0, s——=--—, r p
(11.18)
(the dashed line in Fig. 10.1). Then C
if, and only if,
0< q r,
(11.19)
and
forall 0
(Critical case)
(11.20)
Let
O
(11.21)
Then
if, and onlyif, O
(11.22)
0
(11.23)
and
if,andonlyif, In (11.22) and (11.23) one can replace (iii) (Super-critical case) Let
by
0
(11.24)
11. Classical inequalities
171
(the dotted line in Fig. 10.1). Then
if, and onlyif, O
(11.25)
if,andonlyif, O
(11.26)
and
In (11.25) and (11.26,) one can replace by Lip(R"). Proof (of part (iii)) Detailed of parts (1) and (ii) may be found in [S1T95]. Theorems 3.2.1 and 3.3.1, Remark 3.3.5; short descriptions are given in [RuS96], 2.2, and in [ET96J, 2.3.3. Part (iii) is essentially the lifting of part
(ii) by 1. But this is by no means obvious and must be justified. First we remark that I E Fpq "(W')
if, and only if,
IE
a
and
Of
E
(11.27)
where j =
1,.
..
,n
and (equivalent quasi-norms)
(R")jI
If
1F2(Rn)O
+
.
(11.28)
We refer to [Tri$], Theorem 2.3.8, pp. 58/59. Hence, the if-part of (11.26) (11.23). Conversely, assume
follows from
forsome O
(11.29)
\Ve (onstrmlet special Fourier multipliers and introduce some cones in
= where ol)Viously
=
E
= (h,.
. .
>0,
:
,
1) E
t >0, Let
be a C°° function in
(11.30) {0}
it Ii
if
and
be a COC function in R'2 which is identically 1 if Then, by [Trifl], Theorem 2.3.7 on p. 57, Let
(11.31)
1 and 0
where j=i,...,n,
suppt1?.'.
(11.32)
II. Sharp inequalities
172
and
are Fourier multipliers in all spaces with
Let (11.33)
:
and
=
g(x) =
XE
where we used the notation introduced in 2.8. Since pliers it follows by (11.28) that
(11.34)
are Fourier multi-
(11.35)
and
Hence if we assume (11.29) for some p and q, then it follows for functions f with (11.33) that f E and Ill
IE
can be decomposed in finitely many func, where has the tions of type (11.33) and a harmless function ((1 — above meaning. Then we have (11.36) for those p,q with (11.29). This is the converse we are looking for, and it proves (11.26). Finally we must show that in (11.26) by Lip(IR"). Since Lip(R7') is the larger one can replace space we must disprove (11.37)
C
if p> 1 and 0 <
oo. By the monotonicity of the spaces Fpq with is dense in Fpq (R") and respect to q we may assume q < oc. Then By completion it follows (11.37) with Cl(Rnl) in we have (11.7) for f E But this contradicts (11.26) since p> 1. The proof for the place of B-spaces is the same. q
1
11.5
D.
Remark
is called the differential dimension of the spaces and is the differential dimension of Lr(RTh). This notion In particular, (differencan obviously be extended to the above target spaces Lip(PJ') (differential dimension 1). Continuous tial dimension 0) and C' embeddings between function spaces with the same differential dimension are
Usually, s —
11. Classical inequalities
173
often called limiting embeddings. The above theorem deals exclusively with embeddings of this type in the indicated specific situations: 8ub-crlt2cal:
critical:
differential dimension —
differential dimension 0,
8tLper-cntical:
differential dimension 1.
Furthermore, (10.18) is now an immediate consequence of the above theorem. As explained in Section 10, in connection with the growth envelope in 1+ P) lii (10.20) (10.13) for the critical case and the continuity envelope for the super-critical case, we are interested only in those spaces which are not covered by (10.18), this means by the parts (ii) and (iii) of the above theorem in the spaces 1+11
11
Bq(R"),
with O
(11.38)
with 1
(11.39)
and a
1+a
Fpq
If q =
2
0
in (11.39) then we get by (1.9) the Sobolev spaces a
with
1
(11.40)
On the other hand, we have the famous Sobolev embedding
H;(lIr)cLr(R"), l
(11.41)
as a special case of (11.20). But it is just the failure to extend (11.41) from the sub-critical case 1 < r < cc, to the critical case r = cc, which triggered the search for adequate substitutes. Then we are back to the 1960s. At the same time refinements of sub-critical embeddings according to part (i) of Theorem 11.4 for the Sobolev spaces H and the classical Besov spaces in terms of Lorentz spaces have been studied (in the West inspired by
interpolation theory). Around 1980 further refinements in the critical case and the first steps in the super-critical case for the Sobolev spaces in (11.40) were taken. We shall describe this nowadays historical part of these refined embeddings in Theorem 11.7 below, including in 11.8 the respective references.
Mostly for this reason we discuss in 11.6 the relevant target spaces, whereas later on we prefer to formulate our results in terms of inequalities.
II. Sharp inequalities
174
11.6
Lorentz-Zygmund spaces
For the reasons just explained we restrict ourselves to a brief description. The standard reference for Lorentz spaces and Zygmund spaces is [BeS88]. Their combination, the Lorentz-Zygmund spaces, were introduced in [BeR8O].
Let 0 < e < 1 and let
=
(0,
ej. We use the rearrangement f(t) of a
complex-valued measurable function 1(t) on as introduced in (10.2), (10.3), temporarily with in place of Lorentz spaces Let 0 < r < 00 and 0 < u oo. Then Lru(15) is the set (i) of all measurable complex-valued functions f on such that E
f
i
j
if 0
(11.42)
and sup t*f*(t) <00
if
u = 00.
(11.43)
tE Je
Of course, L,-r(Ie) = Lr(Ie) with 0 < r < oo, are the usual Lebesgue spaces. (ii) Zygmund spaces Let 0 < r < oo and a R. Then Lr(log L)a(IE) jS the set of all measurable complex-valued functions f on 4 such that
I
If(t)Ir logar(2 + If(t)I) dt < 00.
(11.44)
is the set of all measurable complex-valued Let a < 0. Then functions f on 4 such that there is a number A > 0 with
dt <00.
(11.45)
When r < 00, then this notation resembles that in [BeS88], pp. 252 The alternative notation Lexp,_a(4) for is closer to that employed in [BeS88]. The somewhat curious looking expression (11.45) may be justified by the following unified alternative representation, where f E if, and only if,
Ire
(J
/
<00,
when 0 < r < oc,
a E R,
(11.46)
and
when r=oo, a<0.
(11.47)
11. Classical inequalities
175
We refer to [BeS88], p.252, or [ET96], p.66. In [ET96], 2.6.1, 2.6.2, one finds also another unifying representation, further properties and references. This way of looking at these spaces fits also in the scheme developed in the following Section 12. Both (11.46) and (11.47) are quasi-norms. (iii) Lorentz-Zygmund spaces The combination of the above Lorentz spaces and Zygmund spaces results in the Lorentz-Zygmund spaces studied in detail in [BeR8OI. Let
0
is the set of all measurable complex-valued functions f
on 4 such that -L
j
tI
—
<00
if 0
(11.48)
u=oo.
(11.49)
and
if tE Ic
Again (11.48) and (11.49) are quasi-norms. Note that
Lrr(logL)a(Ie) = Lr(logL)a(Ie) where 0 < r < oo,
a E IR,
and
Lru(logL)o(Ie) =
where 0< r <00, 0
For details we refer to [BeS88], p. 253, and, in particular to [BeR8O]. This If reference covers also the interesting case r = oo, hence r = u = oo then (11.49) coincides with (11.47), and one needs a < 0. If r = 00 and 0 < u < oo, then one needs in (11.48) that au < —1, hence a + <0. Otherwise, if r = oo and au —1, then there are no non-trivial functions f with (11.48). Some well-known embeddings between these spaces will come out later on in 12.4 in a natural way. We are not so much interested in the Lorentz-Zygmund spaces for their own sake. We formulate later on our results in terms of inequalities using rearrangements. This will also be done in the next theorem where we collect the historical roots of the theory outlined in the introductory Section 10, although the original formulations looked sometimes quite different. This applies in parin the original versions (11.44), (11.45) ticular when the spaces are involved. Not only the spaces themselves but also their equivalent char-
acterizations via (11.46), (11.47) can be traced back to Hardy-Littlewood,
II. Sharp inequalities
176
Zygmund, Lorentz and Bennett. The corresponding references may be found in the Note Sections in [BeS88J, pp. 288, 180—181, and also in [BeR8O], in con-
nection with Corollary 10.2 and Theorem 10.3. This resulted finally in the Lorentz-Zygmund spaces quasi-formed by (11.48), (11.49) in [BeR8O]. In particular, all the equivalent (quasi-)norms mentioned above were known around 1980 (and often long before). Recall that are the Sobolev spaces, 1 0, according to as special cases. The classical (1.9) with the classical Sobolev spaces Besov spaces are normed by (1.14). The divided differences t) are given by (10.16). Let 0 < e < 1. 11.7
Theorem
(Classical refined inequalities in limiting situations) (i) (Sub-critical case, Lorentz spaces, dashed line in Fig. 10.1) Let s > 0,
l
(11.50)
Then there is a constant c> 0 such that I
f
(I (t*
c
f
(11.52)
(with (11.43) if q = oc). (ii) (Sub-critical case, Zygmund spaces, dashed line in Fig. 10.1) Let s > 0,
l 0 such that 1
(I
I log
fsr(t) dt)
c fif
I
(11.54)
11. Classical inequalities
(Critical case)
(iii)
177
Let
l<ji
(11.55)
Then there is a constant c> 0 such that sup O
a
'
c
a
for all f E H,," (Rn),
(11.56)
and
'p
IC
7'dt\
(
(iv)
forall
(Super-critical case)
(11.57)
Let
l
(11.58)
Then there is a constant c> 0 such that sup
O
11.8
t)
c fif
1W(V)Il for all f E
(11.59)
Historical references and comments
We tried to collect in the above theorem those refined inequalities which we believe are the roots of the programme outlined in the introductory Section 10. "Refined" must be understood in comparison with Theorem 11.4 asking for a tuning of the admissible target spaces finer than used there. The spaces described in 11.6 may be considered as a reasonable choice for such an undertaking. A balanced or even comprehensive history of inequalities of this type seems to be rather complicated. Many mathematicians contributed to this flourishing field of research, and parallel developments in the East (in the Russian literature) have often passed unnoticed in the West, but also vice versa. In a sequence of points, denoted as 11.8(i) etc., we collect related papers, comment on a few topics, and try to clarify to what extent the above inequalities fit in our scheme. We shift more recent references to a later occasion (with exception of a few surveys which in turn describe the history) and restrict ourselves to those papers which stand for the early development of this theory (although this covers a period of some 20 years). 11.8(i) (Sub-critical case, Lorentz spaces) The inequalities (11.51), (11.52) came into being in the middle of the 1960s with the advent of the interpolation
II. Sharp inequalities
178
theory: there was no escape, as we outline in the next point. But there are more direct approaches, especially in connection with a wider scale of Besov spaces (generalized moduli of continuity), vector-valued Besov spaces, and the question about the sharpness of these inequalities. We refer to [Pee66], [Str67], [Her68], [Bru72], [Bru76], [Gol85], [Go186], and the surveys [Ko189], [1
l1.8(li) (Sub-critical case, interpolation) We use without further explanaof two (quasi-)Banach spaces A0 and tions the real interpolation (A0, A1 A1, and 0 < 0 < 1, 0 < q cc. We refer to [Trkll, [BeL76] or [BeS88J, where one finds all that one needs. Recall that
(Lro,Lrj)0,p=Lrp,
1
O<poo, (11.60)
on
or on R", where Lrq are the Lorentz spaces from 11.6(1). Lifting of (11.60)
on RTh gives
l<po
sER. (11.61)
Furthermore, again on (B01,
= Bq,
0 < So <Si <00,
S = (1
—
0)s() +
(11.62)
0 < q ( cc. Now (11.20) with H = F,2, and the interpolations (11.61), (11.60) give (11.51), whereas (11.19), and the interpolations (11.62), (11.60) result in (11.52). 11.8(111) (Sub-critical case, Zygmund spaces) The inequality (11.54) is less satisfactory than the inequalities (11.51) and (11.52). We explain the reason
in the next point. In addition, the above Zygmund spaces are not naturally linked with the spaces and in the reasoning of Section 10 in sub-critical situations, in contrast to the Lorentz spaces. Although we could
not find a direct formulation of (11.54) in the literature, assertions of this type are apparently included (in a somewhat hidden way) in a larger theory of embeddings of the form
lp
(11.63)
stands for an Orlicz is a generalized modulus of continuity and space. This was studied in the 1980s in great detail in the Russian literature. In [Go186], Theorem 5.4, one finds necessary and sufficient conditions for embeddings of type (11.63). A corresponding formulation may also be found in where
11. Classical inequalities
179
[KaL87], Theorem 8.5, P. 27. This paper surveys what has been done by the Russian school of the theory of function spaces. With some modifications it is the English version of [Liz86] where the same result (11.63) may be found in D.1.8. At least in some cases L)a can be identified with an Orlicz space. We refer to [BeS88], p. 266, Example 8.3(e), with 1'(t) = t'jlogtl". Taking together all these remarks then some assertions of type (11.54) for classical Besov spaces are hidden in [Go186], [Liz86], [KaL87]. 11.8(iv) (Sub-critical case, Holder inequalities) As said in the previous
point, (11.54) with (11.53) does not fit optimally in our context. Furthermore, this estimate follows from (11.52) and Holder's inequality: Let again 0 < e < 1,
111
1 0 if 0 < t
r =—+—, U q
(11.64)
—
Then by Holder's inequality,
I
I
=
(/
dt) h(t)f*(t))
I
(/
.1
f*(t))"
The last factor with h(t) =
(11.65)
.
converges if, and only if, au < —1. This
proves the if-part of: 1
1
c
(/ (t*
(11.66)
if, and only if, a < — To disprove (11.66) if a a = — and, as a counter-example, 1(t)
,
—
0< t <
one may choose
e,
(11.67)
assuming that e > 0 is sufficiently small. But this makes clear that I log tI° in (11.54) is a distinguished but not natural choice. There are better functions h(t) such that the last factor in (11.65) converges. Maybe a systematic treatment in this direction would result in problems of type (11.63). Nevertheless we return to inequalities of type (11.54) later on in Corollary 15.4.
IL Sharp inequalities
180
11.8(v) (Critical case) The inequality (11.56) has a rich history reflecting especially well parallel developments in East and West. First we recall that by the equivalence of (11.47) and (11.45) for some A > 0, the left-hand side of (11.56) is finite if, and only if,
Jexp {(AIf(t)I)P'} dt <
for some
oo
A
> 0.
(11.68)
In this version, (11.56) is due to R. S. Strichartz, [Str72], including a sharpness assertion. Corresponding results for the classical Sobolev spaces
l
with
(especially if k =
1)
(11.69)
in the version of (11.68) had been obtained before in
and may also be found in [GiT77], Theorem 7.15 and (7.40) on p. 155. This paper by N. S. Trudinger made problems of this type widely known and influenced further development, in terms of spaces of type (11.68) and more general Orlicz spaces. For classical Sobolev spaces with the fixed norm (1.4), especially jfk = 1, hence it makes sense to ask for the best constant A in (11.68). This may be found in [Mos7l]. As noted above there was a parallel and independent development in the East. We refer in particular to [Poh65] and the even earlier forerunner [Yud6lJ. Best constants for the embedd.ings
of Sobolev spaces according to (11.69) in spaces of type (11.68), extending [Mos7lj to all k E N, may be found in [Ada88]. This paper contains also a balanced history of this subject, including the Russian literature. The natural counterpart of (11.56) for classical Besov spaces normed by (1.14), is given by
l
sup
(11.70)
= 1. Limiting embeddings of spaces in Orlicz spaces were considered by J. Peetre in [Pee66]. If one takes his assertion in [Pee66], Theorem 9.1 on p. 303, and reformulates it in terms of later developments in and
+
the 1980s and 1990s, and which may be found in [ET96], 2.6.2, then one arrives
at (11.70) or equivalently at (11.68) with q' in place of p'. As for (11.57) we first remark that
Ic sup
O
' ',
c
(
I
I
'
" /
\P dt )—I
tj
.
(11.71)
12. Envelopes
181
This is an assertion of type (10.11) and will be considered in detail in 12.4, especially (12.28). Hence (11.57) is sharper than (11.56). This improvement goes back to (Has79] and [BrW8O] (as a consequence of Theorem 2 on p. 781). including sharpness assertion in [Has79]. We refer in this context also to [Zie89]. 2.10.5. 2.10.6, pp. 100—103, where one finds improved and more detailed versions of the arguments in [BrW8O]. 11.8(vi) (Super-critical case) The first direct proof of (11.59) may be found in [BrW8O]. Corollary 5. p. 786. On the other hand, accepting (10.22), its ndunensional counterpart. and that in (11.51), (11.52), (11.56), (11.57), f* can be replaced by then all the inequalities in the super-critical case can be
obtained by lifting from the critical case, in particular (11.59) by lifting of a special case of (11.56), and a stronger and more general version of (11.59) by lifting of (11.57), an inequality which has also been proved in [BrW8O]. Maybe
this connection passed unnoticed not only at that time but also in recent research dealing separately with the critical and super-critical case. This close interdependence is also weU reflected by the parts (ii) and (iii) in Theorem 11.4, and by its short version (10.18). 12 12.1
Envelopes Rearrangement and the growth of flmctions
In this section we introduce the concept of growth envelopes and continuity envelopes as outlined in the introductory Section 10. We restrict ourselves to those spaces and which are of interest for us. On this basis we prove in the subsequent Sections 13—16 the main results of this chapter. The new notion of growth and continuity envelopes makes sense for a much wider range of function spaces and has been introduced and studied recently by D. D. Haroske in [HarO 11. We take over a few results obtained there, including the relevant notation of envelopes and envelope functions. 1'irst we recall again what is meant by rearrangement. For our purpose it is
sufficient to assume that f E Lr(R') with 1 (10.3). the distribution function of f are given by
r < oc. Then as in (10.2),
and the non-increasing rearrangement f
jf(z)I>A}I,
A0,
(12.1)
and
f*(t) =
t},
t 0.
We wish to measure the growth of functions f belonging to
(12.2)
and cit her in the sub-critical situation according to Theorem 11.4(1) or in
II. Sharp inequalities
182
the critical situation for those parameters p, q not covered by Theorem 11.4(ü). Let be such a space and let f E Then the behaviour of the
rearrangement I if t 0 indicates how singular the function f might be Inspired by the spaces in 11.8 we try to on the global scale, the whole of measure the possible growth of I in terms of
(J(f$(t)w(t))u
(12.3)
for some fixed 0 < 1, 0 0 if 0 < t <.e, and j
Recall that we use the equivalence sign in (7.10). Then we have
N, as
j
Jfr).
(12.4)
explained before, for example 1
.1
(f*(2_2)w(23)Y)
,
(12.5)
where the equivalence constants are independent of f. Hence (12.3) behaves including the well-known monotonicity with respect like a sequence space to u. The following reformulation of this type of singularity measurement might be helpful for a better understanding. We assume that the reader is familiar with the basic facts concerning rearrangement. They may be found in [BeS88]. In particular, rearrangement is measure-preserving,
J{t>0 :
(12.6)
:
where 0 < Ti < r0 < 00. We refer also to p. 132. Here "r0 " and/or respectively. Furthermore, can be replaced by "TO>" and/or the Lebesgue measure on is divisible: If M is a Lebesgue-measurable set in R" with 0 < Ml <00 and if A is a positive number with 0 < A < IMI, then there is a Lebesgue-measurable set MA with
Let again f E
(12.7) MACM and IMAI=A. as above. By (12.6), (12.7) there is a set M with
MI = 1 and
Mc{x€R" :
lf(x)If*(1)},
:
(12.8)
12. Envelopes
183
The set M can be decomposed by
1M31=2'ifjEN,
M= UM3,
(12.9)
3=' such
that M3 C {x
E
W'
f*(2_2+1)
If(x)I
j E N.
(12.10)
then we In other words, 11(12.3), (12.4) or (12.5) is finite for all f E get some information on how rapidly If(x)I might grow on a sequence of sets = 2.7+1 where M' = M3 or with IM,I = 2—' or
For a closer look at (12.3) and (12.5) in connection with the above spaces and inspired by the rearrangement formulations of Lorentz-Zygmund spaces according to 11.6, we adopt a slightly more general point of view. Let be a real continuous monotonically increasing function on the interval [0, e], where 0 < e < 1, with
if 0
and
(12.11)
= and let be the associated Bore! measure. We refer to [Lan93], especially p. 285, for details of this notation. In particular, all the integrals below with respect to the distribution function 'I'(t), but also with respect to the other distribution functions needed below, can be interpreted as Riemann-Stieltjes integrals (defined in the usual way via Riemann-Stieltjes is differentiable in (0, then sums). If, in addition, Let
(12.12)
in
be two real continuous monotonically increasing functions (t) and are said to and on [0,ej with the counterpart of (12.11). Then, again, if there are positive numbers Cj and c2 with be equivalent, Let
c1
12.2
0 t e.
(12.13)
Proposition
be a real continuous monotonically increasing function on the interval [0,ej with (12.11). Let 0 < uo
Let
__
II. Sharp inequalities
184
c0 and c1 such that g(t)
sup
ci
O
If
g(t))U1 JL*(dt)
\0 -L
(12.14)
for all non-negative monotonically decreasing functions g on (0, e]. (ii) Let and i,b2 be two such functions which are equivalent according to (12.13), and let 'F3(t) = and be the corresponding distribution functions and measures (j = 1,2). Let 0 < u oo. Then 1
.1.
(12.15) (with the sup-norm if u = co) for all non-negative monotonically decreasing functions g on (0, ci, where the equivalence constants are independent of g. Proof Step 1 We begin with some preparation. Let 0 < u < oo. We claim
0< t
(12.16)
where the equivalence constants are independent oft and of all admitted according to (1). Let be fixed and let
2'
for some
IEZ
(12.17)
(where, as in 2.1. Z is the collection of all integers). Let a3 be a decreasing = sequence of positive numbers, tending to zero with Then
(12.18)
<2k'). Since
=
1
Og
f o
=
J
3=laj+I
(12.19)
12. Envelopes
185
Similarly with aj_1 (respectively e) in place of
on the left-hand side. This
proves (12.16).
Step 2 We prove (i). Let 0
c
,
0
(12.20)
This is even sharper than the first estimate in (12.14). Let 0 < Uo
•
/
/
J
o
.
0
(
sup
\O
(12.21)
Using (12.20) with uo in place of u we get the second inequality in (12.14).
We prove (ii). This is obvious if u = 00 (it is the left-hand side of (12.14)). Let 0 < u < oo. We begin with some preparation. Let H(t) be a Step 3
continuous monotonically increasing distribution function on [0, e] with H(0) = 0. Lot
he
the associated measure. Let G(t) be a bounded non-negative
inonotonically decreasing function on [0, eJ. Then the Riemann-Stieltjes sums —
H(b,)) =
—
G(b,)) + H(e)G(bN_l) (12.22)
tend for suitable subdivisions
0 = b0
(12.23)
to the Riemann-Stieltjes integral
JE G(t) PH(dt).
(12.24)
If one has two such distribution functions H1 (t) and H2 (t) and related measures and which are equivalent, then by (12.22) and G(t) 0 the corresponding Rieinann-Stieltjes sums are also equivalent (with the same equiva-
lence (oust ants. independently of C) and this extends to the integrals (12.24).
