimization problems and monotone operators it is convenient to introduce on .Fr,p(R n ;R) the functionals Er,P(u) == lh\\%rp P
and for / G Z / ( M n ; R ) , ; E^p(u):=Er
=\ l
\W-A^yl2u\pdx
(3.107)
P JR" p - 1'
fudx.
(3.108)
3.2 £»p-Sub-Markovian Semigroups in their Bessel Potential Spaces
259
We assume that J> p (]R n ;]R) is regular, see Definition 3.1.35.B, and p > 2. Note that for p > 2 the function g : K —> K, s H-> g(s) := \s\p is a convex function with strictly monotone increasing derivative, i.e. it is a strictly convex function. Moreover, g has a continuous second derivative and we have g'(s)=ps\sr\
(3.109)
g"(s)=p(p-l)\sr2,
(3.110)
and further the estimate (s\s\p-2-r\r\p-2){s-r)
> 22-p\s-r\p
(3.111)
holds for all r , s £ l , see E.Zeidler [289], p.503. As a shorthand we use sometimes Tr,p:=(id-A^)r/2
and
T*p := (id-A< p ) *) r / 2 -
(3.112)
Definition 3.2.17. Let X be a Banach space. A. For a convex set K C X a functional E : K —+ R is called convex if E{{l-n)u+nv)
< {l-n)E(u)+nE(v)
(3.113)
holds for all ft 6 [0,1] and u,v £ K. It is called strictly convex if E((l-n)u+fiv)
< (l-fi)E(u)+iJ,E(v)
(3.114)
for all /x € (0,1) and u,v S K, u ^ v. B. We call a functional E : G —• R, G c X open, Gateaux differentiable if for each u £ G there exists some a„ £ X* such that l i m ^ + ^ - ^ H ^ f t ) t-»o * for all fcel C. The functional E : H —> K, H C X, is called coercive if oo
as ||w||x —* oo,
u € H.
(3.115)
(3.116)
IMU Lemma 3.2.18. The functional Er,P : .F r ,p(R n ;R) - • R, p > 2, is strictly convex, coercive and Gateaux differentiable. Its Gateaux derivative at u € •7> iP (R n ;R) is given by Aip)u := ( i d - y l W * ) r / 2 ( | ( i d - ^ ( p ) ) r / 2 w | p - 2 ( i d - A ( p ) ) r / 2 u ) .
(3.117)
Chapter 3 Potential Theory of Semigroups and Generators
260
Proof: The strict convexity is obvious and the coercivity follows from E
rAU)
_ 1 ||„||P-1
Now, fix u, h G J>,p(R"; K) and consider on [0,1] the quotient Er,P(u + th) — Er}P(u) t i r |(id-AV>y/2(u +
th)\p-\(id-AV>y/2u\p
=-[ = /
dx
((1 - a)Tr,p(u + th) + aTrjPu)
x |(1 - a)TriP(u + th) +
aTr,p{u)\p-2Tr!phdx,
where we used for the last line the mean value theorem, a E (0,1). For t —> 0 we arrive at v
Er p(u + th)-- Er P(U)
f
t
t->0
(Tr
ich yields i ™
t-»o
^
r
Au
u , + th)-— EA r ) = t
{Tr*tP(\Tr
= (Aip)u,h), and the lemma is proved.
•
Remark 3.2.19. For p = 2 we find for the Gateaux derivative of EV,2(«) at u A^u
= (id - ^ 2 > y / 2 o ( i d -A^y'2u
(3.ii8)
where ( i d - ^ 2 ) ) r / 2 : TT,2 -* L2 and according to Theorem 3.2.16 we have (id-AWy/2:L2->^2 = H-r<2. Corollary 3.2.20. The function E{ is on J-riP(M.n;M.) strictly convex, coercive and Gateaux differentiable with Gateaux derivative at u G J> ) P (R n ;R) given by A{$u:=Aip)u-f.
(3.119)
3.2
L p -Sub-Markovian Semigroups in their Bessel Potential Spaces
261
Proof: The strict convexity of E^p follows since ETtP is strictly convex and -» — JK„ fudx is linear. Furthermore we find
= - N I ? : 1 - n - n — / /«<** P
r P
'
INI*-.,
>^HI^-II/IIL-, implying the coercivity of £7/ . Since
/
fhdx,
it follows from the preceding lemma that E£p is Gateaux differentiable with Gateaux derivative (3.119). • We want to study the operators Ar , Afj • J-r,p —* ^*,P start with some definitions.
more
closely and
Definition 3.2.21. Let Y be a reflexive, separable real Banach space and K cY a, closed convex set. Further let T : K —> Y* be an operator. A. We call T monotone if (Tu-Tv,u-v)
>0
(3.120)
for all u,v e K. B. The operator is called strictly monotone if (Tu-Tv,u-v) >0
(3.121)
for all u, v G K and u^v. C. If there is a strictly increasing continuous function 7 : R+ —> E, 7(0) = 0 and lim i(t) — 00, such that for all u,v £ K t—>oo
(Tu-Tv,u-v)
>7(||u-t/||y)||u-t/||y
(3.122)
holds, then T is called uniformly monotone. D. We say that T is coercive with respect to K if there is an element
= o 0 )
w
£
K
Chapter 3 Potential Theory of Semigroups and Generators
262
Clearly, for a linear space K we may pose the condition lim i ^ l ! ^ Hu||y-K» IM|y
=
oo,
uGK,
(3.124)
instead of (3.123). If K is an unbounded set and T uniformly monotone, then T is coercive as well as strictly monotone, hence monotone. Proposition 3.2.22. Let Ar be as in (3.117). Then for any closed convex set K C J>)P(]R™;R), p > 2, we have the estimate {A{p)u-Arp)v,u-v)
> 22-p\\u-v\\prrp,
(3.125)
implying that Ar is uniformly monotone and coercive on every unbounded closed convex set K. Proof: From the very definition of Ar
we find
,u — v)= / I (|(id— A^)r/2u\p - | (id -A™ )T'2v\p-2(id x ((id -A^)r/2u
2
(id— A^)r'2u
)r/2v)
-A&
- (id -A^)r/2v)}
dx
and (3.111) yields (Aip)u-A{p)v,u-v)
>22~p
[
= 22-p\\u-v\\p:Frp.
\{id-Aip))r/2{u-v)\pdx D
Corollary 3.2.23. For every f e L p '(R n ;R), p' = —^j, p>2 (A{p}u-A%v,
u-v)
> 22-p\\u-v\\pFrp
it follows that (3.126)
holds. Definition 3.2.24. Let Y be a reflexive, separable real Banach space and K C Y a closed convex set. Further let T : K —> Y* be an operator. A. We call T radially continuous if for all u,v G K the function s— i » (T(u + sv),v) is continuous on [0,1].
3.2 Lp-Sub-Markovian Semigroups in their Bessel Potential Spaces
263
B. The operator T is called hemicontinuous if for all u,v G K and h £ Y the function s H-> (T(u + sv),h) is continuous on [0,1]. C. If for every sequence ( U „ ) „ 6 N , UV G K, which converges strongly to u € K it follows that ( T U I , ) „ 6 N converges weakly (in Y*) to Tu, then T is said to be demicontinuous. D. We say that T satisfies condition (S) if for every sequence ('Ui/)i/gN, uu G if, converging weakly to u £ if with the property that lim (Tu„ — Tu,uv—u)
=0,
1/—+00
then u„ converges strongly to u. E. The operator T is bounded if it maps bounded sets into bounded sets. Proposition 3.2.25. Every uniformly monotone operator T : Y —> Y* satisfies condition (S). Proof: Taking 7 from Definition 3.2.21.C we find for (u„)„eN> uv G Y, and u G Y that 7(||u I / -u||y)||'u„-'u||y <
(Tuv-Tu,uv-u).
Hence, if uu —» u weakly and lim (Tuv — Tu, uv — u) = 0 the properties of 7 imply uv —> u in V.
•
Corollary 3.2.26. TTie operator Ar
and Af)
satisfy condition (S).
Proposition 3.2.27. The operator A^] : .Frip(Rn;]R) -+ .F* p (R n ;R), p > 2, is hemicontinuous, hence radially continuous. Proof: The operator (id — A ^ ) r / 2 : J> iP —> L p is continuous and in addition for u , i ) e ^> iP fixed and for 0 < s < 1 it follows for s —+ so that I (id -i4<*>)p/2(u + sw)| p - 2 (id - A ^ ) r / 2 (w + sv) -> \(id-A^)r/2(u
+ s o ^ ) | p - 2 ( i d - A ^ ) r / 2 (it + SQV)
Chapter 3 Potential Theory of Semigroups and Generators
264
almost everywhere. In addition we find for all s e [0,1] 11 (id -A™ ) r / 2 {u + sv) \p~2 (id - A™ Y'2 {u + sv) \ = |(id -A<*))r'2(u
sv)\'-1
+
< cj \(id~ A^y^u^1
+ |(id-A^)r/2v|P-1).
Thus the dominated convergence theorem implies | (id -A^)r/2(u
+ sv)\p-2 (id -A^)r/2(u (p) /2
-> \(id-A y (u
+ sv) p 2
+
in Lp', which yields the hemicontinuity of Ar
Sov)\
- (id-A{p)y/2(
• D
Corollary 3.2.28. A. For every closed convex set K C .7> iP (E";E) the operator Ar • K —> !F*!P(Rn;R) is hemicontinuous. B. For every closed convex set K C ^v )P (E n ;lR) and f 6 L P '(R";R), p' = P\, the operator A^l '• K —> T*]P(Mn;M) is hemicontinuous. The following two fundamental results from F.Browder's and G.Minty's theory of monotone operators are quoted without proof. A proof can be found in E.Zeidler [289],p. 557, and of course in the original papers of F.Browder [38] and G.Minty [218]. Theorem 3.2.29. A. Let T : Y —> Y* be a radially continuous, and coercive operator. Then for every f £Y* the set of solutions of Tu = f
monotone (3.127)
is non-empty, weakly closed and convex. B. Let T : Y —> Y* be radially continuous, strictly monotone and coercive. Then the inverse operator T _ 1 : Y* —* Y exists, is strictly monotone, bounded and demicontinuous. If in addition T satisfies condition (S), then T _ 1 is continuous. Theorem 3.2.30. For f e L P ' ( E " ; E ) C .F* p (E n ;E), p > 2, there exists a unique solution u £ .?v jP (R n ;R) of the equation A(p)u = f.
(3.128)
Moreover, Arp) : ^ r ,p(R";E) -> jf* p (E";R) has a continuous inverse which is given on Z / ( R n ; R ) C ^ * p ( E n ; E ) by f .-> (id-A^Yr/2(\{^-A(p)*)~r/2f\p'~2(^~A{p)*)-r/2f).
(3.129)
3.2 Lp-Sub-Markovian Semigroups in their Bessel Potential Spaces
265
Proof: It remains to prove (3.129). For this we use that for j(x) = x\x\p~2 the associated superposition operator J : Lp —> Lp , p' = p/(p — 1), u — i » J(u) = u|u| p ~ 2 , is invertible and its inverse is the superposition operator J - 1 , where j~1(y) = 2/|2/| p ' -2 , and J - 1 : £ p ' - t ^ t i n J(v) = v\v\p'~2. Thus for g G Lp it follows that ft = #|c/| p ~ 2 G Lp' if and only if g = /i|/i| p ~ 2 G £ p . Now, given / G LP (R n ; R) we find a unique u £ TT,V such that (id-A^*)r^(\(id~AM)r/2u\p-2(id-A{p))r/2u)
= /,
hence |(id-A(p))r/2u|p-2(id-A(p))r/2u = (id-A(p)*)-r/2/, i.e. (id-A^y/2u
= |(id-.4(p)*)-r/2/|p'-2(id-A(p>*)-r/2/,
which leads to (3.129).
•
Let K C Jv ) P (R n ;R), p > 2, be a closed set. minimization problem mi{EltP(u);ueK},
We want to study the
(3.130)
where / G Lp (M.n; R), p' — v_ ^, is a given function. For an open set G e l " A " : = { t i £ Jv, P (K"; R); w > 1 a.e. on G} is a closed convex set. Indeed, the convexity is obvious. Now let (uu)u&^, uv £ K, be a sequence converging in J> iP (M n ; M) to u G Jv, p (R"; R). It follows that (u^)^gN converges in i p ( R n ; R ) , hence a subsequence converges almost everywhere to u, implying u > 1 a.e. on G. Thus determining cap r p (G) is the minimization problem (3.130). We know that E^p is a strictly convex, coercive and Gateaux differentiable function. Thus it follows that (3.130) has always a minimizer on K, compare E.Zeidler [289], Theorem 25.D. However, for us it is not necessary to apply this general result. The proof of Theorem 3.1.31 applies word for word to the new situation. Hence we have a unique minimizer UK to problem (3.130). Now for 0 < t < 1 and
266
Chapter 3
Potential Theory of Semigroups and Generators
follows that d «0 < — EttP((l-t)uK+t
d i
~~ di t=o
{- I
- /
\(id-AWy/2((i-t)uK+t
f.((l-t)uK+t
7R"
>
p)
f-(
= (A^ uK,
=
uK) dx
{J^JuK,f-uK)-
Thus, as it is expected from the general theory of variational inequalities, see for example the monograph [175] of D.Kinderlehrer and G.Stampacchia, we obtain Proposition 3.2.31. Necessary for UK G K to be minimizer of (3.130) is that (Aip)uK,ip-uK)>0
(3.131)
holds for all /
\(id-A^)r/2uK\p-2(id~A^)r/2uK
(id-A^)r/2(
•
/ • ( ¥ > - UK) dx. (3.132)
Remark 3.2.32. In later sections, especially in volume III, when handling the balayage problem, we will work with closed sets of the form {u G Fr,p(Rn; R); u\G = 0 a.e.},
GcRn
open,
or J r r , p (E";E)nC 0 (G;]R)"" r ' p ,
GcR"
open.
These sets are obviously closed subspaces of J> i P (R n ;R). Therefore we may test in (3.131) with ip = ±ip + UK, V> G K, to find ±(A^fUK^)>0,
3.2 Lp-Sub-Markovian Semigroups in their Bessel Potential Spaces
267
which yields for all tp € K (A
f
fipdx,
(3.133)
or /
\(id-A^)r/2u\p-2(id-A^)r/2u-(id-A^)r/2^dx=
f
fi>dx. (3.134)
In order to have a closer analogy to the theory of Dirichlet forms let us introduce the form £
\(id-A^)r/2u\P-2(id-A^)r/2u-(id-A^y/2vdx
f
(3.135) defined on Jv,p(M n ; R). It follows that 8J?\u,u)
= \\u\\vj:rp =pEr 0,
(3.136)
and \4P)(u,v)\
\\\(id-AW)r'2u\>-1\\Lpl\\(id-A<*'>y/2v\\L,
< ,, II
\\PIP
(3.137) II
II
thus £r : TT%P x !Fr,p —* R and it is linear and continuous in the second variable for fixed first variable. Further, from the proof of Proposition 3.2.27 it follows that the mapping s^>
£W(U+SLJ,V),
u^.wefjl";!)
is continuous from R to R. Using £r
(3.138)
we find for u,v £ J> )P (R"; R)
(Alp)u,v)=£W(u,v),
(3.139)
compare also Remark 3.2.19. Remark 3.2.33. Let p = 2 and r = 1 and suppose that (T}'')t >o is a symmetric sub-Markovian semigroup on L 2 (R n ;R). It follows that £[2\u,v)=
[
(id-A^)ll2u-{id-A^)ll2vdx,
JRn
i.e. £[ ' is the symmetric Dirichlet form generating by A^ —id on the Dirichlet space . ^ ( K ^ R ) .
Chapter 3
268
Potential Theory of Semigroups and Generators
Now let K in (3.130) be the set {u £ JrriP(Rn;R);u > 1 a.e. on G} for an open set G and consider ||u||5r = pErtP(u) instead of E^p, in particular / = 0. Then the minimizer becomes the (r, p)-equilibrium potential uG provided that cap (G) < oo, compare Theorem 3.1.31 and Definition 3,1.32. From Proposition 3.2.31 together with the notation (3.135) we derive Corollary 3.2.34. Let G C R™ be an open set, p > 2 and cap rp (C?) < oo. Then the (r,p)-equilibrium potential uG £ ^> iP (M n ;R) of G satisfies (Alp)uG,0
(3.140)
£iP\uG,V-uG)>Q
(3.141)
or
for all y£{v£
^V,p(]Rn; R); v > 1 a.e. on G}.
Proof: First we arrive using Proposition 3.2.31 at (pA{p)uG,ip-uG)
>0
for all ,p(R"; R); v>l
a.e. on G} which implies the corollary. D
Suppose that ip £ ^>, P (R";R) is such that tp\a > 0 a.e. It follows that
1 a.e. on G} implying (Aip)uG,ij)
= £M(uG,i>)>0
(3.142)
for all ip £ J^ r , p (R"; R), tp\G > 0 a.e. Note that (3.142) is part of assertion (iii) of Lemma 2.1.1 in the monograph [102] of M.Fukushima et.al. in case p = 2. In order to improve properties of the (r, p)-equilibrium potential uG we have to use a truncation property. Definition 3.2.35. Let (W(R";R); || • \\H) be a subspace of L P ( R " ; R ) . A. We say that the Lipschitz functions T with constant 1 operate on
W(ln;R)if u £ H(Rn; R)
implies T o u £ W(R n ; R)
(3.143)
and ||Tou|| w < c||u||„
(3.144)
3.2 Lp-Sub-Markovian Semigroups in their Bessel Potential Spaces
269
holds for all u G H(Rn; R) and all T : R -* R such that \T{x) - T{y)\ < \x - y\ and T(0) = 0. B. If the Lipschitz functions operate on W(R n ;R) and (3.144) holds with constant c = 1, then we say that the truncation property holds for H(Rn; R). In the next section we will discuss the truncation property more detailed. But we want to emphasize already now that the validity of the truncation property is a strong restriction for the space J> ]P (M"; R). Now suppose that ,7Y, P (]R"; R) satisfies the truncation property and let G c R n be an open set, cap (G) < oo and let UQ be its (r, p)-equilibrium potential. We want to show that in this situation uG\G = 1 a.e. holds and give a further characterization of uG. First observe that (OVUG) Al G {u G JrrtP(Mn; R); u > 1 a.e. on G} and \\((0WuG)Al)\\^p<\\uGrTrp, implying that UQ = (OVUG) Al, i.e. 0 < UQ < 1 a.e. and UQ\G = 1 a.e. Further we claim that UQ is the unique element in ^>jP(]Rn;]R) satisfying UQ = 1 a.e. on G such that £(p)(uG,v)>0
(3.145)
for all v <= ^v, p (R: n ;M), v > 0 a.e. on G. From (3.142) we know already that for uG inequality (3.145) holds. Now let UQ € Tr,p(Rn;R), UQ = 1 a.e. on G and elP\uG,v) > 0 for all v G ^ ^ ( R " ; ! ) , v > 0 a.e. on G. It follows that w — UQ > 0 a.e. on G for all w G J>, P (R"; R), w > 1 a.e. on G, thus we have
£lp)(uG,w-uG)>0 and the variational inequality characterization of the (r, p)-equilibrium potential yields the result, compare Proposition 3.2.31. In particular, it follows that for v € ^>, P (R"; R) such that v = 1 a.e. on G £ r fr>(u o ,t0=cap P i P (G).
(3.146)
Indeed, for v = 1 a.e. on G we find that ±{uG — v)=Q a.e. on G implying 0<
±£?\UG,UG-V)
±{£M{uG,uG)-£(p\uG,v)),
=
i.e. cap r , p (G) < £lP\uG,v)
< ca Pr>p (G).
Hence we have proved the following
Chapter 3
270
Potential Theory of Semigroups and Generators
Lemma 3.2.36. Let !FrtP(M.n; M), p > 2, satisfy the truncation property and let G C M.n be an open set with cap r p (G) < oo. The (r,p)-equilibrium potential UG fulfills 0 < UG < 1 ffl.e., UG\G = 1 o,.e. and (3.141), (3.145) as well as (3.146) holds. Before we will continue to develop the potential theory of L p -sub-Markovian semigroups in their associated Bessel potential spaces we want to prepare the ground to give concrete examples. For this we will introduce and study in the next section Bessel potential spaces associated with a real-valued negative definite function. Remark 3.2.37. For a metric space X and a finite, fully supported measure \i on X in [172] T.Kazumi and I.Shigekawa studied very systematically the spaces ^> iP and (r, p)-capacities for an £ p -sub-Markovian semigroup (Tf')t>0 with the property that (T}p'*)t>o is also an Lp -sub-Markovian semigroup, p' = £ H . In particular the dual spaces J-* are investigated and positive "distributions", i.e. elements in T*p satisfy (u,ip) > 0 for all 0,
i P , are characterized as measures provided 1 £ J>,p- Further some elements of an L p -potential theory are presented. Moreover, following D.Feyel and de La Pradelle [92], (r, p)-capacities of functions are studied. We will come back to parts of these considerations in later sections.
3.3
Bessel Potential Spaces Associated with a Continuous Negative Definite Function
Let (Tt)t>o be the operator semigroup (defined on S'(E n )) associated to the continuous real-valued negative definite function ip : M.n —» R, i.e. Ttu(x) =
(2TT)-"/2
/
e^e-^^u^dt
(3.147)
As we know from Chapter 1.4, for each 1 < p < oo there is an extension of (Tt)t>o to an L p -sub-Markovian semigroup (Tt )t>o- Our aim in this section is to determine in terms of function spaces and to study for these semigroups the spaces J> ] P (R";E), 1 < p < oo. It is clear that these spaces should also be of greater relevance when studying L p -semigroups generated by a pseudo-differential operator —p(x, D) with a symbol comparable to tjj, i.e. Rep(x,£) ~ tp(£) and |Im;?(x,£)| < Rep(x,£). For p = 2 such results had
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
271
been considered in Chapter 2. Our presentation in this section follows closely the joint papers [86] and [87] with W.Farkas and R.L.Schilling. In Example 3.1.28 we identified already the space !Fr,P(M.n) with the space s H^' (M.n) which was introduces in Section 1.3.10 and intensively used in later sections. However, in the Hilbert space case the spaces H^'3(M") and 5,0 20Rn)> see Section 1.3.10, do coincide since by Plancherel's theorem \\F-\(l+i>(.)y/2u)\)Li
= ||(l+(-)r /2 u|| L2
(3.148)
holds first for u G S(E n ) and then by extension for general u. For p ^ 2 the result is quite different, the spaces B^ (Rn) are distinct from the spaces .Fr,p(]Rn). Note that . ^ ( R " ) must be the domain of id-A^\ hence A(p\ p n A^ being the generator of (T; ) t >o, and on S(R ) the operator A^ is given by A{p)u{x) = -i){D)u{x)
= -(2TT)-"/2 f
eix^{0u{0d^
( 3 -l 4 9 )
compare Example 1.4.6.29 (and note that the arguments given there apply also for S(Rn) instead of C£°(R™)). Thus the norm || • ||^2 p should essentially be the graph norm of A^; having in mind Theorem 3.1.25, we want to identify the norm II • II «• with II
II** r,p
\\(id+iP(D))u\\LP =
\\F-\(l+1,(.))u(.))\\LP.
However the definition of F _ 1 ( ( l + V('))^(")) f° r non-smooth u leads to a serious problem: For example let u £ Lp(M.n) and p > 2. In general we know only u G S'(M.n). A typical continuous negative definite function ip : W1 —> R is not difFerentiable, hence ip(-)u(-) is in general not defined! This is the main problem we have to overcome to define spaces Hps(M.n). Before attacking this question we have to make a further remark. A general continuous negative definite function ip : R n —• R has the Levy-Khinchin representation HO = c+g(0+
f
JRn\{0}
(l-cos(y.O)Hdy)
(3.150)
with c > 0, a non-negative definite quadratic form q and the Levy measure v integrating the function y i—• |y| 2 A 1, compare Corollary 1.3.7.9 and the remark following Definition 1.3.7.11. Further we know that the function v ¥>(£) : = Iun\fo\(^~cos(y'0) {dy) has at most quadratic growth. Hence in any
Chapter 3 Potential Theory of Semigroups and Generators
272
direction £o where g(£o) is non-degenerate, i.e. q(n£o) = H2q{£o) ^ 0 for /J, ^ 0, R (or C) and with respect to the variables y 6 R m i the behaviour of the Fourier transform u(rj, £) is controlled by |r?|2. Therefore, in these directions we may work with classical Sobolev or Bessel potential spaces. For this reason we will restrict our attention to continuous negative definite functions having the representation V>(£)=/
(l-cosfo-OMdj/),
(3.151)
VR"\{0}
where the Levy measure v integrates as before the function y t—> (|y|2 A 1). In order to overcome the first mentioned difficulty we use the observation of W.Hoh [128] discussed in Theorem 1.3.7.13 of our treatise. Let
H0 = M0+^R(0
(3-152)
where for R > 0 the function ipR is defined by l M 0 = / {l-<xx(vt))XBR(0)(y)v(dy). (3.153) VK"\{o} It follows that the Levy measure VR associated with the continuous negative definite function tpR is given by XBR(O)(-)1', hence it has absolute moments of all orders, i.e.
/
\y\lvR{dy) = l
•/R"\{o}
\y\lMdv) = MR,I < oo, i > 2. JBR(0)\{0}
Therefore it follows by Theorem 1.3.7.13 that ipR is for any R > 0 an arbitrary often differentiable function with partial derivatives having at most quadratic growth. In particular it follows that for each u e S'(M.n) the expression ipRU is well-defined and belongs to S'(M.n). Thus for u € Lp(M.n) the expression TJJRU is well-defined in S'(Rn). On the other hand the continuous negative definite function tpR is a bounded function
IVMOI < MBCR(0))
< oo.
(3.154)
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
273
As we noted before S(R") C D(A^p)) for all 1 < p < oo and on S(Rn) we have A^ = —ip(D). We decompose ip(D) according to the decomposition (3.152), i.e. 4(D)u = i>R{D)u+$R{D)u,
ue
S{Rn).
(3.155)
Lemma 3.3.1. The operator ipR(D) extends to a bounded operator in L p (R n ), 1 < p < oo. We will denote the extension again by TPR(D). Proof: Prom (3.153) it follows that
(i-cos(y-0)xB°R(o)(y)v(dy)
^R(f) = / JR\{0}
implying for u € 5(R n ) >R(D)U(X)
= /
(u(x)-u(x-y))i/(dy)
JB%(0)
which yields \WR(D)U\\LP
<
/ J\y\>R
(u-u(--y))v L"
||u — u{- < I J\y\>R
\Lvv(dy) < 2i/(B^(0))||«||iP.
D
Next consider the operator ^R{D) which is defined on S(R n ) by ipR{D)u = F-\i;Ru).
(3.156)
But (3.156) makes sense for u £ S'(Rn), i.e we have a natural extension of ipR(D) to S'(R"). Now we are in a position to extend ip(D) to Lp(Rn) by il>(D)u :=
F-10JRU)+
[
(u(.)-u(-y))
v{dy).
(3.157)
This extension is independent of R: Let Ri > R% > 0 and denote for a moment by tpRl (D) and ipR2 (D) the respective extension of i>(D) using the decompositions belonging to Rx and R?. Since F-1(IPR1U)-F-1(IPR2U)=
[ ./B& !| (0)\B& (0)
(u{-)-u{-y))v{dy)
(3.158)
Chapter 3 Potential Theory of Semigroups and Generators
274
it follows that tpRl (D) = ipR2 (D). For R > 0 let us define the family of norms IMkfl, P := \\(id-i/fR(D))u\\Lr
(3.159)
for those u € Lp{Rn) for which (3.159) is finite. To understand this norm better observe that ipR is associated with an £ p -sub-Markovian semigroup (T t ' p ' )t>o and that S{Rn) C D(A^o which restricted to S(Rn) is just given by I/>R(D)U = F~1(TPR(-)U). It follows from Lemma 3.1.22 that IMIip < \\u\k,R,P n
holds for all u £ S(R ), II • \\ip,R,p is a norm.
(3.160) R > 0 and 1 < p < oo. In particular we conclude that
Lemma 3.3.2. For 0 < R2 < R\ the norms || • ||v,i?i,p and || • ||v>,ii2,p are equivalent on S(Rn). Proof: Let us consider the measure i/fllfl2 = XBRI(O)(-)V ~ XBR2(O){-)V = where v is the Levy measure associated to ip. It follows that fR1R2 is bounded measure and further for u G S(Rn) we find XBRI(O)\BR2(O)(4)V,
\\ipRl{D)u - ipR2(D)u\\LP = II /
(«(.) - u(. -
y))vRlR2{dy) LP
" >/R™\{0}
< /
ll(«(-)-«(- ~ 1 / ) I | L P « ' W < * V ) < 2i/(B« 1 (0)\B f l a (0))||u|| L ,
JK"\{0}
Thus we find using (3.160) \\{id+^R2{D)U\\LP
< ||(id-r^ ft (Z?)t*|| L P +
\WR2(D)U-IPRI(D)U\\LP
Rl(D)u\\i,p as well as \\(id+^Rl(D)u\\LP
< \\(id+^R2(D)U\\LP
+
HRl(D)u-i;R2(D)u\\LP
• R> 0,1
+
\$R{D)u\\LP
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
275
and
IMka,P < ||(id+iMD) + MD))u\\LP + \\MD)u\\LP <(l +
2v(B%(0))\\(id-rt>(D))u\\L»
where we used in particular Lemma 3.3.1. These considerations justify to start with Definition 3.3.3. Let ip : R" —* R be a real-valued continuous negative definite function with representation (3.151). The ip-Bessel potential space of order 2 with respect to L P (R"), 1 < p < oo, is the space Hp2(Rn)
:= {u G Lp(Rn);
\\u\\H^,2 < oo}
(3.161)
where Hjj+.a := ||(id+^(D))u|| L p.
(3.162)
compare (3.157). Remark 3.3.4. A. From our definition it follows that to every u G i^f , 2 (R n ) there exists a unique / G L P (R") such that (id+ip(D))u = f and HttH^.2 =
ll/IUB. It is clear that for R > 0 the norm || • \\H,2 and || • || „i>R,2 are equivalent and that Hp2(Rn) = H$R'2(Rn). Moreover, for u G S(Rn) it follows that 2 n u G Hp {R ) and since (1 + ip)u G flr>i Lr{Rn) l k l l ^ = = ll^" 1 ((l+V'(-))«)l|L,.
(3.163)
Our next aim is to prove that 5(R n ) is dense subspace in Hp2(Rn). need some preparation.
We
Lemma 3.3.5. The operator id+ip(D) maps S(Rn) into f]t^0Ht(Rn) C n n C £ ( R ) and for any R>0 the operator id+i/jR(D) maps S{R ) into itself. Proof: For u G S(Rn) and a G Nft we find ||i?-(id+^(2?))«||i3= /
| r ( l + ^(0)|2|«(OI2de
< c f (1 + |e| 2 ) | a | (l + \Z\2)2m)\2 Jut"-
# = c||«|| W h
Chapter 3 Potential Theory of Semigroups and Generators
276
implying Da((id +ip(D))u) € L2(M") which together with Sobolev's embedding theorem proves the first part of the lemma. Now, for R > 0 and a, /? G NQ we find
iW 7R"
and we arrive at sup \xp
D^R{D)u{x)\
T€R"
sup |ra + i£i 2 )^f[«)^)]|-
m a + i e i 2 )r- ^ ^
Since u £ S , (R n ) and Vfl is an arbitrary often differentiable function which together with its partial derivative has at most quadratic growth, the second assertion follows from the last estimate, i.e. (id+tf>R(D)) : S(Rn) -> 5(R n ). Rerticirk 3.3.6. For a general continuous negative definite function it is not possible to prove that id+ip(D) maps 5(R n ) into itself since for u G S(M.n) the function £ — i > i(>(£)u(£) will not be smooth, i.e. we can not expect that l F~ {ipu) decays at infinity faster than every polynomial. L e m m a 3.3.7. Let ip be a continuous negative definite function with representation (3.151), R > 0 and ipR defined as in (3.153). Further, for A > 0 define on S(Rn) {\\d+^R{D))^u{x):={2«)-n'*
f
e***
)
(fiM£)<%-
(3-164)
Then the operator (\ id+IPR(D))~1 maps S(Rn) continuously into itself and n it is on S(R. ) the inverse operator to Xid+ipR(D). Proof: As in the proof of the preceding lemma we find using the fact that da ( ^ A + A J is polynomially bounded that sup | a ^ £ ^ ( A i d + ^ H ( / ? ) ) - 1 u |
JR* (1 + |£| 2 ) 2
£eR»
i
\l+V>R(£)'
3.3 Bessel Potential Spaces and Continuous Negative Definite Function which implies that (\id+ipR(D))~l Moreover we find for u G 5(R n )
277
maps S(R") continuously into itself.
(Aid+R(£>))-1(Aid+V'H(£>))u = F-1!.
]
.
.F((Md+MD))u))
and analogously we prove that (Aid+^R(D))(Aid+Vfi(£»)) _ 1 =id
on
S(Rn).
D
Now we may prove Theorem 3.3.8. For 1 < p < oo the space S(Rn) is dense in
Hp2(Rn).
Proof: From Remark 3.3.4.A we know for u G Hp2(M.n) and R > 0 that (\id+iPR(D))u = f G LP(Rn), i.e. « = (Aid+V'RCD))- 1 /- Let ( ^ ) „ 6 N be a sequence in 5(R") converging in L P (R") to / . It follows from Lemma 3.3.7 that
vv := F-H(Aid+iM-)r 1 MO) = (Aid+^(D))-V. 6 S(Rn) and that H u - u ^ H ^ ^ p = | | / —y„||zj> which yields that lim ||u — vv\^tntP 1/—»oo
From (3.160) we also conclude that lim ||u —
V„||I,I>
— 0-
p
= 0. The L -continuity
v—»oo
of the operator ipn(D) gives lim ||^#(Z))u — V>£>(-D)iv||z,p = 0. Consequently v —>oo
we find using Remark 3.3.4.B lim \u — vv\H*,i
< lim ||u -tv||y>,fl lP + lim ||^fl(D)u - ^ ( C ^ H i P = 0, l/—+00
which proves the theorem.
•
It turns out that S^R71) is in fact a core for A^.
For this we need
Lemma 3.3.9. Let (Tt)t>o be a strongly continuous contraction semigroup with generator (A, D{A)) and resolvent (Rx)\>o on a Banach space (X, || • ||x)If D C D(A) is a cone such that D is a dense subset in X and R^D C D for all A > 0 then D is an operator core for (A, D(A)).
Chapter 3 Potential Theory of Semigroups and Generators
278
Proof: We write as before ||U||A,X for the graph norm of (A, D(A)), see (1.2.153). We have to prove that ~DHA'X = D{A). Since D C X is dense, we find for every / £ D(A) a sequence (U)v^n such that lim | | / — fv\\x = 0. v—»oo
Since R^ leaves D invariant, we have R\fv
& D implying
\\RxU - Rxf\U,x = \\RXU - RifWx + \\AR£fv - ARff\\x < j||/„-/||x+2||/„-/||x where we used that \\AR$\\ < 2 note that ARX = (A-X)R$ + XRf. It follows that (Rxfv)veN is a sequence in D which converges in the graph norm to R^f and R$f € 2 5 I M | A - \ Since D is a cone we find in addition XRff e DHA-X and therefore \\XRxf-f\\A,x
= IIAitf / - f\\x + \\\AR$f
= / Jo
\e-xt\\Ttf-f\\xdt+
/
-
Af\\x
Ae-A*||TtA/-A/|U^
JO
e—(t|T - / A / - f\\x + \\Ts/xAf
-
Af\\x)ds.
Since / € D{A) and Ta/A is a contraction, an application of the dominated convergence theorem yields for A —> oo that XRxf
—> / 6 -D
convergence is with respect to the graph norm. Hence D(A) C D the lemma is proved. •
where the Ax
'
and
Remark 3.3.10. A semigroup version of Lemma 3.3.9 can be found in E.B.Davies [60], Theorem 1.9. Theorem 3.3.11. Let 1 < p < oo and ip be a continuous negative definite function with representation (3.151). Then D(A^) = H$'2(M.n) and S(Rn) is an operator core for (A^p\ D(A^)). Proof: Consider the operator TJJR(D) where R > 0 and ipu is given by (3.153). We already know that —ipn{D) extends to a generator of a strongly continuous ZAsemigroup and we denote this extension by (Ap/,D(A£)). Prom Lemma 3.3.7 it follows for the resolvent of A£'
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
279
(p)
and each of the operators RXR preserves S(Rn). By Lemma 3.3.9 we conclude that S(Rn) is a core for (A^, DiA^)) and because of Theorem 3.3.8 we get using Remark 3.3.4.B D(A%>) = S O ^ ) 1 1 ' 1 ' ^ . " ? * ' 2 = H^2(Rn)
=
H$'2(Rn).
Since S(Rn) C Lp{Rn) is dense and since
(A^-4 P ) )| s ( K n ) =V; R (£>)| s ( K . ) is (uniformly) bounded for R > 1, we also see that D(A^) = Hp2(Rn). n It remains to prove that 5(M ) is also a core for A^. This follows from A&) = A{p + (A^ - A{p) and the fact that the second operator in this decomposition is an L p -bounded operator. • Corollary 3.3.12. Let ip be a continuous negative definite function with representation (3.151). For the Bessel potential space !F2,p(Rn) associated with the semigroup (T 4 (p) ) t > 0 , where (TJp))t>o is related to ijj by (T t (p) u) A (£) = e _ ** (€ >u(0, « 6 S{Rn), we have F2,P{Rn) = H$'2{Rn).
(3.165)
We want to use the result of the preceding corollary to identify all Bessel potential spaces J> iP (M n ) associated with the semigroup (T^v')t>o, {% u) A (f) = e - t ^ ) { t ( £ ) , with certain function spaces. Therefore we introduce for s > 0 and 1 < p < oo the notation H^'s(Rn)
:= FStP(Rn)
= (id-A(p))-s/2£p(Mn),
(3.166)
and hence by Corollary 3.1.21 D((id -A^y/2)
= Hps{Rn).
(3.167)
For a moment we take on Hps (Rn) the norm || • \\p,tP where as before | | u | | ^ p = | | ( i d - ^ ( p ) ) s / 2 u | | L p . We want to prove (at least on S(Rn)) \\u\\H;.. = \\F-\(l+i>(.)y/2u(.))\\LP.