II. Sharp inequalities
186
With these facts established, one can prove the equivalence (12.15) as follows. We put G(t) = (so far assumed to be bounded) and
/IH(dt) = The corresponding distribution function H(t) is the right-hand side of (12.16) with in place 'I'. By (12.16) this distribution function is equivalent to the distribution function (t). Hence the corresponding integrals of type (12.24) are also equivalent. Since this assertion extends to and we
obtain (12.15). Unbounded functions g(t) can be approximated by bounded ones. 12.3
Discussion
The monotonicity (12.14) is the refined and more systematic version of what follows from (12.5). In the applications in the following sections we identify with the growth envelope function
the continuity envelope function £c
or
(t)
from (10.7) or (10.19), respectively. They have essentially the required prop-
erties. However by our general point of view we do not distinguish between equivalent quasi-norms in a given space With a few exceptions such as or, to a lesser extent, the classical Sobolev spaces there is no primus inter pares among the equivalent quasi-norms. The situation is much the same as in the final slogan in G. Orwell's novel, Animal Farm, [Orw5l], p. 114, which reads, adapted to our situation, as follows, All equivalent quasi-norms are equal but some equivalent are more equal than others. In other words, any notation of relevance must be checked to see what happens when a quasi-norm is replaced by an equivalent one. This is the reason why we included part (ii) in the above proposition. 12.4
Examples
We discuss a few examples which, on the one hand, will be useful for our later considerations, and, on the other hand. shed new light on the Lorentz-Zygniund Spaces mentioned in 11 .6.
12. Envelopes
Let 0
Example 1
187
= t3. By (12.12) we have (12.25)
Let 0 <
u0
sup
g(t)
I
,
g(t))
-I-
uidtt
I
0
—
(t3 g(t))
uodt
0
(12.26)
for all non-negative monotonically decreasing functionsg on (0, eJ. With g(t) = we have the Lorentz spaces Lru(Ie) introduced in 11.6(i), and the wellknown monotonicity with respect to u (for fixed r), [BeS88], p.217.
I
Let b>0 and
Example 2 we have
= Let 0 <
where 0< t e < 1. By (12.12)
=
0
b
< t e < 1.
(12.27)
< u1< oo. Then (12.14) results in sup
0
g(t)
I I g(t) \
dt
I
I
\*
—
(JE \.
o
(
g(t)
dt
b)
tllogtlj
(12.28)
for all non-negative monotonically decreasing functions g on (0, ej. Let a = —b and g(t) = *(t). Then the left-hand side of (12.28) coincides with (11.47). As mentioned there the corresponding spaces L)a(Ie) can be equivalently described by (11.45). The right-hand side of (12.28), say, with u = u1, fits in the scheme of the special Lorentz-Zygmund spaces L)a(Ie) iii 11 .6(iii) with a = —b — Then the requirement mentioned there, au < —1, coincides
I
with b> 0, and looks more natural. Inequalities of type (12.28) in terms of Loou(logL)a(Ie) may be found in [BeR8O], Theorem 9.5, where the notation diagonal comes from the natural combination a + = —b. As stated above, we are not so much interested in the spaces
for their own sake.
We formulate our assertions in terms of inequalities of the same type as in Theorem 11.7. In particular, (11.71) follows from (12.28) with u1 = p and .1.
—
II. Sharp inequalities
188
Let 0< r
Example 3
By (12.12),
O
0
is chosen so small that 1'(t)
is
(12.29)
monotone in the interval [0, e]. Let
0 < u0
a
g(t)
ci
(1 I'1
a
g(t))
,i
\Jrj
fE <
co
(
j
\JO
I log
tlag(t))
—
—J
(12.30)
for all non-negative monotonically decreasing functions g Ofl (0,e].
With g =
we have (11.48), (11.49), and hence the Lorentz-Zygmund spaces Lru(log introduced there. The inequality (12.30) is known and may be found in [BeR8O], Theorem 9.3. 12.5
Growth envelope functions
The concept of growth envelope functions S0
(outlined so far in (10.7), (10.9) modulo equivalences) makes sense for all spaces (where
always means either or which are covered by Theorem 11.2. But we exclude borderline cases where p = cc or a = — 1)+. Hence we always assume
0
(12.31)
Furthermore the growth envelope function is designed to be a sharp instruinent to measure on a global scale how singular (with respect to its growth) a function belonging to Aq(W') can be. Hence it is reasonable to restrict the considerations to those spaces with (12.31) which are, in addition, not embedded in (Rn). To make clear which spaces are meant one must complement Theorem 11.4 by non-limiting embeddings. Since by Theorem 11 .4(u) one has in the critical case both embeddings and non-embeddings in one can combine this assertion with elementary embeddings for with fixed p and variable a, q of type as in [Th/3], Proposition 2 on p. 47, to get a final answer. This results in all spaces in sub-critical situations (11.18) and in those spaces in critical situations a = which are not covered by (11.22) and (11.23). To avoid any misunderstanding we give a precise formulation which spaces we wish to exclude:
12. Envelopes
189
Under the assninption (12.31) the followsng three assertions (1), (ii), (iii) are equivalent to each other: 12.5(i)
A;q(RTh) c L00(R'2),
12.5(11)
A;q(IR") c
12.5(ffl)
either
=
or
with
or
s> 0
S
=
0< q
1,
with
A full proof of this assertion has been given in [SiT95J, Theorem 3.3.1. A short description may be found in 2.2.4, p.32 - 33. Obviously, the concept of growth envelope functions makes sense for a much
larger scale of function spaces than considered here. It has been studied recently in [HarOll. We do not go into detail, but we have a brief look at with I r < oo, obviously normed by (2.1) (we recall the Orwellian confession at the end of 12.3) and put EGILr(t) =
sup{f(t)
:
1},
0 < t <e.
(12.32)
Then
0
I' and g = —
from above by
(12.33)
follows from (12.26) with u1 =
For the estimate from below one can choose the function
(x). where XM (x) is the characteristic function of a set M with MI = t.
As far as the growth envelope function for one of the spaces of interest is concerned, we have first a closer look at with respect to a given quasi-norm . 12.6
Proposition
(B)
Let either
1 0,
0
(12.34)
(sub-critical case, dashed line in Fig. 10.1) or
O
(12.35)
II. Sharp inequalities
190
(F)
or
Let either a, p, q be as in
1
(12.36)
with (B) or
with (F). Let, by definition,
6G
= sup{f*(t)
:
< 1},
hf
0 < t
(12.37)
is a positive, monotonically decrease is a given number. Then ing, unbounded function on the interval (0, ci with where
j=J,J+1,... ,
(12.38)
(where the equivalence constants are independent of j). Furthermore, in the sub-critical case given by (12.34) we have (12.39)
for some c> 0, and in the critical case given by (12.35) or (12.36),
0< t e,
(12.40)
for any > 0 and a suitable constant c,7 > 0. Proof Step 1 Obviously, £c (t) is monotonically decreasing (this means is bounded. non-increasing) and positive for all t > 0. Assume that By (12.2) we have
= f*(Ø)
If for all
fE
with If
Ill Loo(1R1z)fl
sup
( O
£aIAq(t)
(12.41)
= 1, and hence
EGIA,q(t)) hf
IE
(12.42)
However (B) and (F) collect just those cases with (12.31) which are not covered is unbounded if t 0. by 12.5(i)- -12.5(iii). Hence
Step 2 Let. a, p, q be given by (12.34). We prove (12.39). As remarked in 11.8(u), the inequality (11.52) with q = 00, hence sup
f*(t)
c' If
(12.43)
12. Envelopes
191
is very classical, and taken for granted here. The second inequality is an elementary embedding, Proposition 2 on p. 47. This proves (12.39) in all sub-critical cases. As for the critical cases (12.35) and (12.36) we note the elementary non-limiting embedding A;q(R") C Lr(R")
for any
max (p, 1)
(12.44)
Now (12.40) follows from (12.33) and, as a consequence of (12.44),
EcIAq(t) Step 3
We
prove (12.38). Let
<1
with
fE
= f(2*x) where x E R". By (12.1) we have
and let
= I{x = 21{XER" Hence,
(12.45)
f(2*x)I
(12.46)
> .41
:
by (12.2) (and by (12.39), (12.40) ),
= g*(2i+l), j = J + 1
(12.47)
Furthermore with some c> 0 (independent of f)
<1.
II!
(12.48)
Now, by (12.37), and (12.47), (12.48), it follows that
j = J + 1,...
c6cIAq(2'),
(12.49)
wit Ii the same c as in (12.48). Since the converse inequality is obvious we obtain (12.38). 12.7
Equivalence classes of growth envelope functions
If one Puts
w(t) =
(t)
,
0
(12.50)
then (12.38) coincides with (12.4) and we have (12.5). This was one of our moivat ions. The refineji tent of this point of view at the end of 12.1, which resulted
II. Sharp inequalities
192
in Proposition 12.2 and in the discussion in 12.3, requires for the underlying monotonically increasing distribution function w(t) = with (12.11) that it is in addition continuous. However one can circumvent the possibly somewhat delicate question as to whether or not is continuous. First we remark that for two equivalent quasi-norms,
of a given space
(12.51)
II
II
we have (in obvious notation)
0
(12.52)
as an immediate consequence of (12.37). Equivalence must always be under-
stood according to (12.13) adapted to the above situation. This fits in our Orwellian point of view confessed at the end of 12.3. The collection of all positive unbounded monotonically decreasing functions on the interval (0, can be subdivided into equivalence classes, where a class consists of all those admitted functions which are equivalent to one (and hence to all) functions in the given class. By (12.52) all growth envelope functions for a space covered by Proposition 12.6 belong to the same equivalence class, denoted by This class contains also representatives which are continuous on (0, e] (in addition to the other required properties). For example, one can start with a fixed growth envelope function and define (without the midline) as the polygonal line with
j = J, J + 1,...
= and linear in the intervals apply Proposition 12.2 with
t
(modification at
,
(12.53)
Then one can
0
Definition
Let be either Proposition 12.6. Let according to 12.7. Let E
with (B) or with (F) according to be the equivalence class associated to (Rn) be a continuous representative.
(12.54)
12. Envelopes
193
Let
=
and
W(t) =
0 < t e, according to 12.1 and let
=—
(12.55)
be the associated Borel measure on
[O,e]. Let 0 < u oo. Then the couple
= is called the growth envelope for
(12.56) when
(It)f*tnv
(12.57)
(modified as on the left-hand side of (12.14) if v = oo) holds for some c =
>0 and all / E 12.9
if, and only if, u v 00.
Discussion and notational agreement
First we recall that under the restriction (12.31) (excluding borderline cases p = oo or s = (Y,) the conditions (B) and (F) cover all cases for which this concept is reasonable. Furthermore, the definition of the number u in (12.56) makes sense and is independent of the chosen representative This follows from both parts of Proposition 12.2. However we must add a remark. By definition we have always sup
O
f*(t) =
O
for some c> 0 and all I E sense to put
:S c
(12.58)
Hence by Proposition 12.2 it always makes
u = inf{v : (12.57) holds}.
(12.59)
But it is not clear from the very beginning whether (12.57) remains valid with u in place of v. However this will be always the case for all spaces considered here. This may justify the incorporation here of this additional information immediately in the definition. Furthermore we wish to simplify (12.56) by
= where
(12.60)
is a continuous representative according to (12.54). The situation instead of [11 E
is similar to the usual simplification of writing I E
II. Sharp inequalities
194
where [f] stands for the equivalence class of all measurable functions g which coincide with f almost everywhere. This is also justified by the typical examples in (10.14). Hence we prefer, for example,
=
compared with
(12.61)
or even more cumbersome versions avoiding the explicit appearance of the variable t. (Of course the use of [.] is much the same as above in I compared with [1] E Next we collect a few rather simple properties which make clear what type of sharp inequalities can be expected. 12.10
Proposition
be either with (B) or with (F) according to Proposition 12.6. Let 0 <e < 1. Let be a continuous growth envdope Let
function as in (12.54), let, in notational modification of (12.55),
E(t) =
0 < t e,
—
(12.62)
and let PE be the associated Borel measure on (0, e]. (1)
Let
be a positive function on (O,ej. Then there is a number c> 0
such that
fE if, and only if, (ii)
(12.63)
is boundeA
Let x(t) be a positive monotonically decreasing function on (0,
and let
for some00 I
(I for all I
s
(12.64)
Then for some c' > 0, A V
(i for all f E
IL(dt)) if, and only if, x is bounded.
Ill 144q(1R")II
(12.65)
12. Envelopes
195
Proof Step 1 The proof of (i) is simple. On the one hand we have (12.58). On the other hand, if (12.63) holds for some x, then for any fixed t with
0< t e,
c
I
for all
with
1.
Ill
(12.66)
Now by (12.54) and (12.37) it follows x(t) c' uniformly with respect to t. Step 2 We prove (ii). The function g(t) = x(t) I (t) is non-negative and monotonically decreasing on (0, eJ. Hence, by (12.14),
K(t) f*(t) EGApq(t)
c
0
< t e.
(12.67)
Then (ii) follows from (i). 12.11
Discussion
In part (ii) we assumed that , is monotonically decreasing. This is natural in our context, where we ask for (12.65) under the assumption (12.64), and also in connection with the definition of the growth envelope in (12.60). On the other hand, if x is non-negative on (0, and, maybe, wildly oscillating (or monotonically increasing), then at least formally the question (12.65) makes sense without assuming that (12.64) holds. To look at the discretised version of this problem we assume that the numbers have the same meaning as in Step 1 of the proof of Proposition 12.2 with Then the = discrete twin of the left-hand side of (12.65) is given by I
I I
J
1
(12.68)
J
This suggests that not so much the pointwise behaviour of x(t) but the behaviour of the indicated integral means is of interest. However we do not study
problems of this type in the sequel. 12.12
Moduli of continuity
We outlined in Section 10 our methods and results. As explained there in connection with (10.22) we deal with the super-critical case by lifting the results obtained in the critical case. In rough terms, the role played by f (t) in critical (and sub-critical) situations is taken over in super-critical cases by
II. Sharp inequalities
196
t). First we recall what we need in the
the divided modulus of continuity sequel.
where has been introduced in 11.1 as the set of all Let 1(x) E complex-valued, bounded, uniformly continuous functions in R". Then
w(f,t)=
sup XER",IhI
is is
If(x+h)—f(x)I,
0t
(12.69)
called the modulus of continuity. Let I C(R") be fixed. Then w(f, t) a non-negative and monotonically increasing (this means non-decreasing)
continuous function on [0, cc); in particular,
w(f,t)
w(0)
=
0
if
t j. 0.
(12.70)
Furthermore, t) is almost concave in the following sense: Let W(f, t) be the least concave majorant of t). Then w3(f, t)
w(f, t) w(f, t).
(12.71)
We refer to [DeL93J, Ch. 2, §6, pp. 40—44, where one finds proofs of all these properties. Let &5(f,t) =
t > 0,
(12.72)
be the divided modulus of continuity. By (12.71) we have (12.73)
Since W(f, t) is concave and continuous on [0, cc) and w(f, 0) = 0 it follows that the right-hand side of (12.73) is monotonically decreasing on (0, cc). Hence t) is at least equivalent to a monotonically decreasing function. This is sufficient for our purpose. The concept of moduli of continuity has been widely
used in the theory of function spaces. Our goal here is rather limited. We are interested exclusively in the super-critical case according to (10.15), and, even 1+n more restrictive, only in those spaces and which are not continuously embedded in Lip(R'1). This means by Theorem 11.4(iii),
with O
(12.74)
with l
(12.75)
and
12. Envelopes
197
This is in good agreement with (10.18) on the one hand and (12.35), (12.36) on the other hand. We remark that t '—'
< 1}
:
(12.76)
is continuis a bounded function on the interval (0.1) if, and only if, Hence, (12.74) and (12.75) cover just those cases, ously embedded in where (12.76) is unbounded. Now we are very much in the same situation as in Proposition 12.6 with the following outcome. 12.13
Proposition
Let
or the space in (12.75). Let, for be either the space in J+!1 ", be defined by > 0, the continuity envelope function
some
(t) = Then 6c
t)
(R'1)It
:
1},
0 < t <e.
(12.77)
is a positive, continuous, unbounded function on the interval
(0,r1 with
j=J.J+1.... , (where the equivalence constants are independent of j). Furthermore, is equivalent to a monotonically decreasing function, and for any is a number > 0 such that
0
(12.78)
I+n
0 there
(12.79)
is positive, unbounded, and equivaProof By the above remarks. £c lent to a monotonically decreasing function. By [DeL93], p.41, we have
w(f,ti + t2) — w(f,ti)I
and
This proves (12.78) and the continuity of inj > 0, we have the non-limiting embedding
'.
[TriI3J,
(12.80)
Finally, for given in, 1 >
w(f,t) o
w(f,t2).
t
2.7.1, p. 131, formula (12). This proves (12.79).
(1281)
II. Sharp inequalities
198
12.14
DefinItion 1+n
Let A)(J
"(IR")
be either the space Bpq from (12.74) or the space from (12.75). Let 0 < e < 1. Then is the equivalence
class of all continuous monotonically decreasing functions on the iiiterval (0,
which are equivalent to one (and hence to all) continuity envelope function according to (12.77). Let 1+a
ecApq"
"(t)'
=
=
and
log
0
e, according to 12.1 and let [O,eJ. Let 0< n 00. Then the couple
1+n"
is
coiled
be the
([EcApq
the continuity envelope for
associated
1+11
"(t)
(12.82)
Borel measure on
"],u)
(12.83)
(Re') when 1.
I
f
t
—
clif IApq "(R")II
J
/
(modified as on the left-hand side of (12.14) if v 1+ a
(12.84)
oo) holds for some c
>0 and allfEA1,,1"(R") if, andonlyif,uv_
Remark and notational agreement
This definition is the same as Definition 12.8, mutatis mutandis. In particular,
all that had been said before 12.8 in 12.7, but also afterwards in 12.9, in Proposition 12.10, and in 12.11, has respective counterparts which will not be repeated here. But we mention that, much as in (12.60), we (12.83) by
1+a
1+a
= ((cApq "(t), u),
(12.85)
where £cApq" is a representative of "]. As stated above, we reduce later on the super-critical case to the critical case by lifting. If n = 1, then one has (10.22). In higher dimensions the situation is more complicated. In the next proposition we prove what we need later. Recall that
Vf(x) =
IL(x))
,
x E IR",
(12.86)
12. Envelopes
199
and, hence.
IVf(x)I
(12.87)
=
Furthermore we need the rearrangement IVfI*(t) and its maximal function Vft"(t) according to (10.3) and (10.4) with IVf in place of f. Let
and f. t) be the inoduhis of continuity and the divided modulus of continuity introduced in (12.69) and (12.72). Finally, has the same meaning as in (11.6).
Proposition
12.16
LtO<E<1. Thereisanumberc>Osuchthat
(I)
+3 sup for all
U
<
/ <e and all f E
There i.sanumberc>Osuch
(ii) flint
if
c I
by
(12.88)
I
1 We prove (i). Let t with 0 < t < e be fixed. Replacing f(x) f(.r) for sonic p> 0 we may assume that the supremum in (12.88) equals
1. hence (12.91)
where r < t2. Then (12.88) is equivalent to
t' If(z+v) — f(x)I
cIVIl (t2
1)
+3
(12.92)
for all .r E with t. Of course it is sufficient to concentrate and y E on t hose a' and y for which the left-hand side of (12.92) is larger than 3. Without
restriction of generality we may assume z =
0
and y =
= (yi,O,. . .0).
Hence.
A=lf(y')—f(0)I3t
with
(12.93)
II. Sharp inequalities
200
= (O,y') withy' E y=y'+y2 weobtain by (12.93) and (12.91), Let y2 = (O,y2,... —
f(y2)I
1(y1) — 1(0)1 —
>A
—
2t>
and 1y21
= Iii'l =
r t2. With
— f(y')I — 11(y2) — 1(0)1 (12.94)
A
Similarly one can estimate — f(y2)I from above by 2A. By construction y with and y2 differ only with respect to the first component. We fix y' E 12 and obtain Iy'I
—f(y2)l =
J
(12.95)
The left-hand side is equivalent to A. We
integrate
over y' E
with Ii,'I
12. Then we have for some c> 0,
c J IVf(x)I dx,
(12.96)
is a tube in R' with the volume ITI = where T = [0, t] x {y' Iy'I By standard arguments for rearrangements we obtain (switching to arbitrary t and the counterpart of (12.93)) and y E RTh with xE :
f(x+y)—f(x)I
(12.97)
J
This proves (12.88). Step 2 We prove (ii). Let p < oc. By (i) we have di
J
J Here we used that
(12.98)
t) is equivalent to a monotonically decreasing function.
Then application of (12.26) justifies the first term on the right-hand side of
12. Envelopes
201
(12.98). In connection with the second term we used the transformation r = Let g(t) = Vf1(t). Then IVfI**(t) = Mg(t) is the maximal function of according to (10.4). We wish to prove that
(Mg(t)"\"
dt
J0 Let
_Cj ( g(t)
dt
12 . 99 )
0
2' with j J.
E
Mg(23)
Then
Mg(t) =
(12.100)
Let u = vp + 1 and q
Mg(2-i)P
(j
c'
c
j=J
j=J1=0 °° C
"2 k"
j)u
2—jq
gP(2—i—1)
kJ 2—Zq
(k — flu
(12.101)
can be estimated from above by 1 + 1 + 1, it follows that the last factor in (12.101) can be estimated from above by a constant, which is in(lependent of J. Then the right-hand side of (12.101) is equivalent to the right-hand side of (12.99). This proves (12.99). We return to (12.98) and remark in addition that for any > 0 there is an 6o, 0 <Eo < 1, such that Since
if where ii
(12.102)
= vp+l has the above meaning. Inserting (12.99) with g(t) = Vf
in the right-hand side of (12.98), then we have on the right-hand side the desired term from the right-hand side of (12.89) and in addition the same term as on the left-hand side with a factor, say, This We remove this proves (12.89) under the additional assumption 0 < restriction. Let 0 < E < 1 and let 0 < x < 1. Since is equivalent to a monotonically decreasing function it follows that and (12.102) with a small
dt
J0
< —
Cj 0
dt
) dr
j
(12.103)
II. Sharp inequalities
202
This reduces the case 0 <e < 1 to 0 <e <eo. Then we obtain (12.89). If p = 00 then one can follow the above arguments with the necessary modifications
and arrives at (12.90). 12.17
Remark
In the one-dimensional case, (12.88) with n =
1
reduces to
0< t < e, f E C'(R). This follows from (12.95). It coincides with (10.22). The situation in
(12.104)
with
n 2 seems to be more complicated. Whether there is a direct counterpart of (12.104) with IVfI**(tn) is not so clear. On the other hand, the choice of w(f,r) in the second term on the right-hand side of (12.88) is convenient and sufficient for us, but it can be modified. If one replaces by r) where
with x(r)
is a positive, say, monotonically increasing function
if r 0, then one ends up with in place of in the first term on the right-hand side of (12.88) with g(t) —p 0 arbitrarily slowly if t 0. However if one wishes to apply modified versions of (12.88) to get 0
(12.89) one needs a counterpart of (12.102) with x(t) in place of 13 13.1
The critical case Introduction
By the terminology of (10.6) the critical case covers the spaces and
with O
(13.1)
This corresponds to the line of slope n in Fig. 10.1 starting from the origin. Generally in this Chapter II we are interested exclusively in spaces and which are not only subapaces of S'(lR') but also of (and, hence, consist entirely of regular distributions). We refer to Section 10 where we outlined our intentions. Theorem 11.2 clarifies under what conditions and are subspaces of Recall that in all cases considered here (critical, super-critical, sub-critical) we always exclude borderline situations, which means in general
if 0
p=oo and/or
(13.2)
and especially according to Theorem 11.2, with 0 < q
2,
(13.3)
13. The critical case
in the critical situation a =
203
A further distinguished borderline space in
connection with the critical situation not treated in this section is bmo(R'2) =
Here we add at least a brief remark at the end of this section in 13.7. Otherwise as a further restriction of (13.1) we are interested only in those spaces which are not continuously embedded in (or, which is the same, in C(RT')); this means by Theorem 11.4, and as has been detailed in 11.5. especially (11.38), (11.39), we deal only with the spaces
with 0
(13.4)
with i
(13.5)
and
This covers in particular the respective Sobolev spaces mentioned in (11.40). As outlined in the introductory Section 10 we wish to measure the singularity behaviour of functions belonging to the spaces in (13.4), (13.5) in terms of the growth eiivelope as introduced in Definition 12.8. Instead of in (12.56) we use the more handsome version (12.60). In the theorem below we calculate explicitly the growth envelopes for all spaces in (13.4) and (13.5). By Proposition 12.10 it is clear that one gets rather sharp assertions concerning the singularity behaviour of elements of these spaces in a very condensed form. Hence, it seems to be reasonable, after proving the theorem, to discuss what this means in detail. Finally we add references in 13.5 and, as said, a remark about bmo(Wz) in 13.7. Let 1 v oo. As usual v' is given by + = 1. 13.2 (i)
Theorem Let
O
(13.6)
Then
= (ii)
(13.7)
Let
1
(13.8)
Then
=
(13.9)
11 Sharp inequalities
204
We break the rather long proof into 7 steps. Here is a guide. In Step 1 and Step 2 we prove those sharp inequa.litie8 which correspond to the righthand sides of (13.7) and (13.9), respectively. In Step 3 we formulate what this means in terms of the growth envelope functions: They can be estimated from respectively. To prove the sharpness we need above by log extremal functions. They will be constructed in Steps 4 and 5. The outcome
Proof
is of self-contained interest, also in connection with the super-critical case considered in Section 14, and will be formulated separately in Corollary 13.4. In Steps 6 and 7 we prove that log and I log tI are envelope functions, and that q and p, respectively, are the correct exponents according to (13.7) and (13.9).
Stepi
Letpandqbegivenby(13.6),andlet,asalways,0<e<1.We
prove that there is a number c> 0 such that
f*(t) 9dt for all I
(
(13.10)
with the interpretation sup
O
''
iogt
•',
incaseofq=oo.LetO
(13.11)
[Tri/3J, Theorem 2.7.1, p. 129. Hence it is sufficient to prove (13.10) for large values of p, in particular, we may assume
l
(13.12)
We rely on atomic decompositions for the spaces Details (and also references to the original papers) may be found in [THo], Sections 13. (One could also use corresponding quarkonial decompositions according to Definition 2.6 and Theorem 2.9, but atoms are sufficient at the moment.) By [Thö], Theorem 13.8, any I can be optimally decomposed in atoms ajm(x) and complex numbers b2m such that with j=O
f,(x)=
(13.13)
13. The critical case
205
and A
=
\j=0
Ill
( mEZ"
(13.14)
J
(obviously modified by sup3 if q = no). The equivalence constants are independent of f. Recall that the atoms ajm(x) have the following properties:
suppajm c {y
E
for all
hi — 23rn1
:
with
'y
for some d > 0 and all j E No and in E function of the interval j E N0, and I N. For fixed j rangement of bjm with in E then
—
No let
,
(13.15)
+ 1,
(13.16)
Let be the characteristic on = [0,00), where C >0, with I N be the (decreasing) rear-
If C > 0 and c> 0 are chosen appropriately
where t>0 and jENo. Let
1)
—
Since {bk
(13.18)
is monotonically decreasing and 1
1=1
=
=C
c 1=1
(13.17)
(13.19)
mEZ"
1=1
The left-hand side is obvious since b, <
The second estimate is the sequence version of the Hardy-Littlewood maximal inequality and can easily be reduced to the usual formulation of this maximal inequality. (A formulation and a proof of the latter may be found in [Ste7O], p. 5.) Let
C2_l)71
with
kEN.
By the additivity property of f** according to [BeS88], Theorem 3.4 on p. 55. and (13.13), (13.18) we obtain
f**(t) <
+c j=0
(13.20) j=k+1
II. Sharp inequalities
206
b1. If 1
where we used
\Q dt
00
J 'sjlogtl)
If..
)
k
q
q
\
(13.21)
+C2
j=0
/
k=1 \
/
3=k+1
Again we can apply the sequence version of the Hardy-Littlewood maximal inequality to the first sum A1 and obtain 00
A1
(13.22)
If
where we used (13.14). If q = oo then (13.21) must be replaced by
= A1 + A2. (13.23)
+ c2 sup
C2
j=0
—
j=k+1
The term A1 can be estimated from above by sup,
and hence by the right-
hand side of (13.22). We estimate A2. Since for fixed j the sequence bt is monotonically decreasing we have by (13.19),
j
(13.24)
C,, j EN0, I EN.
(13.25)
cC', and, hence,
It follows that
c> j=k+1
c' sup C1.
(13.26)
I
j=k+1
Now we get in both cases, q = oc by (13.23) and 1
A2
c supC,
c'A
If
(13.27)
13. The critical case
207
Here we used again (13.14). Now (13.23) if q = oc and (13.21) if 1 < q < oo, and the estimates (13.22) and (13.27) prove (13.10). Step2 Let p and q be given by (13.8). Let again 0 < e < 1. We prove that
there is a number c> 0 such that C
(i
1.
(fs(t)
p
c
(13.28)
fif
We reduce this case to (13.10) using the following conall I sequence of an observation by Ju. V. Netrusov, [NetS9a}, Theorem 1.1 and there Remark 4 on p. 191 (in the English translation): For any f such that is a function g E for
and
a.e.
If(x)I
(13.29)
c is independent off and g. Since 1
dt 13.30
If
c2IIf where we used in addition
the
the monotonicity of the F-spaces with respect to
q-index. This proves (13.28).
Let p, q be given by (13.6) and let b = we obtain by (12.28) and (13.10), Step S
sup
f*(t)
Ic
\
ciIIII I I
q
in Example 2 in 12.4. Then
dt
I
(13.31) 1
I e
f*(t)
q
dt
T) If p, q are given by (13.8) then it follows in a similar way by (13.28) that sup
f*(t)
O
c If
(13.32)
II. Sharp inequalities
208
Let and be the respective growth envelope functions according to Definition 12.8 and (12.37). Then it follows by (13.31) and (13.32) that
O
(13.33)
l
(13.34)
and
Step 4 To prove the converse of (13.33), (13.34) and to show that q, p are the correct numbers in (13.7), (13.9), respectively, we need some extrernal functions. Let be a non-trivial, non-negative, compactly supported function in R', for example, if
1
xl
1.
(13.35)
Let
l
{b3
(13.36)
be a sequence of non-negative numbers with and
(13.37)
and let
1(x) =
(13.38)
b3
We wish to prove
/
f*(t)
If
(13.39)
where the equivalence constants are independent of b. We remark that (13.38) is an atomic or quarkonial decomposition in according to ITriS], Theorem 13.8, p. 75, or the above Definition 2.6, respectively. With the sequence space given by (2.8), we obtain (in obvious notation) I
If
.
(13.40)
13. The critical case
209
The inequality in (13.40) is covered by the above references. The equivalence in (13.40) follows from the special structure of f in (13.38) and the modifications of fpq indicated in 2.15 (which show that under the above circumstances is unimportant). Next we remark that f(x) with, say, (13.35), is q in non-negative, rotationally invariant, and monotonically decreasing in radial directions. We have
f(x)
if
2_Ic
lxi
where kEN,
(13.41)
where kEN.
(13.42)
and, hence, if
It follows that P
(13.43)
where we used in the second equivalence again the sequence version of the Hardy-Littlewood maximal inequality as in connection with (13.19) and the nioriotonicity of the numbers b, according to (13.37); (the number K is related to E, but otherwise unimportant). Now (13.39) follows from (13.43), (13.28), (13.40). Similarly, but technically more simply, one obtains for 0
i
(f*(t))(i
if
1
(13.44)
and b1 = supb,
sup
f*(t'l iogt/ 1
ill
(13.45)
as follows: One has (13.40) with B in place of F and with q on the right-hand side, 1
II. Sharp inequalities
given by, say, (13.35) and (13.37) with 1
0. We modify
(13.38) by
f(z)
.
(13.46)
Although not really necessary one may choose x0 such that the supports of — x°) are pairwise disjoint. Furthermore the function x(x) satisfies the first moment condition
= 0.
JRN
(13.47)
Otherwise (13.40)—(13.43) remain unchanged and we get (13.39) for those q for which first moment conditions in the related atoms are sufficient. If higher moment conditions for
(13.48)
are needed, then the construction in (13.46) must be modified by
x(x) =
—
—
x°),
(13.49)
where *L is a suitable C°° function with a compact support, say, in the unit and x0 0 chosen in such a way that the supports of t,b1. (2i - — ball in x°) are pairwise disjoint. An explicit construction of such a function may be found in pp. 665-666. We refer also to Corollary 13.4 below and its
proof where we have for later purposes a second and more detailed look at constructions of this type. After this modification we get (13.39) now for all p, q with (13.8). Step 6 We prove the converse of (13.33), (13.34). Let p, q, and be given by (13.36) and (13.35), respectively, and let
XERIZ, JEN.
(13.50)
Then by (13.42) and (13.39), and
(13.51)
uniformly in J. Hence by (12.54) and (12.37), r'..
J EN.
(13.52)
13. The critical case
This proves the converse of (13.34). If q < 1 then one has to replace I in (13.38) a I I as indicated in Step 5. Similarly for Hence I log tj ' and log V are the growth envelope functions for and F,%(R'i, respectively. I
Step 7
We must prove that q and p are the correct numbers in (13.7)
and (13.9). respectively. Since we know already (13.31) and its F-counterpart (13.28) we must prove that q, respectively p. cannot be improved. Assume that there is a number u with v < q and
(
dt
(13.53)
/
Let, according to (13.38),
f(s) =
with
(13.54)
b3
j=2
By (13.44) we have f e B,(R'1). On the other hand, by (13.42), we can estimate the left-hand side of (13.53) from below for some c> 0 by
=00.
(13.55)
We get a contradiction. This proves (13.7). Similarly one obtains (13.9). 13.3
Inequalities
The above theorem covers all cases of interest (excluding borderline situations
according to (13.2)). It describes in a rather condensed way very sharp inequalities. It seems be reasonable to make clear the outconie. We use Example 2 in 12.4, Definition 12.8 and Proposition 12.10. Let 0 <e < 1. 13.3(i) The B-spaces Let x(t) be a positive monotonically decreasing function on (0,E]. Let 0 < u cc. Let p and q be given by (13.6). Then I.
(I
(13.56)
for some c> 0 and all f if, and only if, is bounded and q u 00 (with the modification (13.59) below if u = cc). in particular. if 1
II. Sharp inequalities
212
then
(j ( f*(j)
Ilogtl
O
are
t
(13.57)
j
the two end-point cases according to (12.28). If q = oc then
'
sup
one has
(13.58)
/
o
Let x(t) be an (arbitrary) positive function on (0, e] and again let 1
f*(t)
c
oo.
(13.59)
O
if, and only x is bounded. However the for some c> 0 and all f E between the assumptions for in (13.56) and in (13.59) is rather immaterial. We discussed this point in 12.11. 13.3(il) The F-spaces Let x(t) be a positive monotonically decreasing function on Let 0
difference
()U
.1.
(13.60)
(/ for some c > 0 and all f with the modification
if, and only if,
'
sup
if u =
c
/
O
is bounded and p ( u 00, 00.
(13.61)
In particular, I
sup O
co
'
I
I
I
/
I
logt /
— I t
Ci
(13.62)
j
are the two end-point cases according to (12.28). As above, if x is an arbitrary positive function, then we have (13.61) if, and only if, x is bounded.
Let a E IL Then than or equal to
= max (0, a) and [a] a.
stands for the largest integer smaller
13. The critical case 13.4
(i)
213
Corollary
and let for y E R,
Let 0 < ö
iflyl <6
h(y) =
and h(y) =
0
if
—
1).
6.
(13.63)
Let L E N0 and
hL(y) = h(y)
p,
(13.64)
—
There are numbers
e R such that (moment conditions)
jYkhL(y)dY=o if k=O,...,L.
(13.65)
Let L + 1 E N0 and let hL with (13.65) be complemented by h_1 = h (then (13.65) is empty). Let (ii)
x=
fb(X)=>bjhL(23'X1)
ERTh,
(13.66)
where b =
is a sequence of non-negative numbers with
{b3
b1 b2 ...
b,
(13.67)
Let p, q be given by (13.6) in the B-case, by (13.8) in the F-case, and LB = Let
1,
LF = max
1
(
1
—
j+
—
p
j
.
(13.68)
L + 1 e N0 with L LB in the B-case and L Lp in the F-case. Let If bE€q, then
(I
If
13.69
(usual modification if q = oo) and, if b e £,, then
(I
--'
If
where the equivalence constants are independent of b.
(13.70)
II. Sharp inequalities
214
Proof
Step 1 If one inserts (13.64) in (13.65) then one gets a triangular matrix for from which these coefficients can be uniquely calculated.
By the product structure of the terms in (13.66) we have
Step
for (where
(13.71) is empty if L =
—1):
(13.71)
Since the sequence b is bounded, all
respective sums for converge at least in L3 (R"). Recall that one needs moment conditions (13.71) for atoms in and up to order L, where L max(—1,Lop — sj)
and
L
—
sD,
(13.72)
respectively, with a, and given by (2.20). We refer to [Thö], Theorem 13.8 on p.75. Here we have s = hence L —1 for the B-spaces and L LF for the F-spaces. This formalizes what we said in Step 5 of the proof of Theorem 13.2. Otherwise thc proof of the corollary is covered by Steps 4 and 5 of this proof. 13.5
Further references and comments
We described in Theorem 11.7 and in (11.70) the classical inequalities related to the critical case considered in Theorem 13.2. Recall that
l
(13.73)
are the Sobolev spaces with the classical Sobolev spaces in (11.69) as special cases. In 11.8(v) we tried to collect the historical references of (11.56), (11.57), and (11.70). Obviously all these cases are covered by Theorem 13.2
and by 13.3. In more recent times, inequalities of type (11.57) have again attracted some attention, mostly restricted to the case of classical Sobolev spaces according to (11.69), but in the context of general rearrangementinvariant (quasi-)nornis. We refer in particular to [CwPD8I, [EKPOO], and [Pic99J. The last paper surveys some aspects of embeddings of classical Sobolev spaces (R"), especially of (R"), in rearrangement-invariant spaces. Fur-
thermore, there is a connection between inequalities of type (11.56), (11.57) and capacity estimates in function spaces. Details may be found in [EKPOO] and [Pic991 with references to Maz'ya's results in this direction, especially in pp. 105, 109. As mentioned above, parallel or earlier develop-
ments in the East have often passed unnoticed in the West. In particular, Ju. V. Nctrusov proved in INet87bI, Theorem 3 on p. 108, assertions, which
13. The critical case
215
are related to [CwP98] and [EKPOO], in the framework of spaces of type including optimality of range spaces. He generalizes earlier results in the Russian
literature by Brudnyi, Kaljabin, and especially by Gold'man. A good description of and detailed references to this earlier work may be found in [Liz86], D.1.8 and D.1.9, pp. 398-404. Our own contributions started in [Tri93] and were repeated in a slightly improved form in [ET96], Theorem 2.7.1, p. 82, and Theorem 2.7.3, p. 93. The main new point is the construction of extremal functions f belonging both to and with 1 < p < 00 and having the singularity behaviour
f(x)=
where a>
(13.74)
near the origin. This is now essentially covered by the function f given by (13.38) with, say, b =
j=2,3
(13.75)
Then b E 4, and hence we have on the one hand (13.39) especially for HJ (IR"), and (13.44), especially for On the other hand, if lxi 2_k, then it follows by (13.41),
1(x) .
(13.76)
At the same time it is now clear that the functions in (13.38) improve the earlier developments in [Tri93] and [ET96]. The equivalence (13.39) in Step 4 of the above proof coincides essentially with [EdT99bJ, Theorem 2.1. This paper might be considered as a forerunner of Theorem 13.2, restricted to (IR") and Even worse, we used there (11.57), going back to [Has79] and
[BrW8O], as a starting point and derived the corresponding inequality for this means (13.56) with u = p = q and K = 1, via non-linear interpolation from (11.57). Otherwise the sharpness in [EdT99b] is on the x-level as described in 13.3. All other parts of Theorem 13.2 and its proof are new and published here for the first time. Especially the concept of growth envelopes in the above context came out very recently in collaboration with D. D. Haroske, [HarOl]. Finally we mention the extension of the related results in [Tri93] and in [ET96], Theorem 2.7.1, to spaces with dominating mixed derivatives in [KrS96}, including optimality results via extremal functions.
II. Sharp inequalities
216
13.6
Spaces on domains
Let be a domain in The spaces and have been introduced in Definition 5.3 for all admitted s, p, q. The concept of the growth envelope and
the growth envelope frnction according to Definition 12.8 and the notational agreement (12.60) can be carried over under the same natural restrictions as there to the respective spaces We denote them by (13.77) = In the critical case, considered in this section, Theorem 13.2 can be extended
to spaces on domains: If p, q are given by (13.6), then
=
=
(13.78)
=
(13.79)
and, if p, q are given by (13.8), then
=
To justify these assertions we remark first
6c,nA,q(t) s
0 < t S e,
(13.80)
as a more or less immediate consequence of the definition of spaces on doOn the other hand, the mains by restriction of corresponding spaces on construction of extremal functions in Steps 4 and 5 of the proof of Theorem 13.2 is strictly local. Hence the arguments in Steps 6 and 7 of this proof can be carried over from to Then one obtains (13.78) and (13.79). 13.7
The space bino
We always exclude borderline situations. In our context, described by Theorem 11.2, this means in general (13.2), and with respect to the critical case, (13.3). Furthermore, we excluded in all our considerations so far the spaces If 1
=
C
bmo(RTh) C
if
0
(13.81)
is the inhomogeneous space consisting of those locally integrable functions with bounded mean oscillation for which where
Of
sup
IQI1 QI
If(x)—fQIdx+ sup
IQI>' IQI
If(x)Idx
13. The critical case
217
f with respect
Here Q stands for cubes in R" and
to Q. We refer for details and further information to [Tri/3J, 2.2.2, p. 371 and be a C°° function with a compact support near the 2.5.8, p. 93. Let origin, for example from (13.35). It is well known and can be checked easily that I log In I belongs to bmo(R"). But this is a local matter and can be extended by (13.82) to
f(x)=
(13.83) mEZ"
This makes clear that there is no growth envelope function eabmo according to Definition 12.8 and (12.37), or in other words,
£cbmo(t)
= c,o
for all 0< t
(13.84)
However in sharp contrast to the situation described in 13.6 if p < oo, the growth envelope and the growth envelope function are reasonable for the spaces where is a bounded domain in R" and where bmo(il) is again Let, for example, = Q be a cube defined by restriction of bmo(W') on with IQI = < 1. A detailed study of the spaces bmo(Q) may be found in [BeS88], Chapter 5, Section 7. In particular by [BeS88], Corollary 7.11, on p. 383. we have brno(Q) C
(13.85)
according to 11 .6(u) (again with reference to [BeS88]) where we used that coincides with the space on the right-hand side of (13.85). In particular,
sup O
IlogtI
cllflbmo(Q)ll.
(13.86)
On the other hand, J. Marschall proved in [Mar951, Lemma 16 on p.253 (with a forerunner in [Mar87b])
0
(13.87)
in particular,
0
(13.88)
However by (13.7), 13.6 (and the notation introduced there) and (13.86) one gets (13.89)
In any case in borderline situations one has to distinguish carefully between global and local singularity behaviour.
H. Sharp inequalities
218
14 14.1
Tbe super-critical case Introduction
By the terminology of (10.15) the super-critical case covers the spaces and
1+21
(14.1)
This corresponds to the dotted line in Fig. 10.1. Recall that we always exclude in this chapter borderline situations as described in (13.2). This means in the super-critical case that we do not deal with the spaces and also not with the spaces briefly mentioned in 13.7. As a further restriction of (14.1) we are interested only in those spaces which are not continuously embedded in Cl(Rt2) (or, which is the same, in Lip(P..'1)); this means by Theorem 11.4, and has been detailed in 11.5, especially in (11.38), (11.39), we deal with the spaces
with 0
(14.2)
with I
(14.3)
and 1+11
This covers in particular the Sobolev spaces mentioned in (11.40). As outlined in the introductory Section 10 we wish to measure the continuity of functions belonging to the spaces (14.2), (14.3) in terms of the continuity envelope as 1+21 introduced in Definition 12.14. Instead of in (12.83) we use the more handsome version (12.85). In the theorem below we calculate explicitly the continuity envelope for all spaces in (14.2) and (14.3). Afterwards we describe
what this means in detail. Finally we add a few references. Otherwise we try to keep the presentation of the super-critical case as close as possible in its formulations to the critical case considered in the previous section (this applies also to this introduction compared with 13.1). In rough terms, using Proposition 12.16 as a vehicle, we lift Theorem 13.2 from the critical to the super-critical situation. Let I 5 v oo. As usual, v' is given by + = 1. 14.2 (i)
Theorem Let
0
(14.4)
Then
=
(14.5)
14. The super-critical case
219
l
(14.6)
Let
(ii)
Then 1+""
Proof
Step 1
,p).
(14.7)
Recall that (R")II
II!
Ill
(14.8)
+
Theorem 2.3.8, pp. 58—59. Let p, q be and similarly for given by (14.4). Using (12.87) we obtained by Theorem 13.2 and (13.57) with 0< E < 1,
(obviously modified accordrng to (13.58) if q = cc). Similarly for if p, q are given by (14.6), based on (13.62). We apply Proposition 12.16 and obtain, if q < cc, by completion I
(I
(14.10)
IE
(and again similarly in the F-case). If q = cc then we wish to have
sup
Let f E
logt
'
(R") and let
(IW')II,
fE
(W').