(3.168)
We know already from Lemma 3.1.22 that the continuous embeddings H^'s+t(Rn)
^> Hps(Rn)
«-» L P (R")«-» S'(Rn)
(3.169)
280
Chapter 3
Potential Theory of Semigroups and Generators
hold for s, t > 0. As in Section 3.1 we use the relation 1
(id-AW)-s/2u
roo
= VsWu=-±-
it-VT^udt,
1
the semigroup property on
(3.170)
(2) Jo
Lp(R.n),
(id-A{p))~s/2o(id-A(pYt/2
= (id-A(p))-^,
s,t>0,
(3.171)
and the fact that (id— A^)~3/2, s > 0, is an L p -contraction. Further, since 2 s+r n the operator ( i d - A ^ ) - * / : H^ (R ) -> Hpr(Rn), s,r > 0, is continuous (since it is continuous between the spaces J r s+T . ]P (K") and .7>]p(Mn)) we may apply the calculus for fractional powers of generators of contraction semigroups, compare Section 1.3.3, to find (id~A^)3/2o(id-A^)r/2
= (id-A^)^,
s,r>0,
(3.172)
and \\u\\H+..+r = \\{id-A^)^u\\LP,
(3.173).
moreover we have (id-A(j,))—/2o(id-A(p))r/2
= (id-A(^)!^L
s,r>0,
(3.174)
and \\(id_A(p)y/2u\\Lp
< IKid-AWj^uHLP.
(3.175)
For k G N and u e 5(R n ) (id-A< p >) fc u(z) = ( 2 < T " / 2 /
efa,«(l+^(0)fc«(0^.
implying that 5(R n ) C H ^ f f ^ ' f R " ) . Hp3+r(Rn) to fljf.p(Rn) implies (id~A { p ) ) s / 2 u £ f] Hp\Rn)
The
continuity of (id-.A( p >) s / 2 from
for s > 0 and « G 5 ( 1 " ) .
(3.176)
t>o
Further it follows for all r, s, t > 0 such that s — r > 0 the continuity of the operators ( i d - A ( p ) ) * / 2 o ( i d - A ( p ) ) - r / 2 : Hf't+s-r(Rn)
-> Hpf(Rn)
(3.177)
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
281
and ( i d - A ( p ) ) - r / 2 o ( i d - A ( p > ) s / 2 : H$
-* Hjf- t (R n ).
(3.178)
Hence, for s > 0 the operator (id-A^y/2
: Hpt+s(Rn)
- Hp^W1)
(3.179)
is injective. Therefore for g e #^f>*(Rn) there exists / <= L p (R n ) such that g = ( i d - A ( p ) ) - * / 2 / and setting u := (id-A{p)ys/29 we find (id — A^)s^2u
= (id-A^)'^
f e
Hpa+t(Rn)
= g. Thus we have proved
Corollary 3.3.13. For s, t > 0 and 1 < p < oo the operator ( i d - A ^ Y ' 2 : H^t+s(Rn)
-> ^ ' ' ( r )
(3.180)
is a bijective continuous operator with continuous inverse. Consider for R > 0 the decomposition of ip according to (3.152) and the extension of the operator ipR(D) from L p (R") to S'(R n ) and that from $R(D) as bounded operator on L p (R n ), see the results in the beginning of this section. Let us show that i>R(D) is bounded from Hps(Rn) to H^s(Rn): we have for s n u e Hp (R ), taking into account the translation invariance of A^p\ WR(D)U\\H}"
= H<MWk, P
= 11||(id-A™)'' 2 I
JB%(O) R<°)
(«(•) - «(• - y)Hdy)
{((id-A^y^)(-)
L"
- m-A^yl2u){-
- y)}u(dy)
LP
<2u(BR(0))\\(id-A^y/2u\\LP = 2V{BR{0))\\AF.,P
= 2i/(B£(0))||u|| J J ? ...
Next we observe that {id-A^)s'2u I(JR(D))S/2U, hence
\\(id-A^y/2u\\LP
= (id+ip(D))s/2u
< c(\\(id+^R(D)y/2u\\LP 2
R(D)y/ u\\LP
=
+ \\u\\LP)
((id+ipR(D))
Chapter 3 Potential Theory of Semigroups and Generators
282 as well as
\\(id+TpR(D)y/2u\\LP < c"\\(id+^R(D) +MD))s,2u\W =
c"\\(id-AMy/2u\\L*,
which yields that \\u\\H*,. = ||(id-A( p )) s / 2 u|| L p and \\ (id+IJJR(D))S^U\\LP are 3 n Rs n equivalent norms and Hp (R ) = Hf ' (R ) as Banach spaces. Further, the proof of Lemma 3.3.7 yields that (id+ipR(D))~s/2 maps S(Rn) continuously into itself. Proposition 3.3.14. For s > 0 the space S(Rn) H^s(Rn).
is a dense subspace of
Proof: Take u € Hps(Rn) and for some R > 0 set / = (id+ipR(D))s/2u, thus ||u|| „i,R,s — II/IILP- Given e > 0 there exists v£ E S(Rn) such that "p
11/ - ve\\LP < e. With w£ := (id+ipR(D))-s/2v£ preceding remark and further \\u-we\\
*R,.
we find we e S(Rn)
by the
= | | / - I ; £ | | L P < £•
Since || • || „^ B ,. and || • || H^,3 are equivalent norms the proposition is proved. rip
rip
An obvious consequence of Proposition 3.3.14 is that Hps+t(Rn) is dense in Hpa(Rn) for any t,s>0. To proceed further we need a result generalizing formula (1.4.151) which can be found in U.Westphal [280], [281]. Proposition 3.3.15. Let (A, D(A)) be the generator of a strongly continuous contraction semigroup (Tt)t>o with resolvent (R\)\>o on the Banach space (X, || • | | x ) . For k < s < k + 1, k e N, we have for u € D(Ak+1) (-Ayu
sin7r s =
( ~ f c ) [°° \°-K-iRAAk+1u n Jo
d\.
(3.181)
Theorem 3.3.16. Let ijj: R™ —> R be a continuous negative definite function with representation (3.151) and let s > 0, 1 < p < oo. Then for all u G 5(R n ) we have (id-A^Yu
= F- 1 ((l+V(-)) s "(-))-
(3.182)
Proof: For s = k e N 0 formula (3.182) is already known. Let u € S(Rn), s = k + a, k 6 No and 0 < a < 1. Using Proposition 3.3.15 we find the
•
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
283
representation (id-A<*> )'«(*) = — r°A«'- 1 4 i l ( P > - i d ) (id-AW) f e + 1 u(x)dA i* Jo = ( 2 7 r r
/
™
2
r
/
A-i(A+(i+v,(0))-i
where we used for fc G No and u G S(Rn) the fact that (3.182) holds and for Wx
~'
the representation
*>""~ , d W -'(A7Since u G S(R n ) and since for r > 0 °° ^ d\ A+ r
//o Jo
r ^ r a - ^ ^ - i * ,.-i T(l) sin7rc7
holds, we find further ( i d - A ^ r U ( a ; ) = ( 2 7 r ) - " / 2 ^ ^ /" c f a , « ( l + t f ( 0 ) f c + 1 7T
JR"
A*7"1
r°°
(A+ (1 + ^(0))
7o
7T
S i n 7T<7 ,/jfn
= (27r)-"/ 2 / e«-«(l + V ( 0 ) ' f i ( 0 ^ JR"
which proves the theorem.
•
Prom (3.182) we deduce that for u G S(Rn) \\u\\H*„ = ||^- 1 ((lH-V(-)) s/2 ^(-))IU-, (3-183) i.e. we have proved (3.168) on 5(R n ). Since S(Rn) is dense in Hps(M.n) it follows further that Hps(Rn) is the closure of S(Rn) with respect to the norm | | F - 1 ( ( 1 +^(-))S/2U\\LPFurther, the proof of Theorem 3.3.16 yields Corollary 3.3.17. Forip,s andp as in Theorem 3.3.16 the operator has on S(Rn) the representation (-A^)su(x)
= (27r)- n / 2 f 7R«
eix-S(ip(Z)yu(Z)dt
(—A^)s
(3.184)
Chapter 3
284
Potential Theory of Semigroups and Generators
Applying now Theorem 3.1.26 we arrive at Theorem 3.3.18. For u G S(Rn) /2
70(\\F-\r
u)hr
+ \\u\\Lr)
and 1 < p < oo the estimates <
\\u\\Hi..
(3.185)
^ ^ ( I I F - ^ ^ / ^ I I L P + IIUIM hold and by a density argument (3.185) extends to all u € H^'s(Rn) with an appropriate interpretation of F~l(ips/2u) and F _ 1 ( ( l + ip)s/2u), respectively (see the consideration in the beginning of this section, where we discussed the problem to define F~l(il)s/2u) a n d F _ 1 ( ( l + ip)s/2u), respectively). Remark 3.3.19. We excluded in our considerations the continuous negative definite function £ — i > |£| 2 . However the continuous negative definite func2r tion £ — i > |£| °, 0 < ro < 1, are included in our considerations. The spaces I \2VQ
Hp] •a(Rn) are as Banach spaces equivalent to the classical Bessel potential spaces Hp°s(Rn), compare Definition 1.3.11.9, and (3.185) is the well known comparison of the L p -norm of Riesz and Bessel potentials, i.e. 7o(||(-Ar o / 2 «||Lp+|M|Lp) < \\u\\Hr
<7i(IK-Aro/2H|L,+NM (3.186)
see E.M.Stein [256], Section V.3.2. In Section 3.2, especially Theorem 3.2.15, we gave a description of the dual space !F*p(Rn) of a general (abstract) Bessel potential space Jrr.tP(R.n). We want to make this description concrete for the spaces Definition 3.3.20. Let ip : R™ —> R be a continuous negative definite function with representation (3.151), let 1 < p < oo and s < 0. We define the space Hps(Rn) as the closure of S(Rn) with respect to the norm \\u\\Ht„ = ||F- 1 ((l+V(-)) s / 2 ^(-))lk^
* < 0.
(3.187)
This definition gives us a scale of V'-Bessel potential spaces Hps(Rn) for the range of s £ R with S(Rn) being a dense subspace. This implies, in particular that a continuous linear functional on Hps(Rn) can be interpreted as an element of S'(Rn). More precisely, / e S'(Rn) belong to (Hph<"(Rn))* if and only if there exists c > 0 such that \l(
S(Rn).
(3.188)
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
285
Theorem 3.3.21. Let tp : R n —* R be a continuous negative definite function with representation (3.151) and let s G R, 1 < p < oo and p' = „ £ i • The s
n
(topological) dual space of Hp3(Rn) is the space Hp- (R ), i.e. for any I G (Hps{Rn))* there exists v G Hp~s{Rn) such that for all ip G S(Rn) F-\(l+ij(-)y/2(p)).
JfoO = (F-^l+^r^v,
The norm of I is given by \\v\\H$,-,. element I £ (Hp3(Rn))*
(3.189)
Conversely, for any v G irf'~
s
(R n )
an
p'
is given by (3.189).
Remark 3.3.22. Note that F - ^ l + V - O ) - * 7 2 * 5 ) = (id-A^)-s^v G ip'(Rn) s n : _s/2 p n for v £ Hp~ (R ) implying that F - ( ( l + V ( 0 ) ^ ) e £ '(R ). Therefore (3.189) can be written as convergent integral !(¥>)=/
F-\(l-H'(-))-'/av)(x)F-1((l+^))'^)(x)dx.
(3.190)
Proof of Theorem 3.3.21: Let v G Hp~3(Rn). Since the integral in (3.190) is convergent, an application of Holder's inequality yields
I'(V)I
11*11 HMIfrt"-*p'
Conversely, let us assume that M s a continuous linear functional on 3 n Hp (R ); in particular, we have (3.188). On 5(R") the norm |||
F-\(\+W)Y'2lp{-))(x)u>(x)dx,
¥>eS(R").
(3.191)
JVL"
Since u G Lp'(Rn) we find a sequence (vv)u€N, wu G 5(R"), which converges in LP'(Rn) to w. Clearly u„ := ^ ( ( 1 + V) s/2 w„) G i J ^ ~ s ( R n ) and t a r i / ^ / i we have
IIF-^ci+vc-))^ 2 ^,-^))!!^.-. = K-ovii^,
Chapter 3 Potential Theory of Semigroups and Generators
286
implying that
(V„)V€N
is a Cauchy sequence in Hp
s
( R n ) . Denote by v 6
Hp~"(m.n) the limit of (rv)«/eN it follows that w = ^ - 1 ( ( l + V(.))-/2«), and the theorem is proved.
(3-192)
•
Next we want to study some embedding theorems for the spaces The first elementary result is
Hps(M.n).
Proposition 3.3.23. Let ip : R n —> R be a continuous negative definite function with representation (3.151), 1 < p < oo, and r > 0. For any s € R we have the continuous embedding Hps+r{M.n)
<-> H*'*{Rn).
(3.193)
Proof: For s > 0 the statement is just Lemma 3.1.22. Let s < 0, 1 < p < oo and — s — r > 0. We may use Lemma 3.1.22 to find the continuous embedding Hp-°(Rn)
-> H*'-'-r(Rn),
P'
=
-2-j,
and Theorem 3.3.21 yields (3.193). If s < 0, 1 < p < oo and - s r < 0, we have the continuous embedding Hps+r(Rn) <^> LP(Rn). Since s n n s n Hp~ (R ) <^-> 2 / ( R ) , we find by duality £ P ( R " ) --> Hp (R ) which proves the proposition. • More general embedding result can be obtained by using Fourier multipliers of type (p, q) as they were investigated by L.Hormander in [136]. Definition 3.3.24. Let 1 < p,q < oo We call a distribution m S S'(Rn) Fourier multiplier of type (p, q) if IHiM^^supj1^ ,lyilL*; lit"
° ^ V> € S(Rn)}
< oo.
a
(3.194)
The set of all Fourier multipliers of type (p, q) is denoted by M p>g . We need some properties of the set M p , q . For p = q = 2 we may use Plancherel's theorem to find m
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
287
since the Fourier transform is a bijective on 5(R n ). From the last line we derive immediately that L°°(Rn) c M.2,2- On the other hand, if m € Mi$ it follows that \mp\h*
< c|M| L a
(3.195)
for all tp 6 S(Rn), hence m € Lfoc(Hn). Further, the operator ip t-> imp extends continuously as an operator on L 2 (R n ) such that (3.195) remains valid. Suppose that m is not in L°°(R n ). Then for every N € N there exists AN C R n such that 0 < X^(AN) < oo and \m\ > N on AN. Taking in (3.195) the 2 n function XAN S L (R ) we arrive at NX^\AN)^2
< WmxAjv
< C\\XA„\\L*
=c\^(AN)1/2,
which gives contradiction, hence we have M2,2 = £ 0 O (R n ).
(3-196)
Next we observe that FiL^W1))
c Mp,p
l
(3.197)
Indeed, for m G F(L x (M ra )) the convolution theorem yields F-\mp)
= (2Tr)-n/2
(F-1m)*tp
with F~lm € Ll{M.n). Now (3.197) follows from Young's inequality, Lemma 1.2.3.15. In Proposition 1.2.6.19 it was mentioned that every sequentially continuous and translation invariant operator A : Co°(K n ) —» C°°(]Rn) is a convolution operator, i.e. Au = K * u for some K S V(M.n). A result of the same type holds for bounded translation invariant operators A : Lp(M.n) —» Lg(M.n). As proved in L.Hormander [136] for such an operator exists K £ S'(Rn) such that Au = K * u for all u € Lp(M.n). Moreover, in the same paper it was proved that for q < p < oo a bounded translation invariant operator from L P (E") to L 9 (R") is trivial, i.e. identically zero. If m is a Fourier multiplier of type (p, q) then by ip — i > ir_1(m
= (27r)-n/2m*cp,
m e
S'(Rn).
288
Chapter 3 Potential Theory of Semigroups and Generators
Hence we have a one-to-one correspondence between the set Lp>q of all u G S'(R n ) such that \\(u *
u^O.
The set U(p,q) of points (±, ±) G
2
E such that u G Lp>q is a convex subset of the triangle D :— { ( i , ^) G E 2 ; 0 < j < 1,0 < ^ < 1,A < ^}, and it is symmetric with respect to the line \ + J = 1, i.e. u(p,q) = u(q',p'). Proof: As noted above, necessary for the operator associated with u to be non-trivial is that q > p, i.e. £ < ^ proving that U(p, q) is contained in D. Let u G LPig and define p', q', by ^ + \ = 1 = i + -r- Now, if u G L p>g , i.e. ||u* V|U« < ci|MU* for all G 5 ( E n ) by Holder's inequality |[(u*y>)*V](0)| =
/
(u * p)(y)ip(y) dy
< c\\u * tp\\Lq \\ip\\Lq> < C\\
By basic properties of the convolution we find [(u*
(u*i/j)(y)](0)\ < c\\
7R"
Li'
or \u*ip\\LP' = sup VELP
/ R » ( U * 1>)(v)
which proves the symmetry of U(p, q) stated in the theorem. Finally we may apply the Riesz-Thorin convexity theorem, Theorem 1.2.8.1, to obtain the convexity of U(p, q). D Corollary 3.3.26. For 1 < p < oo we have MPtV C M 2)2 =
L°°(Rn).
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
289
Proof: By the above theorem m G £ p , p implies m G L p ', p ' and therefore by the convexity of U(p,p) it follows rh G Li,i which implies the corollary. • Corollary 3.3.27. For 1 < p < oo the equality Mhp = M p%00 = F(L*(R n )),
p' = - ^ - r . P-l
(3.198)
fto/ds.
Proof: It is sufficient to show that L p ' )00 = Li, p = L p (R n ) holds. The first equality follows from Theorem 3.3.25 and the second equality follows by the same calculation done in Theorem 3.3.25 and the fact that LP (R n ) is the dual space of L P (R"). • Hps(Rn).
Now we may return to the spaces
Theorem 3.3.28. Let ^1,^2 : R n —* R be two continuous negative definite functions both with Levy-Khinchin representation (3.151). Further let s, r G R and 1 < p, q < 00. Then the continuous embedding H^'s(Rn)
-> Hf2'r(Rn)
(3.199)
holds if and only if m := (1+V> 2 ) r / 2 (1+Vi)- S / 2 G M M .
(3.200)
Proof: Assume that (3.200) holds. For u G 5(R") C Hp>s(Rn) v := F _ 1 ( ( l + rpi)s/2u) G L p (R n ) which yields
we find
\\F-\{i+w2u\\Lq
\\u\\H^ =
= \\r\rnv)\\L, m
<\\m\\MrJv\\Lf 1
V>1
I
u
= ll IIMP,Jl llR«l-«) implying (3.199). Now suppose that (3.200) holds and take
\\F-\mp)\\L,
=
\\F-\{l+^)-s/2)v>\\H^ **q
< c\\F-\(l
+ i&i)-/2)0||„*,.. < Up
which gives (3.200) and the theorem is proved.
•
cMLP,
Chapter 3 Potential Theory of Semigroups and Generators
290
Corollary 3.3.29. A. Let 1 < p < oo and let ip : Rn -> R be an unbounded continuous negative definite function with representation (3.151). Then Hps(Rn) ^ Hpr(Rn) if and only ifs>r. B. Let 1 < p < oo, s > 0, and ipi,ip2 '• K™ —* K be two continuous negative definite functions with representation (3.151). Then the embedding Hp's(Rn) <-> H^'s(M.n) implies that there exists a constant c> 0 such that 1+^2(0
^€K".
= Hp's(Rn)
(3.201) we have
< l+^a(0 < c(l+^i(0)-
(3-202)
The converse assertion holds for p = 2. Proof: A. If s > r then the validity of the embedding was proved in Proposition 3.3.23 even for the case that ip is bounded. Assume now that #JM(R") ^ Hpr(Rn). By Theorem 3.3.25 we find that m := ( l + V 0 ( r - s ) / 2 G Mp,p c L°°(R n ). Since ip is unbounded, it follows that s > r. B. We just have to apply Theorem 3.3.25 and to use MPtP c L°°(Rn).
D
It is also possible to prove a Sobolev-type embedding theorem for the spaces Hp (Rn). s
Theorem 3.3.30. Let ip : Rn —> R be a continuous negative definite function with representation (3.151). Further let 1 < p < oo and s €R. The embedding Hps(Rn)
^ CooOET)
(3.203)
holds if and only if F - 1 ( ( l + V ' ) _ s / 2 ) G LP'(Rn),
p' = -E-^,
(3.204)
i.e. if and only if (1 + ip)~3/2 e M P)0O . Proof: Suppose that (3.204) is satisfied and put m(-) := (1 + %p(-))~sl2. Prom the definition of a Fourier multiplier of type (p, oo) we find for ip e
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
291
IMU- = iiF-Ha+^-^F-Ha+v)'^))!!^ <\\rn\\MP,J\F-\(l+ipy^)\\LP =
C\\
Since 5(K n ) is dense in Hps(Rn) each element u £ Hps(Rn) is uniform limit of continuous functions, hence it has a continuous representative and (3.203) follows. Conversely, suppose that (3.203) holds. It follows that l¥>(0)|
This however means that the Dirac measure at zero £o is a continuous linear functional on Hps(M.n). By the duality theorem, Theorem 3.3.21, £Q S Hp~s(Rn). But now the definition of Hp~s(Rn) and the fact that e 0 = 2 ( 2 T T ) - " / yields (3.204). • We want to find some sufficient criterion directly posed on ip in order that (3.203) or (3.200) with V i ( 0 = |£| 2 hold. For this let ip : Rn -> K be a continuous negative definite function with representation (3.151) and assume in addition
i+V-(£)>co(i+|£|2)ro, e e r ,
(3.205)
for some ro > 0. Since for every R > 0 the function ipR is bounded it follows that (3.205) holds also for ipR with CQ = CQ(R), where ipjt and ipjt are given by (3.152) and (3.153). L e m m a 3.3.31. Suppose that the continuous negative definite function tp : R n —> R has the representation (3.151) and satisfies (3.205). Then for all R>0,t>0,l ^-±^ the function (1 + ( J 2 ) *
2
(3.206)
s/2
(i + M-))
is a Fourier multiplier of type (p, p). Proof: Since (3.205) holds also (maybe with a different constant) for IJJR we derive immediately that /1
+
l'lTLe£l(R")
(1+M0)
s/2
fors>—. r
o
(3.207)
Chapter 3 Potential Theory of Semigroups and Generators
292
Further, using Theorem 1.3.7.13 we find for k G N
implying
v-^fcLw*)6"™
fors>i
<3208)
V'
'
Combining (3.207) and (3.208) we conclude that
* , f, (1+ !Txw.>)
e Ll Rn
( )
for s
>
(3-209)
*—•
But (3.209) means by (3.197) that for s > l-tJ± the function f i-> / /
+
i ' [ L/2
is a Fourier multiplier of type (p,p). D Corollary 3.3.32. A. Suppose that the continuous negative definite function ip : R™ -> R witfi representation (3.151) satisfies (3.205). T/ien /or s > ^-±p i/ie space H^'s(M.n) is continuously embedded into the space i7*(R™). B. Suppose that ip is as in part A and that s > yHp (Rn) is continuously embedded into Coo(R n ).
"T^. Then the space
3
Proof: Part A is clear from Lemma 3.3.31 and Theorem 3.3.28. For part B observe that #*(R") C C^R™) for t > ^ see Section 1.3.11. Hence combining this embedding with the result of part A we obtain part B. • Remark 3.3.33. A. A further, more implicit sufficient criterion for (3.203) to hold is the assumption that there is an integer N G N such that with K := n(-V — 5) > N, p' = P\, the function (1 +ipR)~s^2 has (distributional) derivatives Da((l + ipk)'s/2 € L2(Rn) for all \a\ < N. Indeed, in this case the following Carlson-E'eurling inequality
\\F-\(I+M-))-S/2)\\LP>
< cii(i+<M-)r / 2 iiir / w 5 ; ii^(d+^(-))- s / 2 ii^
(3 210)
-
holds. We refer to D.Brenner et.al [36], p.18, or to H.Dappa and W.Trebels [59], Lemma 1.1.
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
293
B. In [87], Theorem 2.3.16, it was proved that if V : Rn -* R has the representation (3.151) and satisfies (3.205) then if 0 < s < 1 is given and the s ace # € ( j 3 7 , l ) then for p = pefi := 1 + u-\)Q P Hp6n/r°(Rn) is continuously embedded into Coo(R"). Note that for 6 —> >> — £i i.e. p —> 2, we get for s -> 0 the continuous embedding Hp"(Rn) <-> Coo(R") for s > ^ . This limiting case follows of course from a direct calculation using Plancherel's theorem. Lemma 3.3.31 is taken from the joint paper [166] with R.Schilling. Corollary 3.3.34. . Suppose that the continuous negative definite function ip : Rn -> R has the representation (3.151) and satisfies (3.205). Then C$°(R.n) is a dense subspace ofH$>s(Rn), s > 0. Proof: As in the proof of Lemma 3.3.31 we find that for t > s + n
(^TWiw)^'^ for all k e N 0 . Hence #£(R") is continuously embedded into Hps(Rn) ing the corollary. •
imply-
An immediate consequence of Corollary 3.3.34 is Corollary 3.3.35. For all s e i and 1 < p < oo Hp'(S.n)
C D'(Rn)
(3.211)
holds. Next we want to determine the complex interpolation spaces for two spaces H£*0(R") and H$Sl(Rn). For doing so we need first some additional considerations to Section 1.2.8. Again we set G:={z€C;0
1}.
(3.212)
For two (complex) Banach spaces (Xo, \\ • \\x0) a n d (Xi, || • ||xi) both embedded into some common Hausdorff space H we set X := Xo + X\ equipped with the norm || • ||x := max(|| • ||x 0 >|| ' llxi), which is obviously equivalent to II' ll-fo + II' ll*i i a n ( l which turns X into a Banach space. Denote by W(G; X) the space of all continuous functions w : G —* X with the following properties: i) u\a is analytic and sup ||w(z)||x < oo;
294
Chapter 3 Potential Theory of Semigroups and Generators
ii) ui(iy) S Xo and u>(l+iy) £ X\ for y g R with continuous maps y i—• w(zy) and y H-> w(l + iy), respectively; iii) ||w||w(G;X) :=max(sup||w(zj/)||x 0 ,sup||a;(H-iy)||x 1 ) < oo. 2/6R
j/e»
By the maximum principle for analytic functions (W(G;X), \\ • ||w(G;X)) is a Banach space. In the situation just described we call the Banach spaces {XQ,X\] an interpolation couple. For any interpolation couple and 0 < 0 < 1 we define its complex interpolation spaces, compare also Definition 1.2.8.5, by [Xo, Xi]e := {u e X;
there exists u> € W(G, X) such that UJ(9) = u} (3.213) and on [Xo, X{\g we introduce the norm ll«ll[jro,Xi]. : = i n f { | M k ( G ; x ) ;
u e W(G; X) and w{6) = u}.
(3.214)
Clearly, ([Xo,Xi]g, || -1| [jf0,-fi]e) is a Banach space. The following lemma, taken from [87] is a modification of Lemma 24.6.3 in H.Triebel [268], and it gives a useful equivalent norm on [Xo,Xi]g. Lemma 3.3.36. Let {Xo,Xi} be an interpolation couple and let 0 < 6 < 1. Then we have ||w|| [ Xo,x 1 ] e -inf{(sup||a;(zy)|| X o ) 1 - 0 .sup||a;(l+iy)||^ 1 }, 3/€R
(3.215)
y€K
where the infimum ranges over all UJ £ W(G; X) such that u(8) — u. Proof: Since by iii) of the definition of W(G; X) we have sup||w(ij/)||x 0 . y€R
sup||w(l+zy)|| X l 3/eK
<\\w\\W(G;X),
it follows that (sup||o;(iy)||x 0 ) 1 ~ e (sup||w(l+iy)|| X l ) e < ||«||[jc0,J!Ci]sy€M
y£R
On the other hand, the function z t—> az~eoj(z), a > 0 and z — i > w{z) as in (3.214), is admissible in (3.214) too, which yields by using iii) NI[Xo,Xi]fl ^
max a
( ~esuP||w(*y)l|x0,a1~esup||w(l+iy)||x1). 3/eK
If ||o;(l + iy)||xi ^ O w e may choose sup||w(i2/)||x0 1/€R
sup||w(l + iy)||x 1
y€R
(3.216)
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
295
to get the estimate IMItXo,*!],, ^ ( s u Plk(i2/)||x o ) 1 _ 0 -sup||w(l+iy)||x 1 ) e . y&S.
If however ||w(l + iy)\\x (3.215). •
j/SR
= 0, then we let a tend to infinity to arrive at
An application at this lemma to LPo(]R") and L P l (M n ), 1 < po,pi < oo, for O < 0 < l a n d l = ! ^ + ! - yields [LPo (IP), LPl (Rn)}9 = Lp(Rn),
(3.217)
||ii||i, =inf{(sup|| 5 (iy)|| L r o ) 1 - 0 ,sup|| 5 (l+i 2 /)|| L I . a ) e },
(3.218)
and
where the infimum is taken over all g 6 W(G; S'(Rn)) such that g{6) — u. In order to apply complex interpolation to the spaces H^s°(Rn) and H^3l(Rn) we need bounds for the imaginary powers of id+i/>(-D) on Lp(Rn). It turns out that this is a rather deep and involved result. We will state it here without a proof, the proof will be discussed in the next section. Theorem 3.3.37. Let ip : Rn —> R be a continuous negative definite function and 1 < p < oo. Then there exists constants 7 > 0 and c 7 > 1 such that ||(id+^(D)) i »«||iP < c 7 e ^ l | | u | | L P
(3.219)
holds for all y £ l . Next we can prove the announced in interpolation theorem which is again taken from [87]. Theorem 3.3.38. Let ip : R —* R be a continuous negative definite function. Further let 0 < po,Pi < 00, so, si G E and 0 < 8 < 1. For 1
s = (1—8)SQ+9S\
and
1
a
- =
P
a
1
Po
Pi
it follows that [H£3°(Rn); holds.
H£31 (Rn)]e = Hps(Rn)
(3.220)
Chapter 3 Potential Theory of Semigroups and Generators
296
Proof: Let X := H^s°(Rn) + H^Sl(Rn) <-> S'{Rn) and G = {z e C; 0 < Re 2 < 1}. Further deno°te for a moment Hg(Rn) := [H£s°(Rn);H£Sl(Rn)}g. First let u <E Hg(Rn) and choose any w € W(G; X) with w(0) = u. We define onG 5 w (z)
= F-1(e(*-*)2(l+^)((1-*)so+^)/2F(u>(z))').
(3.221)
First we prove that gw satisfies the conditions i)-iii) for the interpolation couple {LP0(Rn),LPl(M.n)}. Clearly gu is analytic in G, continuous in G, and the boundedness condition i) follows from the definition of the norms in j = 0,1. In addition we have for y £ R 9u(iy) = e ( i y - e ) 2 ( l + V(£>)) i y ( s i ~ S o ) / 2 F-\(l
+ 1>)'°'2F{u>(iy)))
and e^y+1-e)\i+^{D)yy^-30^2
9u,(i + iy) =
F-\{l+i>yi'2F{u{l+iy))). By Theorem 3.3.37 we find with some 7 = 7(so, s i ) > 0 a n d c^ > 1 that
IM«/)IU™ < |e<^-^2|||(l+V(^))iy(sx-so)/2|| I I F - ^ I + VO'^MHMIILPO <
Cy e-*
a+ a
* eTl«'l||F- 1 ((l + ^)* o / 2 F(w(ty))|| l r o
<M7||F-\(1+V0SO/2^M«/))||L™,
or IMM/)||L>> <M 7 ||a;(ij/)||„«,.o
(3.222)
with M 7 = s u p ^ e - ^ + ^ e 7 ^ ! ) < 00. In particular we have g^iy)
6 L Po (IR n ). An analogous calculation yields
llfl<-(l+*y)l|LPi <M 7 ||w(l+tj/)|| +. M . Moreover we find g„{e) = F-\{l
+
W/*F{u{6))),
(3.223)
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
297
and in the following we set v:=gu(B). Now, using (3.217) we arrive at ||«l|„*,.x = I I F - ^ l + VO^FMflJJIli,, = |M|LP "PI
inf
{sup \\g(iy)\\l^
geW(G;S'(WLn)) g{0)=v
3/GR
• sup \\g(l +
iy)fLP1}
3/€R
< sup \\gu{iy)\\]^o • SUP ||5w(l + iy)\\9Lvi 3/GM
y€R
with gu given by (3.221). Using (3.222) and (3.223) we find H H * . . < M j - ' M * sup I k ^ ) ! ! 1 - ^ -sup | | w ( l + i y ) | | ^ , s i . "
W
3/eR
PO
yeK
-"PI
Since u S W(G; X), w{6) = u, was arbitrary we may pass to the infimum and conclude from Lemma 3.3.36 that H^.. <
Ml-eM°\\u\\H,
implying \H%s°(Rn),H^(Rn)}g
= H6C
H^°(Rn).
Now we will prove the converse inclusion. Note that (id+ii{D)y/2Hps(Rn)
=
Lp(Rn),
i.e. v:=F-1((i+V)'/2u)e£p(Rn) for any u 6 Hps{Rn). We choose now an arbitrary g € W(G;S'(Rn)) with Pfc n n 1 n (R ), fc = 0,1, recall [LP°(R ),IJ> {R )}e = 5(6>) = v and g(k + iy) G L Lp(Rn), i = ~ + ^-. Analogously to the previous considerations, compare (3.221), we define now ojg{z) := F - V Z ~ e ) 2 ( l + V 0 ( Z ~ 1 ) s o / 2 _ 2 S l / 2 ^ ( < 7 ( 2 ) ) ) and calculations analogous to those done before show that ug satisfies i)-iii) for the interpolation couple {H^s°(Rn),H^Sl}. Further, arguing as in the previous case we will find that Hps(Rn) C H9, recall He = [Hff°(Rn),H^Sl]e, proving the theorem. D
Chapter 3
298
Potential Theory of Semigroups and Generators
Finally we will use the (r, p)-capacity in order to examine elements in Hp (Rn) more detailed. Once again the function ip is assumed to have the representation (3.151), i.e. r
V>(£) = /
(l-cosy-£)v(dy).
•/R"\{0}
Since by (3.167) we have Hpr(Rn)
= Tr%p, where Tr,P is the (r,p)-Bessel
potential space associated with the semigroup (Tt Tt{p)u{x) = (2vr)- n / 2 I
)t>o,
e^e-WVuffidt,
we can use all results from Section 3.1 and 3.2. In particular, the capacity ca,pfp in Hpb'r(Mn) is defined as in Definition 3.1.30 and it is a Choquet capacity. Moreover, for r > 0, 1 < p < oo, all the spaces Hpr(Rn) are regular in the sense of Definition 3.1.35.B which follows from Proposition 3.3.14 (or Corollary 3.3.34). In the following we will speak shortly of (r, p)-quasi-everywhere properties, when cap^p-quasi-everywhere properties are meant. Theorem 3.3.39. Let 1 < p < oo,^r > 0 and ip : M™ —* R be a continuous negative definite function with representation (3.151). The following statements hold. A. Ifue H^r(M.n;R) is (r,p) -quasi-everywhere and u > 0 a.e. on an open set G, then u>0 (r,p)-quasi- everywhere on G. B. Each u £ H^'r(Rn;R) admits an (r,p)-quasi-continuous modification u which is unique up to {r,p)-quasi-everywhere equality. Moreover, for p > 0 the inequality
holds. C. let(uk)km, uk G H$
u > 1 (r,p)-q.e. on A}
(3.224)
3.3 Bessel Potential Spaces and Continuous Negative Definite Function
299
and \\uA\\r
= cap* p (A).
(3.225)
E. If a sequence (uk)keN of (r,p)-quasi-continuous functions Uk belonging to Hpr(Rn;R) converges to u € Hpr(Rn;R) in the norm || • \\H*,r, then a subsequence of (uk)keN converges (r,p)-quasi-everywhere to an (r,p)-quasicontinuous modification uofu. Proof: For A we have to apply Proposition 3.1.39 and B follows from Theorem 3.1.41. Further, C is a consequence of Proposition 3.1.49 and Corollary 3.1.50. The existence of the (r, p)-equilibrium potential, part D, is implied by Theorem 3.1.51, and finally, E follows from D and the result proved before. • Without proof we state for our situation the following result due to S.Albeverio and Z.-M.Ma [4]: Theorem 3.3.40. There exists a kernel vf (x,dy) such that for all u G Hf'r(Rn;R) an (r,p)-quasi-continuous version of u is given by x H-> u(x) = I
f(y)v(p) (x, dy)
where u = (id+V>(-D))~ r/2 /, / e L"(R n ;R). Remark 3.3.41. From the formula i
(id-^)-^ = -1- / 1
r°°
r/'-VTtudt
(2) Jo it is clear that (id— .A(p))~r/2 has a kernel representation. The importance of Theorem 3.3.40 is that we may take a single kernel to get for each u 6 Hpr(Rn) an (r,p)-quasi-continuous representation. To complete our discussion of the spaces H^'r(Rn; R) we ask when the space H^' (Rn;R) will have the contraction property, compare Definition 3.2.35.B. For p = 2 this is equivalent to the question when H^ ' r (R n ; R) will be a Dirichlet space and we get the first result: r
Lemma 3.3.42. For any continuous negative definite function ip with representation (3.151) the space H^ r(IRn;IR), r < 1, has the contraction property.
300
Chapter 3
Potential Theory of Semigroups and Generators
Remark 3.3.43. Note that for a continuous negative definite function ip also tpa, 0 < a < 1, is a continuous negative definite function and that H^ = #2 • This shows that there is a principal problem consisting of the tradeoff between the "order" of ip and the order of "differentiability" that makes a sharp formulation of results like Lemma 3.3.42 difficult. An IP-version of Lemma 3.3.42 is much more complicated and requires a more delicate analysis which we will develop in the next sections and volume III. As a result we will obtain the following theorem due to F.Hirsch [119]. Theorem 3.3.44. For a continuous negative definite function ip satisfying (3.151) the spaces Hpr(Rn;M.), 2 < p and 0 < r < 1, have the contraction property. In this case the (r, p)-equilibrium potential UQ for an open set G has the property that UG\G = 1 o,.e. In the paper [270] H.Triebel discussed the problem for which function spaces -Bpq(Rn; R) and Fpq(Rn; R) the Lipshitz function operate on these spaces, compare Definition 3.2.35. Extensions are given in his monograph [271]. It is noteworthy that he could show that the Lipschitz functions operate on these spaces also in some situations where we are out of the scope of Dirichlet spaces. In particular this applies to the spaces Hp(Rn; R ) f o r l < p < o o , l < s < l - | - ^ .