(14.11)
be as in (2.33). Then we can apply (12.90) to 1,
=
in place of f, where the corresponding right-hand We obtain (14.11) with sides can be estimated uniformly with respect to j; hence
fE
(14.12)
II. Sharp inequalities
220
By elementary embedding, f,(x) converges pointwise to f(x). Then (14.11) I+fl t) is (Rn). Since follows from (14.12) and j —* oo. Similarly for equivalent to a monotonically decreasing function we are now in the same situation as in Step 3 of the proof of Theorem 13.2. We get
O
(14.13)
l
(14.14)
and
Step 2
To construct extremal functions we rely on Corollary 13.4 and put
hL(y)=fhL(z)dz, yeR,
(14.15)
where hL has the same meaning as in part (i) of this corollary with L E N0. Then hL is a compactly supported C°° function. Integration by parts and (13.65) prove that
fykhL(y)dy_
ht(z)dzdy=0
if
k=0,...,L— 1. (14.16)
(if L = 0 then (14.16) is empty). We replace fb(x) in (13.66) by
fb(x) =
x = (x1
x,1) ER", (14.17)
is a sequence with b3 0 and (13.67). This can be inter1+2k ' (R") and Fpg where the preted as an atomic decomposition in necessary moment conditions according to (13.72), now with s = 1 + n, may claim be assumed to be satisfied by the above construction. Let 0 < s < 1. that we have in analogy to (13.69) and (13.70), where b =
(14.18)
14. The super-critical case if b
1q
221
(usual modification when q = oc) and
(14.19)
(J Of course we always assume that (14.4) and (14.6) are satisfied. First we remark
that
\* (14.20)
/ in analogy to (13.40). Secondly we claim
where kEN,
if
(14.21)
in analogy (but also in slight modification) of (13.42). Let 17 > 0 be small and k N. Then one obtains by (14.17), and (14.15), (13.64),
fb(0)
k —
fb(_172—k
, 0,. .. 10)
b1 (ht)l(zj,k)?72_k,
(14.22)
< < 0. where 5 has the same meaning as in (13.63). Sinc (ht)l = hL, all factors hL(Z2.k) c> 0 for some c which is independent of j
with
and k. Hence the left-hand side of (14.21) can be estimated from below by its
right-hand side. To prove the converse we note that the terms with j k in (14.17) are harmless. Together with the znonotonicity (13.67) of the coefficients and the converse of (14.22) we get (14.21). But now we are very much in the same situation as in Step 4 of the proof of Theorem 13.2. The counterparts of (13.43) and (13.39) prove (14.19). Similarly one obtains (14.18). We are now
in the same situation as in Steps 6 and 7 of the proof of Theorem 13.2. First we get equality in (14.13), (14.14) and that q and p are the correct numbers in (14.5) and (14.7). respectively. 14.3
Inequalities
The above Theorem 14.2 is the counterpart of Theorem 13.2. Even more, with
Proposition 12.16 as a vehicle, 14.2 is a consequence of 13.2. It covers all cases of interest (excluding borderline situations as described in (13.2), which means here p = ac). It describes in a rather condensed way very sharp inequalities. In
II. Sharp inequalities
222
analogy to 13.3 we discuss the outcome, where now the harvest is even richer, since we have not only inequalities in terms of moduli of continuity hidden in Theorem 14.2, but even sharper inequalities of type (14.9). As in 13.3 we formulate the corresponding assertions for the B-spaces in (i) and for the Fspaces in (ii), but in contrast to 13.3 a few explanations and justifications are needed. This will be done afterwards in (iii). We always assume that 0 <e < 1. 14.3(i) The B-spaces Let x(t) be a positive monotonically decreasing function on (0,e). Let 0 < uS oo. Let p and q be given by (14.4). Then: (i1)
(/
(14.23)
tIl:tI)
for some c > 0 and all f E
(R") if, and only if, x is bounded and
q
(/
(x(tflVfit))U
(14.24)
)
P(Rfl) if, and only if, xis bounded and for some c > 0 and all f E q 5 u 5 oo (again with the indicated modification if u = oo). In particular, if then 1
O
(j
<
Ilogtl
t
j
(14.25)
ci
in place of are the two end-point cases according to (12.28). The two types of inequalities (14.23) and (14.24) are (and similarly with
connected by e
u
I U
dt
<
tllogtl)
—
Co
([(jvfI'(t)\ 1+1L
I U
U
dt
(14.26)
"(Rn), where again q 5 u 5 oc (with for some c0 > 0, c1 > 0, and all f e the modification (14.28) below if u = oc). If q = co then one has sup
O
Iogt(
<
(14.27)
14. The super-critical case
223
Let x(i) be an (arbitrary) positive function on (0, eJ and let again 1 < q
00.
Then
O
(14.28)
for some c> 0 and all I
if, and only if, x is bounded (and the E But as discussed in 12.11 the same assertion with 1Vf1(t) in place of above additional assumption that x is monotone is rather immaterial. The F-spaces Let K(t) be a positive monotonically decreasing 14.3(11) function on (0,cl. Let 0 < u oc. Let p and q be given by (14.6). Then: (ii1)
(/ for some c > p
oo,
cii!
tilogtl) 0
and all f e
(R") if, and only if, x is bounded and
with the modification
sup
and
(14.29)
if
= 00;
(14.30)
(112)
(14.31)
tllogtl)
U
(IR?z) if. and only if. x is bounded and oc with a similar modification as in (14.30) if u = oc. In particular,
for some c > 0 and all f E
p(u
sup O
ilogtV
<
(J
Iogti
ci
t
(14.32)
are the two end-point ca.ses (and similarly with IV! I*(t) in place of according to (12.28). The two types of inequalities (14.29) and (14.31) are connected by e
\
( I
log ti
u
e
dt
ti log tI)
_
(1 I log
dt ti log (14.33)
II. Sharp inequalities
224
'4-fl
forsomec0>0,c1 the modification as in (14.30) if u = oo). As in connection with (14.28) one does not need for the sharpness assertion in (14.30) that is monotone. t) fol14.3(111) Explanations The above inequalities with respect to 2 in 12.4, Definition 12.14 and the modified low from Theorem 14.2, and t) in place of Proposition 12.10 with and I respectively. Or in other words, they simply describe what is meant by a continuity envelope. Furthermore, (14.26), (14.27), and (14.33) are covered by Step 1 of the proof of Theorem 14.2. The only point which is not immediately clear by the above theorem and its proof is the boundedness of in (14.24) and (14.31). By (12.14) this question can be reduced to sup
x(t)IVfI*(t)
(14.34)
O
and its F-counterpart. Hence we assume that we have (14.34) with I
bounded.Letf6begivenby(14.17)withb,=lifj=1,...,Jandbj—Oif
j>
Then by an argument similar to that in (14.22) we have
fVfJ(x) cJ in a cube
with 2'
J E N,
(14.35)
2K where c> 0 and K E N are independent of J. Then cJ and it follows by (14.34) and (14.18)
J*
(14.36)
Hence x is bounded. This proves the x-sharpness also in (14.24) and (14.31). 14.4
More handsome inequalities
In 14.3 we tried to unwrap what is hidden in Theorem 14.2 and, with a switch from f to IVf in Theorem 13.2. This may also be taken as an excuse for the undue length of 14.3 (compared with the lengths of the respective theorems). Nevertheless the formulations remain somewhat involved. But in case of u = oc, this means (14.28) and (14.30) with x = 1, one can convert these assertions into more handsome inequalities which come also near to what is done in the literature. Let v > 0. By the definitions of the moduli of continuity in (12.69)
14. The super-critical case
225
and (12.72) we have with 0 <e < 1, sup O
=
iogt
=
1
sup
sup
O(t<e t iogt
11(x) —
sup
Then (14.28) and (14.30) with x =
If(x+h)—f(x)I
1
(14.37)
can be reformulated as
yER',
I
If(x)—f(y)I
(14.38)
and Ix —
<E, for some c> 0 and all f E
"(Rn), where
O
and Ix— .vI <E, for somec> 0 and all f
P(Rfl) ,where
l
f(x)
—
—
Ilogix
XE R",
Ill
—
(14.40)
and Ix
—
<E. where
=
are the Sobolev spaces, 1 <
7) < 00.
14.5
Borderline cases
This means in our context here p = 00 and s = 1, hence the Besov spaces where 0 < q 00, with the Zygmund class = as a special case. We refer to 1.2(iv), (v), especially (1.11). The extension of (14.38) top = q = 00 is given by
If(x)—f(y)I
,
(14.41)
II. Sharp inequalities
226
It is due to A. Zygand Is — <e < 1, for some c> 0, and all f E mund, [Zyg45], and may also be found in [Zyg77], Chapter II, Theorem 3.4, p. 44. It was apparently A. Zygmund who coined the word smooth functions in this context in his paper [Zyg45]. In [Zyg77J, Notes, p. 375, he mentioned that B. Riemann was the first who considered smooth functions. B. Riemann discussed in his Habilitationsschrift [Rie'54] the possibility to represent a continuous periodic function on the interval [0, 2ir] in terms of trigonometric series: First he surveyed what had been done so far. Afterwards he studied in Sections 7-13 the indicated problem in detail based on the systematic use of second differences. This is just what A. Zygmund called almost 100 years later in [Zyg45] smooth functions. The extension of (14.25) top = 00 (and 1
O
(J
t
j
(14.42)
(with the obvious modification if q = oo which is essentially (14.41)). This has been proved very recently in [BoLOO], Proposition 1. Furthermore, (14.25) with (14.4) and (14.32) with (14.6) follow from (14.42) and the embeddings
Bpq "(IR") c
(14.43) "(IR") c [Ti-iflj, Theorem 2.7.1, p. 129. (In the second embedding we used the first one and [Triflj, (15) on p. 131.) Our own approach is characterized by lifting the
and
critical case, considered in Section 13, to the super-critical one considered here. This results in the sharper inequalities (14.26), (14.27), (14.33), where always
p < oo. But it is unclear whether there is something of this type if p =
oo.
As discussed in connection with (13.3), based on Theorem 11.2, the question itself makes sense at least for the spaces with 1 < q S 2, also for the space bmo(fl), lifted by 1, according to 13.7. By (14.42) and (14.43) the sharpness assertions available for the spaces with p < oo can be carried over to the spaces We get the complement
=
1< q oo,
(14.44)
of (14.5). In particular, the 1b, given by (14.17), are extremal functions also for and we have (14.18) with p = oo. If p = oo then
f(x)=h(x)x1 logIxI, xER'3, with h given by (13.63), for example, is an extremal function in
(14.45)
This
follows from log IxI E bmo(IRIZ) C C°(R"),
IxI
(14.46)
14. The super-critical case
227
the boundedness of the other first derivatives, and elementary calculations. The embedding mentioned is well known. We refer to [RuS96]. p. 33. Other extreinal functions may be found in 17.1. Envelope functions and non-compactness
14.6
This remark applies equally to growth envelope functions and continuity envelope functions and to all cases (critical, super-critical, sub-critical). But it will
be clear what is meant by looking at an example connected with the above considerations. Let fI be a bounded C3° domain in R" (one might think of the unit ball). Then has the usual meaning according to Definition be. by definition, the Banach space of all 5.3. Let u 0. Let (complex-valued) continuous functions in such that = sup If(x)I +
Ill
If(s) — fQi)I
sup
Is — ui I log Iz — ui
(14.47)
We use here the notation introduced in [EdH99]. Then (14.38), restricted to is equivalent to the continuous embedding
O
(14.48)
(where we excluded q = oc). However this embedding is not compact. We prove this assertion by looking at the growth envelope function I log ti q for
(as for spaces on domains we refer also to 13.6). Since q < 'c it follows that — COc(1l). the restriction of is dense in Bpq (cl). Assume that the on embedding (14.48) is compact. We fix a quasi-norm in Bpq "(fi). Then we find for any 6 > 0 finitely many functions
Si with j= 1,...,M(6), such that for any f with Of
if w(f
—
t)
"(f)ii
5.
1,
uniformly for
0
.'
Here functions
(14.49)
is a 6-net. Furthermore we used (14.37) with are smooth one obtains where
(14.511)
2
=
Since
the
(14.51)
II. Sharp inequalities
228
and hence by (12.77), 1+n
ScBpq "(t) C5 +
log tI
,
0
1
(14.52)
If 5> (1 is small one gets a contradiction to ecBpq
when t is tending to zero. Hence the embedding (14.48) is not compact. But it was not so much our aim to prove this specific assertion. We wanted to make clear what happens if both source and target space have the same envelope functions (growth or continuity): The respective embeddings are not compact (at least in those cases where smooth functions are dense in the source space). 14.7
References
First we recall that (11.59) coincides with (14.40) if 1 + = k E N. This inequality in this version is due to [BrW8OJ. We refer also to our remarks in 11 .8(vi). The extension (14.40) from the classical Sobolev spaces W (R") to + may be found in [EdK95). As mentioned in 14.5 the borderline case, in our notation used here, (14.41) goes back to [Zyg45] (and may be found with a new proof in [Zyg77], Chapter II, Theorem 3.4, p. 44). A Fourier-analytical proof of (14.38), at least in some cases, has been given in [Vis98j. The additional point of interest here is the use of spaces of type (on domains
and in IR") according to (14.47) in connection with problems from physics. We refer in this context also to [Lio98], pp. 146, 152. The first full proof of (14.38) and (14.39), including sharpness assertions, was given in [EdH99]. In the context of this paper sharpness means that the exponents and in the log-terms in (14.38) and (14.39), respectively, cannot be replaced by a smaller exponent. The proofs are based on atomic decompositions. The borderline inequality (14.42) (without the middle term) has been derived in [KrS98] using extrapolation techniques. By (14.43), and as has also been mentioned in [KrS98] explicitly, this results in new proofs of (14.38), (14.39). As remarked in 14.5 the decisive improvement concerning the middle term in (14.42) is due to [BoLOOJ. Our own approach which resulted not only in Theorem 14.2, including the x-sharpness as described in 14.3, but also in the sharp assertions in 14.3 concerning IVfI*(t), especially (14.26), (14.27), (14.33), is published here for the first time. Especially the concept of envelope functions and envelopes (here in connection with the super-critical case in the understanding of Theorem 14.2) came out in recent discussions with D. D. Haroske. A more systematic treatment will be given in [HarOlJ. In a different context lifting arguments have also been used in [EdK95J with a reference to [Adm75], Theorem
15. The sub-critical case
229
8.36, pp.254—255. Somewhat different types of envelopes appear in [Net87b] in connection with optimal embeddings of in rearrangement-invariant
spaces, preferably in sub-critical situations which will be treated in Section 15 below. Finally we add a remark in connection with 14.6. As mentioned, there is no hope that the continuous embeddings (14.48) are compact. However the situation is different if one replaces the target space in (14.48) by according to (14.47) with > Then one has compact embeddings. The adequate notation to measure the degree of compactness are entropy numbers and approximation numbers. As for the general background we refer to [ET96], Chapter 1. But later on in connection with the spectral theory for fractal elliptic operators we repeat in 19.16 what is needed. A detailed study of entropy numbers and approximation numbers for problems treated in the present section (in the modification indicated above) has been given in [EdH99] and EEdHOO]. This has been complemented in [HarOOa]. The small survey IHaroob] summarizes these results.
15
The sub-critical case
15.1
Introduction
By the terminology of (10.5) the sub-critical case covers the spaces and (15.1)
Recall our standard abbreviations
\P
1
and
1+
\min(p,q)
(15.2)
0
where
.
1
s>0,
p
r
and 0
(15.3)
We characterized in Theorem 11.4(i) those spaces (15.1) which are embedded in In Theorem 11.7(i) and (ii) we collected the classical more refined inequalities in the sub-critical context and we described their rather rich history in the points 11.8(i)—(iv). Again we are interested only in spaces which consist entirely of regular distributions; this means that they are embedded in A final description has been given in Theorem 11.2. Compared with
II. Sharp inequalities
230
(15.1) we exclude as previously borderline cases, we mean here those spaces and with 0
Theorem
Let
s>O,
with 1
and
(15.4)
(the dashed line in Fig. 10.1). Then
=
(15.5)
and
=
(15.6)
Proof Step 1 Let 0 < e < 1. In 11.8(u) we proved (11.52). The interpolation argument used there applies to all cases covered by (15.4), [Trif3], Theorem 2.4.2, p. 64. Hence, together with (12.26), we obtain always A
sup
If q =
oo
f(t)
(t3 (j \0
(15.7)
t
then one has only the first and the last term. By (12.37) it follows
that
0 < t <E.
(15.8)
15. The sub-critical case
231
As for the F—space.s we use Netrusov's observation described in (13.29). Siini—
larly as in (13.30) it follows that
f(t)
CO
(J
Cl
)
Ill
(15.9)
\0
Hence we have also
0< t <E.
6cFq(t) Step 2
Let t; he given by (13.35). By ç,.(23x).
=
(15.10)
Theorem 13.8. p. 75.
j E N,
(15.11)
(R") where no are atoms in all spaces (R71) and at least in those nioinent conditions are needed, say q 1 (ignoring constants, which may be chosen independent of j). Again by (12.37). ,
j EN.
(15.12)
for some d > 0 and c > 0. Together with (15.8) we obtain
=
t<
(15.13)
Similarly for the F-spaces as far as they are covered. If q > 0 is small then the moment conditions needed for the atoms can be incorporated in the same way as in Step 5 of the proof of Theorem 13.2. Then we have aLso
t < r,
=
(15.14)
without any restricta)n for q. Step .9 It remains to prove that q and p in (15.5) and (15.6). respectively, are the correct numbers. Since we have already (15.7) and (15.9) we must show that q and p. respectively, cannot be improved by smaller numbers. Let v < q and let
(I for some e > 0 and all I E
C If
(15.15)
Let (15.16)
II. Sharp inequalities
232
is large then the supports of the atoms in (15.16) are with x0 E IR". If disjoint. It follows for some d> 0,
where j=1,...,J.
(15.17)
We insert (15.17) in (15.15). Since (15.16) is an atomic decomposition we get
/J \i=i
/j \q
1
.1
)
/
,
)
(15.18)
\i=i /
where c0 > 0 and c1 > 0 are independent of J. But this is a contradiction. In in case of the F-spaces we assume v
15.3
Inequalities
The above theorem covers all cases (15.3). It excludes borderline situations as described in 15.1. Parallel to 13.3 we explain also in the sub-critical case considered now, which is hidden in the above theorem. We use Example 1 in 12.4, Definition 12.8 and Proposition 12.10. Let 0 < e < 1. be a positive monotonically decreasing funcThe B-spaces Let tion on (O,eI. Let 0 < u 00. Let p, q, s be given by (15.4). Then 15.3(i)
I
(/ (M(t)t* ft(t)) for some c> 0 and all f E with the modification sup
O
(15.19)
if, and only if, xis bounded and q u < cc,
f*(t)
(15.20)
if u = cc. Furthermore, (15.7) deals with the two end-point cases according to (12.26). Let x be an arbitrary positive function on (0,e]. Then (15.20) holds if, and only if, x is bounded.
15. The sub-critical case
233
be a positive monotonically decreasing The F-spaces Let function on (0,E]. Let 0 < u 00. Let p, q, s be given by (15.4). Then 15.3(u)
/8
(J
(x(t)t; f*(t))
tzdt
(15.21)
if, and only if, x is bounded and p for some c > 0 and all I in place of when u = oo). Also the u oo (modified by (15.20) with other assertions for the B-spaces after (15.20) have obvious counterparts, in particular the two end-point cases (15.9) according to (12.26). (Ii) were introduced in 11.6(i). The above theorem and The Lorentz spaces the explanations just given can be reformulated in terms of natural and sharp and with (15.4) into We embeddings of the spaces complement these assertions by looking at corresponding optimal embeddings into Zygmund spaces Lr(logL)a(Ic) according to 11.6(u). By (11.46) the original definition (11.44) can be reformulated in terms of rearrangement. Optimal
means here that for given r in (15.4) and in (11.46), (11.54), one asks for all numbers a for which we have the desired embedding, again formulated in terms of inequalities. Corollary
15.4
Let p, q, s be given by (15.4) and let 0 <
e
<
1.
Then
(i)
dt) if, and only if, a 0. r. Then
for some c> 0 and all f E Let, in addition, 0 < q
(ii)
(15.22)
I
dt) for some c > 0 and all f E (iii) Let, in addition, r < 1
q
1
r
q
(15.23)
if, and only if, a 0. oo. Then (15.2.Y) holds if, and only if, a <
II. Sharp inequalities
234
r follow from Step 1 If a = 0, then (15.22) and (15.23) with q Theorem 11.4(1). Let r < q. Then (15.23) with a < — is a consequence of (11.66) and (15.7). This covers all if-parts. Step 2 It remains to prove the only-if-parts of the corollary. First we insert given by (15.11) in (15.22) and (15.23). By the equivalence in (15.12) we get Proof
0.
(15.24)
Hence a 5 0. This completes the proof of (i) and (ii). As for (iii) we modify (15.16) by
1(x) =
—
x°)
with
(15.25)
b, =
This is again an atomic decomposition and we have
/00 Ill
5c \j=2
\.i J
/
<00.
(15.26)
j = 2,3
(15.27)
On the other hand, (15.17) must be modified by
f*(d2_in) Inserted in the left-hand side of (15.23) with a =
J ()
f*(t))r dt
—
we
obtain
= 00.
(15.28)
This proves the only-if-part of (iii). 15.5
Further references
Embeddiugs and related inequalities iii sub-critical cases have a long and rich
history. We tried to collect the relevant papers in 11.8(i), with respect to the Lorentz spaces and in 1 1.8(iii), with respect to the Zygniund spaces This will not be repeated here. In connection with the critical Lr(log case we gave some additional references in 13.5, which apply at least partly also to the sub-critical case considered here. In particular, .Ju. V. Netrusov anticipated in LNct87al, and also in [Net89al, in a somewhat different context, the concept of envelope functions and optimal embeddiugs in rearrangementinvariant spaces. More recent (and independent of each other and of Netrusov's
16. Hardy inequalities
235
work) treatments have been given in [CwP98J and in [EKPOO] (restricted to spaces). As for Sobolev spaces, in contrast to Netrusov, who considered related capacity estimates we refer again to [Maz85] and to the recent paper [Sic99], where one finds also further references. This section is based on [Tri99d]
and might be considered as an improved and extended version.
Hardy inequalities
16 16.1
Introduction
In this book we dealt so far several times with Hardy inequalities. But first we wish to mention that the whole story began with Hardy's note [Had28] and the famous Theorem 330 in EHLP52I, p. 245 (in small print). As a consequence
(ignoring constants) one gets the following assertion: Let 1 < p < oc and
mEN. There isanumberc>Osuch that
j
f
dmu(t)
dt
(16.1)
for all
uES(IR)
with
d'u
for j=0,...,m—1.
In the years after, and especially in the last decades, hundreds of papers and dozens of books have appeared dealing with numerous variations of inequalities of this type. The reader may consult [OpK9OI and the references given there. As far as this book is concerned we refer to 5.7—5.12, making clear how different natural inequalities for F-spaces and B-spaces might be. Of special interest in this section is the following consequence of the previous results. Let be a
bounded C°° domain in R". Let
f=O1, be
D(x)=dist(x,I')= infix—yl, XEW', y€
(16.2)
the distance to r and, for e > 0, r'e = {x
E
:
D(x) <e}
(16.3)
be a neighbourhood of I'. Let
O
(16.4)
There is a number c> 0 such that
j D8'(x)
(16.5)
236
II. Sharp inequalities
for all f E
This follows from (5.104), (5.105). There one finds also the necessary explanations and further assertions of this type. This measures how singular a function f belonging to near 1' = Oil can be. Let r be an arbitrary, say, compact set on Of interest is the behaviour of or near or at r. functions I belonging to a given space There are two different, but closely related aspects: traces on r and Hardy inequalities of type (16.5). If 1' is smooth (maybe I' = Oil as above) then the trace problem is more or less settled and treated in detail in most of the books mentioned in 1.1. Specific references and rather final formulations and (excluding borderline cases) may be found in [Trifl], 2.7.2, 3.3.3, pp. 132, 200, and 4.4.2, 4.4.3, pp.213—221. Sophisticated borderline cases have been treated recently in [JohOO] and [FJSOO]. If r is an irregular, say, compact, set in then the situation is different. We considered this problem in some detail in Section 9 and refer in particular to Theorems 9.3, 9.9, 9.21, and 9.33. There we quoted also the relevant literature. Special attention has been paid to d-sets. Of interest here is Proposition 9.13. One aim in the present section is to complement these trace assertions by a discussion about related Hardy inequalities. We outlined our intentions at the end of Section 10 and added also a warning concerning the outcome. As stated there we are interested with some preference in 1' = {0}, where we get sharp results.