3.4
Stein's Littlewood-Paley Theory for SubMarkovian Semigroups
We remind that reader to our introduction to understand and appreciate the role of E.M.Stein's work [255] for this part of our monograph. Often students in probability theory will hear that the important distinction of measure theory and probability theory lies in the concept of conditional expectation, and the theory of martingales invented by J.L.Doob [69] as a major tool in probability theory dependent from its very definition on the notion of conditional expectation. In volume III of this treatise we will come back to "probabilistic" martingale theory. However, once established in probability theory, the concept of martingales can be seen and used in a plain analytic context. This point of view will be taken in this section in order to discuss certain maximal inequalities for sub-Markovian semigroups. In the following (fi, A, P) is a probability space and for a measurable func-
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
301
tion / : fi —> R (or R) we denote its integral by E(f) := f /(w)P(dw)
(3.226)
and call it the expectation of / provided it exists, i.e. provided / > 0 or / G L 1 (fi, P ) . The following result is a consequence of the Radon-Nykodym theorem, see H.Bauer [19], Theorem 15.1. Theorem 3.4.1. For every sub-a-field J- C A and every measurable function f : Q —> R with existing integral, i.e. either f > 0 or f G L 1 (fi, P ) , t/iere exists a measurable function /o which is in addition J-'-measurable and satisfies [ fodP= Jc for all C G T. /o > 0 a.e.
f fdP Jc
(3.227)
For f G L 1 (fi, P) and / > 0 a.e. it follows in addition that
Definition 3.4.2. In the situation of Theorem 3.4.1 we call /o the conditional expectation of / under T and denote it by E{f\T).
(3.228)
Remark 3.4.3. In probability theory a measurable mapping is called a random variable and the integral of a real-valued random variable is called its expectation. In this context the name conditional expectation makes more sense, but we think that it does not make sense to invent a new name in the analytic context. The following two lemmata summarize properties of conditional expectation. A proof of the first lemma can be found in H.Bauer [19], §15, the proof of the second lemma is taken from E.M.Stein [255], p.90. Lemma 3.4.4. Let (fi, A, P ) be a probability space, f and g real-valued measurable functions on O such that the expectations of f, g and f + g exist (if needed). Further suppose that !F, Q are sub-cr-fields of A and let a G M. Then we have E(E(f\F)) / > 0
= £(/);
a.e. implies E(f\!F) > 0 a.e.
(3.229) (3.230)
Chapter 3 Potential Theory of Semigroups and Generators
302
if f is J7-measurable, then E(f\jr) E(f+g\F)
= E(f\F)+E(g\?)
E(E(f\F)\F)
= E{f\F),
= f a.e.;
a.e. and E(af\F)
= aE(f\7)
a.e.; (3.232)
a.e.;
(3.233)
if g is bounded and T measurable, then E(gf\F) ifGcT
(3.231)
then E{E{f\T)\G)
= E(f\Q)
= gE(f\!F)
a.e.; (3.234)
a.e.
(3.235)
R e m a r k 3.4.5. Clearly (3.230) implies for suitable / and g that / > g a.e. yields E(f\F) > E(g\F) a.e., and further, from (3.231) it follows that E(1\F) = 1
a.e. for all T C A.
Lemma 3.4.6. A. If f,g € L2(£l,P) f E{f\F)g dP= Jn
f fE(g\F) Jn
(3.236) then
dP.
(3.237)
B. For f € LP(n, P) it follows that
IWI-niU" < H/HLP.
(3.238)
Proof: A. Since by (3.229) / E{f\F)gdP Jn
= E(E{f\T)g) =
=
E(E{E{f\F)g\Fj)
E(E(f\F)E(g\f))
and on the other hand / fE{g\F)dP Jn
= E(fE(g\F)) =
=
E(E(fE(g\F)\F))
E{E{f\F)E{g\F)),
equality (3.237) is proved. B. First note that for T c A it follows that D>(Q,,T,P) c LP(Q,
A,P).
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups Now, since E(f\^F) is ^-measurable we find with p' = \\E(f\F)\\Lr
303
^^
= sup{ f E(f\F)gdP;ge
l/(n,F,P)
and \\g\\Lr, < 1}
= sup{ I f(u)g(w)dP;
g G / / (0, JF, P)
and || 5 || t P - < 1}
< sup{ / f(u)g(Lj)dP; g e U''(fi, A P )
and || 5 || LP * < 1}
= II/IIL-, and the lemma is proved.
D
Remark 3.4.7. We may consider the taking of conditional expectation with respect to T as an operator. Since for T C A we have W^l^^P) C LP(Q,, A, P) it follows from the preceding lemmata that E(-\F) : L p (fi, A, P) - • L p (fi, .F, P ) C Lp{il, A, P) is a linear, positivity preserving operator which is idempotent, i.e. a projection. In case p = 2 it is even selfadjoint by (3.237). Remark 3.4.8. We will need a variant of the Marcinkiewicz interpolation theorem, Theorem 1.2.8.9, which is proved in E.M.Stein [255], p.92-93. Let (fi, .4, P ) be a probability space and suppose that T : L 1 (fi, P ) + L°°(Q,, P ) —» L 1 (n, P) + L°°(fl, P) is a sublinear operator which is of weak type (1,1) and bounded from L°°(fl, P ) into itself. Then for 1 < p < oo the operator T maps L p (fi, P ) into itself and is bounded, i.e. ||Tti||iP < cp\\u\\LP.
(3.239)
In the following let (,F„)„eN be an increasing sequence of sub-cr-fields of A, i.e. Tv C J v + i and Tv C A for all f 6 N. For / with existing expectation we define Em(f)
:= E(f\Fm)
(3.240)
and the maximal function / »
:= sup \Em(f)(w)\. m>\
(3.241)
Chapter 3 Potential Theory of Semigroups and Generators
304
Lemma 3.4.9. Let (fv)v=i,...,m be a finite family of functions in L 1 (fi, P) + L°°(Q,P) such that fv is measurable with respect to Tv. Suppose that for v = 1,.. .,m
fv = E{fm\Fv)
a.e.
holds and define / m H :=
SU
P /*( w )-
Then we have the following inequalities P{w;/^(w)>o}<^||/m||il)
a>0;
(3.242)
and ||OL-
(3.243)
where Co and cp are independent of m. Proof: Without loss of generality we may assume fm > 0, otherwise use the representation fm = / + - / " and handle / + and fa separately. Hence we may also assume that fv>Q. For Sa '•= {w; fa(u>) > a} = {w; /„(w) > a
for some 1 < u < m}
we have the decomposition m
Sa=\J Note that S ^
m
W; fv{u) > a n S^
but / M («) < a for /x < v} = ( J SgK
= 0 for
Vl
I fm(w)P(dw) = JT f m
m V-\° ..
e Tv. Now it follows that
fm(„>)P()
„
=£ / *•—'
+ v2 and S^
£(/™|^)MiW
/ o(")
„ °<*
1 J Sn
y =m l
1
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
305
or P { W G fi; / A H >a}<-
f
fm(w)P(dw)
which yields (3.242). Obviously we have H/mlU°°= S U P
SU
P
UJGQ l
< (sup |/m(w)|) uiZil
\E(fm\^)\ SUp £ ( 1 1 ^ ) = | | / m | | i ~ , l
and the Marcinkiewicz interpolation theorem in the variant of Remark 3.4.8 implies the lemma. • Now we turn to our first maximal function estimates: Theorem 3.4.10 (Martingale Maximal Theorem). Let f, (to, A, P) and (3~v)veN be as before. With Em(f) given by (3.240) and f* given by (3.241) we have P{w;/»>a}<^||/||Ll)
(3.244)
i.e. f — i > /* is of weak-type (1,1), and ||/1L»
1
(3.245)
Proof: By Lemma 3.4.9 we know for the sequence (/ m ) m gn, fm = the estimate || SUp / „ | | L P < Cp | | / m | | x , P < CpH/HiP, v<.m
Em(f)
1 < P < CO,
by (3.243) with cp independent of m. Since (sup fv)meN
increases to /* we
obtain (3.245) by monotone convergence. Analogously we have with Co independent of m P{W;/^(W)>a}<|||/m||Li<^||/||L. where we used (3.242). Again we may let m tend to infinity to get (3.245) since P{OJ; f^(u>) > a} f P{w; /*(w) > a}, proving the theorem. • Later on in this section we will discuss a maximal theorem for semigroups involving analogues of the Hardy-Littlewood maximal function or the
306
Chapter 3 Potential Theory of Semigroups and Generators
Littlewood-Paley function. Before doing so we need more results on martingales. We continue with an auxiliary result due to R.Gundy [114] which has a certain analogy to the Calderon-Zygmund decomposition, Lemma 1.3.12.2. Our formulation and proof is taken from E.M.Stein [255]. We need some preparations. Let / G L ^ f y P ) , / > 0 and a > 0. As before we set /„ = Eu(f) = Definition 3.4.11. A mapping T : Q - » N U {OO} is called a stopping time with respect to (.F„)„eN if for each v G N the set {u> G fi; r(w) = u} belong to Tv. Remark 3.4.12. Again, the name stopping time will become clear when we turn to probability theory in volume III. Lemma 3.4.13. For a stopping time r we find with foo(w) := /(w) [ /(w)P(dw) = / fTiu)(u)P((Lj). Jn Jn Proof: For / and r as above we have / /rM(u)P(du)
= JT, [
JQ
„=1
(3.246)
fv{u)P{du)
J{U;T(W)=V}
oo
.
= £ / = ]T /
E{f\Fv)P{) f{u)P{du)
y—j J{OI;T(UI)=I/}
= f f(u)P(ckj).
D
JCl
Lemma 3.4.14. Every k G N is a stopping time with respect to (J-^^ueN and if TU, v G N, are stopping times with respect to (^u)veti> the same holds for T\+k, k G N, sup TV and min TV . i/€N
v
A proof of Lemma 3.4.14 can be found for example in K.L.Chung [52]. Recall that /„ = E{f\Jrv) implying /„ = E ( / v + i \FV) since by (3.235) we have E{E(f\!Fv+\)\Fv) time T /,ArM=/,Ar(o;)N
(3.247) = E(f\Jrl/).
Defining for a stopping
(3.248)
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
307
we claim far = E(f(v+1)Ar\Fv).
(3.249)
For a stopping time r we find £(/(I/+1)AT|-7V)
= -S(/I/+1X{T>I/+1}|^V) + = E(fv
+ lX{T>v}\Fv)
= X{T>v}E(fv+l\F„) = X{T>V}U =
+
E{frX{T
+
E(fTX{T
+
fTX{r
ITX{T
JTAV)
proving (3.249). Remark 3.4.15. The property (3.247) is called the martingale property and (A)f€N is a martingale with respect to (J-„)vgs- The sequence {fVAr)i/e.N is called the stopped martingale, with respect to the stopping time r . The interpretation of (3.249) is therefore that a stopped martingale is once again a martingale with respect to (J-„)V£n. Now we prove the result due to R.Gundy [114] which gives a type of Calderon-Zygmund decomposition for L1 -martingale, compare Lemma 4.12.2. Our proof follows once again E.M.Stein [255]. Lemma 3.4.16. Let f € L 1 (fi,P), / > 0, be given. For A > 0 we may decompose f = g + h-\- k such that P{u> €fi;sup \Ev{g)(u>)\ > 0} < ^ | | / | | L i
and \\g\\Li < c\\f\\Li; (3.250)
oo
^(^(/iJ-E^-i^)! i/=l
||fc|| L - < cX
i
H
< c||/|| L i
and \\k\\Li < c\\f\\Li.
implying also \\h\\Li < c||/|| L i; (3.251) (3.252)
Here we used throughout the notion Ev{g) = E{g\J~v) etc., and the definition E0(g) = 0. Proof: A. Construction of g: Given / > 0, / e L^ft.-P) and A > 0. We claim that P(CJ)
:= inf{n; /„(«) > A}
(3.253)
308
Chapter 3 Potential Theory of Semigroups and Generators
is a stopping time. Indeed we have {w;p(w)=n}
= {u;f1(w),...,fn-i(w)
With tpk := fk - fk-i
<X
but
fn(uj) > A} e JF„.
and / 0 = 0 we find
V
fe=l and define further
The definition of p implies that ev > 0. Next we introduce A>fc=0
Since (.7v)„eN is increasing and £(efc+i|^fc) is ^-measurable, it follows that a is a stopping time. Finally we set T
:= a A/9
which is by Lemma 3.4.14 a stopping time. We prove now P{wGfi;T(o;)
(3.254)
For this note that { w e f l ; p(w) ^ +00} = { u £ 0 ; sup/„(w) > A} = {w € Q.; /* (w) > A}, or by the martingale maximal theorem, Theorem 3.4.10, c P{w e f l ; p(w) < 00} < y | | / | | L i . A Further, since / > 0 / ^(efc+1|^fc)HP(dw) = / •^
fc=0
(3.255)
53efc+iHP(dw)
• ' ^ fc=0
(A+iH-AHJ^dw) 00
< f^f
fk+1(w)P(du,)=
[
< f fp[u){w)P{dw) = f /(W)P(dw) = ll/IUi,
fp(ul)(u>)P(dw)
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
309
where we used in the last step (3.246). From Chebyshev's inequality we deduce oo
P{w;
..
.
oo
£ E ( e f c + 1 | . F f c ) ( w ) > A} < - / \J2E(ek+1\^k)(w) J n
fc=0
P(dw)
k=0
< \\\f\\». But OO
{ueQ,;a(w)
< oo} = {w G
fi;^E(£fc+1|.Ffc)(a;) fc=o
> A},
which leads to P{u;
(3.256)
Thus we find P{UJ
G fi;
T(U)
< oo} <
P{OJ
G Q,; p(uj) < oo} 4- P{u G Q ; a {(J) < oo}
(3.257)
It follows immediately that {w e Q; 5(0;) ^ 0)} C {w G ft; r(w) ^ oo} which gives by (3.254) P{wefi;sH^0}<^||/|Ui. Since £,/(G ft; sup £ „ ( 5 ) ( u ; ) ^ 0 } < y | | / | U i ,
(3.258)
Chapter 3 Potential Theory of Semigroups and Generators
310 and further we have
IM|Li<||/||Li+||/r(.)llLX<2||/|| L i, where we used once again (3.246). B. Preparations to construct h and k: From the definition
sM = /H-/ r ( U )(w) is remains to find h and k such that h((j) + k(w) =/ T (u>)M. We shall construct h and k as L 1 -limits of sequences (/il/)l/gN and (kv)u^. this we look for another representation of / „ A T | namely recalling (fj = fj we start with
For —fj-i
V
fvAr =
/,(PJXJU';T(U')>J}
and using {w' G fi; r(w') > j } = {u/ G ft; p(w') > j} n {u/ € ft; j } we find with £
j '•- >;p(u>)=j}
and
T,- := ¥>jX{w';p(w')>J}
that /I-AT = 2j(7j+£j)X{a;';ff(w')>j}We set V
V
h
:
» = Efe-^l^-l))^"'!^)^'} = X ^ j=l
( 3 - 259 )
j= l
and fc
" = ^(7j+£(£jl^-i))X{">K)>j}-
(3.260)
Since / „ A r =hv + kv by construction, for the L 1 -limit, if it exists, we will find h + k = / r , or 5 + h + k = / . Let us prove that (/i„)i/£N and (fc„)„€N are martingales with respect to (!Fv)vgn, i-e. /i„ = E(hv+i\Fv)
and fc„ =
E(kv+i\?„).
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
311
From k„ = / „ A r — hT and the fact that (/UAT)V€N 1S an (J'v)-martingale, see (3.249), it suffices to prove that (hu)„eN is a martingale with respect to (JV)IVGN- Now, using the notation Ev(l) = E{l\Fv) we find
Ev(hv+1) = £ „ ( £ > , - ) = £„(Vv+i)+5>;, j'-i
j=i
where we used the J^-measurability of if>j. Thus E„(hv+i)
= Eu(ip„+i) + hv,
and it remains to show that Ev(ip„+i) = 0. By the Jv-measurability of {u/ G Q; <J{W') > n + 1} = ( U?=i{ a / / Euiipv+i) = Ev((ev+i =
e
^ i < T ( a ; ')
=
J-})
an
d
our
definition we find
— Ev(Ev+i))x{w'eci\o(u')>v+i})
C ^ f o z + l — £|/(^+l)))^{u'6fl;ff(u')>l'+l}
= ( • £ „ ( £ „ + ! ) — B i / ( £ i , + l ) ) X { a ; ' e n ; a ( w ' ) > l ' + l } = 0>
implying that (/i„)„gN and (kv)vgn are (.F)„eN -martingales. C. Prove of (3.251): We claim that the estimate oo
(3-261)
£||IMILI
will imply (3.251). Since by (3.259) hv = £j = 1 V'j> a h r s t consequence of (3.261) is the L1-convergence of (hv)ven to some limit h G L x (fi, P ) . The martingale property of (f„)ueN yields E„(h) = £?(/i|Jv) = /ij,, thus -Bi/(/i) — E„-i(h) = ^ implying oo
X)II^W-^-IC»)I|LI
u
\ei-E{ej\^J-i)[u)\x{U'€n;a^)>i}(u)P(du)
< c\\f\\Li.
Chapter 3 Potential Theory of Semigroups and Generators
312
Now, £j > 0 and therefore using the measurability properties of Ej we get, recall / > 0, oo
V
-
E(ej\Fj-1)(w)\x{u>M»>')>i}dP
/ fe OO
p,
-Yl
=
dP
/ jX{"'rt"')>J} +J2
-
E £ :F
( i\ i-i)(w)\X{u,';i}dP
/
p
OO
2
OO
£
]C /
£
iX{w'-M"')>i}dP
< 2 / VV(u;)<£P(du;) JaJ=1 f
°°
=
2
/
yi(/jH-/j-iH)XK;p(^')=j}(a')dP(da;)
^
2
/
yZ/i( W )X{a,';p( w ')=i}( W ) rfP ( c(w )
= 2 /
/„(„) (w)P(du) < 2 / / p M (w)P(dw) = 2
•/{w';p(w')
•'O
where we used in the last step (3.246). Thus (3.251) is proved. Note that the proof of (3.259) yields also, due to the L 1 -bound for /^AT, that (fcI/)ygN converges in L1. In fact the argument goes as Doob's convergence theorem for L 1 -martingales which states that L 1 -bounded martingales are almost surely convergent, see H.Bauer [19], § 19. D. Proof of (3.252): It remains to show (3.252) and for this we exploit the properties of the stopping time a. Since k = f — g — /i we know already that l|fe|Ui
k
= X)(7j+£(£ i |.F J _i))-X{u,v(cy)>j}
and we have to show that \k(u)\ < cX
a.e.
a.e.,
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
313
We proceed by proving oo
y^XK; C T (u,')>j}
(3.262)
and | J 3 E{£j irj-iteww)**}
||OO ^
j= l
cA
(3-263)
-
~
Now, since oo
oo W
5 3 ^ ^ { " ' ^ ( " ' ) > J } ( ) = 5Z^'( W ) X { w ';/'( w ')>j}( W )X{u;;a( w ')>i}( a; ) (p(w)-l)A
=
5Z
VJH=/(PH-1)A«M(W),
j= l
and the definition of p yields (3.262). On the other hand oo 0
< 5Z £ ( £ jl^- 1 )( w )X{"';:i}( a ; )
= 53E(ej|^_1)(o;)= j=i
53
E(eI+1|^)(w)
i=i
by the definition of cr, thus (3.263) and therefore lemma is proved.
•
As a further tool we will need the following result on Rademacher functions. For v G No we define on [0,1] the Rademacher functions by r„(t):=(1; \-l;
^ " ' ^ jt
andJeVen
(3.264)
and j odd.
The Rademacher functions form an orthogonal family in L2([Q, 1]) but they are not complete. Thus for a sequence (&v)i/gN0 s u c n that Y^Lo IM 2 < °°> a n element in L 2 ([0,1]) is given by oo
F(i):=5>r„(t) i/=0
Chapter 3 Potential Theory of Semigroups and Generators
314 and
l l ^ = (f>|2)1/2A proof of the following result is given in A.Zygmund [292], Theorem 8.4, p.213. Theorem 3.4.17. Let (bv)vew be a sequence such that Yl^Lo I M 2 < °°- Then the function F(t) := 5Z^Lo^"r"(*) belongs to £ p ([0,1]), 1 < p < oo, and with constants depending only on p we have oo
1/2
° °
1/2
co,p(£|M2)
<||F||LP<
v=0
C
I,
(£|M i/=0
2
P
•
)
(3-265)
With notations of Lemma 3.4.16 we find that (Evf — Ev-if)vg® is an orthogonal family in L2(Cl,P) and therefore, for any sequence (a^)^€N such that Y!,T=o \a"\2 < ° ° > w e raav define for / € L2(Q, P) oo
Taf := J2a»(E»~E-i)f
€ L2(n,P).
(3.266)
k=l
In addition we introduce the Littlewood-Paley
function 1/2
°°
2
G(/)H := ( j ] l ^ ( / H ) - ^ - i ( / H ) | )
(3.267)
Due to the orthogonality of the family {Evf — i?„_i/)„eN it follows that
IIG(/)II2L2= / oo
Y,\E-f-E-if\2dp .
= s/((^-l/)2-(^/)2)^ = f(E0f)2dP=
[ \f\2dP=\\f\\2L2,
i.e. the Littlewood-Paley function give rise to an isometry on L2(Q, P). The following theorem is a martingale version of the Littlewood-Paley inequality (see also below). Again, we follow the representation of E.M.Stein [255] which is based on Lemma 3.4.16 and R.Gundy's work [114]. We use the notations of Lemma 3.4.16.
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
315
Theorem 3.4.18. Let ( a ^ ) , , ^ , W < 1, be a sequence and for f G L2(Q,P) consider Taf as given by (3.266). Then we have \\Tafh*
< cj.il/IU-
for f G L"(n, P) n L2(Q, P), 1 < p < oo; (3.268)
P{w G n;T 0 /(w) > A} <
JII/HLI,
i.e. T a is weak type (1,1); (3.269)
| | G ( / ) | | L P < c p ||/|| L P
2
for f G L*(fi, P) n L (f2, P), 1 < p < oo; (3.270)
II^OO(/)||LP < C P ||G(/)[|Z,P
t/
£ « , / := lim £ „ / .
(3.271)
/n a// inequalities cp> 1 < p < oo, is independent of the particular choice of the sequence (a,,),,^. Proof: Let / G L1(Q, P) n L2(w, P ) , / > 0, and decompose / = g + h + k according to Lemma 3.4.16. We find P{CJ e H; T„/(w) > A} < P{w G fi; To5(o;) > ^ } + P{u; G fi;T«fc(w) > | } + P{w G ft;T0fc(u;) > | } , and estimate each term separately. Obviously we have {w G fi; T a 9 H > ^ } c {a; G Jl; sup \Ev(g)(u>)\ + 0}, which leads to P{o; G fi;To5(o;) > ^ } C P{w G Q;sup |£„( ff )(u,)| ^ 0} < | | | / | | L i . Since Tah(uj) < J2^=i \(EV - Ev-.i)h(u)\
P { w e f i ; W > -}
we find using Chebyshev's inequality
> 3}
L1
Chapter 3 Potential Theory of Semigroups and Generators
316
and since Ta is a bounded operator on L2(Q, P) we get further, again by using Chebyshev's inequality, P{w € Cl;Tak(u) > ^ } < ~\\Tak\\l2
< A|| f c ,|2 2 < £ ^ 1 1 ^
and we arrive at (3.269) for / > 0. For general / we have to use the decomposition / = / + — / ~ , f+,f~ > 0, to obtain (3.269). So far we know that Ta is of weak-type (1,1), and it is continuous on L2(fl,P). Thus by the Marcinkiewicz interpolation theorem, we know that Ta is bounded from Lp(Q,, P) to Lp(£l, P) for 1 < p < 2, and by duality we find that Ta is bounded on LP(Q, P), 1 < p < oo, proving (3.268). In order to see (3.270) note that by (3.268) we have ~
oo
/ £«,(£, f-E^f)
d P < C p | | / | | LP £
for any sequence (a„)i,6N, \av\ < 1, with cp independent of the sequence. For t € [0,1] we may take a„ = rv(t), the vth Rademacher function, to get
/
YsiEvf-E^fy^t)
dP
Jn with cp independent of t. Integrating with respect to t and interchanging the order of integration yields
|(E^(/H)-^-i(/M))r v (t)| P dt)dP
Jn{j0
Now we apply Theorem 3.4.17 with bu = bv(w) = E„(f)(u) find that
I
1 , oo
Y,{Evf{w)
- Ev^f{u;))rv{t)\
dt
1/=1
(^l^/M-^-i/HI2)1' , i/=i
or
/
^^/H-^-i/H^W
Vdt
~ I G (/)MI P .
\\LP-
- Ev-i(f)(u))
and
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
317
where as usual A ~ B means Co < jj < c\ for some 0 < Co < c\. Thus we find with a suitable constant cp r>l
cP f \G(f)\PdP
° °
(| Y,{Evf{u>) - ^ _ i / ( w ) ) r v ( t ) | P
< f (J
dtyp
proving (3.270).
D
We continue to follow E.M.Stein [255]: T h e o r e m 3.4.19. LetT\ C T% C . . . be an increasing family of sub-a-fields of A and denote by Eu the operator E(-\J-„). Suppose that {fv)vm is a sequence of functions on (Q,A,P), fv is not necessary ^-measurable, and let (n„)„ e N be a sequence of positive integers. Then we have
|(X>n„(/,)|2)
|Lp<^|(ElM2)
||Lp.
1
with cp depending only on p. Proof: Denote by Lp(lq) the Banach space of all sequences of functions (fv)v&N for which
ll(/.)IU-(i,) ••= ( / (f>HI*) P / V) 1 / P is finite, with the usual modification for q = oo, i.e. \\{U)\\LHl~) = ( / ( 8 U p | / , ( W ) | ) " d P ) 1 / P . For a linear operator T mapping both LPo(lqo) and LPl(lqi) boundedly into themselves a variant of the Riesz-Thorin theorem, compare Theorem 1.2.8.1, holds stating that T maps also LPt (lqt) continuously into itself with 1 Pt
=
1-t Po
t Pi
+—,
1 qt
=
1-t qa
t q\
+- ,
teo.i,
see A.Benedeck and R.Panzone [23]. Now, consider T defined for a sequence (/z/)i/eN by (/i/)i/gN >->
{En„fv)v
Chapter 3 Potential Theory of Semigroups and Generators
318 Since f
°°
p
v/v
/ ( £ |£n„/„Ml )
°°
f
dP(u) = £ / i^/.Mi'dPM
< £ / \M")\pdP{«>) = / ( £ | / , M I P )
<*pH>
T maps Lp(lp) continuously into itself. Next we claim that T maps Lp(loo), 1 < p < oo, boundedly into itself. Using the maximal function of p*(w) := sup|/„(w)|,i.e. ¥J*(w) =sup|£ fc (y?)(a;)| fc>i
we find / \SUpEnJ„(u,)\PP(du,) < I | SUp £ f c / „ H | P P(dw) < / | y , » | * P(dw) < c p / " ( s u p l ^ H l f P ^ a ; ) , where we used in the last step the martingale maximal theorem, Theorem 3.4.10. The variant of the Riesz-Thorin theorem mentioned above leads to the boundedness of T on Lp(lq), 1 < p < q < +oo, in particular, if 1 < p < 2, T is bounded on L p ( ^ ) , proving the theorem. • Note that for 2 < p < oo we get by duality the result for
Lp(lq).
Remark 3.4.20. Instead of a family T\ C T2 C • • • of sub-cr-fields one may also work with sub-a-fields • • • c Fn+i C Tn C • • • C T\ and the sequence Eu(f\fv), i / e N . This will be a reversed martingale, see H.Bauer [19], p.163, or R.M.Dudley [72], 10.6, for details, especially for the equivalent approach by martingales indexed by —N. For us it is important to note: All estimates done so far are valid for reversed martingales too. Next we want to establish some connections between martingales (/i,)t/€N and one parameter semigroup {Tt)t>o- For this we will need a result due to G.C.Rota [237] on powers of selfadjoint positivity preserving contractions on L p -space.
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
319
Theorem 3.4.21. Let Q be a linear operator having the following properties: \\Qu\\Lv < \\u\\Lv, Q
1 < p < oo;
(3.273)
is selfadjoint on L 2 (0, P);
Qu > 0 foru>
(3.274)
0;
(3.275)
Q\ = 1.
(3.276)
Then there exists a measure space (M, M., /x) and a family of sub-a-fields of M. • • • c J-v+\ C Tv C • • • C T\ C M. and JFQ C M. with the following properties: i) There is an isomorphism j from the measure space (CI, A, P) into the measure space (M,^o,/i) inducing also an isomorphism between the spaces LP(M, Fo, (J.) and Lp(tt, A, P), which we denote again by j . ii) If f € Lp(M,F0,n) then Q2v{j(f)) = EEvf, where E and Ev denote the conditional expectation with respect to TQ and J-v, respectively. The proof of this theorem is very much inspired by the Kolmogorov construction of a canonical stochastic process associated with a transition function and these ideas will be discussed in detail in volume III of these treaties, thus we might be a bit sketchy for some readers now. Proof of Theorem 3.4.21: Set M = fiNo and let M be the a-field generated by the cylinder sets AixA2x-
• -xANxQxClx...,
Aj € A.
(3.277)
For a cylinder set A = AQ X A\ X • • • x AN X fi x fl x . . . we define jl(A) = / XAoQiXAr Q(. • • (XA*-, -QXAN)) • • • ) dP,
(3.278)
and p, extends to a measure n on (M, M) (by the usual arguments, see volume III). Define now TQ := {A0 x Cf; A0 € A}
(3.279)
Tv := {W x A; A e M}.
(3.280)
and
It follows that • • • fv+i
CJr„
320
Chapter 3 Potential Theory of Semigroups and Generators
and by j : M —» O, j(xo, x\,...) = XQ an isomorphism of the measure spaces (M, JFQ, fi) and (Q, A, P) is given, note that for this Ql = 1 is needed. Thus we have proved i). In order to prove ii) we show a) if g : M —> E depends only on x„, i.e. g(xo,xi,...) = gv(xu) then we have E(g)(x0, xi,...)
=
{Qvgv)(xQ)
b) if g : M —> R depends only on XQ, hence is ^o-measurable and g(xo, xi,...) = <7o(#o)! then it follows that E
v(g)(xo, xi, ...) = Qugo{x„).
Suppose that a) and b) hold. It follows for / € LP(Q, A, P) that
EEvu-1f) = rHQ2vf) which implies already ii). We will show now a): We claim that Qvgv{xo) is the conditional expectation of g with respect to J-Q. Clearly, Qug{-) is ^o-measurable, thus we have to prove I gv{xv)ii{dx)= JA
I Qvgv{xQ)ii{dx).
(3.281)
JA
for A € ^o, i-e. A = AQ X Q N . If gv is a characteristic function of B C fi, then both sides in (3.281) are equal to JA QVXB(U) P(dw) by the definition of /i, see (3.278). Hence a standard limit procedure (see again volume III) yields (3.281). Finally we prove b): As in a) this can be reduces to show / go(x0) n(dx) = f {Qvg0){xv)n{dx), JA JA
(3.282)
where A = f i " - 1 x A„ x A„+i x • • • x AN xQx ... and g0 is the characteristic function g0 = \B, B C Cl measurable. Again we use (3.278) to find / 9o{x0) n{dx) = / g0[xo)xAv • XAV+1 XAN Kdx) ^ - / 9o(u)Qv(XAvQ(XA„+1Q(...)...)P(du>). Jn
JA
(3.283)
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
321
Note that the properties of Q imply f (Qh)(u) P(du) = I h{u)P{du>) Jn Jn for h G L1^, P) n L°°(n, P) (for example). Now we find for the right-hand side in (3.282) / (Qv9o)(xu) n(dx) = / (Qvgo)(x0)xA„ JA JM
XAN
Kdx)
= [ Q"(QU9o(x)XA„Q(XA„+1Q(-••)•••)) Jn
dP
•
XA„+1
= f{Q"9o{-)XAMXA„+lQ{. Jn
••)•••)) dP. (3.284)
Setting ip :=
XA„Q{XAV+1Q(-
/ go{xo)ii{dx) = / JA Jn
. . ) . . . ) we obtain from (3.283) that g0(u)Q"
while (3.284) implies / (Q"go)(Xu) V(dx) = / {Qvg0){w)ip{u) JA Jn
P(dw).
Since Q" is selfadjoint, it follows that / g0(x0)n{dx) JA
= / g 0 (w)QV(w)P(clw) Jn = U(?gQ)(u))= Jn
and the theorem is proved.
[ JA
{Qvg0){xu)n{dx),
•
Let us discuss how to use Theorem 3.4.21 for examining sub-Markovian semigroups. Let (Tt)t>o be a symmetric sub-Markovian semigroup which is conservative, i.e. Ttl = 1 for all t > 0. Thus for each t > 0 and 1 < p < oo the operator Tt = T t (p) fulfills (3.273)-(3.276). We may define for k G N Q '•= ^'l/(2 t + 1
Chapter 3 Potential Theory of Semigroups and Generators
322
and find that Q satisfies (3.273)-(3.276). Now, the martingale maximal theorem, Theorem 3.4.10 and part ii) of Theorem 3.4.21 yields ||su P Q 2 "(j(/))(-)||Lp(n,P) = lir 1 (supg 2 "0"(/)))llLP(M,A.M) = ||sup^(/(.))||Lp(^0>M) < \\E{snVEvf)\\LP{Mfoll) < cp\\f\\^{M,fo,n)
<
\\suvEvf\\Lv{M>fo
= Cj»lli(/)ll£»
or for u G L p (fi, P ) ||supQ 2l/ u||i,p < CP||U||LP,
i.e. || s\ipTu/2*u\\Lr
< cp\\u\\LP
(3.285)
with c p independent of k. For A: —> oo we arrive at Proposition 3.4.22. Let (Tt)t>o be a conservative, symmetric semigroup. Then the maximal estimate || supr ( u||z,p < C P ||U|| L P. t>o
sub-Markovian
(3.286)
holds. Now we come to this part of our discussion where we shall follow Stein's work to get better estimates for maximal and Littlewood-Paley functions associated with a symmetric sub-Markovian semigroup, however, from now on we will omit larger parts of the proofs. We need the following ergodic theorem due to N.Dunford and J.T.Schwartz [73], generalizing an earlier result of E.Hopf, which we state without proof. Theorem 3.4.23. Let (T}p')t>o be a strongly continuous contraction semigroup on L p (R n ;R), 1 < p < oo, and a contraction semigroup on L°°(R n ;R). For u € L p (R n ; R) there is a null set N{u) C R™ such that for x $ N{u) MlV)u(x)
:=-
f T(p)u(x)ds t Jo
(3.287)
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
323
and MT,u{x) v
:= sup\M^u{x)\
(3.288)
t>o
are defined and we have \M(p)u(x)\<M{p)u{x), ||M(;)W||LP < 2 ( - ^ y ) 1
x$N{u); P
\\U\\LP,
(3.289) 1 < p < oo,
(3.290)
hence \\M^ U\\LP < c p ||u||i> /or 1 < p < oo, and M{ u converges almost everywhere to u as t —> 0. Note that 1i-> | /„* T s (p) u(x) ds is for all x g JV(«), A(JV(u)) = 0, a continuous function. In order to prove maximal inequalities for t i-+ T t u(x) we need more regularity for this mapping which we may derive from the analyticity of (T t (p) ) t > 0 . Lemma 3.4.24. Let (Tt )t>o be an analytic Lv-sub-Markovian semigroup and let u £ Z P (R"; K), 1 < p < oo. For each t > 0 we can redefine Ttu(x) on a null set such that for every fixed x the mapping t — i > Ttu(x) is on (0, oo) a real-analytic function. Proof: A function t H-> /(£) £ X, (X, || • ||x) being a Banach space, is analytic in (0, oo) if and only if for every to £ (0, oo) there is a neighbourhood U<2e{t0) and elements fk £ X, k £ No, such that f(t) = £ £ 1 0 fk-(t —
a.e.
x £ W1,
t£U2E(tQ).
(3.291)
Now we may redefine Tt u on a null set such that (3.291) holds for every i € l " and ^fc^=o llufc||£»,efe < °°- Next we may cover (0, oo) with countable many neighbourhoods UE(ti) and using the analyticity of (Tt )t>o and the fact that the countable union of null sets is again a null set we are done. • Let {Tt)t>o, 1 < p < oo, be a symmetric, hence analytic, L p -subMarkovian semigroup. For each p we define the maximal function utp)(x):=swp\T{p)u(x)\
(3.292)
324
Chapter 3 Potential Theory of Semigroups and Generators
which makes even pointwise sense by Lemma 3.4.24. Lemma 3.4.25. For (Tt
)t>o as above we have the estimate
||w*2)IU2 < C2IMU2.
(3.293)
Proof: Let us introduce the Littlewood-Paley function for a semigroup by
a
i2
d
°°
\i/2
t -Q-tTtu{x)\ dt) . (3.294) By the spectral theorem there exists a resolution of identity such that
a
00
e~XtdEx)u.
(3.295)
(Note that we have taken now the resolution of identity corresponding to — A^, A^ being the generator of (Ttl )t>o> whereas in the proof of Theorem 1.4.2.12 we worked with that corresponding to T2 .) From (3.295) it follows that
IT^U „-(-jf m.t
Xe^dEx)u,
which yields by a further application of the spectral theorem /
\gi{u){x)\2 dx = /
JM."