But we look also at more general sets. In principle the method to get, for example (10.26) or (10.28), is quite simple. We use (10.23) as a vehicle to reduce
Hardy inequalities to Theorems 13.2 and 15.2, and the related inequalities in 13.3 and 15.3, respectively. A few points should be mentioned. First, in case of r = {0} we deal both with and Bk-spaces, although really satisfactory inequalities for the B-spaces look somewhat different. We refer to (5.77). This may justify that we later on concentrate on the F-spaces. Secondly, if r is the boundary of a C°° domain or (part of) a hyper-plane and if one deals with the full spaces or (and not with appropriate subspaces) then there seems to be a clear distinction between those spaces having traces on r and those spaces with substantial Hardy inequalities. But
if r is irregular the situation might be different. As examples, (16.5) may serve on the one hand and (10.26) on the other hand. In case of irregular compact sets r we have no final assertions, and the later parts of this section might be considered as a discussion of how to shed light on the possibly tricky
interplay between Hardy inequalities, the geometry of irregular sets r and related measures. This justifies our restriction to examples, mostly F2(Rh1). We complement (16.3) by
Ke = {xE]R"
:
(16.6)
16. Hardy inequalities 16.2
237
Theorem
(Cr11 ical case)
< 1 and let x(t) be a positive monotonically decreasing function on
Let 0 (0.E]. (1)
Let.
i
(16.7)
Then
c
IKe
for some e> 0 and all f E (ii)
(16.8)
if, and only ii; X
13
Let
0
(16.9)
Then
x(IxI)f(x)
(16.10)
c
1K.
for some r> 0 and all f E
if, and only if, x is bounded. Proof Step 1 We prove (16.8) with u = 1. We may assume that the e > 0 is so small that 1
fltI(I, hence,
a(t)
(16.11)
c > 0. Recall that a*(t) is the measure-preserving if U < t < rearrangement of a(x). Then (16.11) follows from the behaviour of a(x) and o(t) at si respectively, where j e N is sufficiently large. and t \Ve obtain
L.
= 1K.
I
111
1F2(R")II".
II. Sharp inequalities
238
The first inequality is a well-known property of rearrangement and may be found in [BeS88), p. 44. It goes back to [HLP52] (first edition 1934), Theorems 368 and 378. The last inequality comes from Theorem 13.2 or, more explicitly, from (13.62). Similarly one proves (16.10) with x = 1, where one has to use (13.57). Step 2 We prove that x in (16.8) must be bounded. Let f(x) be a positive
monotonically decreasing function in Kg in radial directions. Since x is also assumed to be monotone it follows in analogy to (13.62) by (12.28) and (16.8), .1
p
(7 \
O
=
j t1
(16.13)
hi
(1K.
q 1. We insert fj with (13.50), (13.51). This proves that x must be bounded. If q> 0 is small, then one has to modify fj as indicated in Steps 5 and 6 of the proof of Theorem 13.2. But this does not influence the above Let
argument. Similarly one proves that c in (16.10) must be bounded. 16.3
Theorem
(Sub-critical case) Let e > 0 and let K(t)
be
a positive monotonicoily decreasing function on (0, ej.
Let
s>O
and
p
r
with
1
(16.14)
(the dashed line in Fig. 10.1). (i)
LetO
for some c > 0 and all I E (ii) Let 0 < q r. Then
if, and only if,
is bounded.
(16.16) IKe
for some c> 0 and all / E
if, and only if, x is bounded.
16. Hardy inequalities
Proof
239
Let a 0. Then (t)
where
t > 0.
(16.17)
This can be applied to the left-hand sides of (16.15) and (16.16) with a = 1 — > 0 and a = 1 — > 0, respectively. Then (16.15) and (16.16) with = 1 follow from the counterpart of (16.12) on the one hand, and (15.21) with p and (15.19) with u = q, respectively, on the other hand. If one inserts f3 (x) given by (15.11) (with the indicated modification for the F-spaces when q> 0 is small) in (16.15) and (16.16), then it follows that x must be bounded.
u=
16.4
Comments and references
First we look at the sub-critical case. Using (16.14), the inequality (16.15) can be reformulated as dx
(
C
(16.18)
IKe q oo. As mentioned in 1.2 if q = 2 and 1 < p < (Rn) are the Sobolev spaces (Rn) with the classical Sobolev spaces as a subclass if, in addition, s E N. Then inequalities of type (16.18) are known although explicit formulations are rare in the literature (especially in higher dimensions). But everything is included in the extensively treated problem of embeddings of Sobolev spaces in weighted spaces, or more generally We dealt in Section 9 with in L9 spaces with respect to Radon measures in questions of this type in the different context of traces. But the references given there apply also to the above case, in particular [Maz851, [AdH96], [Ver99]. the situation is different. Switching to general spaces and
where again 0 < then
The first explicit inequality of type (16.18) for the general spaces may be found in [Tri/3], 2.8.6, p. 155, which covers also the one-dimensional version of (16.5). Such inequalities also have anisotropic counterparts, at least and H We refer to [ST87], 4.3, for anisotropic spaces of type pp. 202—209, and the literature mentioned there. If p q then inequalities of More natural type (16.16) are not optimally adapted to the spaces inequalities may be found in [Tria], p. 319, and more general ones in [Tri99b]. But they do not fit in our scheme here. The above Theorem 16.3 is a modification of [Tri99d]. There one finds also additional discussions concerning the interrelation of rearrangement and Hardy inequalities. In the critical case as considered in Theorem 16.2 there are only very few papers. Restricted to classical Sobolev spaces W? (RTh) inequalities of type (16.8) with log-terms may be found in [EgK9O], Lemma 8, p. 155, and in [So194}. Restricted to H Theorem 16.2 has been proved in [EdT99b]. and
II. Sharp inequalities
240
We reduced the inequalities in the Theorems 16.2 and 16.3 to 13.2, 13.3 and 15.2, 15.3, respectively. It is clear that all the other inequalities mentioned there in 13.3 and 15.3 produce also sharp Hardy inequalities: One has to modify (16.12). Another possibility is to replace r = {0} by more general sets. In principle this does not cause much trouble. But it is unclear to what extent or for which r one gets sharp and natural inequalities. We formulate a few results and complement them by some discussions. It comes out that under some additional geometrical restrictions the outcome is far from being optimal. In other words, the main aim of the rest of this section is to shed light on these problems. This may also justify that we restrict our attention to the critical The first candidates beyond r = (0} and, case and in particular to maybe, compact smooth surfaces are d-sets. Let 0 < d < n. Then a compact and two is called a d-set if there are a Borel measure in set r in positive numbers c1 and C2 such that suppp = 1' and c1td
for all
0< t <1,
(16.19)
t) is a ball centred at y E r and of radius t. Further and -y E r, where details and references may be found in 9.12. 16.5
Proposition
Let 0 < d < n and let r be a compact d-set in
D(x) = dist(x,r),
Let
(16.20)
xE
to r. Let p, q be given by (16.7). Let 0 <e <1 and be the distance of x E let rE be an e-neighbourhood of r as in (16.3). Then
c li
(16.21)
for some c> 0 and all I Proof Let
r,={XEllr Then vol f,
jJ.
:
(16.22)
2'. With a(x) =
I log
one gets
0
(16.23)
16. Hardy inequalities 16.6
241
Discussion
Let F be a hyper-plane in
say,
(16.24)
with x' E and n 2. Letp, q be given by (16.7). For fixed x' E use the one-dimensional version of (16.8) and obtain for 0 < e < 1,
c
I
x' E
(16.25)
If 1
JR2'
log
x,,
we
f
have
(16.26)
where for some c> 0 and allf E is an e-neighbourhood of given by (16.24) according to (16.3). Since for fixed p with 1
Let x(t) be a posztive monotonically decreasing function on (0, e), where again 0 < e < 1, let p, q be given by (16.7). Then
5C
J
(16.27)
I
I
and only if, x is bounded. The if-part is covered by (16.26). We outline how the only-if-part can be proved by modification of previous arguments. Let Si
:
Ix'I<
1,
<2-i}
jEN.
,
(16.28)
We modify (13.50) by J
fi(x) =
—
3=1
x3'1))]
,
(16.29)
1=1
I
(we refer where {••.] are correctly normalized atoms or quarks m to (2.16)) and where stands for suitable lattice-points. (We assume, say,
II. Sharp inequalities
242
q 1, such that no moment conditions are needed. The necessary additional modifications if q > 0 is small have been indicated in Step 5 of the proof of Theorem 13.2.) We have a counterpart of (13.51) with n = 1 in the first 1
equivalence and with
(W') in the second equivalence: We
in place of
note that the arguments in connection with and after (13.40) with a reference to 2.15 apply also to (16.29). Then the desired x-sharpness follows as in Step 2 of the proof of Theorem 16.2. If one compares the sharp assertion (16.26) with (16.21) where flOW d = ii — 1, then it is quite clear that in this special case, (16.21) does not say very much. Even worse: Since for any 5 > 0 the space (—e, E) is continuously embedded in C(—e, e) (in obvious notation and with a reference to, say, 2.7.1) one has an immediate and rather I obvious counterpart of (16.26) with (R") in place of and with —1'• an arbitrary positive integrable function in place of Then, in this special case, (16.21) with n 2 is obvious. On the other hand, the above arguments depend on the special structure of r in (16.24) and on the possibility to apply the Fubini Theorem 4.4. But this is not the case if I' is a general d-set or an arbitrary fractal. In other words, the problem arises under which geometrical conditions for r the inequality (16.21) is substantial and sharp. Finally one can use (16.26) to complement, our considerations in 5.23 and also of (16.5). We formulate the outcome. 16.7
Corollary
Let
be a bounded C°° domain in (16.3). Let 0
and let F,
< < 1 and let K(t)
and D(x) be given by
a positive monotonically decreasing function on (0, eJ. Let p, q be given by (16.7). Then
[ x(D(x))f(x) logD(x)
JFE
be
<
D(x)
f
1630
/
for some c> 0 and all f if, and only if, x is bounded. Proof This follows from (16.27) and standard localization arguments. 18.8
Remark
If p, q, a are given by (5.104), then we have the sharp Hardy inequality (5.105). If now p, q are restricted by (16.7) and s = then
f(s) 1
+ IlogD(x)I
p
dx
C )
This is an itrimediate consequence of (16.30).
.
(16.31)
17. Complements
243
Proposition
16.9
Let p be a finite Radon measure in IR't and let F = suppp be compact. Let p, q be given by (16.7), 0 < e < 1, and
= where
B(x,
J B(x,e)
p(dy)
is a ball centred at x E
f Ip,e(x) If(x)17'dx
E
and of radius e. Then c
for some c > 0 and all f E Proof Let xe be the characteristic function of Ke given by (16.6). Let Then it follows by (16.8) that
I
(16.32)
iogix—yd Xe(x—')
(16.33)
E F.
(16.34)
Integration with respect to p and application of Fubini's theorem results in (16.33). 16.10
Remark
As mentioned at the end of 16.4 the Propositions 16.5 and 16.9 are far from final. This is also clear from the discussion in 16.6 and the more satisfactory assertions in 16.7 and 16.8. We mainly wanted to make clear that there might be a sophisticated interplay between the geometry of irregular fractal sets F and the singularity behaviour of functions belonging to spaces and near r. We restricted ourselves in the course of this discussion to the critical case extending Theorem 16.2. But of course one can deal in the same way with the sub-critical case as considered in Theorem 16.3. 17 1T.1
Complements Green's functions as envelope functions
Looking at (13.7) or (13.9) one may ask whether there are functions I belonging to or F2 (1W') such that f (t) is equivalent to log or log respectively. If q < oc in (13.7), then it follows from (13.57) that this is impossible since in such a case the middle term diverges. Because always p < co,
244
11.
Sharp inequalities
(1W'). Similarly one has by (13.62) a corresponding argument for the spaces for the sub-critical case according to Theorem 15.2 and (15.7). Corresponding questions can also be asked for the super-critical case considered in Theorem 14.2. If q = oo then the situation is different. We deal first with the critical case as covered by Theorem 13.2 and by 13.3. Let S be the usual 5-distribution in 1W' with the origin as the off-point. Then
SE
where
0< p <
oc.
(17.1)
be This is well known and also an easy consequence of (2.37). Let again the Laplacian in 1W'. By well-known lifting properties of — + id it follows
that G = (id —
0
E
00.
(17.2)
where G might be considered as the Green's function of the fractional power (Id — of id — We claim that G(x) is a C°° function in IW'\{0} which decays exponentially if lxi —4 00 and
G(x)—'ilogixli
if
lxi
(17.3)
if 0 < t < e < 1. Hence G(x) is an extremal function for
By
Definition 12.8 and (12.60), and in agreement with (13.7), (13.58) we have
0
(17.4)
We outline a proof. Let
Ie
dt
T'
x€IR
.
and with respect By well-known properties of the Fourier transform of e to dilations x —p cx, c 0, e.g., [Tri92}, pp. 100/101, it follows that
=
=
F = c(1 +
(17.6)
17. Complements
245
for some c > 0. Here we used the Fubini theorem. This is possible since an x [0, oo). integration over in (17.5) results in a convergent integral over Application of the Fourier transform to G, introduced in (17.2), gives
G(z) = for some c >
0.
C
J
0,
xE
We estimate G(x). Let lxi
(17.7)
1. We split the integral in (17.7)
in lxi
IxI
00
=
(17.8)
and in
7eixi e_tC
(17.9)
lxi
IxI
This proves the exponential decay of G(x) if lxi —p 00. (Of course all constants
in the above estimate are positive.) Let lxi > 0 be small. Then by (17.7) and r. t= G(x)
i+
J
(17.10)
Hence by (17.2). the decay assertions, and (17.3) it follows that G(x) materializes the envelope function in (13.7) with q = oo. Furthermore by (17.2) and (13.87), the Green's function C belongs also to for all 0 < q 00, in particular
Finally we mention that (17.3) in case of n = 2 is essentially the well-known behaviour of the Green's function of the Laplacian in the plane. The super-critical case can be reduced to the critical one as follows. Let g be given by (17.5). Then it comes out that
0<poo,
(17.11)
II. Sharp inequalities
246
and
IVhI*(t) I
0< t < e.
(17.12)
1) are the divided differences introduced in (12.72). In particular. h is an extremal function according to (14.27) and we have by Definition 12.14. (12.85) and (14.5) where
llogtI.
(17.13)
We outline a proof. By similar arguments as in (17.6) it follows that (17.14)
Furthermore, as in (17.10) we obtain Oh
j
e
-
(17.15)
This proves (17.12) (one needs only an estimate from below, since the estimate from above is covered by (14.27)).
Finally, formulas like (17.7) originate from heat kernels and their relations to Green's functions. We refer to [Dav89), 3.4, pp. 99-105, for details. 17.2
Further limiting embeddings
In all three cases, critical, super-critical, sub-critical, treated in Sections 13. 14. 15, respectively, we avoided borderline situations. This in the critical and sub-critical case spaces with parameters as described in (13.2). (13.3) as far as they are covered by Theorem 11.2. In the super-critical case we excluded p = oc in the source spaces and concentrated on the target spaces exclusively on .s = 1. p = oo (the dotted line in Fig. 10.1). This might be justified by the history of the topic which we tried to collect in Theorem 11.7 and on which we commented in 11.8. It would be of interest to have a closer look at these
omitted spaces. In the context of a more systematic study it might be even reasonable to modify the sub-division of the spaces covered by Theorem 11.2 in the above three distinguished cases as follows: to extend the sub-critical case as described in (10.5) to those spaces with (i) s = = n.( — 1) + covered l)y Theorem 11.2,
17. Complements
247
to extend the critical
case as described in (10.6) to those spaces with covered by Theorem 11.2, and to call all other spaces covered by Theorem 11.2 super-critical. (iii) Any subdivision of the spaces covered by Theorem 11.2 depends on the ad-
(ii)
p=
oc
mitted target spaces. This means in the sub-critical and critical case spaces with s = 0 according to Fig. 10.1 and in the super-critical case a = 1, p = A somewhat more general case of interest in connection with target spaces is given by
s=1,
(17.16)
(again in the understanding of Fig. 10.1). We give a brief description of the set-up in a slightly more general context. We use standard notation. Let m E N and 1
=
0< t <00,
sup
IhIt
is the iLsual tutu order modulus 2.7, p. 44. Here is given by
of continuity,
(17.17)
[BeS88], 5.3, p. 332 or IDeL93I, Besov spaces
(1.12). The classical
with
described in 1.2(v). can be normed by
= If
If
+
(17.18)
(I
if q < oc and l)V
if q = oc. Here 0 < in
(17.18), (17.19).
+
=
If
< 1, [Tri/3], 2.5.12, R. Then
(17.19) O
t
p. 110. Now we incorporate a log-term
Let b
with
is the collection of all f E
with e
If
= Ill
+
(J
I <00
(17.20)
II. Sharp inequalities
248
(obviously modified if q = oo). These spaces can be characterized in Fourieranalytical terms. Let ç°k be the same functions &s in (2.33) -(2.35). In generalization of (2.37). I
/00
(17.21)
)
/
\j=o
(obviously modified if q = cc) is an equivalent norm in spaces, in their general version of with (2.36) and b
These
go back to H.-G. Leopold in 1998 and may be found in (Leo98] and [LeoOOaj. The point of interest in our context of distinguished target spaces is to replace the second differences in (17.20) by the first differences = and to introduce in this way spaces of Lipschitz type. consisting of
all I E
such that I
+
= Ill
Ill
(17.22)
is finite (with the usual modification if q = oo). Here
(k>-. (with 0 if q = oo). The restriction on is natural. This follows from (with the considerations in 12.12: If < 0 in case of q = oo) then. there are no functions f such that (17.22) is finite. with exception of f = 0. These spaces were introduced in Definition 1. If p = cc in (17.22) then the inequalities in 14.3 for the super-critical case can be reformulated (at least locally) in terms of these target spaces. Hence it is reasonable to extend these considerations from p = cc to, say, 1
+
O
t
(17.23)
We refer to LSte7O]. Proposition 3, p. 139, (Nik77], 4.8, p. 213 (first edition with 1
17. Complements
249
go into detail. A thorough investigation of all these spaces, especially their mutual embeddings, may be found in [HarOOa} with [EdH99] and [EdHOO] as forerunners. We refer also to the small survey [HarOOb]. Finally we mention that limiting embeddings especially in the super-critical case for spaces with dominating mixed derivatives have been considered in [KrS98}. 17.3
Logarithmic spaces
be a bounded C°° domain in R". We assume that = < 1. Let can be introduced much 1
+ If(x)I) dx < 00.
(17.24)
As in (11.46), these spaces can also be characterized as the collection of all I E L1(1l) such that I
(J(Ilogtlaf*(t)Y dt) <00
(17.25)
(equivalent quasi-norms). Details and references are given in 11 .6(u). Based on [EdT95] we proved in [ET96], 2.6.2, Theorem 1, pp. 69/70, another characterization of these spaces with the consequence that they have the same mapping properties with respect to pseudodifferential operators and fractional powers of elliptic operators as the space with 1 < p < oo, itself. In particular one can define logarithmic Sobolev spaces by lifting of in the following way. Let
mEN,
(17.26)
and let Am,Nf = Amf be the corresponding Neumann operator with the domain of definition
domAm,N__{f€Hp2m(1l) where ji
is
,
:
the outer normal with respect to
aER, l
(17.27)
Let
and s=2mr.
Then one can define
H(logH)a(1l) = ATN Lp(logL)a(Ifl.
(17.28)
II. Sharp inequalities
250
We refer for details and explanations to [ET96], 2.6.3, PP. 75-81. In particular. (17.28) imitates the lifting (1.8). If a N, then one obtains, as should he the case,
H(logH)a(I) =
{f E L1(1l)
s}
.
(17.29)
with the equivalent norms .
If a E N and s =
(17.30)
then one is in the critical case with logarithmically
modified classical Sobolev spaces (the dotted line in Fig. 10.1). One may ask for counterparts of Theorem 13.2 and related inequalities in 13.3. Some results of Trudinger type as in (11.56) and 11.8(v) may be found in [FLS96I. Extensions to the fractional case, including Sobolev-Orlicz spaces, have beexi given in [EdK95]. The interest in these logarithmic Sobolev spaces comes also from the regularity properties of the Jacobian. References can be found in [FLS96]. This having a closer look at these logarithmic spaces from the point of may view of sharp inequalities as treated in this chapter. 17.4
Compact embeddings
We proved in 14.6 by geometrical reasoning that the sharp embeddings (14.48) cannot be compact. If one replaces in (14.48) by with then <
one gets a larger space and it turns out that the corresponding embedding is compact. The degree of compactness can be measured in terms of entropy numbers and approximation numbers. Definitive results in this direction have been obtained in [EdH99J, [EdHOOJ. [HarOObJ and recently iii ICoKOO]. This covers also the more general spaces normed by (17.22). and the delicate interplay between these spaces and also in relation to the spaces introduced in (17.18). (17.19). (17.20). As for time latter spaces we refer also to fLeo98] and [LeoOOaJ. The general background may be found in [ET96] and, in connection with weighted and logarithmic spaces as discussed in 17.3, in [Har97j, [Har98), and [HarOOcJ.
Chapter III Fractal Elliptic Operators 18
Introduction
Let r be a compact d-set in
where n N and 0 < d < n, according to (9.1) or (9.67). By our previous considerations, d-sets are especially well adapted to the function spaces and as treated in this book. In particular, by 9.19 and 9.18 such sets satisfy the ball condition with the consequences for traces as described in Theorem 9.21. Furthermore, there is a natural way of introducing function spaces on 1' as mentioned in (9.125) with a reference to [TriSI, Definition 20.2, p. 159. Quarkonial representations of such spaces, in the larger context of more general fractals, have been given in 9.29 9.33. These results pave the way to a substantial theory for fractal elliptic (pseiido)differential operators in continuation of [Triö], Chapter V. We refer especially to [ThÔ]. Section 26, where we discussed in detail our point of view, compared with what has been done in the literature. In particular, there are different interpretations of what is called a fractal drum. The physical background of our approach in [Trio] and also here, as far as spectral theory is concerned, may be found in [TriO], 30.1, and will be briefly repeated in 19.1. As indicated one could continue the studies started in [TriOl, Chapters IV and V. based now on the theory developed in Chapter I of the present 1)00k on the level of (degenerate) fractal elliptic pseudodifferential operators, including fractional powers of elliptic differential operators. But this is not our aim. As far as elliptic operators are concerned we stick to the Laplacian as an oiitst andmg exaniple.
We give a brief description of what follows. Let r be a compact d—set in R" equipped with the Radon measure ,.i such that .SItJ)j)/1
= 1',
/z(B(-). r))
r'1
where
y
r
and 0 <
r<
1,
(18.1)
III. Fractal elliptic operators
252
according to the beginning of 9.12, in particular (9.67). Let be a bounded domain in R" with r C and let be the Dirichlet Laplacian in Il. Let n — 2
(18.2)
:
can be extended by continuity to a bounded into H'(fi) and, as a consequence, into
sending
map from
otr1'
B=
(18.3)
turns out to be a seif-adjoint operator in At least in case of n 2, this operator has physical relevance, describing the vibrations of a drum with the fractal membrane r. A detailed discussion may be found in [Th6}, Section 30. In particular,
keN,
(18.4)
where Qk are the (ordered by magnitude) positive eigenvalues of B. One aim
of Section 19 is to complement these results. In particular we discuss the smoothness of the eigenfunctions and prove that the largest eigenvalue is simple. If n = 2 then by (18.4) we have the classical Weyl behaviour It is the second aim for all d-sets r with 0 < d < 2 and the related measures of Section 19 to discuss under what circumstances for general compact Radon measures in the plane this remarkable property remains valid. Section 20 deals with the Dirichiet problem for in with given boundary data at I', where F is a d-set with F C and n — 2 < d < n. In Section 21 we return to the Riemannian manifolds of hyperbolic type as studied in Section 7, especially to the (fractally bounded) d-domains according to the end of 7.5. We are interested in the so-called negative spectrum of the (seif-adjoint) operator H8 =
+ &iid —
with
0
and
> 0.
(18.5)
where g has the same meaning as in (7.9), g> 0 is sufficiently large, and is the related Laplace-Beltrami operator according to (7.16). Of particuliar interest, and in slight modification of (18.5), is the behaviour of the hydrogen atom in this infinite hyperbolic world. We rely on the techniques developed in Section 7. especially on the quarkonial decompositions from Theorem 7.22. With exception of the end of Section 19 (Weyl measures in the plane) all fractals involved in Sections 19—21 are d-sets. On the other hand we developed
in Section 9 a theory for more general fractals and their relations to function spaces. One may ask to what extent the d-sets in Sections 19-21 can be replaced by more general fractals. In principle this should be possible for some classes
19. Spectral theory for the fractal Laplacian
253
of fractals. But one must have in mind that d-sets are tailored to the spaces and considered so far in this book. Changing the class of admitted fractals then, despite a few rather satisfactory assertions obtained in Section 9, one needs apparently some new optimally adapted function spaces. We describe this method in the remaining two Sections 22 and 23 in case of socalled (d, '11)-sets and their related function spaces. In generalization of (18.1) a compact set r in is called a (d, '11)-set if there is a Radon measure with
=
r,
1i(B('y,r))
where
y El' and 0< r <e,
(18.6)
for some 0 < e < 1. Here might be considered as a perturbation of typically 0 < d < n and '11(r) = I log ri" for some b E IR. Inspired by some previous work of H.-G. Leopold in [Leo98] we formulated this theory in [EdT98] and outlined the proofs in [EdT99a] parallel to the relevant arguments in [Trio). In the meantime there are several papers, preprints and PhD-theses dealing with the subject in detail. This might justify our being very brief here and giving the necessary references. In Section 22 we describe the theory of the underlying spaces including quarkonial decompositions, entropy numbers and spaces on (d, W)-fractals r. The related spectral theory for operators of type (18.3) will be outlined in Section 23, including the music of a rusty drum. 19
Spectral theory for the fractal Laplacian
19.1
The physical background
modify the explanations given in [TriO], 30.1, pp. 233—234. Let Il be a bounded domain in the plane 1R2 with C°° boundary interpreted as a membrane fixed at its boundary. Vibrations of such a membrane in JR3 are measured by the deflection v(x,t), where x = (x1,x2) and t 0 stands We
for the time. In other words, the point (x1, x2, 0) in JR3 with (xi, x2) E
of the
membrane at rest, is deflected to (xi,x2,v(x,t)). Up to constants the usual physical description is given by
ô2v(x t) ôt2'
xEfl,
t0,
(19.1)
and
v(y,t) = 0 if YE
t 0,
(19.2)
and the right-hand side of (19.1) is Newton's law with + = the mass density m(x). To find the eigenfrequencies one has to insert v(x, t) = where
III. Fractal elliptic operators
254
with
A E
JR in (19.1) and obtains
= A2m(x) u(x),
u(y) =
x
0
if y E
(19.3)
where one is interested in non-trivial solutions u(x). Hence one asks for eigenfunctions and eigenvalues of the operator
B=
o
m(.),
(19.4)
We use the notawhere is the inverse of the Dirichlet Laplacian tion DirichLet Laplacian always with the understanding that vanishing boundin the are incorporated into the domains of definition for ary data at
and H(1l) with 1 (this will be specified in greater detail in the next subsection). If p is the related eigenfrequency. We is a positive eigenvalue of B then A = are interested in the problem of what happens when the mass density m(x) shrinks to a fractal set r and a related Radon measure with
(19.5)
Hence we ask for eigenfrequencies and eigenfunctions of drums with a fractal membrane. This is what we call fractal drums and fractal Laplacians (extend-
ing this notation to n E N). Otherwise the term fractal drarns has several meanings in the literature. We gave in [Trio], 26.1, 26.2, pp. 199—201, a short description which will not be repeated here. 19.2
The mathematical background
Let
and be a bounded C°° domain in JR'2. By Definition 5.3 the spaces and respectively, are the restrictions of the spaces
to Il. Of interest for us are especially the Besov spaces
sElR,
(19.6)
with the Hölder-Zygmund spaces C8(fI) =
s E IR,
(19.7)
as a special case, and the Sobolev spaces
1
=
a e R,
(19.8)
with the abbreviation
=
$
e ]R.
(19.9)
19. Spectral theory for the fractal Laplacian
255
Recall that
H'(Q) =
(19.10)
is the very classical Sobolev space which will play a decisive role in what follows. We use standard notation as introduced in 1.2 and 2.1. Let and
(19.11)
Then we have the well-known trace property traci
This may be found in
Bq(II) =
s—I
4.7.1, pp. 329—330, or
(19.12)
3.3.3, p. 200. Since
tXI is an (n — 1)-set, (19.12) is also a very special case of (9.143) with t,, = by (9.120) (complemented now by p = 1 and p = oo). As always (19.13)
is the Laplacian in R" and in ft If p, q, s are given by (19.11), then E B;q(cl)
:
= 0} isomorphically onto
maps
(19.14)
This is a well-known assertion extending mapping properties for classical Soho1ev spaces and may be found in [Tria], 5.7.1, Remark 1, p.402 (with a correction in the 1995 edition), complemented by [Th$J. 4.3.3, 4.3.4. A systematic treat-
ment has been given in [RuS96), 3.5.2, p. 130. If the domain of definition is given by one of the spaces on the left-hand part of (19.14), then we call the Dirichiet Laplacian. In particular, if p. q, s are given by (19.11) and if stands for the inverse of the Dirichiet Laplacian, then (_ 1
(19.15)
:
be the completion of D(Q) in H' (a). This is a hounded map. Let Ii coincides with the spaces on the left-hand part of (19.14) with s = and 1
p=q=
2.
Iii particular. '—p
(19.16)
isomorphic map. Next we clarify what is meant by try' in (18.2) and. in the following subsection. 1w B in (18.3). is
III. Fractal elliptic operators
256
We use the equivalence
in
or
(19.17)
always to mean that there are two positive numbers
and c2 such that
(19.18) clak
results in [TriOJ, Theorems 18.2 and 18.6 on pp. 136 and 139. On this basis we describe what we need in the sequel. Let
i
(19.19)
Then by [l'riO], Theorem 18.6, if
l
(19.20)
complemented by
if p=oo,
(19.21)
where the latter is a well-known embedding which may be found in [Trif3J, 2.7.1, Remark 2, p. 130. It complements (11.22). Let idr be the identification operator introduced in (9.16). By (9.167), (9.168) we have
l<poo,
(19.22)
This coincides with [TriO], Theorem 18.2, p. 136 and applies also to p =
00.
Let tr1' =
and let F C
idr ° trr
(19.23)
Then one can replace R1' in (19.20)-(19.22) by Il and obtains tr1'
B,T
(fI),
I
(19.24)
19. Spectral theory for the fractal Laplacian 19.3
257
The operator B
We clip together the mapping properties of the Dirichlet Laplacian, given by (19.15), (19.11), and of tr1' given by (19.24), (19.19), and put B=
We wish to look at B
as a
(19.25)
bounded operator acting in an admitted space
The smoothness one loses in (19.24) is n — d, the gain in (19.15) is 2.
Hence
n—2
(withO
is a natural restriction. Then we have also n—d 2 n—d with 1 — ji p p .
2—
(19.26)
n—d 2
> 1. (19.27)
We refer also to the Figures 19.1 and 19.2 below. In particular, B is continuous in
l
H'(Il) and g E
(19.28)
We choose
(19.29)
Then we have, by completion of functions belonging to
for functions
IEH'(1l) andgEjf'([l), = = =
(19.30)
We refer for a more detailed justification to [Trio], 28.6, 30.2, Pp. 226, 234. In particular, B is a non-negative self-adjoint compact operator in jj ' (Il), generated by the quadratic form on the right-hand side of (19.30) with
= IIfIL2(r)II.
1
(19.31)
III. Fractal elliptic operators
258
and, consequently, null-space
N(B) =
{i E H'(fI)
trrf = o}
(19.32)
.
Furthermore, if &'k are the positive eigenvalues of B, repeated according to
multiplicity and ordered by their magnitude, then
kEN,
(19.33)
in the sense of (19.17), (19.18), and the explanations given there. Proofs may be found in [TriöJ at the places indicated above. We complement these assertions floW by a closer look at the eigenfunctions belonging to the simplicity of the largest eigenvalue and by the observation that D(fl\F) is dense in N(B) given by (19.32). We prove the latter assertion separately in a larger context in 19.5 which might be of self-contained interest and which is also helpful for the Dirichiet problem in a fractal setting considered in Section 20. Otherwise we refer to Theorem 19.7 where we give a full formulation of all results, known and new. 19.4
Generalizations,
comments
We concentrate in Theorem 19.7 below on the operator B given by (19.25) where i' is a d-set, equipped with a Radon measure p according to (18.1), for example p = Then B can also be written as B = (—s)—'
op,
(19.34)
provided one interpretes p as try' with (19.23). One may ask for generalizations of (—a) ' on the one hand and/or /L with supp = r and tr1' according to (19.23) on the other hand. As far as the inverse (—s)—' of the Dirichlet Laplacian is concerned we remark that almost all of our arguments in [Tri6J, and also many arguments below rely only on qualitative mapSections 27 ping properties as described in (19.15), (19.16), and embedding theorems for B-spaces on and Ofl F. But this can be generalized replacing -' by
b1 ob(x,D)ob2,
b, E Lrj(F),
1<
cc,
(19.35)
where b(x, D) are suitable pseudodifferential operators (of negative order of smoothness) with the special case
0<x<2m,
(19.36)
where AC are fractional powers of regular elliptic differential operators A of order 2m with suitable boundary conditions. The functions b1 and b.2 in (19.35)
19. Spectral theory for the fractal Laplacian
259
indicate that this is a degenerate operator. We refer to [ET96], Chapter 5, where we dealt with operators of this type in a non-fractal setting and to [Trio], Chapter 5, for a fractal setting. In this book we restrict ourselves to the in the above context, with the only exception non-degenerate operator being at the end of this chapter in Section 23 in connection with what we wish to call the sintered drum. As far as the fractal part, so far d-sets, is concerned, the situation is different. We have now an elaborated theory of fractals and their function spaces developed in Section 9 at hand. Subsections 9.24—9.33 are especially well adapted to our approach to a spectral theory for operators of type B in (19.34), now This will not be done in with more general compact Radon measures j.i in a systematic way. But there are two, as we hope, interesting exceptions. First, in Subsections 19.12—19. 18 below we take a closer look at fractal drums in the plane R2. Then
kEN,
(19.37)
in (19.33) for all compact d-sets in R2 with 0 < d < 2. We discuss in some detail for which more general compact Radon measures p in the plane and related operators B in (19.34) the Weyl behaviour (19.37) of the eigenvalues remains valid. Secondly, as indicated in Section 18, we describe at the end of this chapter in Section 23 (with some preparation in Section 22) what happens when one generalizes d-sets and their measures in (19.34) by (d, '11)-sets and their measures according to (18.6). All the operators and all the measures iz, including the indicated generalizations connected with (d, '11)-sets and the measures in the plane treated at the end of this section, are isotropic in some sense: there are no distinguished directions in R'2, or all directions in are equal. However in connection with fractal geometry, iterated function systems etc. it is quite natural to ask what happens if in (19.34) is a related anisotropic or nonisotropic Radon measure in the plane. We studied these problems in some detail in [TriO], Chapter 5, especially Section 30. More recent results may be found in [FaT99]. But
there are no final results comparable with (19.33), (19.37). The situation is somewhat better if one looks at semi-elliptic operators of prototype Au =
—
x = (xl,x2) E R2,
(19.38)
and anisotropic fractals with adapted anisotropies. We refer to [Far98], [Far99], [She99J and [NaSOO]. We do not deal with problems of this type here. But it would be of interest to check what can be said in this direction using the results from Section 9.
III. Fractal elliptic operators
260
As indicated at the end of 19.3 we prove first a density assertion in a somewhat larger context. Let F be a compact d-set in with 0 < d < n. By (19.20),
=
{f
trrf = o}
(19.39)
=
{f E
trr'f = o}
(19.40)
and
with
l
(19.41)
are well defined and closed subspaces of = and respectively. We used this type of notation before in (9.164) and (9.171). 19.5
Proposition
Let F be a compact d-set in
with 0 < d < n. Let
l
and 0<s<1.
(19.42)
Then D(IR"\F) is dense in and in
(19.43)
Proof Step 1 The main point of the proof is to use the deep Theorem 10.1.1 in [AdH96], p. 281, due to Yu. V. Netrusov. For this purpose we need some preparation. Let K be a compact set in RTh. Let 1
0. By [AdH96], Definition 2.2.6, p. 20, complemented by Corollary 2.6.8, p. 44, CQ,p(K) =
:
ion K}
(19.44)
is called the (a, p)-capacity of K. Here the admitted functions are real. By [AdH96], Definition 2.2.5, p. 20, and the explanations given on that page a property is said to hold (a, p)-quasi-everywhere, (a, p)-q.e. for short, if it is with exception of a set E with CQ,p(E) = 0. Recall that a true for all x E point x E IR" is called a Lebesgue point for a locally integrable function 1(x) in IR" if
f(x)=limIB(x,r)1'
f(y)dy, J B(x,r)
(19.45)
19. Spectral theory for the fractal Laplacian
261
where again B(x. r) stands for a ball in U< r <
1.
class [f]
E
centred at x E R" and of radius By [AdH96], 6.1, pp. 157—158, it follows that in each equivalence (1R7') there is an p)-q.e. uniquely determined representative
that (19.45) holds Such a representative f is taken there to define traces on sets. Let now F be the above d-set and let = s + with SuCh
I
IH;(llr)Ik
Itrr
Assume r0 c F and F
=
0.
Let, in addition,
(19.46)
be real and ço 1 on
Then
p(I'o)
(19.47)
= 0. In particular we have (19.45) for the above representative p-a.e. On the other hand this is also the definition of a trace and by (19.44) we get
according to [J0W84], 2.1, p.15. This coincides also with our way of introducing traces via inequalities, first for smooth functions and afterwards by completion. Hence trace assertions from [Ad1196] can be used in our context. We finish our preparat ion by the following observation. Let F be the above d-set. Then
Cap(F)=O,
where l
nd
(19.48)
This follows essentially from an atomic (or quarkonial) argument as used several times in [Trig]. for example on p. 129. We outline the procedure. We cover
F by N3 balls Bk of radius unity
C
and construct a subordinated resolution of
with (19.49)
in a neighbourhood of I'. Here j E N and the equivalence constants in (19.49) are independent of j. Then we may assume that Nj
(19.50) is an atomic decomposition of in, say, (ignoring immaterial constants). We refer to [Trio], 13.3 and 13.8, pp. 73, 75. We have to check the
111. Fractal elliptic operators
262
But
of the N3 terms, each
IP
if
j—'oo.
By embedding we have also —'0
Ike'
if i—p 00.
(19.52)
Now (19.48) follows from (19.44). Step 2 After these preparations we can apply [AdH96J, Theorem 10.1.1, p. s < 1, we have (19.48) for all a 281, with = with k N and a 0. Hence by Step 1 and the explanations given in [AdH96],
p. 234, part (a) of Theorem 10.1.1 in [AdH96] reduces to trr f = 0 and, as a consequence, is dense in Hr(R'1). Step 3 We prove the corresponding assertion for the B-spaces by real interpolation. Let
l
(19.53)
Let 0 < 0 < 1. Then
(H;1'(r),
(R1Z))9q
a=
=
(1
—
O)so
+ Osi.
(19.54)
We refer for details to [Trii3], 2.4.1, 2.4.2, pp.62-64. In particular, I
Ill
(19.55)
,
where K(t, f) is Peetre's K-functional
K(t, f) = inf (lila
+ t 1111
,
(19.56)
and the infimum is taken over all representations
I=
ía + 11
with
ía E
and
=
and ft E
(19.57)
Now let
=
n—d p
+
n—d p
+
with 0 <
< 1. (19.58)
We wish to prove that =
(19.59)
19. Spectral theory for the fractal Laplacian
263
where
a=—+8
with
s=(1—O)8o+Osi.
(19.60)
By [JoW84], Chapter VII, 2.1, Theorem 3, p. 197, there is a common linear extension operator ext from the trace spaces
where j=0,1.
into
(19.61)
Then
P= id—extotrr
(19.62)
is a common projection of
where j=0,1.
onto Now by
(19.63)
1.17.1, p. 118,
(PH;' (R").
P
H'
.
(19.64)
This coincides with (19.59). One can give also a direct argument to prove be optimally decomposed according to (19.57) and (19.64). Let f Then (19.56) with respect to the interpolation couple H0(Rn),
f=Pf=Pfo+Pfi
(19.65)
H'
shows that the K-functionals for the couples and are equivalent (independently oft). Now (19.59) with from (19.54). By the density assertion for interpolation spaces, 1.6.2, is dense in B° Since, by Step 2, is dense in p.39, it follows that is also dense in 19.6
Complements and references
The crucial point of the proof of the above proposition is the use of a special case of the deep Theorem 10.1.1 in [AdH96], p. 281, attributed to Yu. V. Netrusoy. This results iii the density assertion for the H-spaces. As for the B-spaces OflC relies afterwards on the density properties of interpolation spaces and the
substantial assertion concerning the existence of a common linear extension operator under the circumstances of the proposition proved in [J0W84J, p. 197. Density problems in function spaces have a long and rich history going back to the very beginning of the theory of function spaces, [Sob5O] (and
IlL Fractal elliptic operators
264
Sobolev's original papers from 1936-1938). If r = Oil is the boundary of a then we have had final answers for a long time: bounded C°° domain fl in see [Trio], 4.7.1, p. 330, together with references and historical remarks. If r or the non-smooth boundary is not smooth (maybe an arbitrary set in of a bounded domain) then the situation is more complicated and closely connected with the so-called problem of spectral synthesis. What is meant by spectral synthesis may be found in [AdH96J, 9.13, p. 279, and in [Hed84]. In the latter paper the problem was posed (but as mentioned there, partial results had been obtained before, including assertions by the author of this paper and by the author of this book). The final solution of this problem for with m E N and 1
H(W'), l0,
(19.66)
is the subject of [AdH96], Section 10. The version given there was especially prepared after discussions with Yu. V. Netrusov (as acknowledged there). Further references are given in [AdH96], 10.4, p. 303. The considerations are based on INet92], where a solution of this problem was announced for the larger scale of the spaces
s>0, l
(19.67)
Some special cases, based on the techniques developed in LJ0W84), were considered before in [Mar87a] and [Wa191] (where r = Oil is the boundary of a non-smooth domain, Lipschitz or d-set). We refer also to [FaJOO].
to the main subject of this section, the study of the operator B, given by (19.25). We complement the notation introduced in 19.2 and 19.3. We
As there, let r be a compact set in
with r C il, where il is a bounded
is the completion of and Then as usual, and H1(fI\1'), respectively. Recall that we use according to (19.17), (19.18).
C°° domain in
D(i) and D(fl\f) in 19.7
Theorem
Let il be a bounded C°° domain in W' (where n E N) and let r be a compact d-set according to (18.1) such that r C il and n —2
B = (—s)' o
(19.68)
19. Spectral theory for the fractal Laplacian
265
a non-negative compact seif-adjoint openztor in null-space
N(B) = T'(II\r).
(19.69)
Furthermore, B is generated by the quadratic form
I
=
(19.70)
gE
Let Ok be the positive eigenvalues as the scalar product in with of B. repeated according to multiplicity and ordered by their magnitude, and be the related eigenfunctions,
Buk=Oktzk, kEN. The
(19.71)
larqest eigenvalue is simple, Qi
>0223"
.
(19.72)
kEN.
(19.73)
functions in fZ\1', The eigcnfunctions uk(x) are (classical,) harmonic
if
(19.74)
Then
if, and only if,
Uk E
0,
if,andonlyif,
(19.75)
(19.76)
The eigenfunctions u1 (x) have no zeroes in
= cu(x) with
c
C and u(x) > 0 if x E IL
(19.77)
Step 1 The necessary explanations of what is meant by the operator been given in 19.2, 19.3 with a reference for further details to [Trio],
UI. Fractal elliptic operators
266
especially Sections 28 and 30. In particular, the above theorem might be conof Theorem 30.2 in [Trio], p. 234. By this theorem, B sidered as an is a non-negative compact self-adjoint operator in H' (f') with null-space
N(B) = {i E
:
trrf = o}
,
(19.78)
generated by the quadratic form (19.70) and with (19.73). By (19.27) or Fig. 19.1, and Proposition 19,5, (19.78) coincides with (19.69). Hence it remains to prove that the largest eigezivalue m is simple and all assertions concerning the eigenfunctions, including (19.77). Step 2 First we prove
where l<poo, k€N,
(19.79)
with the special case k E N.
tLk E
(19.80)
This is essentially a matter of spectral invariance. Let 1 n—d l
Then s) belongs to the shaded regions in Figures 19.1 and 19.2. By the explanations given in 19.3 and by (19.24) the operator B might also be with 1 q cc. We denote this coxisidered as a compact operator in restriction of B to temporarily by B(p, .9) (the index q is unimportant, and we may treat it as fixed in what follows). By the well-known spectral theory for compact operators in Banach spaces theorem) it follows that any eigenvalue g 0 of B(p, a) has finite algebraic multiplicity. By (19.28) we have (19.82)
U
for any (associated) eigenfunctioii (the upper line in the Figures 19.1 and 19.2). Now let q be fixed and let (pa. SO) and (ps, a,) be two couples with (19.81) and (19.83)
such that not only the couples a,) belong to the shaded regions so) and in the figures, but also as indicated by the triangles iii the figures. a,), Recall
c
c
(19.84)
19. Spectral theory for the fractal Laplacian
269
Using the same notation as in (19.22), introduced in (9.168), we obtain (Q).
E
(19.90)
The Hausdorif dimension of r is d. Then it follows from the distributional characterization of the Hausdorif dimension in [Tri6], Theorem 17.8, P. 130, that the space on the right-hand side of (19.90) is trivial (consists of the null distribution only). Hence Link = 0 in D'(c) and LIUk(x) =
0
in Q with uk(y) = 0 if yE 011,
(19.91)
in the classical sense. We obtain Uk = 0. This is a contradiction. Step 5 Let = be the largest eigenvalue of B. We prove that there is an eigenfunction u = Uj with
Bu = gu,
u(x) >
0
if x
(19.92)
11.
Recall that the seif-adjoint non-negative compact operator B in k is generated by the quadratic form (19.70). By a classical assertion of Hilbert 1
space theory which goes back to Hilbert and Courant, [CoH24], a non-trivial function v 1(Q) is an eigenfunction of B with respect to the eigenvalue g if, and only if,
j
=
Q>j
dx.
(19.93)
In particular, 11, Rev, and Imv are also eigenfunctions (or identically zero). Hence we may assume that v is real. Then (19.94)
As for the first equality we refer to [GiT77J, Lemma 7.6, p. 145 or to [Zie89], 2.1.8, p. 47. Hence, lvi E H'(Q) is also an eigenfunction with respect to the largest eigenvalue p. Then we have
Bw =
gw
with w(x) = Ivi(s) — v(x)
0.
(19.95)
First we assume that w does not vanish identically. Then w is an eigenf unction. By (19.80) it follows that w is continuous in Let G(x, y) be the (symmetric) Green's function for the Dirichlet Laplacian in 11. We have pw(x)
= (—s)' o tr1'w(x) =
f
G(x, 'y) (tri'w)('y)
x
11.
(19.96)
III. Fractal elliptic operators
270
(In 19.8 below we add a comment about this construction.) Since w(x) 0 is non-trivial and continuous, and G(x, > 0 if x E and 'y E 1', it follows that there is a neighbourhood of some point r where w is strictly positive.
Then, by (19.96) we have w(x) > 0 for all x E ft Hence v(x) < 0 for all x and u = —v satisfies (19.92). Next we assume that w(x) in (19.95) is identically zero. Then we have v(x) 0. By the same arguments as above we get v(x) > 0 in Q and, hence, u = v satisfies (19.92). We proved a little bit more than stated: If v is a real eigenfunction with respect to the largest eigenvalue
then v is in ci either strictly positive or strictly negative.