°°
JRI JO
-I
dt * /»oo
J'
0+
and we arrive at HSI(«)||L»
r
/
t
d
{2) n Tt u(x) dt * v ;
dt=
Jo
t
poo
A2/ Jo
dtdx X2e-2Xt(dExu,u)0dt
Jo+ -t
te-2Mdt(dExu,u)0
/.00
= 4 J0+
l(dExu,u)0,
(3.296)
< zhh*-
An integration by parts yields
j\(§-sT^u)ds
=
tT^u-Jj^uds,
(3.297)
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
325
from which we deduce \T™u(x)\ < \\ J*T<M*)ds\
+ | j j\(j-sT^u(x))
ds\
<|iyVi2)^)^ <Mf2)u(x)+gi(u)(x). Now, applying the Dunford-Schwartz-Hopf ergodic theorem, Theorem 3.4.23, especially (3.290), and (3.296) we find
||^ 2) || L2 H|su P |r t (2) u(.)lllL3 < ||M(*a)«||L2 + and the lemma is proved.
||5I(«)||L»
< c 2 ||u|| L 2,
D
As already announced, let us now state partly without proof further auxiliary results. They all are taken from E.M.Stein [255], where also proofs are given. Lemma 3.4.26. For (T/ )t>o as above we introduce the Paley-Littlewood functions f°
9k
„2fc-l
dkrtp\ dsk
Jo
ds,
k£N,
(3.298)
and we claim ||Sfc(w)Hz,2 <
(3.299)
Cfc||u||L2
foralluGL2(Rn;R). Lemma 3.4.27. For (Tt(2)) t>2 as in Lemma 3.4.25 suppose that a
X
+fe-i
^fc_1 ^(2),
ry'u m=i**
<
a,k-i\\u\\L2
(3.300)
L?
and 1 /•* sup- / s t>0 t
I
l ) k_ldf'- T? u k 1
ds ~
ds
L2
< &fc-l|M|/,2
(3.301)
Chapter 3 Potential Theory of Semigroups and Generators
326
hold. Then it follows that sup t>o tJo
dsk
L2
<
(3.302)
Ck\\u\\L2.
n(p) Proposition 3.4.28. Let (T^>) t >o, 1 < p < oo, be an analytic LP -sub-Markovian semigroup. For the maximal function
k
:d
u
k,(P) •= sup t>o
T}p)u(x)
symmetric
(3.303)
ke
dtk
the estimates x
fc,(2)l
(3.304)
< Cfc||u||i2
hold. Combining the Dunford-Schwartz-Hopf ergodic theorem, Theorem 3.4.23, with Lemma 3.4.27 we find the estimates sup k-1 / t>0 I Jo
T(p)u(-)ds LP
p>l,
(3.305)
and k sup t ^T™u t>o 1 dtk *
< Cp||u||jr,p,
k > 0.
(3.306)
The aim is to interpolate (3.305) and (3.306) to obtain supT t (p) u||i,p < cp||tt||z,p, t>o
1 < p < oo.
(3.307)
It turns out that fractional derivatives and integrals which we encountered already in Section 1.4.7 will provide us with the analytic family of operators we are longing for to interpolate (3.305) and (3.306). As before we mention the monograph [243] of S.Samko et.al. as our standard reference for fractional integrals and derivatives. A further source is the book [240] of B.Rubin. For a G C, Re a > 0, we define the fractional integral on L 1 ((0, oo)) by /(Q)u
1
f°°
W = F 7a ^ / r( ) Jo
(ts)a-1u(s)ds.
(3.308)
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
327
Proposition 3.4.29. For u e CQ°((0, oo)) C 5(R) the function a H-> I^u(t) defined on Re a > 0 has an analytic continuation onto C. Further, we have j(-fc) u
^
=
)
feGN,
(3.309)
and j<«) o j(0 = / ( " + « ,
J(°)=id.
(3.310)
Since fractional derivatives are becoming more and more important when dealing with certain sub-Markovian semigroups and Markov processes, we prefer to provide a detailed proof of this proposition, again, following closely E.M.Stein [255]. Proof of Proposition 3.4.29: It is clear that a H-> I^u(t) is for each u £ CQ°((0, oo)) an analytic function in Re a > 0. Now, for Re a < 0 the singularity which may trouble is at point s = t, hence it is sufficient to prove that the operators M(a W ) : = r
a
- ? — [ (t- s ) a _ 1 u ( s ) ds r ( a ) Jo
(3.311)
=fW)l!{i-s)a~lu(st)ds have analytic continuation to C Note that we have now a fixed singularity at s = 1. Therefore we split the integral into two parts M^a)u{t) = M[a)u{t)+M^a)u{t)
(3.312)
where 1/2 r1/*
i
M a)u(t)=
*
(1-a)a~1«*t)d*
T(a)J0
and
M^a)u{t) = - L
f1
r ( a ) Ji/2
(l-sr^uWds.
Clearly, a H-> Mi'u{t) has an analytic continuation to C since there is no singularity at all for Re a < 0, recall that a H-> JTT^T is an entire function. To
Chapter 3 Potential Theory of Semigroups and Generators
328
handle M^
wen
° t e that an integration by parts yields
"*•'*<" " vtki) jj'-Ks^^nkT)
(s)*«<5')-
(3.313) which we may take to get an analytic continuation for a H-> M2" u(t) to the half plane Reoj > —1. Further integrations by part, say Z-times, lead to an expression of M 2 it(i) which allows us to arrive at an analytic continuation for a H-» Mj 'w(i) into the half plane Re a > —I implying the possibility to continue a — i » M2 u(t), hence I^u(t), analytically to C For R e a , Re/3 > 0 we find using Fubini's theorem that I(a)
(^)(r)
=^
^
J\r - 0 - j \ t - sf-iuis)
T(a)T(l3) J0 {r
S)
ds
U{S)dS
'
where B(a, (3) = w , % *s *^ e Beta-function. Thus the semigroup property (3.309) holds for Rea,Ref3 > 0 and extends by analytic continuation to all a, (3 € C. Next, recall the definition of a fractional derivative D " , 0 < a < 1, given in Section 1.4.7. We may apply (1.4.382) t o n e Cg°((0, 00)) C 5(R), and comparing with (3.308) we find D%u(t) = jl^l-a)u{t)
(3.314)
as well as D%u{t) = F^t((-i$)aumt),
(3-315)
which yields for 0 < a, /?, a + (3 < 1 D%D% = D%+13. Furthermore we have for 0 < a < 1 and u € ( ^ ( ( O , °°))i
(3.316) note tnat
u(0) = 0,
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
dtT(l-a)l-aJ0[
> dt{t
(l-a)T(l-a)J0 J0K
T(l -a)
'
$)
329
ds ds
ds
\
ds J
Thus we find C>(1)
= ^ o - . , „ ( t ) = / o-.,(Mf)) ( 1 ) .
In particular we have
=^'-'(|)(.)-i"»(S)(«) and D%I{a)u{t)
= (^/(1_a))(/(a)«)(*)
= |(^M(') = 5 jf «(-)* = «(*)• Thus for 0 < a < 1 we have on C£°((0, oo)) I{a)D%u = D%Iau = u,
(3.317)
which implies by the first part of (3.309) (which is already shown) that /<-Q>u = D%u.
(3.318)
Since (3.314) implies obviously that lim D°[_u = u we arrive at a->0 ( a)
l i m / - u = u,
i.e.
7
(0)
=id.
(3.319)
a—>0
Now, (3.309) follows easily from iW o /(" fc ) = id and jW M (t) =
*
f (t-s^uis)
ds=
f J
...J
u(r)dr...ds,
Chapter 3 Potential Theory of Semigroups and Generators
330
proving the proposition.
•
With the notation (3.311) we have ||supM 1 (Ttu)(-)||LP<4 1 ) ll u IU p '
l
(3.320)
t>o
and ||supM-fc(TtU)0||L*<.4(-fe)H|L2,
fc€N,
(3.321)
t>o
and we want to prove || supM 0 (T tU )(-)||LP < A°p\\u\\LP, 1 < p < oo, t>o by interpolating the operator M£ denned by
(3.322)
M£t;:=sup|M*(t;)(t)|, v = Ttu. (3.323) t>o In order to apply Stein's interpolation theorem, Theorem 1.2.8.3, (or variants of it) we have to overcome two difficulties: We need estimates for Mf+iy and MZk+iy f° r a n V € R, and we have to find reasonable substitutes for M£ by linear operators. Lemma 3.4.30. Let (Tt )t>o, 1 < p < oo, be an analytic and symmetric V-sub-Markovian semigroup. Then there are constants Kp and Ak such that \\M;+iy(Tt(p)u)\\LP
ueL*(Rn;R),
(3.324)
and «6l2(l";M),
\\Mlk+iy(T^u)\\LP
(3.325)
hold. Lemma 3.4.31. Let (T t (p) ) t > 0 be as in Lemma 3.4.30 and let r : R n —> (0, oo) be a measurable mapping. Further define for a £ {7 + iy; I S {1} U —No and
yeR} r
(<*) Jo Then there exists a constant KP such that HS^tilli, = | | T $ u ( . ) | U , < holds.
KP\\U\\L,
(3.326)
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
331
Finally we are in position to discuss Stein's maximal theorem, see [255]. Theorem 3.4.32. Let (Tj ) t >o, 1 < p < oo, be an analytic and symmetric Lp-sub-Markovian semigroup and denote as in (3.292) by ulKF p)(x)^snp\T^u(x)\ ' t>o its maximal function (at u G Lp(M.n;M.)). Then there exists a constant such that \\U*\\LP
< Ap\\u\\LP,
1 < p < oo,
(3.327)
holds and for u G Lp(M.n; R) we have limT t (p) u = u
a.e.
(3.328)
Proof (sketch): In order to get (3.327) one determines a function r such that
l^gn^i^isupir/^)! v
'
& t>0
for each x G R n . Now, (3.326) yields ||su P T t ( p ) u|| L p<2K p ||w|| L P t>0
which is (3.327). It remains to prove that lim T} u = u a.e. Since 11—> T,
u(i)
is real-analytic by Lemma 3.4.24, it follows that lim T t (p) (T s (p) u) = T^p)u for almost all x G Rn and s > 0. Thus we find for u G L2(Rn; R) limsup|T t ( 'u(x) — u(x)\ < lim sup |T t (2) (u - T^){x)\
+ lim sup |T t (2) TJ 2) u(x) - T s (2) u(x)|
t\0
t\0
+ \TMu(x)-<x)\ < sup \T?\u - Ti 2 >)(*)| + \T™u(x) - u(x)\ t>0
= (u- T^u)*(x)
+ \TPu{x)
-
u{x)\,
Chapter 3 Potential Theory of Semigroups and Generators
332
or since the maximal function is Zi2-bounded || limsup | l f >«(•) - ti(.)|||L* < 2||(« -
TM)*\\L,
o is strongly continuous, implying as s —> 0 that limsup|Tt t\o
u(x)— u(x)\ — 0 a.e.,
or T^2)u{x) -> u(x) a.e. for u € L 2 (R n ; R). Now, let u € L P ( R " ; R), 1 < p < oo, and for £ > 0 given, take v € L 2 (R"; R) n L P (R"; R) such that ||w - V\\LP < e. (2)
As before we find using that Ttv u —> u a.e. limsup|T( t\fl
u(x) — u{x)\ <
< \Tt{p)u(x) - Tt{p)v(x)\ +limsup|T t (2) w(x) - v(x)\ + \v(x) - u(x)\ t\o ip < limsup \Tt \u -v)(x)\ + \(u- v)(x)\ < (u - v)*(x) + \(u -
v)(x)\.
p
The L -boundedness of (u — v)* yields further ||limsup|T t {p) u(-)-M(-)ll|Lp < A"||U-W||LP < Ke.
t\o
For e —> 0 we get finally limsup|T( t\fl
u{x)— u{x)\ = 0 a.e.,
which means that Tt u —» u a.e. as t —> 0.
•
We will state now E.M.Stein's Littlewood-Paley inequality in the general case. For this let (Tt)t>o be a symmetric and conservative sub-Markovian semigroup defined consistently on all spaces L p (R n ;R), 1 < p < oo. We denote the corresponding Littlewood-Paley function once again by t ^Ttu{x)\
dtj
.
(3.329)
3.4
Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
T h e o r e m 3 . 4 . 3 3 ( E . M . S t e i n ) . If u € Lp(Rn;R) and \\gi(u)\\Lp
then g^u)
333 €
LP(Rn;]
(3.330)
< C P ||U||X,P
holds for 1 < p < oo. C o r o l l a r y 3 . 4 . 3 4 . For all k e N the ||3fc( u )l|L" < C P ||W||LP,
estimate (3.331)
1
holds. C o r o l l a r y 3 . 4 . 3 5 . Denote by EQ the orthogonal projection {h G L 2 ( R " ) ; Tth = h for all t > 0 } . Then the estimate \\U\\L*
< CP\\9I{U)\\LP
+
EQ :
L2(Rn)
(3.332)
\\EQ{U)\\LP
holds. P r o o f ( s k e t c h ) : First we have t o show t h a t ||.Eo( u )||i> makes sense which will follow from II•£<)(«)||L" < c p ||w||i,p, Since in addition (T t spectral theorem yields
1 < p < oo.
)t>o is symmetric its generator is selfadjoint a n d t h e
OO
Ttu=
(3.333)
/-OO
e'xtdExu = E0(u)+
/
Jo
e-XtdExu Jo-\ /o+
and further a
/-oo
jrTtu ot
(-X)e-xtdExu
= / J'0+ 0+
which gives
Ttu{x)
/ \i
>
/-oo
' dx=
X2e-2Xt(dExu,u).
\ Jo+
jR" i or r
Therefore 2 / we \9l(u)\find dx
/•
r°o
= /
Qrp
t /•OO
-^u{x)
dxdt
/-OO
= / A2 / e~2Xttdt(dExu,u) Jo+ Jo 1 f°° = - / ld(Exu,u), 4 Jo+
334
Chapter 3 Potential Theory of Semigroups and Generators
or |2
_ A \ \ „ / „ . \ \ \ 2
I M I k = 4 | | p i ( u ) | | i a + ||^(«)||L*.
(3.334)
We m a y polarize (3.334) if EO(UJ) = 0 t o get /
u\{x)u2{x)
dx = 4 /
Now, for u € L2(Rn)nLP(Rn)
/
t—jj-
^—dxdt.
(3.335)
and « e L 2 ( M 2 ) n L p ' ( R " ) , p' = j ^ j , E0(u) =
EQ{V) = 0, we find SU
IMILP =
P
/
u(x)v(x)dx
h\\Lpl
f°° /
sup
f /
,dTtudTtv. ,. t— —— dx dt
HI LP <<1 <4
sup veL2nL"' \N\LP,<1
/ ^
gi(u)g2(v)dx,
J n
where we used t h e Cauchy-Schwarz inequality in t h e last step. Now, Holder inequality leads t o \\U\\LP < ( 4
sup
||5I(U)||LP')|MU)||LP,
veL2nLp' or since ||5i(v)|| L P / < Cp/||u||iP we find \M\LP
for u G V>{Rn) n L 2 ( R " ) , E0(u) = 0. T h e general case follows when substituting uby u — EQU a n d t h e observation t h a t 4rTtEQU = mEou = 0: ||U||LP - | | £ 0 ( U ) | | L P < \\u ~ E0(U)\\LP
= which is (3.332).
< Cp\\gi(u -
E0(u))\\Lr
Cp\\gi(u)\\LP,
D
We continue t o use t h e spectral decomposition of Tt, i.e. t h e representation Ttu=
I
I
e~XtdExu.
3.4 Stein's Littlewood-Paley Theory for Sub-Markovian Semigroups
335
More generally, we define operators for bounded measurable function on (0, oo) by /•OO
Smu := /
(3.336)
m(A) dE\u
and we call m of Laplace transform type if e~MM{t)dt,
i(A)
(3.337)
A>0,
Jo
holds with a bounded measurable function M on (0, oo). Again we follow E.M. Stein [255] to get Proposition 3.4.36. Let m be of Laplace transform type. bounded operator on Lp(M.n), 1 < p < oo.
Then Sm is a
Proof: We will prove gi(Smu) < cg2(u).
(3.338)
for all u £ LP(R") with c independent of u. Once the estimate (3.338) is proved the proposition follows from the above corollaries: \\Sm(u)\\Lp < cp\\gi(Smu)\\
+
\\Eo(Sm(u))\\Lr
where we also used the fact that Eo(Smu) = 0 which is due to the representation (3.336). To prove (3.337) we need some preparation. For u such that EQ(U) = 0 it follows that
1^-T.u) \ds2'
! * •
-i;
ds2
Tsu
ds ds
ds2
T3u
1/2
l
ds)
,-1/2
where in the last step we used the Cauchy-Schwarz inequality. Now the last estimate and Fubini's theorem yields {9l{u)f
i: i:
Ttu
dt d2Ts' ds2
d^T°U
s2 ds) dt ds = g2{u)2,
Chapter 3 Potential Theory of Semigroups and Generators
336
or (3.339)
9i(u) < 92(") for E0(u) = 0. Next observe that /»oo
r°°
d
/ -TtuM{t)dt= Jo ot /»oo
= / Jo
f°°
J0
8
f°°
e-XtdExuM(t)dt
— / at J0+
/»oo
poo
/ -Xe-xtdE\uM(t)dt= Jo+
/»oo
-Xe~xtM{t)dtdExu
/ / Jo+ Jo
/»oo
= — / m(A) dE\u = —Smu, Jo+ i.e.
Sm(u) = -J
^TtuM(t)dt.
(3.340)
Now we may use exactly the same calculation leading to (3.339) to find gi(Smu)2
I"
= Jo
3 -^Ts(Smu)
and the proposition is proved.
D
ds < c
f Jo
92
T
TtU
dt = cg2(u)2,
W
This proposition has an important corollary: Corollary 3.4.37. Let (Tt)t>o be as above, in particular let A^ be a selfadjoint operator. Then for every real number r G 1R the operator (—A)ir is a bounded operator from Lp(Rn) into itself, 1 < p < oo. Proof: We only have to observe that Xir is of Laplace type.
•
Corollary 3.4.38. Let ip : R™ —> R be a continuous negative definite function satisfying tp(0) = 0. Then for all r 6 l the operator (ip(D))ir is bounded on Lp(M.n), 1 < p < oo.
3.5
Global Properties of ZASub-Markovian Semigroups
Let us suppose that we may associate to a given L p -sub-Markovian semigroup (Ttp ) t >o, 1 < p < oo, a Markov process (f2, A, {Xt)t>o, Px)x^w- We may ask
3.5 Global Properties of Lp-Sub-Markovian Semigroups
337
for properties which are common for all the measures Px, x £ R n , or for almost all paths 11-> Xt{uJ), XQ{W) = x € K n . Following M.Fukushima, Y.Oshima and M.Takeda [102] we will call these properties global properties of the process. In this section we will study properties of (T}p')t>o which will imply global property of the associated process. Therefore we call these properties global properties of the semigroup. Some of our considerations depend on whether a given L p -sub-Markovian semigroup (Tt )t>o has an extension to a positivity preserving L 1 -contraction. This however is equivalent to the fact that (T/ p )t>o is an IP -sub-Markovian semigroup as we may deduce from the next theorem which is taken from our joint paper [166] with R.L.Schilling. Theorem 3.5.1. Let 1 < p < oo and T : L p (R n ;R) -> L P (R";R) be a linear (bounded) operator. Then T* : Lp'(Rn;R) -> Z p '(R n ;R) is a (bounded) U>' sub-Markovian operator, p' = p/(p — l), if and only ifT is positivity preserving and extends to a contraction T : L : (R";R) -> L ^ R ^ R ) . Hence T and T* are sub-Markovian if and only if both operators extend to positivity preserving contractions onto all spaces L 9 (R;R), 1 < q < oo. Proof: Throughout the proof (Bk)k€N denotes a sequence of Borel sets in R n increasing to R n , i.e. Bk | R™, each having finite measure, i.e. X^(Bk) < oo, and we set as before p' = ^ j . Assume that T : L P (R";R) - • L P (R";R) is positivity preserving. Forw € Lp'(Rn;R), v > Oa.e., we set Ak := Bkf\{T*v 0}. It follows that XAk e L p (R n ;R) and / {Tv){x)dx= I JBkn{T*v<0} -MR" = /
<
{T*v){x)XAk{x)dx v(x)T(XAk)(x)dx>0
implying that \<-n)(Bk n {T*v < 0}) = 0 for all k € N, i.e. T* is positivity preserving if T is positivity preserving. Interchanging the role of T and T* we find that T* is positivity preserving if and only if T is positivity preserving. Now assume that T : L1(Rn;R) -> i 1 ( R n ; R ) is positivity preserving and an L 1 -contraction. For v G Lp (R";R) such that 0 < v < 1 a.e. we set Cfc := Bk n {T*v > 1}. We find that / {T*v){x)dx= JBkn{T-v>n
[ JR"
{T*v){x)XCk{x)dx
Chapter 3' Potential Theory of Semigroups and Generators
338
= [
v(x)T{xcKx)dx<\\v\\00\\Txck\W<\\xc,\\^ \(n\Bkn{T*v>l})
=
which yields A(n)({T*ii > 1}) = 0. Since T is by assumption positivity preserving it follows that for 0 < v < 1 a.e., we find 0 < T*v < 1 a.e., i.e. T* is sub-Markovian. Suppose now that T* : £ p '(R n ;R) -> l / ( R * ; R ) is sub-Markovian. In particular it follows from the first part of this proof that T is positivity preserving. Therefore, for u £ Lp(M.n; R) n L ^ R " ; R) it follows that /
\{Tu){x)\XBk{x)dx<
[ = [ JRn
T\u\{x)XBk{x)dx \u(x)\(T*XBk)(x)dx<
f
\u(x)\dx.
JR
By monotone convergence we find as k —> oo that \\Tu\\Li < ||u||ii
for all u e Lp(Rn; R) n L x (R n ; R),
which clearly implies that T extends to an L 1 -contraction. Finally, if T and T* are both sub-Markovian operators, the above proof and the consequences of Remark 2.6.15.B give that both T and T* extend to sub-Markovian contractions on all the spaces L 9 (R n ; R), 1 < q < oo. • Corollary 3.5.2. Suppose that for some 1 < p < oo an l?-sub-Markovian semigroup (Tt )t>q is given such that its dual semigroup (Tt )t>o is Lp-subMarkovian. Then (T^ )t>o and (T^*)t>o extend both to Lq-sub-Markovian semigroups (Tt )t>o and (Tt )t>o for all 1 < q < oo. In addition we have foJ, M u e Ll(Rn. R j n £,oo(R». R ) a holds T(q)* = fW) md TtMu = Tt{p)u
and
Tt{q)*u = Tt{p)*u,
t > 0.
(3.341)
Moreover, (Tj )t>o is a semigroup of sub-Markovian operators. Remark 3.5.3. A. The relations (3.341) justify that in the following we will write just Ttu or T*v for T^'u or Tt v, respectively. B. In the situation of Corollary 3.5.2 we may also consider for all 1 < q < 00 the corresponding resolvent operators R\ = /0°° e~XtTt dt which form for 1 < q < oo strongly continuous contraction resolvents, whereas for q = 1 a contraction resolvent is given.
3.5 Global Properties of Lp-Sub-Markovian Semigroups
339
C. Note that in general (T{ )t>o will not be strongly continuous, see J.Voigt [278]. Let T : Lp(Rn; R) -> Lp(Rn; R) be a bounded linear and positivity preserving operator. The operator norm ||T|| of T is given by |
m
,
= s u p
f * > .
(3.342)
The next lemma taken from E.B.Davies [60] shows that we may restrict the supremum to non-negative functions u e L p (R n ;R). Lemma 3.5.4. For a linear, bounded and positivity preserving operator T : £P(R n ;R) -» L P (R";R) the operator norm equals \\T\\ = sup ( ^U}LV
;
u e L p ( l " ; R ) andu>0
a.e.\.
L \\U\\LP
(3.343)
J
Proof: Clearly we have that CT := sup ( ^U}L"
;
u € L p ( R n ; R ) and u > 0 a.e.) < ||T||.
L ||W||LP
J
p
Now, for u G L (R";R) it follows that - | u | < u < \u\ and therefore - T | u | < Tu < T\u\ implying ||Tu||iP < | | T | « | | | L P < C7 r |||«||| L p =
i.e. ||T]| < CT.
CT\\U\\LP,
D
We arrive now at a first central definition in this section. Definition 3.5.5. Let T : Z , P ( R " ; R ) -> Z,P(R n ;R), 1 < p < oo, be a linear bounded and positivity preserving operator. A. A Borel set G C R n is called T-invariant (or invariant with respect to T)if TXGU
= XGTXGU
a.e.
(3.344)
holds for a l l u G L P ( R n ; R ) . B. A Borel set G C R n is called strongly T-invariant if TXGU = XGTU
a.e.
(3.345)
holds for all u £ L P (R";R). Using XGU instead u shows that every strongly T-invariant set is T-invariant.
340
Chapter 3 Potential Theory of Semigroups and Generators
L e m m a 3.5.6. Let T : Lp(M.n;R) -> Lp(Rn;R), 1 < p < oo, be a linear bounded positivity preserving operator. A. If G is T'-invariant then Gc is T*-invariant. Further, arbitrary intersections and countable unions of T-invariant sets and a countable intersection of T*-invariant sets are T-invariant. Thus an arbitrary union of T*-invariant sets and a countable intersection of T* -invariant sets are T-invariant. B. If G is strongly T-invariant then Gc is strongly T-invariant. Further, G is strongly T-invariant if and only if G is T-and T*-invariant. C. An arbitrary intersection and union of strongly T-invariant sets are strongly T-invariant. Proof: A. For a T-invariant set G we find for u 6 L P '(R";R) and v <E Z , P ( R " ; R ) that /
xavT*
{XG°V)
dx =
XG"uTxGdx=
(XG°U)XGTXGV
dx = 0
which means that (l-XG')(T*XG"u)vdx
= 0,
ve
so T*XG"U = XG"T*XG"U. Next let ( G J ) J 6 J be an arbitrary family of Tinvariant sets. Since T is positivity preserving it is monotone (in the sense that u\ < U2 implies Tui < TU2) implying TxnGi u < TxGj u = XGj TXG, U
for all j e I.
Hence (TxnG,u)(z) = 0 for x <£ f]jeiGj o r ^XnG," = XnG,TxnG, for all u S -L p (K n ;R). For a countable family of mutually disjoint T-invariant sets (Gj)je® we find oo
T(xuGjU) =
^/TxGju, 3=1
Dut
Y^jLi TxGjU(x) = 0 for x € (L)Gj)c, implying that oo
oo
TixuGiU) = Y^T(xGjU)
= XuGj (53 T (XG?i«)) =
3=1
XOGJTXUGJU.
3=1
If G\, G% are two arbitrary T-invariant sets we have trivially XGiUGzTxduGiU <
T'XG 1 UG 2 W
if
u > 0,
3.5 Global Properties of £p-Sub-Markovian Semigroups
341
but in addition we find TXGXUG2U
= Txdu =
TxG1nG2u
+ T\a2u -
XG1TXG1U
+
- XGinG2T'xG1nG2'".
XG2TXG2U
Further, since TxGtnG2u = XG1nG2TxG1nG2u + XGl\jG2TXG1nG2u
w e nj
id
TxGiUG2u = XGiuG2 (TxGiu + TXG2U - TxG x nG 2 ") = XGIUG 2 T'XG 1 UG 2 W,
where we used for u > 0 the trivial fact XG1UG2TXG1UG2U < T^GiUGa^. In order to prove the general case we just have to pass from an arbitrary countable family of T-invariant sets to an appropriate family with mutually disjoint Tinvariant sets. B. For a strongly T-invariant set G we deduce from (3.345) that TXG<=U = TU-TXGU
= TU-XGTU
=
XG-TU,
i.e. Gc is strongly T-invariant. We prove now that the T-invariance and T*invariance of G imply the strong T-invariance of G. For G we have TXGU = XGTXGU or XG"TXGU = 0- Further, since G is also T*-invariant, Gc is Tinvariant and it follows that Txcu = XG"TXG<=U or XGTXG<=U — 0. Thus we find XGTU = XGTXGU
+ XGTXG-U
=
XGTXGU
= XGTXGU
+ XG°TXGU
=
TXGU.
On the other hand, if G is strongly T-invariant it follows that Gc is strongly invariant, hence Gc is T-invariant implying that G is T*-invariant. (Note for p = 1 we have to use the fact that T**|n = T). C. Part B of the lemma says that the family of strong T-invariant sets is the intersection of the two families of T-and T*-invariant sets. Therefore part A and B yields part C. Corollary 3.5.7. Suppose thatT^v^L'1 «s symmetric on L2(M.n;M). Then G is T-invariant if and only if it is strongly T-invariant. Definition 3.5.8. We say that a Borel set G C M.n is (Tt)t>o-invariant or invariant with respect to an L p -sub-Markovian semigroup (Tt)t>o if it is Ttinvariant for all t > 0. Analogously strongly (Tt)t>o-invariant sets are defined.
Chapter 3 Potential Theory of Semigroups and Generators
342
Lemma 3.5.9. Let (Tt)t>o be an Lv-sub-Marhovian semigroup with resolvent (R\)\>o- Then G is (Tt)t>o-invariant (strongly (Tt)t>o-invariant) if and only if it is (Rx)x>o-invariant (strongly (Rx)x>o-invariant), i.e. invariant (strongly invariant) for all Rx, A > 0. Proof: Since Rxu = /0°° e~xtTtudt we find for a (Ti)t>o-invariant (strongly invariant) set G C R" and u £ L p (R n ;R) immediately /•OO
R\XGU=
l-OO
e-XtTtXGudt
Jo
= XG
Jo
e-xtTtXGudt
= xoRxXGU
and y»oo
RXXGU
= / Jo
/>oo
e~XtTtXGu
dt = XG
Jo
e~xtTtu dt =
XGR\U,
respectively. On the other hand, since the Yosida approximation of the generator A of (Tt)t>o is given by Ax = XARx = X(ARx — id) it follows that a (i?,\)-\>o-invariant (strongly invariant) set G, is also invariant (strongly invariant) for all operators Ax implying the invariance (strong invariance) of G with respect to the operators °°
e\AR,t
=
^
fk
-(\ARx)k,
t > 0.
(3.346)
fe=0
But eXARxt -> Tt strongly in LP(R"; R) implying the invariance (strong invariance) of G with respect to Tt for all t > 0. • Remark 3.5.10. Clearly the results of Lemma 3.5.6 carry over to semigroups and their resolvents. The following result relates strongly invariant sets to the Dirichlet form associated with an L 2 -sub-Markovian semigroup and it is taken from the work [226] of Y.Oshima. Theorem 3.5.11. Let (T$)t>o be an L2-sub-Markovian semigroup defined on L 2 (R";R) with dual-semigroup (T*)t>o being also L2-sub-Markovian. Denote by {£, D{£)) the corresponding non-symmetric Dirichlet form, compare Definition 1.4.7.21 and Theorem 1.4.7.23. Then a Borel set G C R n is strongly (Tt)t>o-invariant if and only if for all u, v G D{£) it follows that XGU, XGV E D(£) and £(u, v) =
£(XGU, XGV) +£(XG-U,
XG'v).
(3.347)
3.5
Global Properties of L p -Sub-Markovian Semigroups
343
Proof: Suppose that G C ttn is an (Tt)t>o-invariant set. It follows from the (i?A)A>o-invariance of G for u £ D(£) that {(id-XRx)xau,XGu)o+((i^-^^)XG'=u,XG<'u)o.
((id-A.RAKu)o =
Using Lemma 1.4.7.18 we deduce that XGU,XGCU € D(£) and (3.347). Conversely, suppose that XGU,XGV € D(£). Then of course XGc^ = u — XG - " S D(£), hence X G ° " € .D(
£(XGU,XG°«')=0.
Indeed, now (3.347) implies by a straightforward calculation £(XGU, XG*V)+£(XG°U,
XGV)
=0
(3.349)
for all u, v £ D(£) and taking XGU instead of u the second term in (3.349) vanished which yields (3.348). Since for A > 0 and v € D(£) £\(XG-U,
R\V)
= (v, XG-u)0 =
£\(XG°U,
R\XCV)
we arrive at £\{XG-=u, R\v -R\(XG"V)) R\(XGCV))
and with u := R\v — £\{XG"U,u)
=0 we have
= 0.
Now (3.348) yields £X(XG-U,XG-U)
= £X(XG<=U,U) = 0,
and therefore X G C " = 0, or XG<=R\V = we have XGRXV = XGRXXGV. Thus we find RXXGV
= XGRXXGV
+
XG'R\(XG<>V).
Taking G instead of Gc
XG'RXXGV
= XGRXV + XG"Rx(v - XG'V) =
XGRXV+XG°RXV
- XG'RxXG'V =
XGR\V
proving the (i?A)A>o-invariance of G which implies the theorem.
•
344
Chapter 3 Potential Theory of Semigroups and Generators
Remark 3.5.12. Since (3.347) is symmetric in G and Gc, accordingly to Lemma 3.5.6 we should not expect (3.347) to be equivalent to the (Tt)t>oinvariance of G. Definition 3.5.13. An ZAsub-Markovian semigroup {Tt)t>o is called irreducible if for any (T t ) t >o-invariant set G either X™(G) = 0 or \(n\Gc) = 0 holds. Since for G c R " such that X^(G) = 0 or X^(GC) = 0 it follows that TtXGU = XGTU
a.e. and TtX&u = XG"Tu a.e.,
a set G of measure zero as well as its complement are always (Tt)t>o-invariant. Suppose that a Dirichlet form (£, D(£)) has the property u,v £ D{£) and supp u n supp v = 0 implies £{u, v) = 0.
(3.350)
Then (3.347) holds for all u,v € D(£). Dirichlet forms satisfying (3.350) are called local Dirichlet forms. Typically they are associated with a differential operator, i.e. they have the (typical) structure £(u,v) = V
/
^Lp-vkl(dx)
(3.351)
ifjti ^R" dxk dxi
with some suitable measures Vki{dx). In case that Vki{dx) — a,ki(x)\(n\dx) 2 2 a n da a n with //|£| < E*,i=i Ofc/0»0&& < £l£l *i = ik € Cl(R ) (for simplicl n ity), then D{£) = H {R \R) and typically XGU $ i7 1 (R n ;E) for a Borel set such that \(n\G) or A^™^(GC) is strictly positive. This observation is just a special case of Lemma 3.5.14. Suppose that the Lp-sub-Markovian semigroup (Tt)t>o has a kernel representation Ttu(x)=
[
Pt(x,y)u(y)dy
(3.352)
with a kernel function pt(x, y) > 0 for almost all x, y 6 M.n and t > 0. Then (Tt)t>o is irreducible. Proof: Suppose that G c M.n is an invariant Borel set with X^(G) > 0. Take u G LP(M"; R). It follows that XG{x){TtXGu){x) = (TtXGu)(x) = /
Pt(x,y)xG{y)u{y)dy,
3.5 Global Properties of Lp-Sub-Markovian Semigroups
345
i.e. JR„ pt(x, y)XG(y)u(y) dy = 0 for all x £ Gc implying that JGpt(x, y)dy = 0 for all x G Gc which is a contradiction to pt(x, y) > 0 unless where \(n\G) = 0
or\(n\Gc)=0.
0
Example 3.5.15. Since the Gauss kernel is strictly positive we find that the Gaussian semigroup is irreducible. Further, if L(x, D) is a second order differential operator generating an L p -sub-Markovian semigroup {Tt)t>o and satisfying all the assumptions needed to prove Aronson's estimates (2.442) then the semigroup (Tt)t>o is irreducible. Moreover, all semigroups subordinate to (Tt)t>o with respect to a convolution semigroup (i]t)t>o, suppVt C [0, oo), admitting for each r)t a strictly positive and continuous density on (0, oo) are irreducible too. In particular, since it is possible to prove with some effort, compare V.M.Zolotarev [291], Section 2.7, that the one-sided stable semigroup (Vt )t>o, 0 < a < 1, has smooth and strictly positive densities, the semigroups generated by fractional powers of L(x, D) are all irreducible. Let (Tt)t>o be an L p -sub-Markovian semigroup with generator (A, D(A)). The elements in the kernel of A, Ker(^4) := {u G D(A);Au = 0} are called A-harmonic functions (belonging to D(A)). It is interesting to note that for an irreducible semigroup we can obtain easily information about Ker A. The following results are borrowed from E.B.Davies [60]. Proposition 3.5.16. Let (Tt)t>o be an Lp-sub-Markovian semigroup. A. The space Ker A is a closed sub-lattice of L p (]R n ;R), i.e. u £ Ker (A) implies \u\ € Ker (A). B. If(Tt)t>o is irreducible £AendimKer(.i4) < 1. Further if dimKer(A) = 1 then there exists a strictly positive element UQ G Kei(A), i.e. UQ > 0 a.e. Proof: A. If u € D(A) and Ttu = u for all t > 0 then u € Ker(A). conversely, since by Lemma 1.4.1.14
Ttu-u=
I TsAuds,
Jo
But
u£D(A),
it follows for u € Ker(^4) that Ttu = u for all t > 0. Thus we find Ker(,4) = f]{u t>o
€ Lp(Rn; R); Ttu = u}
and it remains to show that each of the sets {u G Lp(Rn;W);Ttu = u} is a closed sub-lattice of LP(M.n;R). Clearly {u G L p (R";R);T t u = u} is a closed
Chapter 3 Potential Theory of Semigroups and Generators
346
sub-space of
Z,P(E";
E). Now, for u G {u G £P(R n ; R); Ttu = u} we find
o<|«| = |rt«|oB. That 0 G Ker(A), hence dimKer(.A) > 0 is trivial. Hence suppose that dimKer(A) > 0. Since Ker(A) is a sub-lattice, u G Ker(^l) yields \u\ G Ker(A), thus Ker(A) contains at least one non-trivial non-negative element v > 0 a.e. For u) G L P (E"; E) such that \LJ\ < av a.e. for some a > 0 it follows that \Ttu)\ < Tt\u\ < Tt(av) = av a.e.. Thus we find that M := {u G L p (E n ;R); |w| < av a.e. for some a > 0} is invariant under the action of Tt for all t > 0, i.e. TtM C M. Since the closure of M is the set of all w G L P (E"; E) such that {a; G E n ; u)(x) ± 0 a.e.} C {x G R n ; n(x) 7^ 0 a.e.}, we deduce that {x G E n ; v(x) ^ 0 a.e.} is (T t ) t >o-invariant implying by the irreducibility assumption that {x G R n ; v(a;) ^ 0 a.e.} = R n . Therefore, if v G Ker(A), v > 0 a.e. and u ^ 0, then {x G E n ; v(x) ^ 0 a.e.} = R". The sub-lattice property implies further that v+ = ^(\v\ + v) a.e. and v~ = TJ(\V\ — v) a.e. belong also to Ker(A), but {x G R"; v{x) > 0 a.e.} D {x G E n ; v(x) < 0 a.e.} C {x G E"; v(x) = 0 a.e.} which means that either v+ or v~ must vanish a.e. Therefore either v > 0 a.e. or — v > 0 a.e. Finally take two positive functions u,v £ Ker(-A). Then for all A G Q the function u + Xv must be either almost everywhere positive or negative and the exeptional set is independent of A G Q. Hence we may construct some Ao G E such that u + XQV = 0 a.e. proving that Ker(A) is one dimensional. • Next we will discuss transient and recurrent semigroups. In order to get an idea of the probabilistic relevance of these notions and some intuition for analytic formulations we start with a rather heuristic discussion. Suppose our semigroup has an extension to L 1 (R";R) as a semigroup (Tt)t>o of positivity preserving contractions which is not necessarily strongly continuous. For a (bounded) Borel set G e l " the term pt(x, G) := TtXG{x) is the probabilistic interesting quantity, giving the probability for being at time t in the set A if the process associated with (Tt)t>o starts at t = 0 in the point
3.5
Global Properties of L p -Sub-Markovian Semigroups
347
x. Thus the sum M
M
M
J > * ( * , G) = YJTtkXG{x) /c=0
=
fc=0
^J*XG(X) fc=0
gives information to be at the time instants 0, t, 2t,..., Mt in the set G. Fix t and M e N, divide [0, Mt] into N equally long disjoint intervals, i.e. consider the partition , Mt r „ , Mt {0,1- — , . . . > — , . . . , M i } and consider the sum N-l
E
N-l
Tkjnxoix)
T
= E
k=0
^XG(X).
k=0
It follows that N-l
(id-%) ^
T L X G W = (id-T^)XGCr) - (id-TMt)*G(z)
fc=0
or
(TT1) E ^ ^ V N
= (id-TMt)xG(z).
fc=0
With the substitution Hr = s w e find
( ^ )
E *txo(x)
=( ^ )
k=0
E - ^ ^ W =
(id-TMt)XG(x).
fc=0
But Sfello1 sTksXG{%) is a Riemann sum for JQ Tt>XG(x) dt'. Suppose now that lim pr{x, G) = lim TrXG(x) = 0 and let N tend to infinity such that Mt/N = r—»oo
i—>oo
s remains constant, so Mt —> oo too. Assuming that J2k=o sTksXG(x) will tend to the improper integral J"0°° Tt>XG(x) dt' we will arrive at (^T^)
jfTt'XG(x)dt'
=
XG(X).