Step 6 We prove (i) and (iv). In Step 1 we mentioned that (19.73) is known. is simple. Let us assume that p is not simple. Then by We prove that = Step 5 there are two real '(ci)-orthogonal eigenfunctions v1 (x) and v2(x) with
=
=
= 0.
(19.97)
By the end of Step 5 we may assume that v1 and v2 are strictly positive continuous functions in ci and hence on I'. But this contradicts (19.97). Hence p is simple and we have by Step 5 also (19.77). The proof is complete. 19.8
Two comments
We comment on two points of the above proof. First, as we have noted, (19.84) is known. But the two embeddings follow also from the quarkonial representations for B-spaces. By (19.83) the normalizing factors for the po)-13-quarks and the (sj,p1)-$-quarks in (2.16) are the same. Then the first embedding in
(19.84) follows from (2.7), Definition 2.6(i) and the monotonicity of the 4As for the second embedding we remark that one needs .-.' balls of radius to cover ci in connection with the (s1,pj)-/3-quarks in (2.16). Furthermore, by Holder's inequality applied to )q E C, spaces.
P1
(19.98)
for some c' > 0. By Definition 5.3 quarkonial decompositions for B-spaces on ci can be restricted to the above-indicated balls. The first factor on the right-hand side of (19.98) compensates the normalizing factors for ,Pi )-f3quarks and (si,po)-f3-quarks in (2.16). Then one gets the second embedding in (19.84) as above as a consequence of (2.7) and Definition 2.6. Secondly, we add then the right-hand side of (19.96) is well a comment on (19.96). If x E
19. Spectral theory for the fractal Laplacian
271
defined, since both G(z. and (trr are continuous and the OUt(OflW COincides with the left-hand side: The latter follows by definition for the Sobolev mollifications (trrw)h (x). The rest is a matter of completion. However these observations remain also valid if x E 1'. Let n 3. Then G(x.
—
near
x.
This singularity is well compensated by (18.1), since d > n — 2. Similarly if n = 2. We return to this point in greater detail in Section 20 in connection with single layer potentials. 19.9
Discussion
As said above. Theorem 19.7 is the continuation of Theorem 30.2 in [TriS], p. 234. We described in 19.3 what was known so far. We repeated the corresponding assertions in Theorem 19.7 with references to [Triol. Detailed discussions and interpretations connected with the physical background described in 19.1 may be found in 30.3. 30.4. pp. 235—236, including the very few references dealing with problems of this type. We took over the crucial equivalence (19.73). Its proof in [Trio] is based on quarkonial representations. entropy numbers and approximation numbers. At that time the attempt to prove assertions of type (19.73) was the decisive impetus to develop the theory of quarkonial decompositions for function spaces. We return to this technique in the later parts of this section and also in some other sections of this chapter. In comparison with the spectral theory for, say. the (Dirichiet) Laplacian in bounded domains with smooth or fractal boundary one may ask whether (19.73) caii be strengthened by
Qk=ek'r(1+o(1)), k€N,
(19.99)
where e > 0 is a suitable constant and o( 1) is a remainder term tending to zero if k tends to infinity. But this cannot be expected. There is even a counterexample. We refer to [TriO], 30.4, PP. 235 236, and the literature mentioned there. 19.10
Nullstellenfreiheit
Part (iv) of Theorem 19.7. including the simplicity of the largest eigenvalue is the fractal version of Courant's classical assertion for the Dirichlet Lapladan in
(1924). Courant's strikingly short elegant proof on less than one page
may be found in [CoH24], pp. 398 -399, where the title 'Ciiarakterisierung der ersten Eigenfunktion durch ihre Nullstellenfreiheit' indicates what follows in a few lines. Based on quadratic forms Courant relies (in recent language) on
III. Fractal elliptic operators
272
H'-arguments. But lie did not bother very much about the technical rigour of his few-lines-proof. Problems of this type have a long and rich history at lea.st since that time. More recent versions may be found in [Tay96), pp. 315-316. We refer also to ITai96i for generalizations and to IReS78I, Theorem X1I1.43, for an abstract version.
19.11
Singular perturbations; the case n =
1
First we (leal with n =
1. Then we have by Theorem 19.7 for B given by (19.68) and d-sets F with 0 < d < 1,
kEN.
and
(19.100)
Of course, we have = —t/'(x) in this case and (19.74) means that the eigenfunctions uk(x) are linear in In this case, 13 makes sense for any finite Radon measure z with, say,
=
rc
= (—1,1).
(19.101)
First we remark that (19.102)
1
again + = 1. The first embedding is Holder's inequality, the last embedding follows again from [Tri[3j, 2.7.1, p.129. As for the middle embedding where
we refer to (9.9)-(9.11), extended to L1(f') (complex measures). All spaces in the shaded region in Fig. 19.2, now applied to n 1, are continuously embedded in C(1), the space of continuous functions on the interval (—1, 1). Hence there are pointwise traces for all spaces of interest in Theorem 19.7 and its proof. We have for the eigenfunctions Uk,
We have a quick look at the case F=
with
— 1
<1.
(19.104)
19. Spectral theory for the fractal Laplacian
Then d dim11 I' = this case we have
0.
273
be the 5-distribution with the off-points a3. In
Let
Bti(x) =
(19.105) 1
N
= JG(x,y) —1
=
xE(—1,1),
where G(r. y) is the Green's function for the Dirichlet Laplacian = —u" on (—1,1) with respect to the off-pointy E (—1,1). Recall that G(x,y) for
fixed y is continuous on [—1,11, linear in [—l,y) and in (y, 1], with G(—1,y) = C(1, y) = 0 aiid G"(., y) = Hence Bu is a polygonal line, the dimension of
the range of B is N, and the N eigenvalues of B are the N real eigenvalues of the symnietric matrix (G(a3, ak . The physical interpretation is a vibrating string where the whole mass is evenly distributed in the N points a3.
We add some comments. As mentioned, in the one-dimensional case B makes sense for any finite Radon measure with (19.101). In case of Lebesgue measure, say, restricted to we have d = 1 in (19.100), which means Pk k2. One can extend this assertion to
k2
if,
and only if,
IF1
>0,
and
k
00
if,
and only if,
ri
= 0.
We do not go into detail. We only mention that these observations can be proved using entropy numbers for estimates from above and approximation numbers for estimates from below. The abstract background will be described in 19.16 below and used later extensively. However this phenomenon is not new. It can be found in a larger context in [BiS74] and in [Bor7OJ.
If n 2 then it is not possible by our method to deal with arbitrary finite Radon measures with compact support. On the other hand, in the slightly different but nearby context of quantum mechanics, it is of interest to study, say, ± S in R", or related operators with strongly singular measures. This attracted a lot of attention since the 1960s. The state of the art may be found in [AIKOO] with almost a thousand refereines. The methods there and here are different. Nevertheless the question arises of whether they can complement each other.
III. Fractal elliptic operators
274
19.12
Fractal drums in the plane
In 19.11 we discussed fractal strings, where n = 1. Also the cases ii = 2 and n = 3 are of physical interest. As explained in 19.1, if n = 2 then one might think of vibrations of a drum C R2 with a fractal membrane 1' C Let be a bounded C°° domain in the plane R2, let r be a compact d-set with F C fI and 0 < d < 2 and let B be given by (19.68). Then we have by Theorem 19.7,
Qk"-'kt
and
ukECd(Ifl,
kEN.
(19.106)
In the plane, —1 is the classical Weyl exponent concerning the distribution of cigenvalues of the inverse of the Dirichiet Laplacian (—s) in bounded smooth domains. We discussed this problem in greater detail in fTriöl, 26.1 26.3 and 28.9-28.111 pp. 199-202 and 230, respectively. In particular, the expo-
nent —1 in (19.106) is independent of d. This observation can be immediately extended (based on the techniques developed in [Trio], explained and used later on in this section and in the following sections of this chapter) to the following situation: Let fl he a bounded C°° domain in the plane, let (—s)-' be the inverse of the related Dirichiet Laplacian, let F3 where j = 1,. . , N be pairwise disjoint compact dy-sets with 0 < d3 <2 and .
(19.107)
Naturally, 1' is equipped with the Radon nieasure p = p,, where = is the restriction of the Hausdorif measure to F3. Then B, given by (19.68), is well defined and the distribution of the positive eigeuvalues is again equivalent to The question arises for which more general Radon measures p in the plane, B makes sense and the positive eigenvalues are equivalent to (Weylian behaviour). 19.13
Definition
Let p be a finite Radon measure in the plane 1R2 with compact support. Then domain !l in R2 with p is called a Weyl measure if for any bounded
=
rc
(19.108)
the quadratic form (19.70) generates a non-negative seif-adjoint compact opwith erator B in
kEN,
(19.109)
19. Spectral theory for the fractal Laplacian
275
where Qk are the positive eigenvalues of B, repeated according to multiplicity and ordered by their magnitude. 19.14
Discussion
Let n =
and let 1', be as in the above definition. We ask under what is a Weyl measure. By (19.70) and 9.2, especially (9.14), one has first to check whether there is a constant c> 0 such that 2
conditions
dx)
=c for all
E
(19.110)
Then, by completion, the trace operator trr,
trç
L2(r).
:
(19.111)
exists, where L2(I') must be understood with respect to the given measure p according to the notation (9.14). A refined version of what is meant by traces has been given in Step 1 of the proof of Proposition 19.5. Recall the classical duality assertion =
H'(fl)
(19.112)
in the understanding of the dual pairing (D(fl), 4.8.2, p. 332. Then we have by for the identification operator idr.
idr Let again (19.16) that
(19.113)
be given by (19.23). Then we obtain by (19.111), (19.113) and
B=
otrr
'—i
:
(19.114)
is a bounded operator and as indicated in 19.3 the generator of the quadratic form (19.70). To make clear what is going on we remark that we have for any finite compactly supported Radon measure p in the plane
idr L2(r) c
c
(19.115)
This follows from (9.9), (9.12). extended to complex measures p (in particular = the spaces in (19.113) and (19.115) f L2(T)). Since differ only by the third index. This makes clear that we have a rather delicate
276
III. Fractal elliptic operators
limiting situation. In particular by 13.1 or Theorem 11 .4(u), especially (11.23), always with n 2, there is no continuous embedding of H' (1k) in C(Ifl. As a consequence a measure p with (19.111) must be diffuse (or non-atomic; as for
notation we refer to 9.17). By Theorem 13.2 with n = 2 the measure p with (19.111) must compensate the singularity behaviour of functions belonging to H' (1k) expressed by the growth envelope
=
(19.116)
.
On the other hand, by Theorem 9.3, we have a necessary and sufficient criterion under which circumstances the trace operator (19.111) exists. Let f,,,,1 be given by (9.26). Then trr in (19.111) exists if, and only if,
fvm<00,
(19.117)
VEND mEZ2
where the supremum is taken over all
IEL2(F) with fO and This follows from (9.29) with 8 = 1 r = suppp in (19.117) one gets that
and n =
p
< oc
=
2.
Inserting f =
1
on
(19.118)
yEN0 nzEZ2
is necessary for (19.111), whereas by Corollary 9.8,
sup p(2Q,,m) <00
(19.119)
mEZ2
is sufficient for (19.111). It might be of interest to have a closer look at the interplay of the growth envelope (19.116), the conditions (19.117), (19.118), (19.119) and the the operator B in (19.114). This will not be done here in a systematic way. We wish to find out sufficient conditions ensuring that p is a Weyl measure according to Definition 19.13. 19.15
Strongly diffuse measures
Let p be a finite compactly supported Radon measure in the plane R2. If p has an atom then (19.118) is violated. Hence any Weyl measure must be necessarily diffuse. As for notation we refer to [Bou56], §5.10, p. 61 or [Die75], 13.18, p. 215, where diffuse means non-atomic. But one easily finds diffuse measures
19. Spectral theory for the fractal Laplacian
277
which do not satisfy (19.118). We describe an example. Let p be concentrated on the line segment
i=
0
:
in R2 and let the restriction ph
be
y =
x
o}
(19.120)
given by (19.121)
Let
be
the square
=
x [—2",
[0,
2"l.
1/
N.
(19.122)
ii
N.
(19.123)
Then we have, by definition,
=
If 0 <
b
J
dt
1çb,
then (19.118) is wrong; if b > 1, then the sufficient condition
(19.119) is satisfied. In any case if p is a Weyl measure, then it must be diffuse in a qualified way. We describe the class of measures we have in mind. Let p be a finite compactly supported Radon measure in the plane. Then, by definition: p satisfies the doubling condition if there is a number c 1 such that (i)
p(B(y,2r)) ep(B('y,r)),
(19.124)
for all 'yE 1' = suppp. allr, 0< r < 1, and all circles B('y,r) centred at and of radius r. p is strongly diffuse if it satisfies the doubling condition and if there is (ii) a number x, 0 < x < 1, such that (19.125)
r = supp p and of side-length for any square Qo centred at some point Qi C Qo centred at some point with r with 0 < r < 1 and any sub-square 'yi E 1' and of side-length Of course, in (ii) all squares Qo and for all points E r, 'yi E r, and all r. 0 < r < 1, with the indicated properties must be checked. Incorporating the doubling condition in the definition of strongly diffuse measures is convenient
278
III. Fractal elliptic operator8
for us. One may call p (not necessarily satisfying the doubling condition) to be directionally strongly diffuse if one has (19.125) for the squares Qo, Qi as indicated, now, in addition, with sides parallel to the axes of coordinates. Then the two conditions (doubling and directionally strongly diffuse) are independent. The Dirac measure satisfies (19.124), but not (19.125). Conversely, let where is generated by the restriction of the Lebesgue line meaP= sure to I in (19.120) and P2 is the restriction of the Lebesgue measure on R2 to a square with I as a base-line. Then p is directionally strongly diffuse, but does not satisfy the doubling condition. If one admits in the definition of directionally strongly diffuse measures not only squares with sides parallel to the axes, but arbitrary squares, then there is no x> 0 such that (19.125) is always satisfied with respect to the above measure P = P0 + Hence the definition of directionally strongly diffuse measures depend8 on the axes of coordinates. This might be of some interest in connection with anisotropic problems. But for our (isotropic) problems the above definition of strongly diffuse measures which includes the doubling condition is at least convenient. 19.16
Entropy numbers and approximation numbers
The second main result of this section is to prove that strongly diffuse measures in the plane are Weyl measures according to 19.13. This will be done in Theorem 19.17 below. We rely on entropy numbers and approximation numbers and their relations to the spectral theory of compact operators in quasi-Banach spaces. We developed this theory in [ET96J, 1.3, pp. 7—22, in detail with com-
plete proofs and with references to the literature. In
Section 6, pp. 33-35, we summarized the main assertions needed, mostly following [ET96], and complemented the references. The proof of (19.73) given in [Trio], Sections 28 and 30, is (as we hope) an outstanding example of this technique. We give here a very brief description of the bare minimum needed to provide an understanding of the relations between entropy numbers. approrimation numbers, and eigenvalues,
on an abstract level (restricting ourselves to Banach spaces, in contrast to (ET96] and [TriO], where we developed this theory in quasi-Banach spaces). The interplay of this abstract theory with quarkonial decompositions as developed in Chapter I of this book produces equivalences of type (19.73) and (19.109).
But this will be explained in the course of the proof of Theorem 19.17 below and in the following sections of this chapter. Let A, B be complex Banach spaces. The family of all linear and bounded operators T: A '—p B will be denoted by L(A, B) or L(A) if A = B. Let
U8= {bEB
IlbiBil
l}
19. Spectral theory for the fractal Laplacian
279
be the unit ball in the Banach space B. Then the entropy numbers are defined as follows.
Let A, B be complex Banach spaces and let T L(A, B). Then for all k E N entropy number Ck(T) ofT is defined by the (
2'
e >0
ek(T)
T(UA) C U
for some b1,...,b2k-I E B
(b3
3=1
I..
(19.126)
Of course ei(T) =
that
11Th. Recall
T e L(B) is compact if, and only if, ck(T)
0 for k
oc.
(19.127)
Furthermore, if T E L(B) is compact, then the spectrum of T, apart from the point 0, consists solely of eigenvalues of finite algebraic multiplicity. This is Riesz's theory. Let { (T) } kEN be the sequence of all non-zero eigenvalues of the compact operator T E L(B), repeated according to algebraic multiplicity and ordered so that (19.128)
If T has only m (< oc) distinct eigenvalues and M is the sum of their algebraic multiplicities we put = 0 for all ii> M. The crucial observation in our context is Garl's inequality
kEN,
(19.129)
proved in [Carl8l] and [CaT8O] for Banach spaces and extended to quasiBanach spaces in [ET96J. We followed [Trio], Section 6 and [ET96]. 1.3. There one finds further results and references to the extensive literature. Furthermore we need the approximation numbers which are defined as follows. Let A. B be complex Banach spaces and let T L(A, B) . Then for all k E N approximation number ak(T) of T is defined by the
ak(T) = inf{hIT— Lu
:
L E L(A.B). rankL < k}
(19.130)
where rank L is the dimension of the range of L.
Of course ai(T) =
11Th. Let
now H be a Hilbert space and T
L(H) be a
compact self-adjoint operator. Let be the sequence of all eigenvalues of T, repeated according to their geometric multiplicity and ordered so that IA1(T)I
.
. .
.
(19.131)
III. Fractal elliptic operators
280
Then
IAk(T)I=ak(T),
kEN.
(19.132)
This is a well-known classical assertion. References, further information, and also comments about the relations between ek(T) and ak(T) have been given in [ET96], 1.3, P. 7—22, and [Trio], 24.3—24.7, p. 191—192. We do not repeat these assertions with exception of some rather elementary inequalities which are needed later on.
Let A, B, C be complex Banach spaces, let S E L(A, B), T E L(A, B) and R E L(B, C). Let hk be either the entropy numbers ek or the approximation numbers ak. (i) 11Th = h1(T) (ii)
h2(T)
For all k E N, 1 E N,
hk+1l(R oS)
hk(R) h1(S),
(19.133)
hk(S) + hz(T).
(19.134)
and
hk+,_j(S + T)
To avoid a misunderstanding we remark that above either all h3 are entropy numbers or all h, are approximation numbers (no mixed inequalities). 19.17
Theorem
A finite, compactly supported, strongly diffuse Radon measure in the plane 1R2 as defined in 19.15 (ii), is a Weyl measure according to 19.18. Proof Step 1 We begin with some preliminaries. Let be a bounded C°° domain in the plane R2 with
supp,z=r c
(19.135)
is a finite Radon measure with (19.124), (19.125). We may assume We must prove that the right-hand side of (19.70) is a bounded quadratic form in H1 (i)). Then, by the previous considerations, the generator B, defined by (19.70), is non-negative and seif-adjoint and can be represented by (19.68) in the interpretation of (19.114), based on (19.113), and the explanations given in 19.3. Hence first one has to prove that where
,z(r) =
1.
trr
L2(r)
(19.136)
19. Spectral theory for the fractal Laplacian
281
is bounded. As before, L2(r) is the L2-space with respect to the given measure We begin with a closer look at the interplay of ji, the geometry of its support r = supp ft and the quarkonial set-up in 9.24. Step 2 Let be the above measure with i(F) = 1 and the compact support r = suppji. We use the quarkonial set-up in 9.24, where we now replace the
j&.
balls
by the open cubes Qk,m
(19.137)
E r and with side-length c2 with sides parallel to the axes, centred at This is immaterial for the quarkonial approach described there. Otherwise we use the same notation as in 9.24. We may assume in addition that :
m=1
'1(k) c
in
= 1,... ,Mk+l} ,
kEN0, (19.138)
and that for any cube Qk+lm there is a cube Qk,z with Qk+1,mCQk.1,
kEN0.
(19.139)
The additional assumption (19.138) does not cause any problems. If (19.139) is not satisfied one can replace all squares Qk,m for all admitted k and rn by It follows by geometrical 2Qk,m, centred at and of side-length C2 reasoning that these modified cubes have the desired property. Hence, we assuine that also (19.139) holds from the very beginning. Again this modification is immaterial for the quarkonial approach in 9.24 and what follows afterwards. We may aLso assume that only one cube with k = 0 is needed to cover r. Let K2 be the number of all cubes Qk,m with
j E N0.
(19.140)
We wish to estimate the number of these cubes. Let x = with L E N in be a cube with (19.140) and let Qk÷L,1 be a sub-cube of 19.15(u). Let in iteration of (19.139). Then by (19.125) and (19.140), (19.141)
p(Qk+L.1)
be the number of cubes with k N0 and (19.140). By the above construction (19.139) and (19.141) these cubes are disjoint for different values of k. Together with the controlled overlapping of cubes with the sanic k it follows that Let
(19.142)
111. Fractal elliptic operators
282
where c > 0 is independent of j, the sum is taken over all k and in where the respective cubes have the property (19.140). Similarly one can estimate with I = 0, 1,. . , L — 1, being the number of cubes QkL+I,m with the property .
(19.140). We get K5
=
K,,1
= N,
(19.143)
for some c> 0 which is independent of j. Step 3 After these preparations we first observe that there is a number D > 0 (diffusion number) with
j E N,
(19.144)
for some c> 0: If L has the above meaning and (19.145)
then, using (19.139) and iteratively (19.125),
(
li(Q,,m)
(19.146)
with D = L1. We apply Corollary 9.8(11) with
p=r=n=2, .s=1 and d=D>0,
(19.147)
and obtain by (9.28) that
trr
H'(R2)
:
L2(F)
(19.148)
is a bounded operator. Since F C Q this is the same as (19.136). Let t = > 0 he the typical number introduced in Definition 9.25, where the t2 l)alls I3krn in (9.109) can be replaced by the above cubes Qk,m (since we have the doubling condition it (loes not. matter whether we choose or Here D has the same meaning as in (19.146). We apply Theorem 9.33 to = and p = q = n = 2 and obtain
=
Ht(F).
(19.149)
We denote temporarily the embedding operator from Ht (F) in L2 (1') by id,
H'(F) :: L2(r).
(19.150)
19. Spectral theory for the fractal Laplacian
can be estimated by
We wish to prove that the entropy numbers of
ek(idt)ek2,
283
kEN,
(19.151)
for some c> 0. We rely on the quarkonial representations in Definition 9.29. By Definition 9.27 the (t, 2)-fl-quarks, responsible for Ht(r), are given by = 2k1131 — (19. 152) E 1',
now with respect to the above cubes Qk,m in place of the balls
By
Definition 9.29,
Ht(r) with
gE
IHt(T)II
(19.153)
1
can be represented as oo
MA,
=
(19. 154)
k=0rn1
E C with
where the fl-quarks are given by (19.152), and
/00
MA,
sup /iEN0
Here
\k=Orn=l
/
2.
(19.155)
0 has the same meaning as in 9.29 and can be chosen arbitrarily large.
By Remark 9.30 and Proposition 9.31 all representations of type (19.154), (19.155) converge absolutely and unconditionally and can be rearranged as one wishes. In particular, one can re-organize the collection of all cubes by
: k€No; m=1 (19.156)
where Qjm are the cubes with (19.140), and Hence, g can be represented as
call be estimated by (19.143).
30
gey)
(19.157)
= k=O m=1
where
are the 13-quarks related to Qk,m and
coefficients with
/30 (
Isk
I
are corresponding
III. Fractal elliptic operators
284
We prove (19.151) by factorization through sequence spaces. For this purpose we introduce some notation in modification of Sections 8 and 9. Let
5ER, ,>0,
and LkENwherekENo.