But now we may also pass to the limit s —> 0 to find -A(J°°
Tt,XG(x) dt') = XG(X),
(3.353)
Chapter 3 Potential Theory of Semigroups and Generators
348
where A is the "generator" of (Tt)t>o (but note that there is of course a problem since we did not assume (T t ) t > 0 to be strongly continuous on L 1 (E n ;R)). Assuming further that we may pass from (3.353) to all / £ L 1 (R n ;E), / > 0, we arrive at -An°°Tt/fdt^=f.
(3.354)
Now let us try to interpret some of our heuristic arguments. First, note that lim pr(x,G) — 0 means that for large r there is "no" probability to be in the set G. Suppose that this condition yields both, the existence of Jo TtXG(x)dt < oo and the convergence of Y^kLo^tXG{x) < °°. The latter has the interpretation that the process may visit the set G with some positive probability finitely may often, but for large k visits to G are very unlikely. This phenomenon is called transience of a process . In addition (3.354) has also an interesting analytic interpretation. Since /
Ttf{x) dt = lim Rxf(x)
J0
A-*0
= lim /
e'xtTtf(x)
dx,
A-»0 7o
(3.354) means that in the "transient case" for A —> 0 the resolvent of the semigroups (Tt)t>o should tend to an inverse of —A. Because the inverse of —A is the classical Newton potential, / 0 Tt dt should be considered as an abstract potential operator. It is also worth to have a look when pks{x, G) > rjo > 0 for infinitely many k. Clearly, in this case the series Y^kLo TSXG{X) will diverge and we should expect the same for the improper integral fQ TtXG{x) dt. But pks(x,G) > 7]0 > 0 for infinitely many k has the interpretation that the process associated with {Tt)t>o will visit the set G infinitely often, such a phenomenon will be called recurrence of the process. In volume III of this treatise we will be more precise with these interpretations and arguments. But now we are prepared and motivated to develop the analytic theory of transient and recurrent semigroups. For this we assume that the L p -sub-Markovian semigroup (T t ) t >o has a sub-Markovian dual semigroup. From Theorem 3.5.1 we know that in this case Tt, t > 0, extends to a sub-Markovian contraction Tt : ^ ( M ^ R ) -> L ^ R ^ R ) . Therefore the operators Stu:=
/ Tsuds, Jo
t>0,
(3.355)
3.5 Global Properties of Lp-Sub-Markovian Semigroups
349
satisfy for 1 < p < oo the estimates ||S , t«||Lp<*N|Lp,
ueP(ln;R).
(3.356)
In addition St is positivity preserving and since Stl+t2u=
/ T3uds= / Tauds+ Tsuds, Jo Jo Jti the fact that Ts, s > 0, is positivity preserving yields for 1 < p < oo and u£ Lp(M.n;R),u>0 a.e., Stu < St'U for t < t'. According to Remark 3.5.3.B the resolvent R\ = J0 e~xtTt dt is also well denned on Lp(Rn; R) for 1 < p < oo. Definition 3.5.17. Let (Tt)t>o be an L p -sub-Markovian semigroup, 1 < p < oo, with dual semigroup being sub-Markovian. For u 6 L^M^IR), u > 0 a.e., we define the potential operator associated with (Tt)t>o by Gu(x) = lim
SNU(X)
N-+oo
= sup SNU(X).
(3.357)
NeN
Remark 3.5.18. A. Note that the limit in (3.357) exists but might be infinite, i.e. in general we only have Gu{x) < +oo a.e. B. Since f sup SNU(X) = sup sup / e~ tTtu dt JVeN NeN\>oJo rN
= sup sup // 'e \>0N€NJQ i Jo
Ttu dt = sup R\u, A>0
we have lim SNU(X) N->oo
= lim R\u(x)
a.e.
A->0
C. Often G is also called the Green operator associated with (Tt)t>oDefinition 3.5.19. Let (Tt)t>o be an L p -sub-Markovian semigroup, 1 < p < oo, with dual semigroup being sub-Markovian too. A. We call (T t ) t >o to be transient if Gu(x) < oo a.e. for all u £ L ^ E " ; R), u > 0 a.e. B. The semigroup (T t ) t >o is called recurrent if for all u £ £2(]R™; 1R), u > 0 a.e., Gu(x) £ {0, oo} for almost all x £ R™.
Chapter 3 Potential Theory of Semigroups and Generators
350
Remark 3.5.20. A. Note that for a general semigroup (Tt)t>o as in definition 3.5.19 transience-recurrence does not form a dichotomy, see however Example 3.5.23 where translation invariant sub-Markovian semigroups are discussed. B. A semigroup as in Definition 3.5.19 extends by Theorem 3.5.1 to an Lqsub-Markovian semigroup for all 1 < q < oo. The definition of transience and recurrence however depends on the L1 extension, hence if (Tt)t>o is transient (or recurrent) for some 1 < p < oo it is transient (or recurrent) for all 1 < p < oo. To proceed further we need Hopf's maximal ergodic inequality the proof of which we have adapted from M.Fukushima et.al. [102]. Proposition 3.5.21. Let (Tt)t>o be an Lp-sub-Markovian semigroup with dual semigroup being sub-Markovian too and let St, t > 0, be defined by (3.355). For u g L ^ R ^ R ) and h > 0 define Eh := {x g Rn; sup Skhu(x) > 0}. Then fc€N
we have
I Shu(x)dx>0.
(3.358)
JEh
Proof: We define El := {x g W1; max Svhu{x)
> 0} = {x g R"; max (Svhu)+(x)
i
> 0}
i
where as usual / + denotes the positive part / V 0 of / . For x g E^ and v = 1,.. .,k we have Suhu - Shu = ThS^_i)hu
<
{ThS(v_l)hu)+
< max {ThSvh)+ = max (Sru+i)hu l
Shu)+,
l
hence for all v = 1 , . . . , k we get Svhu < max (S/v+i)hu-
Shu)+ + Shu,
l
or max Svhu < max (Sfu+1)hu-Shu)+ l
+ Shu,
l
implying Shu(x)+
max (S(u+i)hu-Shu)+(x) l
> max ( 5 ^ u ) + ( x ) . l
3.5 Global Properties of Lp-Sub-Markovian Semigroups
351
Since Tt is positivity preserving we get further max (Srv+nu
Shu)+(x)
-
Kv
tax ( / v
*h
T3u(x)ds—
/ JO
+
Tau(x)ds) '
{Svhu)+)(x),
l
where we used calculus results for semigroups. Hence /
Shu(x)dx>
( max (S„hu)+(x)-Th(
/
JE*
max.
JE£ M
> || max (Svhu)+\\Li l
(Suhu)+)(x))dx
l
- \\Th( max (Svhu)+)\\Li
'
> 0,
l
1
where we used that Th is an L -contraction. Now we let k tend to infinity and arrive at (3.358). D Lemma 3.5.22. Let (Tt)t>o be an Lp-sub-Markovian semigroup with dual semigroup being sub-Markovian too. The semigroup (Tt)t>o *5 transient if and only if there exists g € L 1 (R n ; R) such that g > 0 a.e. and Gg(x) < oo. a.e. Proof: Let g 6 L ^ R ^ R ) such that Gg(x) < oo a.e. and g > 0 a.e. If u € L1(Wl; R), u > 0 a.e., and a > 0, /i > 0 we consider the sets A := {a; 6 R"; supS feh (u-ag)(a:) > 0} fceN
and 5 = {z e R n ; Gu(z) = oo}.
With a set C of measure zero we clearly have B C AUC. By Proposition 3.5.21 it follows that / Sh(u — ag)(x) dx > 0, ./A
and therefore h
udx>
Sh,udx > a I Sh,gdx > a /
S^gdx.
Thus for all N 6 N and any compact Borel set Kv C W1 we find - / a
JH"
udx>-
l h JBnKu
Sh(gAN)dx.
Chapter 3
352
Potential Theory of Semigroups and Generators
Now \Sh{g A N) converge in Z, p (R n ;R) (or any L«(R n ;R), 1 < q < oo) to g A N as h [ 0 and this implies - / a
udx> I
JR"
(gAN)dx.
JBr\K„
Finally, if Kv \ R™, N | oo and then a f oo, we arrive at / g(x) dx = 0 JB
implying \^{B) = 0, i.e. Gu(x) < oo a.e. The converse is trivial.
•
E x a m p l e 3.5.23. Let (^t)t>o be a convolution semigroup on R™ with a corresponding continuous negative definite function tp : R™ —> R. Hence there corresponds a symmetric L 2 -sub-Markovian semigroup to (fJ,t)t>o given by Ttu = fxt * u. Further let us assume that ip(£) > 0 for £ ^ 0 and that ip(€) > Co in 5f(0). We claim that in this situation (Tt)t>o is transient if JBlm\ •JJTTT d£ < oo. In this case we find for the resolvent of (Tt)t>o
By Lemma 1.3.1.4 we know that g(x) = e~lxl / 2 is a fixed point of the Fourier transform and we need to prove lim(2
/s1(o)v^y^<00f eix< JUn
A + V(0
eix'«—-^—e-l«l2/2^
e-l«l 2 / 2 d£= I
eix-t
7K"\SI(O)
/
-
e~^/2di
A + V(0 22 / „IIM
e -f__L_ e -l«l 1
JB,
we may examine each term separately. Now for A —> 0
/2^
3.5 Global Properties of Lp-Sub-Markovian Semigroups
353
and
|xiWfle^ r - , « , ' a
< I e -iei 2 /2 j CO
and further YR <^(£)eix-Z XB l( o)U;e A+ ^
e -l^l e
2
/2 _>
,m(^e«-«_Le-l«|2/2 XBx(o)^;e ^ e
almost everywhere on JBj(O) as A —* 0 and in addition
^wK) e t e - € rrW e " l € | , / a l- X f t ( 0 ) ( 0 ^)Thus Lebesgue's theorem of dominated convergence implies the result. Note that a much deeper result is valid: A convolution semigroup (IM)t>o with associated continuous negative definite function ip : W1 —» C give rise to a transient sub-Markovian semigroup if and only if Re 4; e L}0C(M.n), compare [27] or [244]. Let (Tt)t>o be as in Lemma 3.5.22 and suppose that {Tt)t>o is transient. It follows that {Gg(x) — oo}, g € L1(M™;R) and g(x) > 0 a.e., has measure zero. Hence it is an invariant set with respect to (T t ) t > 0 , and the same holds true for {Gg(x) < oo} = {Gg(x) = co} c since \<-n^{Gg(x) = oo} = 0. Our aim is to determine further invariant sets and for this we need some preparation. Let (Tt)t>o be an L p -sub-Markovian semigroup such that its dual semigroup (T£)t>o is sub-Markovian too. As before we may associate with (Tt)t>o its potential operator G, but of course we may also associate with (T*)t>o the corresponding potential operator which we will denote by G, i.e.
rN Gfl(s):=lim
/
T?g(x)dt,
jeL^R";!),
5
> 0 a.e.
(3.359)
JV—>oo Jo
Clearly G is not the adjoint or conjugate operator to G. However for u,v £ L ^ R ^ R ) n L°°(R n ;R), u,v > 0 a.e. we have J (Gu)(x)v(x) dx = lim / R"
N—>oo JO
— lim
/
iV-»oo JO
/ (Ttu)(x)v(x)
/ u(x)TfV(x)dxdt= JK"
dxdt
Jun
/ JK«
u(x)(Gv)(x)dx,
Chapter 3 Potential Theory of Semigroups and Generators
354
i.e. /
(Gu)(x)v(x)dx=
f
u{x)Gv{x)dx.
(3.360)
Moreover, if (Tt)t>o is transient, then (Tt*)t>o is transient too. For this note that Gu(x) < +oo a.e. for all u G L1(Rn;M), u > 0 a.e., implies by Lemma 3.5.22 the existence of g G . ^ ( R ^ R ) , g > 0 a.e., such that ||<2g||L°° < M. (Note added in the proofs: We follow in these considerations the lecture notes [226] of Y.Oshima, where the existence of g is taken for granted. R.Schilling quite recently pointed out that this might not be the case and provided a proof in a forthcoming paper dealing with global properties of Z/p-sup-Markovian semigroups.) Therefore we can find h G L 1 (R n ;R), h > 0 a.e., such that / (Gg)(x)h(x) dx = /
g{x)Gh{x)dx < oo
implying Gh(x) < oo a.e. and again Lemma 3.5.22 yields the transience of G. Lemma 3.5.24. Let (Tt)t>o be an Lp-sub-Markovian semigroup such that (T*)t>o is sub-Markovian too. For any g G L 1 (R n ;R), g > 0 a.e., the sets {x E Rn;Gg(x) = oo}, {x e Rn;Gg(x) < oo}, {x e M.n;Gg(x) > 0} and {x € Mn; Gg(x) = 0} are invariant with respect to (Tt)t>oProof: For g G L ^ R " ; R), g > 0 a.e., and Bk := {x G R"; Gg(x) < k} we find for any u G L x (R n ; R), u > 0 a.e., / JRn
Tt(xBku)Ggdx=
/ JRn
XBkuT*Ggdx<
I
XBkuGgdx
v/R"
udx JM"
(3.361) where we used that for g > 0 />00
fOO
Tt*Gg = Tt*
Jo
T*sgds=
Jt
/.OO
Tr*gdr<
Jo
Tr*gdr = Gg.
From (3.361) it follows that JM„Tt(xBku)Ggdx is finite, which implies that n TtXBku = 0 on {x G R ;Gg(x) = oo}, or X{Ggoinvariant. By the same type of argument we see that {x G R"; Gg(x) < oo} is (Tt*)t>o-invariant which yields that its complement is (Tt) t > 0 -invariant, i.e.
3,5 Global Properties of Lp-Sub-Markovian Semigroups
355
{x G R"; Gg(x) = 00} is (T t ) t >o-invariant. Next observe that T
/ JRn
t(x{Gg=o}u)Ggdx
= / XIGg=o\uT?Ggdx JRn
< I XiGg=o\uGgdx
= °-
JR"
Thus, TtX{Gg=0}u = 0 on {Gg > 0} or TtX{Gg=0}u = X{Gg=0}Ttx{Gg=0}u which is the invariance of {x G R"; Gg(x) = 0} with respect to {Tt)t>o- As before we conclude that {x G R n ; Gg(x) = 0} is invariant with respect to (Tt*)t>o- Thus its complement {x G R n ; Gg(x) > 0} is (Tt)«>0. D Corollary 3.5.25. A. If (Tt)t>o is as in Lemma 3.5.24 but in addition symmetric, then for g G L^R";!*), g > 0 a.e., the sets {x G Rn;Gg(x) < 00}, {x 6 R"; Gg{x) = 00}, {x G R"; Gg{x) > 0} and {x G R n ; G5(a;) = 0} are invariant with respect to (Tt)t>oB. If (Tt)t>o is as in Lemma 3.5.24 but in addition transient, then for every g G L ^ R ^ R ) , g > 0 a.e., the sets {x G M.n;Gg(x) = 00} and {x G M"; Gg(x) < 00} are invariant. In addition up to a set of measure zero for all g G L 1 (R";R), g > 0 a.e., these sets are independent of g. Proof: A. In this case we have obviously G = G and the assertion follows from Lemma 3.5.24. B. There is some h > 0 a.e. in L 1 (R";R) such that Gh(x) < 00 a.e., i.e. A (n) ({x G R"; Gh{x) < oo}c) = 0 and from Lemma 3.5.22 it follows that A({ar € Rn;Gg(x) < oo}c) = 0 for all g G L ^ R ^ R ) , g > 0 a.e. Hence for all g G L ^ R ^ R ) , g > 0 a.e., the set {x G R ra ;Gg(z) < 00} is invariant according to the remark following Definition 3.5.13. The remaining statements are trivial. • Let (Tt)t>o be an L p -sub-Markovian semigroup with dual semigroup being sub-Markovian too, and let A C R" be a (T t ) t >o-invariant set. Since LP(A) = XA • £ p ( R n ) and for u G LP(A) we may consider XAU a s an extension of u to R n , we find for t > 0 that Tt\LP^A)u = TtXAU = XA%XAU G LP(A), i.e. TtXA-" £ ^(-A)- Therefore the family (Tt\Lp(A))t>o form an L p -sub-Markovian semigroup on V{A). Further, for u G L p (R n ;R) and v G £ p ' ( R n ; R ) we have / (Tt\Lp(.A)u)XAvdx= JA
/ JRn = /
(xATtXAu)XAvdv (xAu)T*XAvdx=
/
XA«r t *| L ,' (il) («)dar
Chapter 3 Potential Theory of Semigroups and Generators
356
i.e. (Tt]Lp(A))* = T*\LP'{A)- Moreover, for h G L1{A;R), ft. > 0, it follows that GxAh = f£°TtXAhdt = f£°Tt\Li(A)hdt = G|LI(.A)/I, i.e. G restricted to the non-negative functions of L1(A;B) is the potential operator associated with (Tt\Lr(A))t>0Let g G L ^ R ^ R ) , g > 0 a.e., and set A := {x G Rn;Gg < oo}. Prom Lemma 3.5.22 it follows that (Tt\Lp(A))t>o is transient implying that {x G Wl;Gg(x) < oo} is invariant with respect to (Tt\LP(A))t>o- Moreover, if 31,32 are two strictly positive functions in L1(Bn;R) and Aj := {x G Rn;Ggj < 00} then A\ U Ao is transient implying Ggj < 00 on A1UA2 which yields that A\ = A<2. Thus we have proved a Hopf-type decomposition Lemma 3.5.26. Let (T t ) t >o be an Lp-sub-Markovian semigroup such that its dual semigroup is sub-Markovian too. Then for every g € L 1 (R n ;R), g > 0 a.e., the set {1 £ 1 " ; Gg(x) < 00} is invariant and up to a set of measure zero independent of g. For an L p -sub-Markovian semigroup (Tt)t>o with (Tt*)t>o being subMarkovian too and for g S ^ ( R " ; ® ) , g > 0 a.e., we decompose R"=XdUXc where Xd := {x G R n ; Gg{x) < 00} and Xc := {x G R n ; Gg(x) = 00} and call Xd the dissipative part and X c the conservative part of R n with respect to (Ti)t>o- Up to a set of measure zero these two sets are uniquely determined. Lemma 3.5.27. Let f G L ^ R ^ R ) , / > 0 a.e., and set B := {x G Xc; Gf(x) < 00}. Then f = 0 and Gf = 0 a.e. on 5 . Proof: Let 0} where Skh is as in Definition 3.5.17 and h > 0, ken a > 0 are arbitrary. The Hopf maximal ergodic inequality, Proposition 3.5.21, yields n
- / Shfdx< a- / gdx h JB •/R" implying first fB Shf dx — 0 and therefore j B Gf dx — 0, hence Gf = 0 a.e. on B. An argument similar to that in the proof of Lemma 3.5.22 yields also
/ = 0onB. •
3.5 Global Properties of Lp-Sub-Markovian Semigroups
357
Proposition 3.5.28. Let (Tt)t>o be an Lp-sub-Markovian semigroup with dual semigroup being sub-Markovian too. A. The semigroup is transient if and only if Xd = R™ up to a set of measure zero. B. The semigroup (Tt)t>o is recurrent if and only if Xc = R™ up to a set of measure zero. C. If (Tt)t>o is irreducible then we have the dichotomy: (Tt)t>o is either transient or recurrent. In the second case Gf(x) = oo a.e. for any non-negative measurable function f with \(n)(x G R n ; f(x) > 0}) > 0. Proof: Part A is trivial by Lemma 3.5.22. Suppose that (Tt)t>o is recurent. As in the proof of Lemma 3.5.22 it follows that / G L ^ R ^ R ) , / > 0, must vanish a.e. on {x G R n ; Gf(x) = 0}, thus \(n\{x G R n ; Gf(x) = 0}) = 0 by the recurrence of (Tt)t>o implying R n = Xc up to a set of measure zero. Conversely, suppose that R n = Xc up to a set of measure zero. Then Lemma 3.5.27 implies the recurrence of (Tt)t>o- Finally, if (Tt)t>o is irreducible then Xc = R n or Xc = 0 up to sets of measure zero, i.e. (Tt)t>o is either recurrent or transient. The last statement in part C follows as the first implication of part
B. • Note that it is possible to prove that if (Tt)t>o is given by Ttu = \it * u with a convolution semigroup (/J,t)t>o then it is irreducible. Since we will not need this result later on we do not supply its proof, for this we refer to K.-I.Sato [244] for a proof. We want to relate transience and recurrence to conservativity. As in the case of Feller semigroups, compare Section 1.4.8, we may extend each operator Tt of an L p -sub-Markovian semigroup by taking monotone limits to the space L°°(R n ;R) and this extension is independent of the approximating sequence. Indeed, if {uk)ken, uk G L p (R n ; R), uk > 0, increases to u G L°°(R"; R), u > 0, we first may define Ttu := lim Ttuk and for two sequences (u^')keN, ujjf > 0, each in L p (R n ;R) and increasing to u we have lim Ttu(k1} = lim lim Ttiu^
/\uf])
= lim lim Tt(u£] Au{ 2) ) = lim Ttu^ I—>oo k—*oo
a.e.
l-+oo
As usual, for general u G Z.°°(Rn;]R) we define Ttu — Ttu+ — Ttu~. We call an L p -sub-Markovian semigroup (T t ) t >o conservative if Ttl = 1 a.e. for all t > 0. (Whereas in Section 1.4.8 it was important to distinguish the
Chapter 3 Potential Theory of Semigroups and Generators
358
extended operators from the original ones, we do not introduce now a separate notation for the extension of Tt to L°°(R n ; R).) Lemma 3.5.29. Let (Tt)t>o be an Lp-sub-Markovian semigroup with dual semigroup being sub-Markovian. Then we have Ttl = 1 a.e. on X*, where X* and X% are defined with respect to (Tt*)t>o- Further «/(T(*)t>o is recurrent then (Tt)t>o is conservative. However, ifTtl < 1 a.e., then (Tt*)t>o is transient. Proof: We always have Ttl < 1 a.e. due to the sub-Markovian property. Thus for / £ X^R"; 1 ) n X°°(R n ; R), / > 0 a.e. it follows /
(S*Nf)(l-Ttl)dx=
f ( f Jun v Jo
JR"
=f f f
'
T:fds)(l-Ttl)dx
T.(l-Ttl)dsdx
JO
JR"
= I f( [ T.lds- [ JR"- ^ Jo
<
JR"
f JO
Tslds)dx
JN
Tsldsdx
'
I
fdx.
JR"
In the limit N —* oo we arrive at
/ (Gf)(l-Ttl)dx
f
fdx
JR"
Since Gf{x) — oo a.e. on X* we find Ttl = 1 a.e. on X*, so on R n a.e. if (Tt*)t>o is recurrent which shows conservativeness. If Ttl < 1 a.e. for alH > 0 then \(n\X*) = 0, hence Xj, — R n up to a set of measure zero implying that (Tt*)t>o is transient. D Example 3.5.30. Let (T t ) t >o be a symmetric L p -sub-Markovian semigroup. Define the new L p -sub-Markovian semigroup T/ 'U := e~xtTty, A > 0. Since T t l < 1 a.e. we find clearly for all t > 0 that T t (A) l = e~XtTtl < e~xt. Hence if (.A, D(A)) generates a symmetric L p -sub-Markovian semigroup, then (A — A, D(A)), A > 0, generates a transient semigroup. Our next aim is to have comparison criteria for recurrent or transient semigroups. It turns out that for this purpose we have to introduce the notation of an extended Dirichlet space. First let us state an auxiliary result.
3.5 Global Properties of Lp-Sub-Markovian Semigroups
359
Lemma 3.5.31. A. Let (£,D(£)) be a non-symmetric Dirichlet form in the sense of Definition 1.4.7.21.B. Then (£sym,D{£)) is a symmetric Dirichlet form. B. Let (£,D(£)) be a symmetric Dirichlet form. Then every normal contraction operates on (£,D(£)), i.e. if T : R —> R such that T(0) = 0 and \T(s) - T(t)\ < \s - t\ then Tone D(£) for every u £ D{£) and £{Tou,Tou) <£(u,u). Proof: A. The only non-trivial assertion to prove is the contraction property, i.e. £{u+ A 1, u+ A 1) < £(u, u), which, however, follows from adding the two inequalities in (1.4.354). B. We prove the following statement: If m € N and T : R m —> R satisfying m
\T(x)\<^2\xk\
m
and \T(x)-T(y)\
< ] T \xk-yk\
k=i
for x,y G W1, then for all u\,...,um D(£) and
fc=i
€ D{£) it follows that T{u\,...,um)
€
m
£{T(UU ..., Um),T(Ul, ..., Um))Xl2 < Y, S(uk, Ukf'2. k=l
Clearly this statement implies that every normal contraction operates on (£, D(£)) from which it follows that (£,D(£)) satisfies the contraction property. Thus it remains to show that the contraction property of {£, D(£)) implies the above statement. We denote the resolvent corresponding to (£,D(£)) by (•RA)A>O and by Lemma 1.4.7.18.B and D it is sufficient to prove that ((id -Afl A )T(wi,..., um), T(m,..., m
i/ 2 < ^((id-Afi A )w f c ,ufc)J'
um))l/2 (3-362)
jfe=i
for all ttfe S L 2 (R"; R) which may be reduced further to be proved only for all Wfc = D j i i UkjXAkj with Nk € N, akj <E R and for each k, Akjl n Akj2 = 0 for ji =^ J2 where \^(Akj) < 00. In fact, changing the sets Akj, 1 < k < m, 1 < .7 < -Nfe, we may even assume N
Uk = 5 3 0!kjXAj j= l
Chapter 3 Potential Theory of Semigroups and Generators
360
with akj e M, Aj C\Ai = 0 for j ^ i, and A( n) (A,) < 00. We set bij :=
((id-XRx)xAi,XAj)o
and (3.362) becomes N
• 1/2
f 2 J ^(aiii • • •
J<Xmi)T(atij,...,a m j )bij)
i• ,j j = 1i
'~
(3.363)
m •»
"N
fc=l
i,j = l
V
xl/2
^
With fa = / R „ XA„ «te and a^ := (Ai^A^> XA, )o we find that % = /?,% -a f j -, 1 < hj < N, and therefore by the symmetry of R\ we have for all x\,..., XN £ R N
N
i,j=l
i<j
j=l
where rrij = fc — £ } i = 1 ay. Substituting these terms into (3.363) we arrive at /
y
atj(T , (ait, • • •, otmi) - T(aij,...,
amj))2
i<3 N
(3.364) j=l
i=l
m
k=l
i<j
m
N
j=l
i=l
and taking into account the basic property of T, it follows that (3.363) holds if a^ > 0 and rrij > 0. Since \R\ is sub-Markovian we have indeed a^ > 0. Further, for A = \J?=1 At it follows that N
f
J2 a^ = A /
< /
f XARXXAJ
dx = X
(RXXA)XAJ
XAjdx = Pj,
implying rrij > 0 and the lemma is proved.
D
dx
3.5 Global Properties of Lp-Sub-Markovian Semigroups
361
Remark 3.5.32. We borrowed the proof of part B from the monograph [204] of Z.-M.Ma and M.Rockner, see also the monograph [35] of N.Bouleau and F.Hirsch. Let (Tt)t>o be an L p -sub-Markovian semigroup such that its dual semigroup is sub-Markovian too. It follows in particular that, on L p (R n ; M) n L 2 (R"; R) it coincides with an L 2 -sub-Markovian semigroup which we denote again by (Tt)t>o and whose dual semigroup is also sub-Markovian. Considering Lp-subMarkovian semigroups whose dual semigroup are again sub-Markovian seems to be the correct framework if one wants to work with non-symmetric Dirichlet forms, compare Definition 1.4.7.21 and Theorem 1.4.7.23. In particular for these Dirichlet forms the sector condition, see Definition 1.4.7.12, does hold. For shortness we will call such a sub-Markovian semigroup a Dirichlet semigroup. We know that every Dirichlet semigroup is in fact a family of ZAsub-Markovian semigroups (T^p')t>o, 1 < p < oo, each having a dual semigroup being subMarkovian too. Furthermore, these sub-Markovian semigroups are analytic. Theorem 3.5.33. Assume that (Tt)t>o is a Dirichlet semigroup and denote by (£,D(£)) the associated non-symmetric Dirichlet form. If (Tt)t>o is transient then there exists a function g £ L 1 (R n ;M) which is a.e. strictly positive and bounded, and moreover it satisfies f \u\gdx < {£{u,u)f2
= (£sym{u,u)Y'2
(3.365)
for all u e D(£). Proof: Let / € L ^ R ^ R ) n C ^ R ^ R ) be strictly positive and {Kk)ke® a sequence of compact sets Kk C R™ increasing to R". We define Ak := {x G Kk', Gf(x) < k} and without loss of generality we may assume that \(n\Ai) > 0 and we set ck := inf f(x) > 0. Since (Tt)t>o is transient it x€Ak
follows that A(™)((|Jfe€Nylfc)c) = 0. Clearly, we have f GXAk dx<— JIA, A,
f
Gfdx<
C
^k JAi JAi
The function
^)-E2HAS(^)^(X)
~X^n\Ai) c
k
< oo.
Chapter 3 Potential Theory of Semigroups and Generators
362
Cl
is clearly strictly positive a.e. and moreover g(x) <
a.e. In addition
(tl)
we have
and by the transience of (Tt)t>o it follows that Gg < oo a.e. Moreover .
0
0
jjGgdx
0
0
»
= £ £ 00
m ( : < ) ( A 0 2kJ*){Ak)
jAi
GxMtodx
00
°l
< V^ V^
~ £lti
<
Cl
2'2*AW(^fc) - A(»)(^i)
°°-
Now we define g
91 ••= —
9
(3-366)
—-TH,1/2
(SRn§Ggdx)
and find /
giGgidx=-T—.
/
gGgdx = l.
Therefore it follows using first the very properties of the resolvent compare Theorem 1.4.7.10, and then the sector condition
(\u\,gi)o = £\(R\gi, \u\) = £(R\gi, \u\) + \(R\gi, \u\)0 < co(£sym(Rxgi,Rxgi))1/2(£3ym(\u\, |«|))1/2 + (\Rm, Now, as A —> 0 it follows that R\g\ | Ggi < 00 implying that (\R\gi, as A —> 0. Further we have Ssym{Rxgu
Rx9l)
= £(Rx9l,
Rxgi)
< (R\gi,gi)o
= (Rxgi,gi)o
- X(R\gi,
|«|)0. \u\)o —> 0
Rxgi)o
< (Ggi,gi)o < 1,
which yields using the fact that normal contractions operate on Saym (M,0i) < c«)(C?ffi,fli)5/2(f " ^ ( | « | , M ) ) 1 / 2 < <*(£'«">, u ) ) 1 ^ . Thus the function g := ^-gi will satisfy
(|«|, g)0 = (|u|, i f l l ) 0 < - ( H , 51) < (*"">, ")) 1/2 Co
•
Co
It is important to note that (3.365) already implies the transcience of (Tt)t>o, i.e. we have
3.5 Global Properties of Lp-Sub-Markovian Semigroups
363
Corollary 3.5.34. An Lp-Dirichlet semigroup (Tt)t>o is transient if and only if for the associated Dirichlet form (3.365) holds for some g € L 1 (R n ; R), g > 0 a.e., and all u £ D{£). Proof: It remains to show that (3.365) implies the transience of (Ti)t>oSuppose that g G V-(Rn; R) n I ° ° ( R " ; R), g > 0 a.e., is such that (3.365) holds for all u E D(£). Since u := Rxg € D(£), u > 0, we derive from (3.365) /
(Rxg)g dx < (S(Rxg, Rxg))1'2 <(£x(RXg,RXg))l/2
= (£x(Rxg, Rxg) - \(Rx9, =
Rxg))l/2
(Rxg,g)10/2,
or
o<(Rxg,g)o<(Rxg,g)l/2 implying (Rxg, g)o < 1, and as A —> 0 it follows by monotone convergence that
f (Gg)gdx 0 a.e. we arrive at Gg < oo a.e., i.e. (Tt)t>o is transient.
•
Remark 3.5.35. Note that in (3.365) only the symmetric part of £ enters, i.e. if (Tt)t>o and (St)t>0 are two Dirichlet semigroups and if the symmetric part of the corresponding Dirichlet forms coincide then (3.365) holds for both semigroups with one and the same function g provided one of them is transient. In particular in this situation (Tt)t>o is transient if and only if (5 t )t>o is transient. Definition 3.5.36. A. A (non-symmetric) Dirichlet form is called transient if and only if the corresponding (L 2 -)sub-Markovian semigroup is transient. B. Let (Tt)t>o be a Dirichlet semigroup and denote by {£, D(£)) the associated non-symmetric Dirichlet form. By (Ttsym)«>o we denote the symmetric jL2-sub-Markovian semigroup associated with the symmetric Dirichlet form (£"vm,D(£)). According to Remark 3.5.35 the transience of a Dirichlet semigroup depends only on (T*ym)t>o- Therefore we will now have a closer look at symmetric L 2 -sub-Markovian semigroups and their corresponding Dirichlet forms. More precisely we will have to discuss extended Dirichlet spaces. They arise from the observation that the symmetric part (f8^"1, D{£)) of a non-symmetric Dirichlet form in general will not be a scalar product on D(£).
Chapter 3 Potential Theory of Semigroups and Generators
364
Definition 3.5.37. Let (£, D{£)) be a symmetric Dirichlet space , D(£) C L 2 (R";R). The extended Dirichlet space Te associated with {£, D{£)) is the family of all measurable functions u : R n —> R, \u\ < oo a.e., such that there is a sequence (uk)k&N, Uk € D(£), which converges a.e. to u and forms a Cauchy sequence with respect to £, i.e. £(uk — U(,Wfc — uj) —» 0 as k, I —> oo. The sequence (ufc)fcgN is called an approximating sequence for u € ^"eBefore we can study ^ e we need some additional results on symmetric Dirichlet forms. For these results we will follow closely the monograph [102] of M.Fukushima, Y.Oshima and M.Takeda. First we look for a different representation of the form £^{u, v) = X(u — \R\u, v)o approximating £. Lemma 3.5.38. A. If S is a symmetric sub-Markovian operator on L 2 (R n ; R) then there exists a unique positive symmetric Radon measure a on R n x R™ such that for all u, v £ L 2 (R n ; R) (u,Sv)o=
u(x)v(x)o(dx,dy)
(3.367)
holds. In addition we have cr(R" x A) < A (n) (A)
for all Borel sets A.
(3.368)
B. For all u € L 2 (R ra ; R) such that u = v a.e. and v is a Borel function it holds f W(«, u) = ^A / /
(v(x) - v{y)fax{dx,
JJmnxUn
+ A/
dy) (3.369)
v2(x){l-sx(x))dx,
where ax is the measure on R™ x R™ associated with the operator \R\ accordingly to part A and s\ is the Radon-Nikodym density ofo\(M.n, dy) with respect to XM(dy) satisfying 0<S\(x) j £ C0(M.n;R) and set N
*(/):= X^StyJo.
(3.370)
3.5 Global Properties of Lp-Sub-Markovian Semigroups
365
Let us consider I on the space C:={/eC0(RnxRn;R); N
f(x,y)
= ^2uj(x)vj(y),
N e N
and Uj,Vj
£ C 0 (R n )}.
3=1
Obviously / i s a linear form on C and / ( / ) is independent of the representation of / . Therefore if we can prove that / is a positive functional on C, we may extend it as a positive linear functional on Co(lRn x M";R) and the result follows from a variant of F.Riesz' representation theorem, compare Theorem 1.2.3.4. So suppose that f(x,y) > 0 and put K := Uj=i supp'iij. On K each Uj is uniformly continuous and therefore there exist a finite decomposition K = U fe=1 Ek of K into Borel sets and points £* £ Ek such that with iij(x) := J2k=i uj(£k)XEk (x) w e have sup \u(x) — uj(x)\ < e. It follows that N
N
I J ( / ) - ] [ > , • , S«,-)o| ^ «lZ(Xif, \SVJDO. In addition, with f(k(y) N
Y,(UJ,
(3.371)
:= f(£k,y) we have M
Svj)0 = J2(XEk, Sfa)0 > 0,
j=l
fe=l
which combined with (3.371) yields / ( / ) > 0 and part A is proved. B. Since XRx is sub-Markovian there exists a symmetric positive Radon measure o\ such that \{u,Rxv)o
= \ if
u(x)v{y)ax(dx,dy).
(3.372)
Now, for A £ #(") we find ax (Mn, A) = ff
Xm™
(x)XA (y)
hence ax(R.n, A) = (XA, Sl)0 < Xin)(A) by the sub-Markov property of S, i.e. ax(Rn,dy) is X^^dy) continuous implying by the Radon-Nikodym theorem that ax(M.n, dy) = s\(y)X^(dy) with a suitable measurable function sx which
366
Chapter 3 Potential Theory of Semigroups and Generators
has to satisfy 0 < sx < 1. Therefore we find for u G L2(Rn; R) £'' '(u, u) = X(u — R\u, u)o =
oA / /
( u ( x ) ~~ u(y)2(T^(dx>
dy) + >* /
u2(x)(l
- s\(x))
dx,
which implies of course (3.369). D Further we have Lemma 3.5.39. Let (Tt)t>o be a symmetric L2-sub-Markovian semigroup with corresponding symmetric Dirichlet form (£, D{£)). For every g G L1(Rn; R) n L 2 (R n ;R), g > 0 a.e., the equality SU
P
iH'u°/2 = ( /
9Ggdx)1/2
< +oo
(3.373)
holds where G is the potential operator associated with (Tt)t>oProof: If
SU
P ~cj—^' *\ 1/2 — °°> then nothing is to prove by (3.365). As before we denote by St the operator Stu = / 0 Tsuds. According to Lemma 1.4.1.14.A for all u G L 2 (R";R) we have Stu G D{A) and since D(A) C D{{-A)1'2) = D{£) it follows that Stu G D(£) for all u G L 2 (R";M). In addition, for u E D(A) and v G D(£) it follows that lim -{u — Ttu,v)o = (—Au,v)o = t-»o *
£(u,v).
For g G L ^ R ^ R ) n L 2 (R n ;R), g > 0 a.e., we find
Trgdr+I
Trgdr
which yields for all u G £>(£) 1 / 1 /"*+s lim - ( 5 t f l - TsTtg, u) 0 = lim ( - - / Trgdr+s->0 S S 7t s_0 V
=
(9-Ttg,u)0,
but also
lim -{Stg-TsStg,u)o 5—*0 ^
= S(Stg,u)
1 f Sj0
\ Trgdr) /O
3.5 Global Properties of i7-Sub-Markovian Semigroups
367
implying £(Stg,u)
= {g-Ttg,u)Q.
Now assume that
sup
(3.374)
j j " ' ' ^ 2 = c < oo. Taking u = Stg > 0 and observ-
ueD(£) £(u>u)
'
ing that by (3.374) we have £{Stg, Stg) < (Stg,g)0, (St9,g)o < c£{Stg,Stg)ll2
<
we derive the estimate
(Stg,g)l/2,
or {Stg, g)l/2 < c. Thus
0
'
gGgdx)
= sup{S t g,g) 0 < c = sup
}
^,/2-
Conversely suppose that J*K„ gGgdx < oo. Since {Tsg,g)o = {Ts/2g, Ts/2g)o = ||Ts/25||o a n d ll^s/2fl'|lo i s non-increasing as s —> oo due to the contraction property of (Tt)t>o it follows that {Tsg, g)o —> 0 as s —» oo because J 0 (T s g, g)orfs— / R „ gGgdx < oo. Now (3.374) yields (|u|,fl)o = £(|u|,Stfl) + (|ti|,rts) 0 < f (5tfl, 5 t 5 ) 1 / 2 £ ( u , U ) 1 / 2 + ( U ) K ) J / 2 ( r t 5 , T t 5 ) J / 2 < {Stg,g)10/2£{u,u)^2
+
\\u\\0{T2tg,g)l12,
where we used as before £{Stg, Stg) < {Stg, g)o which follows also from (3.374). Thus for t —+ oo we arrive at {\u\,g)0<(
gGgdx)1'*£{u,ufl2
[
implying SU
P
{\u\,g)o ^( f
proving the lemma.
_ , \i/2 aGgdx)
•
Remark 3.5.40. For a (symmetric) transient Dirichlet semigroup estimate (3.373) sharpens Theorem 3.5.33. Lemma 3.5.41. Let {£,D{£)) be a symmetric Dirichlet form with associated resolvent {R\)\>o ond semigroup {Tt)t>o- Then £<<x\u,u) = \{u \Rxu,u)0 is non-decreasing as X tends to infinity and lim £^x\u, u) = £{u,u). Further A—»oo
y{u — Tu, u)o is non-decreasing as t tends to 0 and lim j{u—TtU, u)o — £{u, u). t-»o
368
Chapter 3 Potential Theory of Semigroups and Generators
Proof: Using the spectral projection (£^) M associated with the generator of (£, D{£)) we find
£^\u,u)
=
Jo
f°°'-^Ld(E„«,u) A + /i
and 1 f°° 1 - e~t>x -(u — Ttu,u)0= / * Jo t proving the lemma. •
d(E^u,u)
Theorem 3.5.42. Let (£, D{£)), D(£) c L 2 (R"; R), be a symmetric Dirichlet form with associated sub-Markovian semigroup (Tt)t>oA. For every u G J-e and approximating sequence {uk)k& for u the limit lim £(iik,Uk) exists and is independent of the choice of the approximating sequence. We will denote this limit once again by £(u,u). B. Ifu£j-e and u = v a.e. for a Borel function v then (v(x)-v(y))2o-x(dx,dy)+\
- ff n
n
A J jR xWL
f
v{xf{l-sx{x))dx
JU"
increases to £(u,u) as X tends to oo. C. It holds D{£) =fen L 2 (R n ; R). Proof: First note that there is no problem to extend the right-hand side of (3.369) by monotone convergence to all Borel functions. Since £1/<2 is a seminorm the triangle inequality yields |£(ujfc,ti fc ) 1/2 -£(u,,uj) 1/2 | <
£(uk-ul,uk-u,)1/2
implying that lim £(uk,Uk) exists. Now let uk and u be some Borel modik—»oo
fication of uk and u respectively satisfying lim Uk{x) = u(x) for all x G R™. k —too
Fatou's lemma yields £^(v.k
-u,uk-u)
< l i m i n f £ w ( u k -ui,v,k
-ui)
l—>oo
< lim £(uk — ui, Uk - ui), l—>oo
and since (uk)k€N is an £-Cauchy sequence it follows that lim fc—+ 00
£(xXuk,Uk)=£(X)(u,u)
3.5 Global Properties of Lp-Sub-Markovian Semigroups and £^(u, that
369
u) is non-decreasing with A by Lemma 3.5.41. Moreover it follows
\£(u,u)V2-£W(u,u)V2\ <\£{u,u)1'2
-£{uk,uk)l'2\
+ \£{uk,uk)ll2
-£W{uk,uk)ll2\
£^\uk-u,uk~u)1'2
+
implying that lim £^x\u,u)
= £(u,u),
which gives first part B and further,
A—*oo
if u = 0 a.e. then £^(u, u) = 0, hence £(u, u) = 0 implying, with the previous remark that lim £(uk,Uk) exists, also part A. Part C is clear from part B, k—>oo
compare also Lemma 1.4.7.18.
•
Remark 3.5.43. Suppose that (Tt)t>o is a general Dirichlet semigroup, i.e. not necessarily symmetric, with associates Dirichlet form (£, D{£)). We may construct Tlym as before by using (Tt3ym)t>0 or £sym. Then the sector condition implies that we may extend £ to Te (= J r * a m the extended Dirichlet space associated with £ s " m ) by £(u,v) := lim £{uk,vk) with approximating k—*oo
sequences (uk)k^
and (vk)k^
for u and v, respectively.
Let (£, D(£)) be a symmetric Dirichlet form with corresponding subMarkovian semigroup (Tt)t>o- Since (T t ) t >o is analytic, compare Stein's theorem, Theorem 1.4.2.12, D(£) is invariant under Tt, i.e. TtD(£) C D(£) = D{{-A)1/2) for alU > 0 and in addition for all u € D{£) £{Ttu - u, Ttu - u) = \\{-A)x'2{Ttu
=
- u)\\2
\\Tt{{-Af'2u)-{-Afl2u\\l
implying \im£(Ttu-u,Ttu-u)=0. t-»o
(3.375)
Moreover, for u G D{£) £(Ttu,Ttu)
= \\{-Ayl2Ttu\\2
= \\Tt{-Afl2u\\2
<
\\^Af'2u\\l
i.e. £(Ttu,Ttu)
<£(u,u).
(3.376)
Chapter 3 Potential Theory of Semigroups and Generators
370
Lemma 3.5.44. Both (3.375) and (3.376) extend to Te. Proof: Take an approximating sequence (uk)keN, Uk G D(£), for u G J-e. Prom (3.375) it follows that (TtUk)keN is a Cauchy sequence with respect to £. Now, for u G L 2 (R n ;R) the contraction property of (Tt)t>o implies (TtuJ, Ttuj)o < (Tt/2W, Tt/2^)o < (w, Ttuj) and therefore it follows that -\\(uk-ui)
-Tt(uk
= T((U* ~ u 0
-W/)||Q _ T u
t( k ~ u0> uk - ui)o
- r((uk - u{) - Tt(uk - ui),Tt(uk
- ui))0
< -((uk -ui)-Tt{uk-ui),Uk-ui)o
<£{uk
-ui,uk-ui),
where in the last step we used the second part of Lemma 3.5.41. It follows that (ufe — TtUk)keN is a Cauchy sequence in i 2 ( R n ; R ) . Thus a subsequence of this sequence converges a.e. and since (uk)k&i converges a.e. to u it follows that a subsequence {TtUk,)ieN converges a.e. to some function v G L 2 (R n ;M) and this limit is independent from the choice of the approximating sequence. If we define Ttu := v then Ttu G J-e, (TtUki)ieN is an approximating sequence for Ttu and u - Tu G Fe D L 2 (M n ; R) = D{£). Since £{Ttu - u, Ttu - u)1'2 < £(Ttuk - uk, Ttuk + £(Tt(u - uk), Tt(u - uk))1'2 + £(u-uk,u-
uk)1/2 uk)1/2
we find if first t —> 0 and then k —• oo that (3.375) holds and (3.376) follows by an analogous argument. • Now we may start to relate T& to the transience of (Tt)t>oProposition 3.5.45. If the symmetric Dirichlet form (£, D{£)) is transient then its extended Dirichlet space J-e is complete with respect to the scalar product £. Proof: For u G J-e and an approximating sequence (wfe)fcgN we have £{u -Uk,uuk) = lim £{ui - Uk, ui - uk) implying that (ufc)fceN converges with respect to £ to u. Now let (£,D(£)) be transient and (uk)k&N be a Cauchy sequence in Te. From the above remark follows the existence of a sequence (i'fc)fcgNi Vk G D(£) and {vk)k€W is a Cauchy sequence with respect to
3.5 Global Properties of Lp-Sub-Markovian Semigroups £ such that lim £{uk — vk,uk—vk)=0,
371
but now Theorem 3.5.33 yields that
(uk —Vk)keN is a Cauchy sequence in the weighted L 1 -space L1(Wl;g\^). Therefore a subsequence (vkl)i€n converges a.e. to a function u. Thus t i £ f e and furthermore £(uk -u,uk
-u)1/2
+ £(ukl -vkl,ukl
< £(uk -uknuk -vkl)l/2
+£(vkl
-ukl)1/2 -u,vkl
-u)l/2.
If we let first I tend to +oo and then k to +oo it follows that (ufc)fceN converges with respect to £ to u. • Now we are able to characterize the extended Dirichlet space of a transient Dirichlet space. Theorem 3.5.46. Let (£, D{£)) be a symmetric transient Dirichlet form on L 2 (R"; R). Then for its extended Dirichlet space Te with scalar product £ we have i) J-e is a Hilbert space with inner product £; ii) there exists a bounded strictly positive function g G L 1 (E n ; R) such that /
\u(x)\g(x)dx<£(u,u)1/2
holds for all u G Te; Hi) Te n L 2 (M n ;R) is dense in L 2 (R";R) and {Te,£); iv) every normal contraction acts on (Te,£~), i.e. if T is a normal contraction then Tu € Te for all u G Te and £(Tu, Tu) < £(u, u). In addition Te l~l L 2 (R";R) = D(£). Conversely, suppose that for a pair (J-e,£) i)-iv) are fulfilled. Then {Fe,£) is the extended Dirichlet space of the transient symmetric Dirichlet form (£, Te ("1 Zr2(Rn; R)). Proof: If (£, D{£)) is a symmetric transient Dirichlet form and Te its extended Dirichlet space we have already proved i)-iii). To see iv) take u € !Fe and an approximating sequence (uk)ken- Further let T be a normal contraction and define ujk := (TuA\uk\)+ e i 2 ( R n ; R). As usual let us denote a Borel modification of a function v G L 2 (R n ; R) by v. It follows that
< £W(&, u) + fW(u f c , uk) < £(u, u) + £(uk,
uk).
Since w 6 L 2 (R"; R) it follows that uk G D{£) and since £(uk, uk) converges to £(u, u) it follows that £(u)k, u>k) < c uniformly in k. Thus by the Banach-Saks
372
Chapter 3
Potential Theory of Semigroups and Generators
theorem, Theorem 1.2.7.2, the arithmetic mean of some subsequence of (u>k)keN converges with respect to £ to some u £ Te. Since (Tu)+ = lim w^ = u> £ J-e, i.e. (Tu)+ £ !Fe, and analogously (Tu)~ £ Te we find first Tu £ J-e and then the estimate £(Tu,Tu) < £(u,u) follows from the extension of (3.369) to Te, see Theorem 3.5.42. Suppose now that i)-iv) hold. Then it is clear that (£, Te n L 2 (R n ; R)) is a symmetric Dirichlet form on L 2 (R n ; R) and ii) implies the transience of this Dirichlet form. Further it is clear that the extended Dirichlet space associated with (£, J-e D L 2 (R n ;R)) must be J-e. • Definition 3.5.47. A pair (.Fe,^) satisfying i)-iv) of Theorem 3.5.46 is called a transient extended Dirichlet space. We need Lemma 3.5.48. Let (Tt)t>a be a symmetric transient sub-Markovian semigroup with generator A. Then lim Ttu = 0 for all u £ L2(Rn; R). t—+oo
Proof: Suppose that there is some UQ £ L 2 (R n ; R) such that lim Ttuo ^ 0. t—+oo
Using the spectral projections associated with the selfadjoint operator A, the spectral theorem yields /»oo
lim /
e-^dE^uo
^ 0
t-too Jo
which is only possible if Ttu0 = u0 for all t. But for any u £ i 1 ( R n ; R ) n L 2 (R"; R) it follows from the transience of the semigroup that rt+r
lim / t-»oo7t
Tsuds = 0,
r > 0,
and since Ll(Rn; R) n L 2 (R"; R) is dense in L2(Rn; R) such a u0 £ L 2 (R n ; R) can not exists proving the lemma. • Most important is now Theorem 3.5.49. There is a one-to-one correspondence of all transient extended Dirichlet spaces (J-e,£) and all symmetric transient L2-sub-Markovian semigroups (Tt)t>o on L 2 (R n ;R): For any non-negative measurable function f such that JjR„ fGf dx < oo we have Gf £ J-e
and £(Gf, v) = (/, v)o for all v £ Te.
(3.377)
3.5 Global Properties of Lp-Sub-Markovian Semigroups
373
Proof: Let (Tt)t>o be a symmetric transient jL2-sub-Markovian semigroup with corresponding extended Dirichlet space (Fe, £). We have to show that for every non-negative measurable function / such that JRn fGf dx < oo we have Gf £ Te and £(Gf,v) = (f,v)o for all v € !Fe, and that (Te,£) is uniquelydetermined by (Tt)t>0. For / e L2(Rn;R), / > 0 a.e. and / R „ fGf dx < oo it follows from (3.374) that for t > t' > 0 £(Stf
~ St,f, Stf - St.f) = {Tt,f - Ttf, Stf = [
f-(
< [
f [
f
Svf)0
Tsfds-
f
Tsfds)dx
Tafds dx.
Since / K „ / J , Tsfdsdx^>0 as t, t' —» 0 it follows that (Skf)keN is a Cauchy sequence with respect to £ which converges a.e. to Gf, i.e. Gf e Te- Now, (3.374), i.e. £(Stg,u) = (g — Ttg,u)o, and Lemma 3.5.48 yield first for all « 6 l 2 ( i " ; R ) n f e that £(Gf,v)
=
(f,v)0
and now the density of Te n L 2 (E";R) in Te and L 2 (R n ; R) implies (3.377). Next if / is any measurable function such that JK„ fGf dx < oo then we choose g G L ^ R ^ R ) n L 2 (R";R), g > 0 a.e., and consider the sequence fk •= f A (kg) € L2(Rn; R). For fk we may apply (3.377) to find £(Gfk - Gfu Gfk - Gfi) = (fk,Gfk < f
- Gfi)0 + (fi,Gfk - Gf,)0
fkGfkdx-
j
fiGfidx.
Thus forfc,/ —> oo it follows that £(Gfk — Gfi, Gfk — Gfi) —> 0 implying first that Gf 6 Fe and then we may pass to the limit in £(Gfk,v) = (fk,v)0 to extend (3.377) to all measurable / , / > 0 a.e., such that J R „ fGf dx < oo. Finally, since for (Tt)t>o given, the set of all Gf, / > 0 and measurable such that f&n fGf dx < oo, is sufficiently rich to approximate elements in Te (after decomposing into positive and negative part), it follows that (Tt)t>o determines uniquely the extended Dirichlet space. Conversely, let a transient extended Dirichlet space (.Fe,"?) be given. We have to show that (3.377) determines (Tt)t>o uniquely. Since Gf =
Chapter 3
374
lim Ri/Nf
a.e. for / G L1^;
Potential Theory of Semigroups and Generators
R) DL 2 (R n ; R), / > 0 a.e., we may extend the
N—too
resolvent equation with RQ := G by Gf = R0f = Rxf + XGRxf. If in addition JR„ fGf dx < oo then
/" (Rxf)GRxfdx<\
f < i /
RxfGfdx fGRxfdx
< 4
•A 7R"
/
fGfdx<
oo.
•* J R "
It follows from (3.377) that GRxf G Te and €(GR\f, v) = (Rxf, v)0 for all v £ J-e- By the extended resolvent equation we obtain that Rxf G Te and further
£(Rxf,v)=£(Gf,v)-XS(GRxf,v)
=
for all i ) £ f e . Therefore we may deduce Rxf
(f,v)0-X(Rxf,v)0 e f = f e n L 2 ( R n ; R ) and
f A ( B A / , «) - (/,«)o for all « G Z?(£).
This relation holds for all / = h • g, g being a reference function in Theorem 3.5.33 and h any bounded function. By a density argument it follows that (Rx)\>o is uniquely determined by {Fe, £) proving the theorem. • It is now straightforward to extend basic properties of symmetric Dirichlet forms to transient extended Dirichlet spaces. In particular we have u,v G Te
imply u V v, u A v, u A 1 G !Fe\
(3.378)
u G Te, and Ufc := ((—k) V u) A k imply Uk G Te and £{uk — u, Uk — u) —> 0 as k —• oo; u G Fe and us :=u— ((—e) V u) A e for £ > 0 imply u£ G J-e and £(u £ — u, u£ — u) —> 0 as £ —» 0.
(3.380)
Further, if y?: R -> R is such that y?(0) = 0, \ip(t) - 0 for all v G J^-
(3.381)
3.5 Global Properties of ZASub-Markovian Semigroups
375
Theorem 3.5.50. Let (J-e,£) be a transient extended Dirichlet space with corresponding semigroup (Tt)t>o- A necessary and sufficient condition that Te C Ljoe(Rn; R) and that for any compact set K C R n / \u{x)\dx
(3.382)
JK
holds for all u € J-e is
f fGfdx=
I (Ttf,f)0dt
JR"
(3.383)
JO
for all f eCo(M.n;Rn),
/ > 0.
Proof: We only have to combine Lemma 3.5.39 and Theorem 3.5.49.
•
Now we are in a position.to discuss further examples. Theorem 3.5.51. Let (/it)t>o be a symmetric convolution semigroup with corresponding continuous negative definite function ip : Rn —> E and symmetric Dirichlet form {£^, D(E+)) = ( f ^ i T ^ Q R " ; ! ) ) . Further denote the associated semigroup by (Ttf)t>o- The following assertions are equivalent: i) (T^)t>o is transient; ii) for any compact set K C Rn it holds /•OO
K.(K) := / Jo
IH{K) dt < oo;
(3.384)
Hi) for all u G Co(R n ; R), u > 0, we have /»oo
/ (Tfu,u)0dt Jo
iv) the function 4 belongs to
(3.385) Ljoc(Rn;R).
Proof: If ii) holds then /•0O
/
Jo
T?XK{X)
dt = K{K-x)
< oo
for every compact set K c Rn and almost all i £ R " which implies / Jo
/ Jun
(T?u)udxdt
376
Chapter 3 Potential Theory of Semigroups and Generators
for every u € Co(K n ;R), u > 0, i.e. ii) implies iii). Of course iii) implies i) as well as i) implies ii). Now assume that iii) holds. Let if C M" be compact and u e C0(Rn;R), u > 0, such that |u(£)| > 1 for all f € K (clearly such a function u exists). It follows that
II J0
(Ttu){x)u(x)
JRn
dx dt < oo,
= J ^ y , the fact that (T?u)A(0
where we used /0°° e'^^dt
= e - * « ) * u ( 0 and
Plancherel's theorem. Thus iii) implies iv). On the other hand with u as before and v(£) := (2n)~n/2(u * u)(£). We find first that suppu C supptz + suppu C K', where K' c R" is a compact set. Further it follows that K{K)<
lim / T-»oo Jo
= lim / T->oo JO
= (2TT)-"/2
{ / l
|fi(OlV*(de))dt J
JMr-
{( /
v(0Md£))dt J
lim /
{ /
v/E"
«(OA*(dO}d*
where we used in the last step the non-negativity oiv and fit, hence iv) implies ii) and the theorem is proved. • Corollary 3.5.52. Let ip : W1 —• C be a continuous negative definite function satisfying |ImV(OIa Example 1.4-1-5, is transient.
(3-386) associated with i/>, compare
Proof: We know from Example 1.4.7.32 that (Tt )t>o corresponds to a nonsymmetric Dirichlet form if and only if (3.386) holds. Hence (T?)t>o is a Dirichlet semigroup which is transient if and only if (T^ y m ) t >o is transient. But (T?'sym)t>0 = (r t ReV ') t >o and the Corollary follows. •
3.5 Global Properties of Lp-Sub-Markovian Semigroups
377
Remark 3.5.53. Since (3.386) implies that 1
(l+c2)ReV>-
1
ip ~ Reip
it follows that (T?)t>0 is transient if and only if Re X € Ljoc(Rn; R) Example 3.5.54. Consider the Poisson semigroup on K n given by fit = 53^10e_txT£/vfc, h € R™. According to Example 1.3.6.20 it corresponds to the continuous negative definite function V>(£) = (1— e~lh^) = (1— cos/i£)+isin/i£. Since ^
1 1 — cos h£ 1 2 2 V^) ~ (1 - cos h£) + (sin h£) ~ 2
it follows that the operator semigroup associated to (/it)t>o is transient. Since for 0 < £ • h < | we have sin h£ > §/i • £ and 1 - cos /i£ < ^ | ( / i • £) 2 it follows that for 0 < f • /i <
|
ImV»(0 _ sin/i£ ReV>(£) 1-cos/i^
_
47T 1 V2/i-^'
Hence we can not fulfill (3.386) showing that this is not a necessary condition for the transience of (Ttf)t>oExample 3.5.55. Let I/J : R™ —> R be a continuous negative definite function such that 4- G Ljoc(Rn; R). The symmetric Dirichlet space {£, D(£)) associated with ip is given according to Example 1.4.7.28 by
and
£{u,v)= [ M)*(Z)W)d£The corresponding extended Dirichlet space Te is now Te = flJ'/fR"; R) := {u G S'(Rn; R), u 6 L L ( K n ; R) and «V 1 / 2 G i 2 ( R 2 ) } Note that u ^ 1 / 2 £ L 2 (R 2 ) if and only if u belongs to the weighted L 2 -space L2(Rn;i/>A(™)). Clearly the extended form is given once again by
378
Chapter 3 Potential Theory of Semigroups and Generators
In general we do not know whether Hf;} (Rn;R) C L P (R";R) for some p > 1. The following discussion of the space Hf, (R™; R), 0 < s < 1, shows the difficulties we may encounter. Example 3.5.56. Consider the semigroup generated by —(—A)Q, 0 < a < 1, i.e. T2au(x)
=
(2TT)-"/2
e fa *e-*l*l aa «(Od£
/
with corresponding continuous negative definite function "02a : R™ —* R, £ •—• V>2a(£) = |£|«, 0 < a < 1. Since
L.KF*-*^'-* it follows that - 4 - G L/oc(]Rn;]R) if and only if 2a < n, i.e. the semigroup (^t )t>o is transient and if n = 2 only the Brownian semigroup is not transient. Since the Gauss kernel is strictly positive it follows that the Brownian semigroup in R 2 is recurrent. For n = 1 it follows first by the strict positivity of the Gauss kernel that all of the semigroups (Tt )t>o a r e irreducible since they are obtained from the Brownian semigroup by subordination with respect to the one-sided stable convolution semigroup of order a, 0 < a < 1, compare Example 3.5.15 , thus for 0 < a < ^ these semigroups are transient whereas for ^ < Qf < 1 they are recurrent. Now, suppose 2a < n. We will characterize the extended Dirichlet space associated with (T 2 a ) t >o in terms of Riesz potentials. First note that for 2a < n and u G S(R"; R), u > 0, it follows that /.oo
roo
G2au{x) =
T?au{x)dt Jo
e ^ e ' ^ " u^) d^ dt Jo
= {2-K)~n'2 f JR"
/•
= {2n)-n'2
eix< ^"e-'KI2"
Jun (ft«(£K
JO
or by the convolution theorem G2au =
R2a*u
where R2a(x) = c 2 Q , n | £ | _ n + 2 a is the Riesz kernel of order 2a. In the following we will write Rpu for RP*u. Note that from Theorem I.3.3.6.C it is clear that
3.5 Global Properties of Lp-Sub-Markovian Semigroups
379
R2a(x) must be homogeneous of degree —n+2a. Thus we find for H?eJRn;R), the extended Dirichlet space associated with (T t 2a ) t >o, or equivalently with Ha(Rn;R), tf(ae)(Rn; R) := {u G LJoc(Rn; R);
u = Raf
for some / G L 2 (R n ; 1 ) ,
and of course, the corresponding form is given by E(u,v)= with u = Raf
f
u(OW)\tfad£=
Raf(x)-Rag(x)dx
[
and v = Rag.
Prom our general theory we do only know that H?,(Rn; R) is a subspace of •kJoc(^™' ^ ) - But the homogeneity of the kernel Ra allows us to prove a variant of Sobolev's embedding theorem to find H£e)(Rn; R) C L p (R n ; R), p = ~ ^ . Before proving Sobolev's theorem let us follow a consideration given in E.M.Stein [256]. We want to prove WfU*
<
(3-387)
C||/||L«
for some p and q. For / > > 0 w e find with fp{x) — f(px) that \\fP\\L« =
p-n/q\\f\\L*
and
\\Rafp\\L*=p-a\\(Raf)P\\Lr=p-a->\\Rafhr. Thus (3.387) would imply p-a-?\\Rafhr 0. Thus we shall expect that a+jj = ^ , o r ^ = i — | | . In particular, for q = 2 we should have p =
w
_ ^a.
Theorem 3.5.57 (Sobolev's inequality). Let 1 < q < p < oo, 0 < a < n n n and I = I _ 2 > 0. T/ien /or a// u G 5(R ; R) ('or « € L"{R ; R)) \\Rau\\LP < ZioJds.
CO||«||L«
(3-388)
Chapter 3 Potential Theory of Semigroups and Generators
380
Proof: We use a variant of the Marcinkiewicz interpolation theorem, compare Theorem 1.2.8.9: If T is as in Theorem 1.2.8.9 and 1 < qi < pi < oo, i = 0,1, q0 < gi, po ^ pi, then, if T is of weak type (q%,Pi) i = 0,1, compare Definition 1.2.8.8, then it satisfies | | T U | | L P < ||U||Z,Q for | = 1-=^- + ^ and
?=T
+
^
O < 0 < 1
-
Without loss of generality we may assume u > 0 a.e. We decompose Rau for r > 0 according to Rau(x)=
[
\x-y\a-nu(y)dy+
JB2r.(x) = Ui(x) +U2(x).
dy
JB^r(x)
Since aq < n, Holder's inequality yields with q' =
u2(x) <( [
\x - y\a-nu{y)
[
a
l\
\x- j,|<«-">«' dy) 1/9 •( f
that
Hy)\« dy) ^
0 take r > 0 such that Cir((a-n)«'+n)/«'||ti||i.g=M
which implies that A(n)({a; G Rn;Rau(x)
> 2/x}) < A (n) ({x 6 R n ;«i(a:) > p})
Thus Ra is of weak-type (q, p) and the same calculation yields that it is always of weak-type (l,p). Thus an applications of the Marcinkiewicz interpolation theorem yields the result. • Remark 3.5.58. The proof of Theorem 3.5.57 is essentially that given in E.M.Stein [256] but we used much the presentation in Y.Mizuta [219]. Corollary 3.5.59. For 2a < n we have Hfe)(Rn;R) n
^-> Lp(Rn;R),
p =
™ia, and the estimate ||u|| L , < c ( £ ( u , u ) ) 1 / 3
holds for all u 6
H£e)(Rn;R).
(3.389)
3.5 Global Properties of Lp-Sub-Markovian Semigroups Proof: We just have to apply Sobolev's type inequality to u £ Raf, the equality /(£) = \Z\au(£). •
381 note
Corollary 3.5.60. Let ip : R™ —> R be a continuous negative definite function such that coj^| 2 a < ip(£) holds for some c 0 > 0, 0 < a < 1 and all £ € Rn. If2ao corresponding to ip is transient and the extended Dirichlet space Hf^}(Rn;R) is continuously embedded into Lp(M.n; R),
p-n-^zaProof: First we observe that / JBR(0)
~T7E\ d£^~ nO
T7T2^ d £ < °°> CQ JBR{0)
\£\2<*
irther we have
II«II£P < c /
\z\2a\m\2di < c [ v(oi«(oi 2 de,
i.e.
\\u\\LP
1S t n e
scalar product in the extended
The idea behind the theory of pseudo-differential operators and its applications to stochastic processes is to reduce properties of operators (or stochastic processes) to properties of the corresponding symbol. Often we will use comparison of a given symbol q(x,£) with a fixed continuous negative definite function ip(£) in order to derive results for —q(x,D) (and the corresponding stochastic process) analogously to those for —ip(D) (and the associated Levy process). We want to apply this idea to the study of the transience and recurrence of a semigroup for this we need additional preparations. Recall that Xc := {x 6 Rn;Gg(x) = oo} where G denotes the potential operator associated with (£; D(£)). Lemma 3.5.61. Let (£,D(£)) be a symmetric Dirichlet form on L 2 (R";]R) with corresponding sub-Markovian semigroups (Tt)t>o- Further suppose that g e L1(Rn;R) D L°°(Rn;R) such that g\Xc > 0. Then the semigroup (T/) t > 0 associated with the symmetric Dirichlet form (£3,D(£)) defined by £9{u,v)
:=£(u,v)+
/
u(x)v{x)g(x)dx
(3.390)
382
Chapter 3 Potential Theory of Semigroups and Generators
is transient. Moreover, G9g < 1 a.e. on R n and, on Xc we have G9g = 1 where G9 is the potential operator associated with (£9,D(£)). Proof: We denote by (R9^)\>o the resolvent corresponding to {£9, D(£)) and by (R\)\>o the resolvent corresponding to (£, D(£)). Since £x(R9xf, u) = £9(R{f, u)-f
((R9xf)u)g dx = (f-gR{f,
u)„,
(3.391)
JM.n
we find R9xf = Rx(f-gR9J).
(3.392)
In particular we have for / € L 1 (E n ; R), / > 0 a.e.,
R{f
A>o.
It follows now that every (T t ) t >o-invariant set (which is by symmetry strongly (7t)t>o-invariant) is also (T t 9 ) t >o-invariant. In fact, for A cRn being (7t)t>oinvariant we have R9XXAU < RXXAU
=
XAR\U
implying invariance, i.e. R9XXAU = XAR^XAU, and by symmetry the strong invariance follows. Moreover, for Xd defined as before by Xa := {x G R n ; Gg(x) < oo} it follows that Xd C X9 := {x e R n ; G9f(x) < oo} except for a set of measure zero. Since (R9xg,gR{g)o < £9x{Rxg,Rxg)
= (g,R9xg)o
^[f^Rlgfgdxf'^j^gdxf2 we find that /
(R9xg)2gdx<
f
gdx
and monotone convergence for A J. 0 shows that a.e. on Xc we have G9 < oo. Now for any h € L1 (R n ; R), h > 0, we consider the function / = hxxd +9XXC G L 2 (R"; R) which is strictly positive a.e. It follows that G9f = G9(hXXd+gxxc)
= XxdG9h+XXcG9g
< oo
3.5 Global Properties of Lp-Sub-Markovian Semigroups
383
implying by Lemma 3.5.22 that (T?)t>o is transient. Next we prove G9g < 1 a.e. on M n . Since for e > 0 £(R9(ef+fg),
v) + f (R9(sf + fg))v(g + e) dx
= £9(R9(ef
+ fg),v) + e f
R9(ef +
fg)vdx
JRn
= (ef + fg,v)o=
I (fv)(g + e)dx, JRn
we may consider R9 ((e + g)f) as the resolvent at 1 of the Dirichlet form (£,D(£)) with reference L 2 -space L 2 (M n ; (g + e)\W). By sub-Markovianity we have R9((e + g)f) < 1 a.e. for any / 6 L 2 (R";R) such that 0 < / < 1 a.e. Now we first let e decrease to 0 and then letting / increase pointwise to 1, we obtain G9g < 1 a.e. on Rn. Finally observe that (3.392) implies G(g(l - G9)) = G9g < oo a.e. Since Xc C {x € Mn;5(a;) > 0} we find with Lemma 3.5.27 that g(l — G9g) = 0 a.e. and now g > 0 a.e. on Xc yields that G9g = 1 a.e. o n I c . • Corollary 3.5.62. A. Let g € ^ ( R ^ M ) n I < x > ( R n ; 8 ) be such that g > 0 a.e. on Xc and g = 0 on Xd- Then the function Uk := R^/k £ D(£) satisfies lim Uk{x) — xxc{x) k—»oo
a e
--
an
d
n m
£{uk,Uk) = 0.
(3.393)
k—too
B. We have xxc € J"e and S(xx.,XxJ
= 0.
Now we may characterize transience and recurrence by using the extended Dirichlet space. Theorem 3.5.63. Let (£, D{£)) be a symmetric Dirichlet form with associated semigroup (Tt)t>o- The following conditions are equivalent: i) (Tt)t>o is transient; ii) if u G !Fe and £(u,u) — 0 then u = 0 a.e.; iii) Te is a (real) Hilbert space with inner product £. Proof: We know already that the transience of (Tt)t>o implies iii), compare Theorem 3.5.49, and from iii) it follows obviously that ii) holds. But ii) implies by Corollary 3.5.62.B that xxc = 0 a.e. i.e. the transience of (Tt)t>o- D
Chapter 3 Potential Theory of Semigroups and Generators
384
Theorem 3.5.64. Let (£, D(S)) be a symmetric Dirichlet form with associated semigroup (Tt)t>o- The following conditions are equivalent: i) (Tt)t>o is recurrent; ii) there is a sequence (uk)k€N, uk £ D{£), such that lim Uk = 1 a.e. and fc—>oo
lim £(uk,Uk) = 0; fc—>oo
iii) the constant 1 belongs to Te and £(1,1) = 0. Proof: The equivalence of ii) and iii) is clear. Further, Corollary 3.5.62.B guarantees that i) implies iii). We prove that ii) implies i). Suppose that (Tt)t>o is not recurrent, i.e. \(n\Xd) > 0, and that ii) holds. By passing to unit contractions we may assume that 0 < Ufc < 1 a.e. Define D := {w|xd; u G D{£)} and £D(u\xd,v\xd) '•= £(xxdu,Xxdv). From Theorem 3.5.11 it follows that (£D,D) is a symmetric Dirichlet form on L2(Xd,M.) and the associated semigroup is given by (St)t>o, St(u\Xd) := Tt{uxxd) for u e D{£). Clearly (St)t>o is sub-Markovian and transient on L2(Xd\ M). Hence by Lemma 3.5.22 there exists g £ L1(Xdm, M), g > 0, such that / ukgdx < (£(xxduk, Jxd
Xxduk))1/2
< (£(uk,
uk))1/2,
where we used for the last inequality once again Theorem 3.5.11. Thus, as k —> oo it follows that Jx g dx = 0 which is of course a contradiction, hence the semigroup (T t ) t >o is recurrent. • Now we have arrived at a comparison theorem for transient and recurrent Dirichlet forms. Theorem 3.5.65. Let {£^\D{£^)), i = 1,2, be two symmetric Dirichlet 2 n forms on L (M. ;M) with associated semigroup (Tf)t>o, i = 1,2. Suppose further that D(£^)
C D(£^)
and £^(u,u)
< c£i2\u,u)
for all u £
D(£W). (3.394) If (Tt)t>o is transient, then (T2)t>o is transient too, and if (T2)t>o is recurrent, then (Tt)t>o is recurrent too. Proof: If (T/)t>o is transient there exists g G L ^ R ^ R ) , g > 0 a.e., such that
/
\u\gdx<{£^{u,u))1'2
3.5 Global Properties of Lp-Sub-Markovian Semigroups
385
implying /
\u\-^dx
< ^(E^^u))1'2
i.e. the transience of (T 2 ) t >oD(£M)e c D(EW)e and
<£^\u,u), If however (T 2 )t>o is recurrent then 1 G
5(1>(l,l)o-
•
Finally we give a characterization of the conservativeness of a symmetric L 2 -sub-Markovian semigroup which is due to Y.Oshima [227]. T h e o r e m 3.5.66. Let (Tt)t>o be a symmetric I?-sub-Markovian semigroup with associated Dirichlet form (£,D(£)). Then the following conditions are equivalent: i) (Tt)t>o is conservative, i.e. Til = 1; ii) there exists a sequence (uk)ken, Uk G D{£), such that 0 < Uk < 1 a.e., limk-,ooUk = 1 o-e., and lim £(uk, v) = 0 for all v G Te; fc—too
Hi) there exists a sequence (uk)k&N, Uk S D(£), satisfying 0 < Uk < 1 a.e., lim uk = 1 a.e., and for some v = Rxf, \>0andf£ L1(Rn;R)nL2(Rn;R), k—too
/ > 0, a.e., it follows that lim £(uk,v) = 0. k—too
Proof: Suppose that (Tt)t>o is conservative and that (fk)ken, fk G L 2 (R n ;lR) is a sequence satisfying 0 < fk < fk+i < 1 a.e. and / f e l l a -e. For uk := Rifk we find uk G D(£), uk = f™ e_tTtfk dt | 1 a.e. since Ttfk T 1, and therefore for any v G D(£) D L 1 (M n ;R) it follows (using XR\1 = 1 which holds for a conservative semigroup) that lim £{uk,v) k—>oo
= lim (fk-Rifk,v)o
= 0
fe—too
proving that i) implies ii) and it is obvious that ii) implies iii). Now suppose that lim £(uk,R\f) = 0 holds for some A > 0 and / G L 1 ( R n ; R ) n L 2 ( R " ; R ) , k—too
/ > 0, and Uk G D{£), 0 < Uk < 1 a.e. and lim £(uk, Uk) = 0. Passing to the fc—too
limit fc -> co in £(uk, R\f) = (uk - \R\uk, /)o we get (1 - XR\l, f) = 0, or XR\1 = 1 a.e. From the resolvent equation it follows that [iR^l = 1 a.e. for all /i > 0 implying the conservativeness of (Tt)t>o. •
Chapter 3 Potential Theory of Semigroups and Generators
386
Now we are going to discuss these results in the light of several examples of sub-Markovian semigroups generated by (non-local) pseudo-differential operators with negative definite symbols. For examples of (local) second order differential operators we refer to M.Fukushima, Y.Oshima and M.Takeda [102]. Example 3.5.67. In Chapter 2 we encountered many pseudo-differential operators q(x,D) with a symbol q(x, £) being comparable to some continuous negative definite function ip : R" —> R. In addition we often could prove a Garding inequality B{u, u) > KolMl^.i -Ao||«||2
(3.395)
for all u e fl^^R^R) with some K0 > 0 and A0 > 0. Furthermore 1 n (BA,i?*' (R ;R)) often was a non-symmetric Dirichlet form for A > Ao- We may rewrite (3.395) as V^K£)|2^-(AO-KO)|M|3.
B(u,u)>K0[
(3-396)
Suppose that \ G Ljoc(M.n;M), KQ > Ao and that B is symmetric. Then it followsthat (B, ^ - 1 ( R n ; M ) ) is a transient Dirichlet form. Of course,
Bx(u,u),
A > A0 is always transient since the translation invariant Dirichlet form JRn (1 + V'(£))l"(£)| 2 ^£
ls a w a
^ Y s transient.
Next we continue Example 1.4.7.31. Example 3.5.68. For 1 < j < n let ip? : R —> R be continuous negative definite functions and set ip2(£) = X)" = 1 V^fo)- Further let bj : R " _ 1 -> R be functions independent of Xj belonging to L ° ° ( R n - 1 ; R ) and being bounded from below by A0 > 0, i.e. bj(x) > A0 for all x e R n _ 1 , 1 < j < n. Then n
£{u, v) = X i t o O V ^ X - ) , ^(Z?,-M-))o is a symmetric Dirichlet form with domain H^ ' J (R n ;R) satisfying n
Further the transience of the Dirichlet form
3=1
J n
^
(3.397)
3.5 Global Properties of Lp-Sub-Markovian Semigroups
387
will imply the transience of £. It is clear that Co°(Rra; R) is a dense subset of D(A), the domain of the generator of £, and on C Q ° ( R " ; R) we have Au{x) = -q(x, D)u{x) = -(27T)-"/ 2 /
e»« £ ^ - ( x ^ f o M O
#
(3.398) where Xj = (xi,..., £j_i, Xj+i,..., xn) and we consider fy as an element in L°°(R n ;R) with 1 ^ = 0 . We will now construct a special Dirichlet form belonging to the class considered in Example 3.5.69 and we will apply to this example Y.Oshima's conservativeness criterion. This example is taken from our joint paper [132] with W.Hoh. Example 3.5.69. Let £ be the Dirichlet form (3.397) with generator — q(x, D) given by (3.398). Suppose in addition that for all j , 1 < j < n, we have V>|(0) = 0. Then the semigroup (Tt)t>o associated with £ is conservative. Proof: Let tp G C£°(R n ;R) such that 0 < C Bi(0) and oo
(3.399)
holds. For this note that Sap\q{x,D)
x6R"
sup(27r)-"/2 / x6K™
<(27r)-"/ 2 c/
=
eil€f>(*i)$&)&(0#
,/R"
.
= 1
f>2(0)l^(OI^
i>2(okn\ip(ko\dt+c' [
^2mnmko\dc
Chapter 3 Potential Theory of Semigroups and Generators
388 It follows that
f
tf2(0*nW*OI#<(
^ p i>2(0) f *"l#*OI# = ( SuP ^ ( 0 ) f lvK*0l#| « | <-Tfc 4r
•/»"
Since sup tp2(£) —» 0 as k —> oo and the integral of |<£| is finite, we get
/
V
2
n
(^
|^0l^^°
as A; ^ oo.
(3.400)
The function for all £ 7^ 0. Thus we get using that V 2 ( 0 < c(l +1£| 2 ) for any continuous negative definite function that
<4/ fc
• / ^ir
ier ( B + s ) «+4/ fc
• / lfl>i
ir(n+1)#
which gives /
ip2(Z)kn\
asfc^oo.
(3.401)
From (3.400) and (3.401) we get (3.399). Since C£°(R n ; E) C i ^ 2 , 1 ( R " ; E) and fc lies in the domain of the Dirichlet form £ and 0 < ipk < 1 as well as lim tpk = 1. It remains to show that for all u € H^ ^ ( R ^ R ) n L ^ E ^ R ) it follows that £( 0 as fc -> oo, then Theorem 3.5.66 will imply the conservativeness of (Tt)t>o- Since y>fc € Co°(R ra ;R) we find £{
(q(x,D)tpk,v)0
3.5 Global Properties of Lp-Sub-Markovian Semigroups
389
and using (3.399) we get \£{Vk,v)\ = \(q(x,D) 0 for k —> oo, which proves the conservativeness of (Tt)t>o-
D
Remark 3.5.70. A. The proof of Example 3.5.69 uses much ideas from the thesis of W.Hoh [121] and in volume III we will come back to these techniques when dealing with the martingale problem for pseudo-differential operators. B. Example 3.5.69 fits very well to the observations that a Feller semigroup generated by a pseudo-differential operator with symbol q(x, £) is often conservative if q(x,0) = 0 for all x £ K n . For a detailed discussion of this point we refer to the paper [249] of R.Schilling. Let L(x,D) = Y^k i=i Tr~(akl{x)~fcr') be a second order elliptic differential operator satisfying all assumptions in order that the Aronson estimates (2.442) hold, i.e. the operator extends to a generator of a symmetric sub-Markovian semigroup (Tt)t>o having a symmetric density thus Ttu(x) = JRn T(i, x, y)u(y) dy and Kil^ 1 (t, x, y) < T(t, x, y) < K2V^ (t, x, y)
(3.402)
holds where V* is given by
v
'
'
i*-i/l 2
1
T1Ht,x,y)
= -
yj
—jz-e
4 1
^
.
2
(4TT7^)"/ )
Now let / : (0,oo) -» M b e a Bernstein function and (f]t)t>o the associated convolution semigroup with suppr^t C [0,oo). For the subordinate semigroup (Tt )t>o we have, compare Proposition 2.9.16 Ttfu(x)=
/ JR"
T(s,x,y)r)t(ds)u(y)dy= JR
n
JO
and the estimates Kir*'f(t,x,y)
< Tf(t,x,y)
r»>t(t,x,y)=
/ Jo
<
K2T^f(t,x,y)
with /•OO
rv(s,x,y)r,t(ds).
Tf(t,x,y)u(y)dy
Chapter 3 Potential Theory of Semigroups and Generators
390
We set
T?Ju(x)
= f
r^(t,x,y)u(y)dy.
If n > 3 then we know that (T t 7i,/ )t>o, j = 1,2, is transient hence it follows for g £ Z ^ R ^ R ) , g > 0 a.e., that /•OO
/ Jo
/-OO
Ttfg{x) dt
T?Jg{x)
dt < +oo
Jo
i.e. (T/)t>o will be transient too. For n = 2 and / satisfying / ( s ) > Cos", a < 1, for 0 < s < s0 it follows that (Tt7i )t>o is also transient. Indeed, the corresponding continuous negative definite function is given by /(7j|£| 2 ) > co7°l£| 2a and Li^ < oLiik. for 0 < |£| < ^ ^ T implying that (T?3 )t>o is transient, hence (T/)t>o is transient by the same argument as before. In case n = 1 the assumption / ( s ) > cos a , a < 2, for 0 < s < so, implies that (T ( 7j ' )t>o, hence (T/)t>o is transient. However for /(s) < cos™, a > A and 0 < s < so, it follows that /(7i|£| 2 ) < cxTf |£| a or ^ p < ^ p j for 0 < |£| < ^ , and therefore (T^J )t>o is recurrent, implying the recurrence of (T/)t>o by /•OO
/
/>OC>
T?'fg(x)dt<
/
T/ fl (x)dt.
The considerations just made give rise some new problems, for example 1. Investigate whether subordinating a transient sub-Markovian semigroup will always lead to a transient semigroup. 2. Find estimates corresponding TtXA{x) with fit(x — A) where (/it)t>o is a suitable convolution semigroup. It turns out that a satisfactory answer to some of these questions is related to a different problem related to the potential operator which will lead us to K.Yosida's theory of abstract potential operator, see [284]-[288], and the commentary of S.Watanabe in [279]. For an Z p -sub-Markovian semigroup we have defined its potential operator by Gf:=
[ Jo
Ttfdt=
lim / Ttf dt, Jo
N-KX
/ > 0,
(3.403)
3.5 Global Properties of Lp-Sub-Markovian Semigroups
391
and clearly the definition extends to Feller semigroups. Further for those / G Lp(M.n; M) (or / G Coo(Rn; R)), / > 0 a.e., for which Gf G D(A), A being the generator of (T t ) t >o, we should expect AGf = GAf = —/, i.e. — A'1 = G. But so far (3.403) gives not a linear operator mapping from Lp(M.n; W) into D(A) (in the case of Feller semigroups replace ip(M™;M) by Coo(]Rn;]R)). Thus we may ask when there is a linear operator coinciding with G for "good" functions. This was exactly K.Yosida's approach to define abstract potential operators. Many of his results can be formulated for arbitrary strongly continuous contraction semigroups on some Banach space (X, \\ • \\x)- We will follow in our discussion K.Yosida [288] as well as Chr.Berg and G.Forst [26]. Let (X, || • \\x) be a Banach space and (Tt)t>o be a strongly continuous contraction semigroup on X with generator (^4, D{A)). Since now we have not necessarily an order structure on X we define the potential operator associated with (Tt)t>o by D(G) := \u e X; lim *•
/
Ttudt
exists in x).
(3.404) J
AT->oo Jo
and r-OO
G = / J0
rN
Ttudt:=
lim /
Ttudt.
(3.405)
JV-KX) J o
First we aim to extend some results of Lemma 1.4.1.14 to G. Lemma 3.5.71. Let (Tt)t>o be a strongly continuous contraction semigroup on (X, || • ||x) with generator (A, D(A)) and potential operator (G, D(G)). The domain D(G) of G is invariant under Tt for all t > 0 and for all u € D(G) it follows that TtGu = GTtu. Moreover, for u G D(G) we have TtGu-Gu
= - f Ttuds. Jo
(3.406)
Proof: For t > 0 and u G D{G) we find rN
*N
I Ts(Ttu) ds = Tt Tsuds, Jo Jo
N G N.
As N tends to infinity Tt(fr0N Tsuds) -> TtGu, implying first TtD{G) C D(G) and then TtGu — GTtu. Now, a straightforward calculation for u G D(G)
Chapter 3 Potential Theory of Semigroups and Generators
392
yields rN
TtGu=
lim
/
rN
T3Ttuds=
N—>oo JO
= lim
/
Ts+tuds
/V-»oo Jo
/
Tsuds=
N->oo Jt
lim ( / N->oo
= Gu — I Jo proving (3.406).
lim
Wo
Tsuds—
I
Tauds\
Jo
'
Tsuds
•
Lemma 3.5.72. Let (Tt)t>o, (A,D(A)) and (G, D(G)) be as in Lemma 3.5.71. Then it follows that D(G) C R(G), and lim ||T t u|| = 0 t-»o
for
uG R(G).
(3.407)
Moreover, R(G) C D{A) and AGu = -u
for u € D(G),
(3.408)
implying in particular the injectivity of G. Proof: Using (3.406) we find for u € D(G)
G(i(u-T t «)) =\ I Tauds which yields as t —* 0 that u is the limit in (X, \\ • \\x) of elements in R(G), i.e. D(G) = R(G). Next observe that {u £ X; lim Ttu = 0} is closed which follows t—HX>
from the estimate ||T t u||x < ||u — uu\\x + \\TtUv\\x- Further, (3.406) implies for u £ D(G) that lim Tt(Gu) = 0, hence R(G) C {u € X; lim Ttu = 0}, t>oo
t—>oo
i.e. (3.407) is proved. Finally, again using (3.406) it follows for u G D(G) and t>0 \{Tt{Gu)-Gu) t
= - \ f Tsuds t Jo
implying as t —• 0 that Gu G D(A) and AGu = -u for u G D{G).
•
3.5 Global Properties of .Lp-Sub-Markovian Semigroups
393
Lemma 3.5.73. In the situation of Lemma 3.5.71 we have for u £ D(G) and N>0 / Tsuds€D(G). Jo
(3.409)
Proof: For t > 0 it follows that Tr(
Tsuds\dr= = /
(
( Trudr\ds
Tr+Suds\dr = I
( j
Trudr-
j
Trudr\ds.
Now, given e > 0 choose to > 0 such that Gu— I Jo
Tsuds
x
<e
for i > to
and observe that for t >to rN
.
fs+t
1 ( 1
Trudr)ds-NGu ii rN+t i r \ = / ( / Trudr-Gu)ds\\ "Jt Wo '
implying that lim /„* Tr ( f" Tsu ds) dr £ D(G).
II < Ne, I'x •
The next result, due to Chr.Berg [25], gives conditions in order that (3.407) does hold for all u e X. Theorem 3.5.74. Let (Tt)t>o be a strongly continuous contraction semigroup on(X, \\-\\x) with generator (A, D(A)) and potential operator (G, D{G)). Then the assertions i) D(Gl= X; ii) R{G) = X; Hi) lim ||Tttt|| = 0 for allu 6 X t—*oo
are equivalent. If any (hence all) of these conditions hold then G is a densely defined closed operator on X, A is injective and satisfies G = -A_1
and
A = -G'1.
(3.410)
Chapter 3 Potential Theory of Semigroups and Generators
394
Proof: We know already from Lemma 3.5.72 that i) implies ii) and that ii) implies iii). Now suppose that iii) holds. By Lemma 1.4.1.14 we have for u € D(A) that t
TaAuds =
Ttu-u
and therefore iii) implies that lim / TsAuds = -u,
(3.411)
t—•oo./O
i.e. Au 6 D{G) and GAu = —u. Thus A is injective and maps D(A) into D{G) which together with Lemma 3.5.72 yields (3.410). The closedness of A implies now that of G and from (3.411) we deduce by Lemma 3.5.73 that D(A) C ~D(G). Since D(A) is dense in {X, || • \\x) we find D~{G) =X. D We can now introduce K.Yosida's abstract potential operator. Definition 3.5.75. Let (Tt)t>o be a strongly continuous contraction semigroup on a Banach space (X, \\ • \\x) with generator (A,D(A)), potential operator (G,D(G)) and resolvent (R\)\>o. The abstract potential operator or zero order resolvent (Ro, D(Ro)) associated with (Tt)t>o is the operator with domain D{R0) := {u € X; lim Rxu A—o
exists}
(3.412)
and R0u := lim Rxu, u G D(Ro).
(3.413)
A-»0
First let us prove some results for R0 analogous to those for G. Lemma 3.5.76. Let (Tt)t>o be as in Definition 3.5.75. A. For all\>0 we have Rx{D{Ro)) C D(RQ). B.IfX>0andue D(R0) then Rx(R0u) = R0(Rxu). C. The resolvent equation extends in the sense that Rou = Rxu + XRx(Rou) holds for A > 0 and u e
D(RQ).
(3.414)
3.5 Global Properties of Lp-Sub-Markovian Semigroups D. It holds D(R0)
C
395
R(RQ).
E. For u 6 R(Ro) it follows that lim XRxu = 0
(3.415)
A—0
F. It holds R(RQ) C D(A) and A(Rou) = —u for u G D(RQ) implying that Ro is one-to-one. Proof: The assertions A and B clearly follow from RxR^ = R^Rx, A, \i > 0, and (3.414) is an obvious consequence of the resolvent equation. Further, the strong continuity of (XRx)x>o implies C, and (3.414) that Ro(X(u-XRxu))
= XNxu -> 0,
i.e. D(RQ) C -R(-Ro) and D is proved. Again as in the proof of Lemma 3.5.72 it is easy to see that {u G X; lim XRxu = 0} is closed and using C we find for A-»0
u G D(R0) lim XRx(Rou) = Ron— lim Rx = 0, A->0
A-»0
or R(RQ) C {U G X; lim XRxu = 0} and hence E is shown.
Note that
A—o
Rx = (Aid —A) - 1 . From this it follows that A(Rxu) = —u + XRxu
for u G X,
and for u € D(Ro) it follows now lim ARxu = —u. A->0
But the operator (A, D(A)) is closed implying that lim Rxu = Rou G D(A) X-,0
and
A{RQU)
= —u and the lemma is proved.
•
Corollary 3.5.77. The abstract potential operator RQ satisfies \\XRou\\x < ||u+A/Zou|| x , for
A > 0,
(3.416)
allueD(Ro).
Proof: The extended resolvent equation yields XR0u = XRx(u + XRou). implying (3.416) since \\\R\\\x < 1. • Now we may prove K.Yosida result, see [288].
396
Chapter 3 Potential Theory of Semigroups and Generators
Theorem 3.5.78. The following conditions are equivalent: i) D{RQ) is dense in X; ii) R{RQ) is dense in X; Hi) for all u € X it follows that lim XR\u = 0. A->0
If one, hence all of, i)-iii) hold then RQ is a densely defined closed operator on (X, || • ||x) and Ro = -A'1
and
A = -R^1
(3.417)
hold. Proof: Prom Lemma 3.5.76 we deduce immediately that i) implies ii) and ii) implies iii). Now assume that iii) holds. For u £ D(A) we find AR\u = R\Au — —u+\R\u,
A > 0,
and it follows that lim R\(Au) = —u, i.e. Au £ D(RQ) and RQ(AU) = — u, A-»o thus A is one-to-one from D(A) into D(Ro) which yields by Lemma 3.5.76 that (3.417) holds, in particular it follows that RQ is closed. Moreover, for u £ D(A) we find lim A(R\u) = -u, thus D(A) C D(R0) since A(R\u) £ D(R0). But A-»o D(A) C X is dense, hence D(RQ) is dense in X, i.e. iii) implies i) and the theorem is proved. • Finally we may relate the potential operator G and the abstract potential operator or zero order resolvent Ro to each other. Proposition 3.5.79. The abstract potential operator (Ro,D(Ro)) is an extension of the potential operator (G, D(G)). However, if lim TtU — 0 for all t—*oo
u £ X then they coincide, i.e. G = RQ. Proof: For u 6 D{G) we have to show that u £ D(RQ) and RQU = Gu. For A > 0 and t > 0 we find for u £ D(G) from an integration by parts I e-XsTsuds
= e-Xs
f Trudrt
+ f Xe~Xs(
f
Trudr\ds
and for t —> oo we get R\u=
J
Xe~Xs( J Trudr)ds=
f
e~r(f
Tsuds)dr.
3.5 Global Properties of Lp-Sub-Markovian Semigroups
397
Now passing to the limit A —• 0 we arrive at lim R\u = Gu A—0
proving that (R0,D(RQ)) extends (G,D(G)). By Theorem 3.5.74, lim Ttu = 0 for all u G X implies that D(G) is dense t—*oo
in X, hence D(G) C D(Ro) = X, and therefore Theorem 3.5.78 gives the second assertion of the proposition. • Corollary 3.5.80. Let (Tt)t>o be a symmetric transient sub-Markovian semigroup on L2(M.n;M.) with generator (A,D(A)). Then the corresponding potential operator and the abstract potential operator as Li2-operator do coincide and satisfy A = — G _ 1 as well as G = —A-1. Proof: We just have to combine Lemma 3.5.48 with Proposition 3.5.79.
•
Let us return to our investigation on the transience of sub-Markovian semigroups. We will now discuss the case of translation invariant sub-Markovian semigroups, in particular subordinated semigroups which originate from convolution semigroups. We will follow partly the presentation of Chr.Berg and G.Forst [26]. Let (/it)t>o be a convolution semigroup on R n . The resolvent of (nt)t>o is then defined by /•oo
P\= I e~xtnt dt Jo with the interpretation px(u) = (p\,u) = / u(x)px(dx) = / JR" Jo
-L
oo
o
e
xt
Jwtn
u(x)fit(dx) dt
e~MAt. fj,t(dx)dt
for all u G C0(Mn;]R), u > 0, compare Remark 1.3.9.11. We call (pt)t>o a transient convolution semigroup if lim px{u) < oo
for all
u G C 0 (R"; R),
u > 0,
A->0
and in this case we may define the kernel K K(U) := lim px{u), A-»0
u G C0(Rn; R),
u > 0.
Chapter 3 Potential Theory of Semigroups and Generators
398
The kernel K is called the potential kernel associated with the convolution semigroup (/it)t>oSince for y £Rn the function uy(-) = u(- - y) G C 0 (K") if u G C 0 (R"), we find that (nt)t>o i s transient if and only if the associated operator semigroup {T?)t>0 defined by Ttu{x) = / K „ u{x - y)nt{dy), £t(£) = e - ' * « \ is transient. Next let (r]t)t>o be a convolution semigroup on R with supp vt C [0, oo) and associated Bernstein function / . The following result is taken from Chr.Berg and G. Frost [26]. Proposition 3.5.81. Let (r]t)t>o and f be as above. Either f(x) = 0 for all x G (0, oo) implying r)t — eo for all t > 0, or f(x) > 0 for all x S (0, oo). Then, in the latter case, (r]t)t>o is transient and its potential kernel r is characterized by C(T) = I
(3.418)
Proof: Since / is increasing and concave it follows that f(x) = 0 for all x if it is 0 for some x > 0. Now, if f(x) > 0 for all x > 0 then / Jo
C(r,t)(x)dt= Jo
t
e-
^dt=-—, f{x)
(3.419)
where we used the defining identity C(r]t){x) = e~tf(~x\ Given 0, we find c > 0 such that for s > 0 we have ip(s) < ce~s. Therefore we have ' o
/
(C(fH))(l) dt =
JM.
c
/(I)
JO
< CO,
which implies that the transience of (7jt)t>o and then by (3.419) that £ ( r ) =
In the following we will assume that (r)t)t>o is associated with / , f(x) > 0, and denote its potential kernel always by r . Since r is a positive measure we may define for g £ i 1 ( ( 0 , oo); R), g > 0 a.e., poo
/ J0
poo
g(s)T(ds) := lim /
gk(s)T(ds),
k—>oo Jo
where gk > 0 is a sequence in Co(M;K) increasing to g. Thus if (Tt)t>o is a transient Dirichlet semigroup on L 2 (R";R) and h G ^ ( R " ; ^ , h > 0, then we
3.5 Global Properties of Lp-Sub-Markovian Semigroups
399
find /•OO
/
/'OO
T/hdt=
.
/«0O
U
/>O0
Tthr)t(ds))dt=
(Tsh)(r{ds)),
(3.420)
i.e. we have Lemma 3.5.82. Let (Tt)t>o be a transient Dirichlet semigroup and {r]t)t>o a convolution semigroup supported in [0, oo) and associated Bernstein function f, f(r) > 0 for all r > 0. Denote the potential kernel associated with (r]t)t>o by T. If for some h £ S(Rn; R), h > 0, it follows that /•OO
/ (T3h)(x)T(ds) Jo
< oo
(3.421)
for all x £ M.n, then (T/)t>o is transient. To our knowledge there are no general results assuring (3.421), but if Ttu(x) = fRn u(x — y)fj,t(dy), then there is a complete satisfactory theory due to F.Hirsch [115] and M.Ito [146] and we will discuss their results immediately following Chr.Berg and G.Frost [26]. However, having in mind the Aronson estimate (2.442), then we may formulate the following corollary. Corollary 3.5.83. Let (Tt)t>o be a transient Dirichlet semigroup and (rjt)t>o a convolution semigroup supported on [0, oo) associated with the Bernstein function f, f(r) > 0 for r > 0. If there is a convolution semigroup (fJ.t)t>o such that with some non-negative constant CQ we have TtXA{x)
(3.422)
for all bounded Borel sets, then (T/)t>o *s transient J// 0 °° /x s (/i)r(ds) < oo for some h £ S(Rn;M), h>0. Remark 3.5.84. A. If (T t ) t > 0 is generated by — q(x,£) and q(x,£) ~ ip(£), where if) is a continuous negative definite function it would be natural to seek for some (/i*)t>o such that (it(£) = e~tCa^c^ and (3.422) holds. B. Clearly we do not need a convolution semigroup in (3.422), any estimate of the type Tth(x) < gt(%) such that J0°° gs(x)r(ds) < oo is sufficient. Hence we are longing for estimates for the kernel pt(x, A) = TtXA(x), this will be the theme of the next section. As stated before our aim now is to investigate /0°° fxsr(ds) for a convolution semigroup (fit)t>o and T as in Proposition 3.5.81.
Chapter 3 Potential Theory of Semigroups and Generators
400
As we know many properties of (fit)t>o are characterized by properties of its associated continuous negative definite function ip : R" —> C, and the set T^ := {£ G Rn;V>(£) = 0} as well we r * 6 ^ : {£ G R";ReV>(£) = 0} are of importance when considering questions like conservativeness, characterization of domains or dealing with transience. In the rest of this section unless it is stated otherwise let us assume that T^ = r R e , / ' C {0}, i.e. the periodicity group of (fit)t>o is either empty or trivial. (In fact we prove this condition only to avoid a discussion of periodicity which is never of relevance when having an assumption as (2.302).) It is known from Lemma 1.3.6.27 that each of these sets form a closed subgroup of (R™, +). For a set A C R n we denote for a moment by Ax := {£ G (R n )*; eix* = 1 for all x G A} its group-theoretical orthogonal complement. Note that under the general assumptions made above we have G 0 = (vp)±
= Rn,
where G0 C R n is the smallest closed subset of (R n , +) with the property that (J t > 0 (supp/i ( ) c G0, compare Chr.Berg and G.Forst [26]. For the following the reader should recall that measures are distributions of order zero, hence for u G Co(R n ) we may write (/i, / ) instead J Rn f(x)fi(dx). The following result is again taken from Chr.Berg and G.Frost [26]. Theorem 3.5.85. Let (Tt)t>o be the L2-sub-Markovian semigroup associated with the convolution semigroup (fJ-t)t>o and denote the corresponding continuous negative definite function by ip. A. The abstract potential operator (Ro, D(RQ)) is densely defined if and only if II>\K ^ 0 a.e. for any compact set K C M n . B. The potential operator (G,D(G)) is densely defined if and only if (Reip)\K 7^ 0 for all compact sets K C K". Proof: We denote by (i?A)A>o the resolvent of (T t ) t > 0 and by resolvent of (^t)t>o- If tp(0) > 0 then \(Ttu,v)o\
(/9A)A>O
< e -t^(0)
IMIolMlo implying that ||T t || L 2_ L 2 < e - ^ ( o ) . Thus for u G L 2 (R";R) we find f°°
II II
the
3.5 Global Properties of Lp-Sub-Markovian Semigroups
401
which implies that /0°° Ttudt exists in L 2 (R n ; R), thus rN
lim / N-*oo
i >Jo
PC
Ttudt=
I/
Ttudt
Jo
and lim R\u = / A-»o
Jo
Ttu dt
do exists in L 2 (R n ;R). Thus in this case R0 and G are defined for all u S L 2 (R"; R) and in fact they coincide as operators being bounded with operator norm less or equal to -TTQT- Now suppose that V'(O) = 0. For u 6 Co(R"; R) it follows that F-'iXRxiF-'u))
= j^yuO),
A > 0.
As A —> 0 it follows that \ , >u —> 0 a.e. and since \\\_ nated convergence theorem yields now lim WF-^XRxiF-^h,
= lim f
—A—utf) \
i u\ < \u\ the domi-
= 0.
Since S(Rn;R) C {F^uiu € C 0 (R";R)} is dense in L 2 (R";R) and {u G L 2 (R n ;R); lim XRXn = 0} is closed it follows by Theorem 3.5.78.iii) that A—>oo
RQ is densely denned. If there is a bounded set A, X(n\A) > 0 such that V>|A = 0, then T^ must be an open subgroup of (R™, +) and (T^)-1 must be a compact subgroup of (R n , +) since in any locally compact Abelian group the orthogonal complement of any open subgroup is compact, compare H.Reiter and J.Stegeman [235], Proposition 4.2.21. Hence the measure (Xpx) * (XSpx), Sx = —x, are concentrated on (F^)- 1 . Now, with u € Co(R n ;R), u > 0, and u * Su > I o n the compact set (r^)- 1 we find for all A > 0 1 < (Xpx*XSp,u*Su)
= /
\Xpx*u\2dx = \\XRxu\\2L2
and from Theorem 3.5.78 we deduce that D(RQ) is not dense in L 2 (R n ; R) and part A is proved. Now, Re tp is a continuous negative definite function too and therefore YRe^ is again a closed subgroup of R n . For u e Co(R") we find F-^TtiF^u))
= e_tV
t > 0,
Chapter 3 Potential Theory of Semigroups and Generators
402
and if Re^lii" ^ 0 a.e. it follows as above that lim \\TtF-lu\\2L* = lim / t—too
e~2tKe^\u{0\2^
= 0
t—>oo JU"
which by Theorem 3.5.74 yields that G is densely defined. Conversely, if (ReVOU ¥= 0) A bounded and A(")(A) > 0, we may argue as before that ( r R e V , ) x is a compact subgroup on R n and with u G C 0 (R n ;R), u > 0, such that u * Su > 1 on ( r ^ ^ ) 1 we find for all t > 0 1 < (/j,t*SiJ,t,u*Su) = /
|/x t *u| 2 (x)dx = ||Ttu||| 2 ,
and by Theorem 3.5.74 deduce that D(G) is not dense in L 2 (R n ; R).
D
Corollary 3.5.86. Let (Tt)t>o be an L2 -sub-Markovian semigroup associated with a convolution semigroup (fit)t>o- Its abstract potential operator is densely defined except when fit = SQ for all t > 0, whereas its potential operator is densely defined whenever (/it)t>o is not a translation semigroup. Proof: If Ro is not densely denned then T^ is an open subgroup of (R™.+) as it follows from the arguments in the proof of Theorem 3.5.85. Hence (T^)-1 C K n must be compact, hence ( r ^ ) x = {0} being the only compact subgroup of (R n , +), thus r ^ = R n , implying that \it = s0 for all t > 0. If G is not densely defined then r R e , ( ' is an open subgroup of (Rra, +) and as before we conclude that r R e ^ = R" hence Reip($) = 0 for all £ G R n , i.e. V(£) = « ( 0 and, by Example 1.3.6.19,1: R n -> R must be linear, i.e. V ( 0 =ib-£ for some b G R n and fit is the translation semigroup /zt = £bt- O Remark 3.5.87. If (Tt)t>o is an i 2 -sub-Markovian semigroup associated with a convolution semigroup (fit)t>o then whenever {fJ-t)t>o is not of translation type (£bt)t>o, b G R", then the abstract potential operator and the potential operator related to (Tt)t>o coincide and they are densely defined. Remark 3.5.88. A careful inspection of the proof of Theorem 3.5.85 shows that we do not really use much L 2 -norms, but the representation of all operators using the Fourier transform. From the early examples in Section 1.4.1 we may deduce that the assertions of Theorem 3.5.85 indeed do hold for all L p -sub-Markovian semigroups (T t (p) ) t >o, 2 < p < oo, and for p = oo we have to take the Feller semigroup associated with (/xt)t>o- In addition, since ip is
3.5
Global Properties of L p -Sub-Markovian Semigroups
403
also a continuous negative definite function we may use the standard duality argument to find that Theorem 3.5.85 holds for all the L p -sub-Markovian semigroups, 1 < p < co, associated with (n)t>o- In view of our transient-recurrent observation (p-independence) this should not be surprising. To proceed further we need Lemma 3.5.89. Let (Tt)t>o be a Feller semigroup. Then the following conditions are equivalent i)C0(Rn;R)cD(G); ii)C0(Rn;R)cD(R0). Proof: Since by Proposition 3.5.79 (RQ, D(RO)) is an extension of (G, D(G)) it is obvious that i) implies ii). Suppose that Co(R n ;R) C D(Ro) and take u £ Co(M n ;K), u > 0. Since (Tt)t>o is a Feller semigroup it follows that Rou e Coo(Rn; R) and R0u > 0. Moreover, for every x € Rn we have RQU(X) = lim R\u(x) A->o
= I Jo
Ttu(x) dt
implying fN
lim /
Ttu(x) dt = {R0u)(x)
for x E Rn.
(3.423)
N->ooJ0
We claim that the limit in (3.423) is even uniform. For e > 0 given, take K cRn compact such that 0
inKc.
(3.424)
From Dini's theorem, compare Theorem 1.2.1.6, it follows that the convergence in (3.423) is uniform on K, i.e. there exists No such that for all N > No
f'
Ttu(x) dt — Rou{x) < £ for all x € K,
)o
Jo
and together with (3.424) the lemma follows.
•
We want to emphasize that in Lemma 3.5.89 we do not assume (Tt)t>o to be associated with a convolution semigroup, i.e. to be translation invariant. Denote by r x , x e Rn, the right translation TX : l n -> R", y >-> y + x.
Chapter 3
404
Potential Theory of Semigroups and Generators
Definition 3.5.90. A. A Borel measure \i on R" is called shift-bounded if the set {TX\X; x £ R n } is vaguely bounded. B. A Borel measure p, is said to vanish at infinity if /i * u £ C00(M.n; R) for allusC0(R";R). C. A Feller semigroup (Tt)t>o is said to be integrable if one of the equivalent conditions of Lemma 3.5.89 holds; a convolution semigroup (/J.t)t>o is called integrable if its associated Feller semigroup is integrable. Recall that, by definition, a family M of positive Borel measures is vaguely bounded if u(x)fi(dx)
sup /
for a l i u s Co(R n ;R).
(3.425)
Thus if /i is a shift-bounded positive measure and K C R" is compact then x i-> p,{x + K) is bounded on R n , which follows from (3.425) by taking u e C 0 (R n ;R), U\K = 1. Conversely, if the function x i-> /j,(x + K) is for all compact sets K C R" bounded, then \x is shift-bounded which follows when taking supp u C K. Now let (/-0t>o be a convolution semigroup on R™ with associated continuous negative definite function ip : R n —> C, resolvent (PA)A>O and potential kernel K. Lemma 3.5.91. Let (fj,t)t>o be a transient convolution semigroup on R™ with potential kernel K. Then the measure TJ(K + SK), SX = —x, is positive definite and shift bounded. Proof: Recall that a measure is positive definite if the corresponding distribution is positive definite, compare Definition 1.3.5.8. For the resolvent (PA)A>O we find !/
o
>A,„
T,
1
A + Ret^)
implying
hence ^{px + Sp\) is positive definite. Next we prove that every positive definite measure is shift-bounded. Indeed, given u G Co(M"; R), u > 0, we find g G C 0 (R n ;K) such that f
n*g*Sg.
3.5 Global Properties of Lp-Sub-Markovian Semigroups
405
Since fi * g * Sg is a positive definite function it is bounded. Hence fi is shiftbounded. As we already proved, ^(K + SK) is positive definite, the lemma follows. • Now we may relate the transience of a convolution semigroup to its integrability. Proposition 3.5.92. A convolution semigroup (n)t>o is integrable if and only if it is transient and its potential kernel K vanishes at infinity. Proof: Let (Tt)t>o be the Feller semigroup associated with (fit)t>- If (nt)t>o is integrable it follows that the positivity preserving translation invariant operator i?o|c0(K™;K) is a convolution operator, i.e. Rou(x) =
(K*U)(X)
for all u G C 0 (R n ;R) and x G R". The Feller property of (T t ) t > 0 implies that RQU £ C 00 (R n ;]R), i.e. « vanishes at infinity. Moreover for the resolvent (PA)A>O and for u G C 0 (R"; R), u > 0, it follows that lim (pA, U) = lim Rx(Su)(Q) = Ro(Su)(0) A->0
A-*0 = (K* SU)(0)
=
(K,U),
implying that (/x)t>0 is transient with potential kernel K (vanishing at infinity). Conversely, suppose that (/i)t>o is transient and has a potential kernel K vanishing at infinity. For u £ Co(R n ; R), u > 0, we have lim Rxu(x) = lim(p\,Tx(Su)) A^O
=
(K,TX(SU))
=
(K*U)(X)
A-»0
as a pointwise, increasing limit. But by assumption K*U belongs to Coo(Rn; R) implying with the arguments in the proof of Lemma 3.5.89 that lim R\u = K*U uniformly on R", A—o i.e. u G D(RQ) and RQU = K * u. Thus C 0 (R n ;R) C D(R0) and consequently (Vt)t>o is integrable. D Finally we return to the question when a subordinate convolution semigroup is transient.
Chapter 3 Potential Theory of Semigroups and Generators
406
Proposition 3.5.93. Let (fa)t>o be a convolution semigroup on R" and (rjt)t>o o, convolution semigroup supported on [0, oo) associated with the Bernstein function f. The subordinate semigroup (/4)t>o is transient if and only if as a vague limit /•OO
KT : = /
fisT(ds)
(3.426)
Jo exists where r is the potential kernel associated with (r)t)t>oProof: From Proposition 3.5.92 we deduce for u £ C70(Rn; R), u > 0, that /•OO
/
/»00
{Ht>u)dt =
/»0O
(/
implying the proposition.
/»00
(fa,u)nt(ds)jdt
=
(^s,u)r(ds)
•
Lemma 3.5.94. Let (/i)t>o be a transient convolution semigroup with potential kernel K. For every shift-bounded measure T on R with suppr C [0,oo) the vague integral /•OO
KT :=
/ Jo
fj,tT(dt)
exists. The measure KT is shift-bounded and in addition it vanishes at infinity if K does. Proof: From the comments following Definition 3.5.90 we know that c :— supr([fc, k + 1]) < oo. Define fc>0
Htdt Jo
as a vague integral which gives clearly a positive bounded measure on M.n and cr(Rn) > 0. Now given h e C 0 (R n ;R), h > 0, there exists g e Co(K";R), g > 0, such that h < a * g. Indeed this follows from the fact that for every K CW1 compact there exists / e C 0 ( l n ; l ) such that fi * f < 1 on K. For k > 0, s £ [k, k + 1] and x G R n we find further (Us * h)(x) < (/is * a * g)(x) = I {fis * fa * g)(x) dt Jo
s+l
/.fc+2
/ {Ht * g)(x) dt<
(fa
*g)(x)dt,
3.5 Global Properties of Lp-Sub-Markovian Semigroups
407
and therefore /•fc+i / (fis * h)(x)r(ds) Jk
< r([k, k + 1]) max (fia * h)(x) s€[k,k+l] rk+2
Thus we find /•oo
\
Jo
oo ..fc+l "J rK+l (fit * h)(x) r(ds) = V / {lit* h)(x) r(ds) Jk k=o OO
rk+2
{nt*h){x)dt
/•OO
< 2c / (/it * h)(x) dt = 2C(K * g)(x). Jo Taking i = 0 w e arrive for every h e C 0 (R n ; R), /i > 0, at
I
(/xt, Sh)r(ds) < 2C(K, Sg) < oo,
Jo
showing that the vague integral exists and that K*h < 2c K*g. By Lemma 3.5.91 the measure K is shift-bounded implying that nT vanishes at infinity if K vanishes at infinity. • Finally we arrive at the result already announced and which is due to F.Hirsch and M.Ito. Theorem 3.5.95. Let (/i)t>o be a transient convolution semigroup with potential kernel K and let (rjt)t>o be a convolution semigroup supported on [0, oo) with associated Bernstein function f, f(x) > 0 for x > 0, and potential kernel T. Then the subordinate semigroup ((i{)t>o is transient and has the potential kernel /•OO
K=
/ Jo
iitT{dt).
(3.427)
If (fit)t>o is integrable then (/if)t>o in integrable. Proof: From Lemma 3.5.91 we know that T is shift-bounded and suppr C [0, oo). Therefore by Lemma 3.5.94 the integral (3.427) exists and by Proposition 3.5.93 it follows that {fi{)t>o is transient with KT being its potential
408
Chapter 3 Potential Theory of Semigroups and Generators
kernel. Further Proposition 3.5.92 and Proposition 3.5.93 yield the integrability of (n{)t>o if (Mt)t>o is integrable. D Let us recollect where we are by now in terms of examples. We have a complete characterization for the transience or recurrence of translation invariant Dirichlet semigroups. Further we can deduce the transience of semigroups of the type TtA = e~xtTt> X > Ao, when (T t ) t >o is generated by a symmetric pseudo-differential operator — q(x, D) satisfying a Garding inequality (q(x,D)u,u)0
> COIMIVM-AONIO-
We know special examples related to symbols of the form X)j=i ^j(^j)V'|(^i)) Xj = (x\,..., Xj-i, Xj+i,..., xn). But apart of these special cases we are not aware of general theorems of the type: conditions on q(x,£) imply (T t ) t >o is transient or recurrent. However we have quite satisfying conditions on q (x, y) in order that the Feller semigroup (Tt)t>o generated by — q(x, D) is conservative, namely we need to have in particular q(x,0) = 0, compare R.Schilling [248] and [249]. Such a general theory does in fact not exists by now, however more can be proved when we take more care on the operator Tf. We will use Lemma 3.5.82 to explain this remark. Before doing so we should make a comment on the deep work by F.Hirsch upon which our discussion is based. As Lemma 3.5.82 is stated, the potential kernel r, not the convolution semigroup (r]t)t>o is clearly the central object, i.e. one may ask for kernels operating on semigroups or their potential operators. This is essentially the theme of F.Hirsch's papers [115][118], see also the discussion in Chr.Berg and G.Forst [26]. We cut this discussion short by posing the following question: Are there potential kernels r related to convolution semigroups supported on [0, oo) with associated Bernstein function / such that estimates of some operator norms of {Tt)t>o will imply the transience of (T/)(> 0 ? This question is already a bridge to the next section. Let us make our point clear by an example. Example 3.5.96. Since £(^—r)(s) = jL, 0 < a < 1, it follows from Proposition 3.5.81 that r(ds) := X(fttO0)(s)^L-sa-1\M{ds)
(3.428)
is the potential kernel associated with the one-sided stable convolution semigroup (rj")t>o of order a. Let (T t ) t >o be a transient Dirichlet semigroup on
3.6 Nash-Type and Sobolev-Type Inequalities — a Short Outline L 2 (R n ; R) and for some h € S(Rn; M),h>0, /•OO
409
consider
/>00
/ (Tsh)(x)T(ds) Jo
=c
Jo
(Tsh)(x)sa-1
ds.
Suppose that in addition Tt is an L°°-contraction and maps also L 1 (R n ;R) boundedly into L°°(R"';R) with some operator norm estimate ||7t||i,i_£,oo < g(t). We find now /»1
/•OO
/ Jo
Tsh(x)sa-1
yOO
ds = / (T,ft)(x)s a - 1 ds + / (T s /i)(x)s a - 1 ds Jo J\ < Halloo / \\H O O
^-1ds+\\h\\LxJ
+ \\h\\Li /
a Thus whenever the integral J^° g(s)sa is transient.
1
g{s)sa-1ds
a 1a 1 g(s)s 9{s)s ~ >- ds
ds is finite then (T/" )t>o, /a(»") = ra,
In [117] F.Hirsch gave much more examples of potential kernels r(ds) having a density with respect to X^(ds) and being associated to convolution semigroups supported on [0, oo). Definition 3.5.97. A function A; : (0, oo) —• R not being identically zero is called logarithmically convex if k(x) > 0 for all x > 0 and log k is convex. Theorem 3.5.98 (F.Hirsch). Let k : (0,oo) -> R be decreasing and logarithmically convex. Then there exists a Bernstein function f, f(x) > 0 for x > 0, such that X(o,oo){s)k(s)^1\ds) is the potential kernel of the convolution semigroup supported on [0, oo) one? associated with f. For a proof of Theorem 3.5.98 we refer to F.Hirsch [118] or [26]. Note that every Stieltjes transform in the sense of Definition 1.3.8.17 (with a = 0 in (1.4.220)) is a decreasing logarithmically convex function.
3.6
Nash-Type and Sobolev-Type Inequalities — a Short Outline
In our treatise we do not handle the analysis of local generators of L p -subMarkovian semigroups or Dirichlet forms, and the related diffusion processes.
Chapter 3
410
Potential Theory of Semigroups and Generators
Our interest lies in non-local operators which generate jump-processes. Therefore we do not make any attempt to discuss "The geometry of Markov Generators", a headline we borrowed from M.Ledoux [192]. For this highly interesting and rapidly developing topic relating analysis, geometry and stochastics we refer beside to Ledoux's paper to N.Varopoulos, L.Saloff-Coste and T.Coulhon [277], the survey of A.Grigoryan [110], the most recent book of L.Saloff-Coste [241] and the references given therein. Of interest is also the survey [262] of K.-Th.Sturm and the recent paper [108] of V.Gol'dshtein and M.Troyanov. (It would be interesting to know whether the approach of A.Parmeggiani [231], see also [230] and the short, but very informative survey of C.Fefferman [89], may provide in some situation a suitable setting for handling the operators we are interested in.) In this section we want to handle only one very special topic, namely how to obtain bounds for some operator norms of Tt, (Tt)t>o being a symmetric subMarkovian semigroup, by using estimates for the associated Dirichlet form. Estimates for the resolvent associated with a Dirichlet form were first derived by M.Fukushima [98] using lower L p -bounds, p > 2, for the Dirichlet form by taking up ideas of G.Stampacchia's work [254] on second order elliptic differential operators. His ideas had been taken up by M.Tomisaki [264], in [150] and more recently by M.Kassmann [170] and [171]. It seems that N.Varopoulos [276] was the first who used consequently Nash-type and Sobolev-type inequalities to get bounds for the norms ||T t ||LP-L" for symmetric sub-Markovian semigroups (Tt)t>Q-
The following fundamental result is taken from N.Varopoulos et.al., [277]. Theorem 3.6.1. Let (Tt)t>o be a symmetric conservative sub-Markovian semigroup on L 2 (R";K) with corresponding regular Dirichlet form (£,£)(£)). Further let p > 2 and N := _f n > 2. The following estimates are equivalent \\U\\IP < c£(u, u) \\ufL+4/N
for all u e £>(£);
lirtlUi-L- < c't~N'2
forallueD(£)nL1(Wl);
for all t > 0,
(3.429) (3.430) (3.431)
where | | B | | x - y denotes the operator norm of B : X —> Y\ Note that (3.429) is a Sobolev-type inequality and (3.430) is a Nash-type inequality.
3.6 Nash-Type and Sobolev-Type Inequalities — a Short Outline
411
An application of the Dunford-Pettis theorem, Theorem 1.2.6.20, to (3.431) yields that Tt has a kernel representation, i.e. Ttu{x)=
f
u(y)Kt(x,y)dy,
Kt(x,y)>0,
(3.432)
and esss\ipKt{x,y)
< c\t~N?2
(3.433)
Clearly, when we have instead of (3.429) a Gdrding-type inequality \\ufLP < c2£\(u, u) = c2(£(u, u)+X(u, u) 0 ),
A > 0,
(3.434)
we obtain instead of (3.431) ||T«|U.-L~<<&(A)p^
(3.435)
ewsupK^x,y) < c 4 ( A ) - ^ .
(3.436)
and
Obviously, (3.431) and (3.435) (or (3.433) and (3.436)) are diagonal estimates (for Kt(x,y)). Now, in the case of diffusion semigroups, i.e. sub-Markovian semigroups with a local generator, from (3.431) or (3.435) one may deduce using Dames' method, see E.B.Davies [61] or [62], to derive off-diagonal estimates for Kt(x,y). In [44] E.Carlen, S.Kusuoka and D.Stroock gave a certain extension of Davies' method to non-local generators, i.e. non-diffusion semigroups. But to our best knowledge nobody was able to solve in a non-trivial concrete situation of interest to us the final optimization problem. Thus in case for non-local generators we have so far only diagonal estimates to our disposal and they follow almost easily from a well documented theory. For this reason we do not provide proofs for these results here. A straightforward application of Theorem 3.6.1 to symmetric L 2 -subMarkovian semigroups generated by a pseudo-differential operator —q(x, D) is possible when using (3.434). For a lot of examples in Chapter 2 we have proved the Gdrding inequality ||«||ff*.i
AO>0,
(3.437)
Chapter 3
412
Potential Theory of Semigroups and Generators
for some continuous negative definite function i> : M" —> R, and B given on C£°(M n ;R)by B(u, v) = (q(x, D)u, v)0.
(3.438)
Thus if i T M ( R n ; R ) is continuously embedded into L P (R";R), p > 2, i.e. if we have for some p > 2 the estimate IM|ip, i.e. iKO £ cblf I** for |£| large, CQ > 0, r 0 e (0,2],
(3.439) However,
(3.440)
immediately implies (3.439), compare Theorem 1.3.10.12. It seems that this is the best we can at the moment get from the general theory. However, there is a fine comparison result due to M.Tomisaki [265] and interesting examples can be provided. We state first Tomisaki's result in the formulation of [132]. Theorem 3.6.2 (M.Tomisaki). Let (£W,D(£^)) and (£<2>,D(£(2)J) be two regular, symmetric Dirichlet forms on L 2 (R n ;R) and denote by (T; ')t>o and (Tj )t>o, respectively, the corresponding L2-sub-Markovian semigroups. Assume that D{£^>) C D(£^) and that for CQ > 0 we have £w(u,u)
(3.441)
Moreover, suppose that
\\TtW\\L^L~
(3.442)
for allt > 0 with a right continuous nonincreasing function g such that
H(s):=
f Jw)d'<0°
(3 443)
'
for all s > 0, where G is the left continuous inverse of g. Suppose further that {Tl )t>o is conservative, i.e. T{ '1 = 1 a.e. for all t>0. Then we have ||Tt(2)||L1-L~<2/i(^), where h is the inverse function of H.
(3.444)
3.6 Nash-Type and Sobolev-Type Inequalities — a Short Outline Let us discuss this result first with g(t) = 'jot The function g has the inverse function
413
K
, t > 0, with some K > 0.
G(t)=iy*t-v* and the function H becomes
H{t) =
K^r1^,
leading to h(t) = KKj0t~K = KKg(t). Thus we have the two corollaries: Corollary 3.6.3. In the situation of Theorem 3.6.2 suppose that for some K > 0 it holds F t ( 1 ) | k i - L ~ < 7ot" K
for all t > 0.
Then it follows that ||2f'Uii.i- < 2 ( ^ ) \ t - «
(3.445)
holds for all t > 0. Corollary 3.6.4. In the situation of Theorem 3.6.2 suppose that for some K > 0 we have -t-K < ||r t ( 1 ) || L 1 _ L ~ < l0t-K 7o for all t > 0. Then it follows that
(3.446)
||T/ 2) || L1 _ L0 o < 2 ( ^ - ) K 7 o 2 | | T t ( 1 ) | | L 1 _ L ~ holds for
(3.447)
allt>0.
Clearly, if we have in addition a lower bound for ||Tt tion 11—> 71 t~K, 71 > 0, we also get the estimates m\\Tt{1)\\Li-L-
< \\Tt{2)\\Li-L-
< V2\\Tt{1)hi-L°°,
||LI-I,°°
by the func(3.448)
with suitable constants r/i,T]2 > 0, provided (2^ )t>o is also a conservative semigroup. We want to apply these result to a special example which fits to the class introduced with W.Hoh in [131], compare Example 1.4.7.31. Our example is taken from the joint paper [132] with W.Hoh. We need some preparations.
Chapter 3 Potential Theory of Semigroups and Generators
414
Lemma 3.6.5. Assume f : R+ —• IR+ is an increasing function such that f(x) > 0 for all x > 0. Moreover, assume that the Laplace transform £/(*) = Jo e~txf(x)dx exists for all t > 0. Then there is a continuous negative definite function ip : K —> K+, V'(O) = 0> suc/i that the corresponding semigroup (Tt)t>o of operators associated with the convolution semigroup (vt)t>o °f measures given by vt(£) = {2^)-1'2e~t^)
(3.449)
satisfies ||T t ||jri_ L « = £ ( / ) ( « ) ,
t>0.
(3.450)
Proof: Let F(x) = /Qx /(s) ds for all x > 0. Then F is continuous, increasing and convex on [0, oo). Furthermore, F(Q) — 0 and lim F(x) — oo and therefore F maps [0, oo) bijectively onto itself. Define ^(0 = -F"1(^),
ee».
(3.451)
Then ^ is continuous, even and restricted to [0, oo) it is increasing and concave, hence by Polya's theorem, Theorem 1.3.5.22 combined with Corollary 1.3.6.10, it follows that ip is a continuous negative definite function. Moreover, C f{t) = J" e~txf(x)
dx = - f"
e-tp"
«/»> de = ^ - / c-**(« d£. (3.452)
Thus, £ H-> e - ' ^ is integrable for all t > 0 and | | r t | | L i _ L - = sup J - / e ^ e - * ^ dZ = ±- ( e-**«> d* = £ /(*).
•
Now, for p > 0 consider the function fp : R+ —> R+ given by rP*
(3 453)
^=£ifeif§fe+i)
-
which converges for all x > 0 and for the Laplace transform of fp we find oo
„k
-,
£
°°
1
1
1
'««) = ,£_ -H ^r(^ (R3+ii) + i), w " E_ s'""" - V'-
<3-454>
3.6 Nash-Type and Sobolev-Type Inequalities — a Short Outline
415
Denote by ipp : R —> R+ the continuous negative definite function associated with fp by Lemma 3.6.5. Suppose pi,.. .,pn > 0 are given. Then a Dirichlet form is given on L 2 (R"; R) by «
£(u,v) =
n
$>ft.&)fi(0«(iK,
(3.455)
and n
D(£) = H^\Rn;R),
l^(0 = £lMfc)-
( 3 - 456 )
i=i
Since ipPj (0) = 0 the corresponding semigroup is conservative. Now, let ipj : R —> R, j = 1 , . . . , n, be continuous negative definite functions satisfying 1>i ( 0 ) > 1>Pj & )
for all fc G R, Vi(0) = 0.
(3.457)
It is clear that the Dirichlet form (£^, £>(£*)) given by
£*(u,v)= [ J2mj)H0W)d!;,
(3.458)
and D(£*)=H+>1{Rn;R),
tf(0
= £ > & ) .
(3-459)
give rise to a semigroup (Tt)t>o we may apply Theorem 3.6.2 too. Hence we find \\1?hi-L-=
n<-)-'/R. J
|(2TT)- A
n
/
e " ^ ^
.
d£j
n^/«w=n(K"'i)'
J=I
||2?||
J=I
416
Chapter 3 Potential Theory of Semigroups and Generators
Of course, it is now straightforward to produce an example with variable coefficients: take the Levy-Khinchin, i.e. the Beurling-Deny representation of 5 * and insert bounded coefficients bounded uniformly from below by a strictly positive constant.
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Author Index Adams, D. 10 Agmon, S. 4, 33 Albeverio, S. 299 Aronson, D. 7 Baldus, F. 8, 147, 150, 198, 204 Bass, R.F. 7, 8, 214, 218 Bauer, H. 301, 318 Beals, R. 151 Benedeck, A. 317 Bennett, C. 135 Berg, Chr. 12, 391, 393, 397-401 408 Bertoin, J. 8 Bliedtner, J. 214 Blumenthal, R. 8 Bochner, S. xix, xxi, xxii Boettcher, B. viii Bokobza, J. 204 Bouleau, N. 361 Brenner, D. 292 Browder, F.E. 4, 9, 33, 46 Burckel, R. 9, 214 Calderon, A.R 5, 113 Caps, O. 151 Carasso, A. 245 Carlen, E. 411
Carleson, L. 214 Casteren, J.,van 166 Chernikov, V.M. 65 Choquet, G. 8, 214 Chung, K.L. 306 Clarkson, J.A. 232 Coulhon, T. 410 Courrege, Ph. xxi, 5 Cowling, M. 10 Danilov, V.G. 147 Dappa, H. 292 Davies, E.B. 11, 278, 339, 345, 411 Davies, I.M. viii de La Pradelle, A., see Pradelle, A., de La Dellacherie, C. 8, 9, 214, 220 Demuth, M. 166 Doob, J.L 312 Doob, J.L. 300 Doughs, A. 4, 33 Dudley, R.M. 318 Dunford, N. 322 Dynkin, E.B. xix, 176 Edmunds, D.E. 4, 17, 23 Egorov, Yu. 23 Evans, W.D. 4, 17, 23
444
Fabes, E.B. 7, 176 Faradzheva, M.R. 7 Farkas, W. vii, 6, 9, 10, 12, 134, 224, 271 Fefferman, C. 65, 151, 410 Feller, W. xix Feyel, D. 270 Forst, G. 12, 391, 397, 400, 408 Frehse, J. 9 Friedman, A. 143, 176, 195 Frolovitchev, S.M. 147 Frost, G. 398-400 Fukushima, M. xix, 8, 9, 11, 12, 220, 238, 244, 245, 268, 337, 350, 364, 386, 410 Gagliardo, E. 17 Getoor, R. 8 Gianazza, U. 65 Gilbarg, D. 23, 27 Gol'dshtein, V. 410 Gramsch, B. 147 Grigoryan, A. 410 Grubb, G. 199 Gruschin, V.V. 62 Gundy, R. 306, 307, 314 Hansen, W. 214 Havin, V. 10 Hedberg, L. 10 Hirsch, F. 12, 300, 361, 399, 407-409 Hoh, W. vii, viii, 5-8, 10, 81, 86, 88, 119, 128, 151, 155, 161, 163, 165, 198, 199, 204, 272, 387, 389, 413 Hopf, E. 322 Hormander, L. 4, 5, 61, 65, 147, 230, 286-288
Author Index
Ikeda, N. 4 Il'in, A.M. 176 Ito, M. 12, 399, 407 Ito, S. 143, 176 Ito, K. xix Iwasaki, C. 143, see also Tsutsumi, C. Jacobs, K. 214 Jacod, J. 6 Kalashnikov, A.S. 176 Kassmann, M. 410 Kato, T. 245 Kazumi, T. 270 Kikuchi, K. 8, 204 Kinderlehrer, D. 27, 266 Knopova, V. viii Kochubei, A.N. 143-146 Kohn, J.J. 5, 61, 63 Kolmogorov, A. xix Kolokoltsov, V.N. 144, 146, 147 Komatsu, T. 5, 8 Kondratiev, V. 23 Krengel, U. 11 Kumano-go, H. 97, 139, 140, 143 Kusuoka, S. 411 La Pradelle, A., de see Pradelle, A., de La Landkof, N.S. 8 Lax, R 254 Ledoux, M. 410 LeJan, Y. 11 Leopold, H.-G. 8, 12, 204 Levi, E.E. 139, 143, 195 Long, R. 10 Louhivaara, I. 208
Author Index
445
Lunardi, A. 3 Levy, P. xix
Rudin, W. 232 Rockner, M. viii, 11, 361
Ma, Z.-M. 11, 299, 361 Malliavin, P. xix, 4, 238 Martinez Carracedo, C. 7 Maslov, V.P. 141 Maz'ya, V. 10, 17 McKean, H. xix Menaldi, J. 5 Meyer, P.-A. 8, 9, 214, 220 Meyers, N. 15 Mikulevicius, R. 5 Minty, G. 9, 264 Mizuta, Y. 380
Saloff-Coste, L. 65, 410 Samko, S.G. 145, 146, 326 Sanz Alix, M. 7 Sato, K.-I. 8, 357 Schechter, M. 4 Schilling, R.L. vii, viii, 6-11, 134, 153, 161, 201-203, 224, 271, 293, 337, 354, 389, 408 Schwartz, J.T. 322 Serrin, J. 15 Shaposhnikova, T. 17 Sharpley, R. 135 Shigekawa, I. 270 Shiryaev, A. 6 Shishmarev, LA. 141 Simader, Chr. 33, 208 Sogge, Chr. 23 Stampacchia, G. 27, 266, 410 Stegeman, J. 401 Stein, E.M. 9-11, 61, 65, 213, 245, 284, 300, 301, 303, 306, 307, 314, 317, 325, 327, 331, 332, 379, 380 Stroock, D.W. 5, 7, 10, 65, 176, 411 Sturm, K.-Th. 166, 410
Nagel, A. 65 Nash, J. 192 Negoro, A. 8, 204 Nirenberg, L. 4, 5, 17, 33, 178 Oleinik, O.A. 70, 176 Oshima, Y. 8, 9, 11, 12, 238, 337, 342, 354, 364, 385-387 Ouhabaz, E.M. 254 Panzone, R. 317 Parmeggiani, A. 410 Phong, D.H. 65 Pradelle, A., de La, 270 Pragarauskas, H. 5 Radkevic, E.V. 70 Reiter, H. 401 Rogers, C.A. 215 Rota, G.C. 318 Rothschild, L.P. 61 Rozovskii, B. 5 Rubin, B. 326
Takeda, M. 9, 11, 12, 238, 337, 364, 386 Tokarev, A.G. viii, 199 Tomisaki, M. 410, 412 Trebels, W. 292 Triebel, H. viii, 12, 167, 294, 300 Troyanov,M. 410 Trudinger, N.S. 23, 27 Truman, A. viii
Author Index
446
Treves, F. 61 Tsuchiya, M. 8 Tsutsumi, C. 143, see also Iwasaki, C. Unterberger, A. 204 Vaillancourt, R. 113 Varopoulos, N. 410 Vespri, V. 65 Vodop'yanov, S.K. 65 Voigt, J. 339 Wainger, S. 65
Watanabe, S. 4, 390 Westphal, U. 282 Wiener, N. xix Xu, C.-J. 65 Ye Qiang viii Yosida, K. xix, 12, 232, 254, 390, 391, 394, 395 Zeidler, E. 9, 259, 265 Ziemer, W. 25, 178 Zygmund, A. 5, 314
Subject Index 7-admissible weight function, 148 7-slowly varying function, 147 V'-Bessel potential space, 275 abstract potential operator, 394 .A-harmonic function, 345 analytic set, 214 approximating sequence, 364 Aronson estimates, 176 Aronson's theorem, 191 Banach-Saks theorem, 232 Besov spaces, 167 Bessel potential type space, 226 bounded (nonlinear) operator, 263 Calderon-Vaillancourt theorem, 116 capacitable set, 217 capacity -(r,p),232 - Choquet, 217 - inner Choquet, 217 - outer Choquet, 217 Carlson-Beurling inequality, 292 chain rule in Sobolev spaces, 27 Chapman-Kolmogorov equations, 142, 175 Chebyshev-type inequality, 238
Choquet capacity, 217 - inner, 217 - outer, 217 class, see also symbol class -A, 94 - A , 89 classical symbol, 92 coercive - functional, 259 - operator, 261 commutator - field, 58 - iterated, 60 - of operators, 21 complete Bernstein function, 195 complex interpolation space, 294 condition - H.ro, 139 - Hormander, 60 - (5), 263 conditional expectation, 301 conservative - part, 356 - semigroup, 357 contraction regular space, 236 convex function - logarithmically, 409 - strictly, 259
448
convex functional, 259 convolution semigroup - potential kernel, 398 - transient, 397 core of an operator, 127 Cotlar-type lemma, 113 Davies' method, 411 demicontinuous operator, 263 derivative - fractional, 326 diagonal estimates, 411 Dirichlet form - local, 344 - transient, 363 Dirichlet semigroup, 361 Dirichlet space - extended, 358, 364 - transient extended, 372 dissipative part, 356 double symbols, 97 embedding, see Sobolev's embedding theorem equation - Chapman-Kolmogorov, 142, 175 equilibrium potential, 233, 242 estimate - Aronson, 176 - diagonal, 411 - off-diagonal, 411 - sub-elliptic in the strict sense, 54 - uniformly sub-elliptic, 54 exceptional set, 237 expectation, 301 - conditional, 301
Subject Index
extended Dirichlet space, 358, 364 - transient, 372 Feynman-Kac formula, 166 Fourier multiplier of type (p, q), 286 fractional - derivative, 326 - integral, 326 Friedrichs - mollification, 15 - symmetrization, 106 function - (r, p)-quasi-continuous, 237 - (r, p)-quasi-continuous in the restricted sense, 237 - yl-harmonic, 345 - 7-admissible weight, 148 - 7-slowly varying, 147 - invertible sub-7-admissible weight, 148 - Littlewood-Paley, 314, 324, 325, 332 - logarithmically convex, 409 - maximal, 303 - quasi-continuous, 237 - quasi-continuous in the restricted sense, 237 - sub-7-admissible weight, 148 functional - coercive, 259 - convex, 259 - Gateaux differentiable, 259 - strictly convex, 259 fundamental solution, 141, 175 Gagliardo-Nirenberg theorem, 17 gamma-transform, 224
Subject Index
Garding inequality, 121, 411 - sharp, 120, 140 - type, 411 Gateaux differentiable functional, 259 generalized Holder inequality, 178 global property of a semigroup, 337 Green operator, 349 Gruschin operator, 62 Hamiltonian, relativistic, 165 hemicontinuous operator, 263 Holder inequality, generalized, 178 Hopf's maximal ergodic inequality, 350 Hormander - condition, 60 - uniform, 60 - metric, 148 - multiplier theorem, see Michlin-Hormander - theorem (sums of squares), 61 - type operator, 54, 57 hypersingular integral operator, 145 - characteristic of, 146 inequality - Carlson-Beurling, 292 - Chebyshev for capacities, 238 - Garding, 121, 411 - sharp, 120, 140 - type, 411 - generalized Holder, 178 - Hopf's maximal ergodic, 350 - Nash-type, 192, 410 - Poincare, 22 - Sobolev, 177, 379 - type, 410
449
- Stein's Littlewood-Paley, 332 integrable, 404 integral - fractional, 326 - oscillatory, 94 integral operator, hypersingular, 145 intermediate estimate for Sobolev spaces, 17 interpolation - couple, 294 - space complex, 294 - theorem of Marcinkiewicz, 380 invariant - set w.r.t an operator, 339 - strongly, 339 - set w.r.t. a semigroup, 341 - strongly, 341 invertible sub-7-admissible weight function, 148 irreducible semigroup, 344 Kac, see Feynman kernel - potential, of a convolution semigroup, 398 - Riesz, 378 Kohn-Laplacian, 63 Kolmogorov, see Chapman Ka-set, 214 K„s-set, 214 Laplace transform type, 335 Leibniz rule in Sobolev space, 16 lemma of Cotlar-type, 113 Lipschitz function, operate, 268 Littlewood-Paley function, 314, 324, 325, 332
450 Littlewood-Paley inequality, E.M.Stein's, 332 local Dirichlet form, 344 logarithmically convex function, 409 LP-solution, 42 - weak, 42 Marcinkiewicz interpolation theorem, 303, 380 martingale, 307 - maximal theorem, 305 - property, 307 - stopped, 307 maximal - function, 303 - Hopf's ergodic inequality, 350 - martingale theorem, 305 measure - shift-bounded, 404 - vanishing at infinity, 404 metric - Hormander, 148 - on R m , 147 - slowly varying, 147 - splitting, 147 Meyers-Serrin theorem, 15 Michlin-Hormander multiplier theorem, 36 mollifier, see Friedrichs mollifiers monotone operator, 261 - strictly, 261 - uniformly, 261 multiplier - Fourier of type (p, q), 286 - Michlin-Hormander, 36 Nash-type inequality, 192, 410
Subject Index negative definite symbol (continuous), 66 Nirenberg, see Gagliardo non-locality of an operator, 88 normal contraction, 359 off-diagonal estimates, 411 operator - abstract potential, 394 - bounded (nonlinear), 263 - coercive, 261 - commutator, 21 - core of, 127 - demicontinuous, 263 - Green, 349 - Gruschin, 62 - hemicontinuous, 263 - Hormander-type, 54, 57 - hypersingular integral, 145 - characteristic of, 146 - Kohn-Laplace, 63 - monotone, 261 - potential, 349 - pseudo-differential, 93 - pseudo-differential of Weyl-type, 149 - pseudo-differential with negative definite symbol, 5 - radially continuous, 262 - strictly monotone, 261 - sub-elliptic, 54 - sub-Laplacian, 63 - uniformly elliptic, 32 - uniformly monotone, 261 - Weyl-type pseudo-differential, 149 Ornstein-Uhlenbeck semigroup, 185 oscillatory integral, 94
Subject Index Paley, see Littlewood part - conservative, 356 - dissipative, 356 partition of unity, 33 Poincare's inequality, 22 Polish space, 213 potential - kernel of a convolution semigroup, 398 - operator, 349 - abstract, 394 - space, Bessel-type, 226 process - recurrent, 348 - stable-like, 8 - transient, 348 property - global of a semigroup, 337 - martingale, 307 - truncation, 269 pseudo-differential operator, 93 - Weyl-type, 149 - with negative definite symbol, 5,66 quasi - continuous function, 237 - in the restricted sense, 237 - continuous modification, 237 - everywhere, 237 - uniform convergence, 242 Rademacher function, 313 radially continuous operator, 262 recurrence of a process, 348 recurrent - process, 348
451 - semigroup, 349 regular - nest, 239 - space, 236 - weakly, 236 relativistic Hamiltonian, 165 resolvent, zero order, 394 reversed martingale, 318 Riesz kernel, 378 (r,p) - capacity, 232 - capacity function, 233 - equilibrium potential, 233, 242, 298
- exceptional set, 237 - nest, 239 - quasi-continuous function, 237 - in the restricted sense, 237 - quasi-continuous modification, 237 - quasi-everywhere, 237 - quasi-uniform convergence, 242 - regular nest, 239 scaling, 189 semigroup - conservative, 357 - Dirichlet, 361 - global property of, 337 - irreducible, 344 - Ornstein-Uhlenbeck, 185 - recurrent, 349 - strong L p -sub-Markovian, 246 - transient, 349 - transient convolution, 397 set - (r, p)-exceptional, 237
452 - (Tt) t >o-invariant, 341 - strongly, 341 - Ka, 214 - KoS, 214 - T-invariant, 339 - analytic, 214 - capacitable, 217 - invariant w.r.t an operator, 339 - strongly, 339 - invariant w.r.t. a semigroup, 341 - strongly, 341 - Souslin, 214 sharp Garding inequality, 120 shift-bounded measure, 404 simplified symbol, 98 singular support, 61 Slobodeckij, see Sobolev slowly varying metric, 147 Sobolev - embedding theorem, 24 - inequality, 177, 379 - type, 410 - Slobodeckij spaces, 167 - space, 14 solution - LP, 42 - fundamental, 141, 175 - strong, 85 - variational, 84, 125 - weak Lp, 42 Souslin sets, 214 space - V'-Bessel potential, 275 - Besov, 167 - Bessel potential type, 226 - complex interpolation, 294 - contraction regular, 236
Subject Index - extended Dirichlet, 358, 364 - transient, 372 - Polish, 213 - regular, 236 - weakly, 236 - Sobolev - chain rule, 27 - intermediate estimate, 17 - Leibniz rule, 16 - Sobolev-Slobodeckij, 167 spectral invariance, 150 splitting metric, 147 stable-like processes, 8 standard symplectic form, 147 Stein theorem, 333 Stein's Littlewood-Paley inequality, 332 Stein's maximal theorem, 331 stopped martingale, 307 stopping time, 306 strictly convex functional, 259 strictly monotone operator, 261 strong - (Tt)t>o-invariant set, 341 - L p -sub-Markovian semigroup, 246 - T-invariant set, 339 - solution, 85 sub-7-admissible weight function, 148 sub-elliptic - estimate, 54 - operator, 54 sub-Laplacian, 63 sum of squares of vector fields, 57 support, 258 symbol - classical, 92
Subject Index - double, 97 - negative definite (continuous) 66 - simplified, 98 symbol class -A, 94 - A , 89 - 5 ( M , 7 ) , 149 -S?'*(Rn),90 - S f t (R n ), 92 -S^(R»),139 - S ^ > * ( R n ) , 89 symmetrization - Priedrichs, 106 symplectic form - standard, 147 theorem - Aronson, 191 - Calderon-Vaillancourt, 116 - Gagliardo-Nirenberg, 17 - Hormander (sums of squares), 61 - Marcinkiewicz, 303, 380 - martingale maximal, 305 - Meyers-Serrin, 15 - Michlin-Hormander multiplier, 36 - Sobolev embedding, 24 - Stein, 333 - Stein's maximal, 331 time, stopping, 306 transform - gamma, 224 - Laplace-type, 335
453 transience of a process, 348 transient - convolution semigroup, 397 - Dirichlet form, 363 - extended Dirichlet space, 372 - process, 348 - semigroup, 349 truncation property, 269 (7<)t>o-invariant set, 341 - strongly, 341 T-invariant set, 339 - strongly, 339 Uhlenbeck, see Ornstein uniform - elliptic operator, 32 - Hormander condition, 60 - monotone operator, 261 - quasi convergence, 242 - sub-elliptic estimate, 54 uniformly convex, 232 uniformly elliptic operator, 32 Vaillancourt, see Calderon vanishing at infinity measure, 404 variational solution, 84, 125 weak //-solution, 42 weakly regular space, 236 weight function - 7-admissible, 148 - sub-7-admissible, 148 Weyl-type pseudo-differential operator, 149 zero order resolvent, 394
PSEUDO-DIFFERENTIAL OPE AND MARKOV PROCESSES Generators and Their P o t e n t i a l Theory
Volume II
In this volum
ion of Feller and L -sub-Markovian semigroups by starting with a p
pseudo-differential operator, and the potential theory of these semigroups and their generators. The first part of the text essentially discusses the analysis of pseudo-differential operators with negative definite symbols and develops a symbolic calculus; in addition, it deals with special approaches, such as subordination in the sense of Bochner. The second part handles capacities, function spaces associated with continuous negative definite functions, Lp -sub-Markovian semigroups in their associated Bessel potential spaces, Stein's Uttlewood-Paley theory, global properties of Lp-sub-Markovian semigroups, and estimates for transition functions.
ISBN 1-86094-324-1
Imperial College Press www.icpress.co.uk