(19.159)
Then
1p
(19.160)
is the Banach space of all complex-valued sequences
kEN0;
(19.161)
with p
too ILk = sup \k=o
(
\nz=1
/
(19.162)
in (19.150) by
Now we factorize
(19.163)
with
S
:
Ht(F)
idt
:
£oo,xI k2
(19.165)
T
:
{e1
(19.166)
,
(19.164)
by (19.153). (19.157), (19.158) in the Here S maps the unit ball in indicated sequence space now with x1 > 0 sufficiently large and Nk. given by (19.143). Of course, if Kk < Nk, then the remaining components under the map S are zero, by definition. Furthermore, idt in (19.165) is the identity between the indicated sequence spaces with > 0. Finally, Tin (19.166) > is given by
Tx=
(19.167) k=Oin=1
19. Spectral theory for the fractal Laplacian
285
where a is the (19.161) with Lk = Nk, terms with Kk < m Nk are simply neglected, and are the /3-quarks in (19.157). First we have a look at T. For fixed j3 E we have K,
1L2(r) k=O m=1 2
Kk
krO
if >
ji(d'y)
m=1
/14
00
k=O
2
2_k)
(
.
(19.168)
\m=1
The first estimate is simply the triangle inequality. The second estimate comes from (19.140). the support properties of the involved cube Qk,m described in
Step 2 and r > 0 in 2n181 is the same constant as in 2.4 and 2.5 adapted to the above more general situation (19.152), and c> 0 is independent of We always assume
Xl >
>r>0
(19.169)
in (19.164) (19.166) is linear and l)ounded. We have (19.163), where S is bounded (not necessarily linear) and
T is linear and bounded. As for idt we can apply [Thö], Theorem 9.2, p. 47, and obtain
kEN.
(19.170)
Here we use (19.143). Now (19.151) follows from (19.170) and (19.163). Step 4 We summarize what is known so far. We always assume that r and ci are related by (19.135). By (19.149) and the compactness of ide, (19.150), (19.151). it follows that
trj'
:
L2(r)
(19.171)
is compact. We have (19.113) for the identification operator idr and hence,
tr"=idrotrr :
(19.172)
is compact. Together with (19.16) it follows that
B=
ot?'
is compact in
I°f'(fl).
(19.173)
III. Fractal elliptic operators
As discussed in 19.14 we have as in (19.70), based on (19.29), (19.174)
and the counterpart of (19.78),
: trrf = o}
N(B) = {i E
(19.175)
for null-space. Hence B is a non-negative, compact, seif-adjoint operator in By the usual Weyl decomposition and (19.149) we have
= N(B)
= N(B) ® trr
Ht(r).
(19.176)
with and We identify the orthogonal complement of N(B) in is a positive seif-adjoint denote the restriction of B to Ht (1') by Br. Then compact operator with the same eigenvalues, if
k—'oo,
(19.177)
as B, and the related orthonormal eigenfunctions Uk,
BrUk=PkUk,
kEN,
UkEHt(I'),
(19.178)
span Ht(1'). Fltrthermore, by (19.174),
IHt(r)II = Ill
(19.179)
and {uk} is also an orthogonal system in L2(f). By construction, Dir, the restriction of D(R2) on r, is dense in Ht(r). Since is a Radon measure, is also dense in L2(r). This follows from the proof of Theorem 3.8 in [ThöJ, p.7 (with a reference to [Mat95J, Theorem 1.10, p. 11). Hence any p can be approximated in Ht(r) and consequently also in L2(r) by linear combinations of {uk}. In particular, {uk} is a complete orthogonal system in L2(r). Let Br be the extension of B' to L2(JT). It has the same eigenvalues and (19.150) we and eigenfunctions. By (19.179) with ..JBj in place of have in obvious notation, (19.180)
By (19.129) and (19.151) we get
kEN.
(19.181)
According to Definition 19.13 it remains to prove the converse inequality.
19. Spectral theory for the fractal Laplacian
287
Step 5 We use again the coverings of r by cubes as constructed in Step 2. By the doubling condition (19.124) there is a number C> 0 such that for any two cubes with (19.139),
C/t(Qk+1,m),
k E N0.
(19.182)
j E N0,
(19.183)
Let j E N0. We ask for all cubes Qk,z with
2'
such that there is no larger cube in which Qk,1 is contained according to the hierarchy (19.139) having this property. We denote these largest cubes by Qm• \Ve claim
rcUQ;,m.
(19.184)
Let -', E I' and
This follows from E Qk,m. If k is large then JL(Qk,m) (19.144). Stepping iteratively from k to k — 1, then by (19.182) there must a nunhl)er k with (19.183). There might be even smaller k's with (19.183). But in any case there is a largest (or several largest) cubes with this property. This (19.184). Let Pj be the number of cubes Q,m• By (19.184) we have ? e23 for some c> 0. Two such cubes = Qk,m with different values
of k arc disjoint by construction. For the same level k we have the above controlled overlapping. In any case there are C°° functions
= 0 if
fl
C 1
and
1L2(r)II
24
with
m,
1
(19.185) (19.186)
,
where all equivalence constants are independent of j. In the last equivalence we used again the doubling condition. We obtain for linear combinations of t IleSe functions. F,
Pi
19.187 •
and.
(19.174), P;1
1k1(I)D P2
(EIAJ.112)
.
(19.188)
III. Fractal elliptic operators
288
We wish to estimate the approximation numbers ap,
defined by (19.130).
Let L be a corresponding operator in (19.130) with rankL < Pj. Then one finds a non-trivial linear combination as used in (19.187), (19.188) with Lço = 0. Inserted in (19.130) one gets
iEN. Using P,
(19.189)
and (19.132) we obtain for some C> 0,
kEN.
(19.190)
The proof is complete.
Problems and comments: Weyl measures
19.18
19.18(i)
B=
We proved that
otr1'
exists and that
Ok ".'
k'.
k EN,
(19.191)
for its positive eigenvalues Ok according to Definition 19.13 if p is strongly diffuse. This assumption is isotropic (there are no distinguished directions). In Section 30 and in [FaT99] we dealt also with anisotropic and nonisotropic measures p in the plane and related fractals (ferns, grasses etc.). For corresponding fractal drums and related operators B we obtained only estimates,
kEN, 0
(19.192)
for the (ordered) positive eigenvalues, c1 > 0, > 0. It is unclear whether these estimates are sharp. The main problem in this context is to characterize all Weyl measures
in the plane. In particular, the question arises to what extent isotropy assumptions (doubling condition, strongly diffuse) are really necessary. With respect t.o the just indicated anisotropic fractal beauties, monsters, grasses and ferns one can rephrase (in a somewhat vague Hamletian spirit):
Is the music of the ferns Weylian or alien: that is the question. The situation can be described as follows: Let r (and the related measure) be a genuine anisotropic, but otherwise quite regular fractal, for example the PXTfractal from [Trio], pp. 14/15 and 20. The question is of whether the positive where eigenvalues Ok behave like k' (Weyl measure) or at least like
19. Spectral theory for the fractal Laplacian
289
is a tame function; or whether the clash of anisotropic fractals and measures on the one hand with isotropic elliptic operators on the other hand creates a might be chaotic distribution of eigenvalues. Here a distribution function called tame, when
jEN.
In this case one has the desirable equivalence
kEN. Some related details may be found in 19.18(vii) below. Finally we refer to the end of 9.34(viii), where we described a characterization under which circumpreserves the Markov inequality. These sets play stances a compact set r in a significant role in the theory of (isotropic) function spaces on F as has been developed by A. Jonsson and H. Wallin, [JoW84], [J0W97]. It is quite clear that such sets F are isotropic in the indicated sense. This sheds additional light on the above problem when genuine anisotropic fractals collide with isotropic function spaces and isotropic elliptic operators. 19.18(11) We refer in this context also to [NaS94J, [NaS95]. These papers deal with eigenvalue distributions for operators of type (18.3), (18.2), where ji is a self-similar measure defined by probabilistic IFS (Iterated Function This set-up goes back to [Hut8l) and may also be found in Systems) in [Fa190], Chapter 17, especially p. 263, [Fa197], 11.2, p. 192, and is connected with multifractal measures. Restricted to n = 2 one has always (19.191). Hence, all the considered multifractal measures in the plane R2 are Weyl measures according to Definition 19.18.
There are always some separation conditions. But then it follows from the construction in [Fal97}, pp. 192, and 36, 37, that all these measures fit in the scheme of 19.15 or at least in the reasoning of the proof of Theorem 19.17. One has always (19.125) . Hence, Step 2 and Step 5 of the proof of Theorem 19.17 can be applied ensuring the estimate ck1, c> 0, from below. The doubling condition need not be satisfied in any of these cases. But this is also not necessary in this very regular construction. One can follow the arguments in Step 3 of the proof of Theorem 19.17 directly and gets finally the estimate c k1 from above. On the other hand on random isotropic (but otherwise 9k special) fractals the situation is less favourable. We refer to [HamOO). There one finds spectral assertions for Laplacians on some types of fractals. Although the set-up is different from our approach it makes clear that spectral theory of generators of quadratic forms on fractals seems to be rather complicated. 19.18(iii) There are further examples of Weyl measures where the doubling condition is not necessarily satisfied. We refer to Theorem 23.2 and in particular, to the end of the discussion in 23.3. In any case the roles of the two
HI. Fractal elliptic operators
290
assumptions (19.124) (doubling condition) and (19.125) seem to be different. We used (19.125) in a decisive way. On the other hand the doubling condition was convenient in the general case but apparently it is not needed if one has additional information about the nature of the underlying measure (probabilistic IFS, or as at the end of 23.3). 19.18(iv) Furthermore, the estimate Qk ck1, c > 0, is a local matter (it is sufficient to have such an estimate in a neighbourhood of some point E F), whereas the estimate Qk
k'
The example of a measure = at the end of 19.15 which + is not strongly diffuse and does not satisfy the doubling condition is also a Weyl measure. Obviously, and are Weyl measures. Let B1 and B2 be the related operators according to (19.191). Then we have by (19.134), 19.18(v)
e2k(B1 +
B2) ek(B1) + ek(B2) ck'
(19.193)
as the desired estimate from above. Hence Qk < c k'. As mentioned in 19. 18(iv) the converse is a local matter which applies to the case considered.
As for the estimate from above one has the following rather satisfactory assertion: 19.18(vi)
sup kêk
supkek(B)
(19.194)
k
k
This is a consequence of (19.129) on the one hand and of = Ok ak(B) in (19.132)
and the relations between approximation numbers and entropy numbers as described in fET96], Theorem 1.3.3(u), p. 15, on the other hand. 19.18(vli) The assertion of the previous point can be complemented as follows. Let
If
be given by (19.150).
is a Weyl measure according to Definition 19.13 then
kEN.
(19.195)
As for justification we first remark that by (19.180), ek(jdt)
Recall that
"s-
are the eigenvalues of
k E N.
(19.196)
We have
kEN,
(19.197)
19. Spectral theory for the fractal Laplacian
291
where the first inequality and the equality come from (19.129) and (19.132), respectively. The crucial second inequality is covered by [ET96], Theorem 1.3.3(i). p. 15, where one needs that
Q2i-z "Q2i, jEN,
(19.198)
(t he equivalence constants are independent of j). Together with (19.196) and
one gets (19.195). There is the following converse assertion: Let B he compact and assume that we have in addition (19.198) for its positive eigenvalues. Then p is a Weyl measure if, and only if, (19.195) holds.
This claim is a consequence of (19.197) which, in turn, follows from the additional hypothesis (19.198) and the quoted literature. 19.18(viii)
F\irther discussions about the Weylian behaviour of positive with n
eigeiivalues of more general fractal elliptic operators in found in [Trio], 28.9—28.11, 30.3, pp.230, 235. 19.19
N may be
The degenerate case
\Ve illustrate Theorem 19.17 and also the problems in 19.18 by glancing at the (legc'nerate (non-fractal) case in the plane. Let fl be a bounded C°° domain in the plane R2 and let
{xER2 Let
:
lxi
sfl =FCfI.
(19.199)
be the Lebesgue measure in R2 and let b be a weight function with
he L1(R2),
b(s) >Oa.e. in 1',
b(s) =Oif lxi>
(19.200)
\Ve equip r with the Radon measure p = b(x) PL and ask, in our previous notation, for existence, boundedness, compactness, spectral properties, of
B=
o
tr" as an operator in
(19.201)
Recall that (—s)-'
is the inverse of the Dirichlet Laplacian in Specializitig 19.14 to the above situation, (19.110) reduces to the Hardy inequality
/
b(x) 14r)12 dx
c EJ
dx,
q
D(fl).
(19.202)
III. Fractal elliptic operators
292
If there is a number c> 0 with (19.202) for all D(1l), then B is a bounded operator in ill which, in our context, and with an obvious interpretation can be written as ob,
B=
(19.203)
justified by
=
f
=
o
(19.204)
E D(cl). The boundedness of B is naturally and intimately E D(cl), related to the sharp inequalities in the critical case as described in Theorem 13.2 and detailed in 13.3. Let b be the rearrangement of b, and let for some where
c>0, b*(t) I
log
t2'
0
Then it follows by (13.62) with n = p = q =
/
b(x)
<
f
(19.205)
2,
W*(t)2 dt < c
IH'(R2)i12.
(19.206)
As for the first inequality we refer to (16.12) and the comments and references afterwards. We specify b in (19.200) by b(x)
=
ixI2 I log lxi 12'
lxi
1
(19.207)
where is a positive monotone (decreasing or increasing) function in the If x is increasing we put x(0) = limx(t). With respect to this interval (0,
special choice of b we have for B, given by (19.203), in the above interpretation as an operator in the following assertions: B
is bounded if, and only if,
B
is compact if, and only if,
is bounded,
(19.208)
and
x(0) = 0.
(19.209)
Here (19.208) is a special case of the sharp Hardy inequality in Theorem 16.2(i)
with n = p = q =
2.
19. Spectral theory for the fractal Laplacian
293
1 is not compact. Let 0 be monotonically decreasing in radial directions from the origin. Then we get by (13.62), Theorem 16.2(i) with n = p = q = 2, and (16.11),
We prove that B with c =
< —
[ (f \O
dt (19.210) 1
c'
This applies in particular to the extremal functions in (13.50)—( 13.52) approx-
imating the growth envelope function EGH' (t). But now we are in the same situation as in 14.6. If the embedding related to the last inequality in (19.210) is compact, then the set of the above extremal functions in the space quasinormed by the left-hand side of (19.210) is pre-compact. This is a contradiction by the same arguments as in 14.6. Let x(0) = 0. Then x(ö) 0 if c5 0. We decompose r in lxi 5 and in and get S < lxi
/
Ixl2Iiog lxi 2
x(5)
If(x)12 dx
dx + c5 J lf(x)12 dx lxix 2 /
(19.211)
for some > 0. Now it is clear that the embedding of If' (Il) in the space related to the left-hand side of (19.211) is compact: the first term on the right-
hand side creates a x(S)-net and for the second term one has the classical compact embedding of H'(fl) in L2(1l). This completes the proof of (19.209).
There remains the problem under which circumstances for b in general, and is a Weyl for b given by (19.207) with x(0) = 0 in particular, = b(s) measure according to Definition 19.13, and hence
kEN,
(19.212)
for the positive eigenvalues of B, given by (19.203). This is not so clear so far then we have (19.209), but (to the author). IfS> 0 and K(t) =
ii =
I
log lxi
12L,
lxi
(19.213)
III. Fractal elliptic operators
294
is not strongly diffuse according to (19.125), and hence we cannot apply Theorem 19.17. On the other hand, jL
e >0,
= lxi
(19.214)
lxi
is strongly diffuse. Hence, by Theorem 19.17, it is a Weyl measure, and we have (19.212).
In tET961, Chapter 5, we developed a spectral theory for degenerate elliptic operators, especially the examples on p. 211 of [ET96] are related to the above considerations. There one finds also further references.
20 20.1
The fractal Dirichiet problem Introduction
where, temporarily, n 3, and let Let fl be a bounded C°° domain in r = Ofl be equipped naturally with a Radon measure equivalent (or equal) in R" to I'). The to ir (the restriction of the Hausdorif measure single layer potential G,
(Gh)(x) =j
xE
lx
(20.1)
makes sense both in RTh and on r (using the same letter G) if, for example, h is bounded. Since 1' is a compact C°° manifold,
H8(I') =
s E IR,
(20.2)
can be introduced in a canonical way via local charts. It turns out that C (restricted to F) makes sense for some spaces H8(I'). In particular,
=
(20.3)
is an isomorphic mapping. This has the consequence that the uniquely determined solution u(x) of the (almost) classical Dirichlet problem
xEIl, uEH1ffl),
(20.4) (20.5)
for given g, can be uniquely represented by (20.1) as u = Gh with some hE
It is the main aim of this section to extend these observations
20. The fractal Dirichiet problem
295
to fractals. We rely on the techniques developed in Section 19. We describe what can be expected. First we remark that the boundary F = of the above C°° domain is an (n — 1)-set according to (18.1). In particular, the singularity x y E F, is well compensated by (18.1) with d = n — 1. In addition it is quite clear that this argument applies to any compactly supported d-set F with d> n — 2. One obtains the generalization
n—2
(20.6)
of (20.3). Then we are precisely in the context of Section 19, especially The-
orem 19.7. Let the fractal F, the domain Il and the related inverse of the Dirichiet Laplacian be as there, and let the trace operator trr and the identification operator idr as in 19.2. Then
=
(20.7)
combined with (20.6) (where G stand now for the Green's operator with respect to is the concise version of the uniquely determined solution u E H' of the Dirichiet problem,
=
=g
u = 0,
in
0
(20.8) (20.9)
E
including its (uniquely determined) representation as a single layer potential. The plan of the section is the following. First we need some preparation: some duality assertions; how to introduce the spaces HS(F) with s E R; mapping properties of the operator B, given by (19.68), restricted to F. This will be done in 20.2—20.6 The main result of this section, concerning the Dirichlet problem (20.6)—(20.9), may be found in Theorem 20.7. Afterwards we return in Corollary 20.10 to the Dirichiet problem as described at the beginning of this subsection, but now under the assumption that the boundary = F of a bounded domain in is a d-set with n — 1
Some notation
We collect some notation partly repeating earlier definitions. Let 0 < d < n. A compact set F in R" is called a d-set if there is a Radon measure in and two positive numbers c1 and c2 such that sup-p = F and
p(B('y,r))
for all
0< r <1,
(20.10)
and all -y E F, where B('y, r) is the ball centred at 'y E F and of radius r. Up to equivalences,
is uniquely determined and may be identified with the
elliptic operators
III.
296
restriction of the Hausdorif measure in IR" on r. Comments and references may be found at the beginning of 9.12. Let
=
H8(R") =
8 E R,
(20.11)
be the (special) Sobolev spaces as introduced in (1.9). If Cl is a bounded domain in R", then by Definition 5.3.,
=
H6(1l) =
8 E R,
(20.12)
are the restrictions of H8(R") on Cl.
Let r C Cl, where r is a compact d-set in Ill?' with 0 < d < n and Cl is a bounded domain in R". The trace operator trr' has been introduced in 9.2 and studied in detail in 9.32 and 9.34 in connection with rather general compact sets r and related Radon measures As mentioned in 9.34(i) for d-sets with 0 < d < n we have t,, = in (9.143). We are interested here only in the special case (20.11), (20.12) and we extend this notation to 1' by
!13(r) =
8 > 0.
(20.13)
Then, by (9.143),
8>0.
=
H8(F) =
(20.14)
Some further information is given in 19.2, including references to [Trio], Theorems 18.2 and 18.6 on pp. 136 and 139. The identification operator has been introduced in (9.16). We rely in particular on the duality assertions (9.20), (9.21).
Let w be an arbitrary bounded domain in RIL. Of special interest for us are
H'(w), and (20.15) Recall that H'(w) and H'(w) are defined by restriction of H'(]R") and respectively, on As usual, p1(w) is the completion of D(w) in H'(w). We have in H1(]R") the explicit norm (1.4) with 8 = 1 and p = 2. Then it follows by standard arguments that for any bounded domain the space can be equivalently normed by
In Ill (with a corresponding scalar product). As usual, (complex-valued) distributions in w, the dual of
2\2I )
(20.16)
/
is the collection of all
20. The fractal Dirichiet problem 20.3
Proposition
Let
be
297
an arbitrary bounded domain in IR". Then
= H1(w)
(equivalent norms)
(20.17)
with respect to the dual pairing (D(w). D'(w)). Proof This &ssertion is well known if w is smooth and also if w is replaced by IR",
(H'(R"))'
(20.18)
with respect to the dual pairing
trary domain. Let yE By the explicit norm (1.4) of
Now let w be the above arbi-
according to the dual pairing (D(w),D'(w)). we have
II.qII 11w"' (w)II =
Interpreting I°J'(w) as
a
II
E D(w) .
.
(20.19)
closed subspace of H'(R"). we find by (20.18) an
with
element h E g(w) = (Ii. w)
IIh
1H' (R't)lI
E
D(w)
.
(20.20)
With G = hlw (restriction to w) we have = (G.w),
where c > element h
IIGIH'(w)U
is independent of q. Conversely, let G E H '(w). There is an H_1(Wl) with G = hlw and
0
lhlH'(R")ll Let
(20.21)
=
IIGIH'(w)II.
(20.22)
if ço E D(w). Then we have
= <
IIh 1H' (R")Il 11w lii' (R")ll c hG 1H'(w)II 11w IH'(w)hI, w E D(w). I(h, w)l
(20.23)
Hence. g
(?i1(w))'
and
Together with (20.21) and the usual interpretation we have (20.17).
(20.24)
III. Fractal elliptic operators
298
20.4
Remark
did not use in the above proof that w is bounded. In other words, in extending Definition 5.3 to arbitrary domains in (20.17) remains valid. But in connection with (20.16) we used that is bounded. Checking the proof it is clear that it depends on the special nature of the norm (1.4). An extension We
to more general spaces is not clear. The right way to look at duality in (smooth or non-smooth) domains is indicated in (5.157). We refer also to 4.8.1, p.332 (in the second edition, 1995), which works also for non-smooth domains. 20.5
Spaces on d-sets
Let r be again a compact d-set in IR'2 with 0 < d < n and let D(F) = Since
fl H5(fl.
(20.25)
s>O
with a E R, it follows by (20.14)
S(IR'2) is dense in any space
that D(I') = trp In particular,
(20.26)
is a locally convex space, naturally equipped with norms,
k E N. It is dense in H8(f) with s > 0. The latter assertion is also an immediate consequence of the quarkonial representation iii Theorem 9.33 and Definition 9.29. Recall that is also dense in L2(r). This follows from the properties of Radon measures and has been mentioned with some references in Step 4 of the proof of Theorem 19.17. But in case of d-sets we have also for example fi
.
O
(20.27)
as remarked in (9.124) with a reference to [Tri6], Theorem 18.6, p. 139. Hence, the density of D(JT') in L2(I') is also a consequence of Theorem 9.33, Definition 9.29 and (9.135). Imitating the definition of S'(R'1) as the dual of we introduce in an obvious way D'(r) by
D'(r') = (D(r))'.
(20.28)
As usual we identify the dual of L2(f) with L2(F) itself and we use the scalar product of L2(F) as the dual pairing of and D'(r). In this way we can introduce
H8(r) =
,
<8 <0,
(20.29)
20. The fractal Dirichiet problem
299
and one obtains
(Hs(r))l =
E
R,
with
H°(r) = L2(r).
(20.30)
and
U w(r).
D'(r) If
r=
is, say, the boundary of a bounded
domain in
then as
have a direct meaning. In this case. (20.30) mentioned in 20.1, the spaces is an assertion which can be checked easily. 20.6
The operator B and single layer potentials
We collect what has been said before for the operator B aiid complement these and let be assertions. Let again be a bounded COC domain in the inverse of the Dirichiet Laplacian with
H'
:
(20.31)
'(fl)
according to (19.16). including the explanations and references given there. Let G(x. y) with x and y E be the classical Green's function for the Dirichiet Laplacian. in Particular. G(x. y) = 0 if x E 0f1.
if n 3 (and x E Q, y n = 2. Furthermore,
y E
ci
and
G(x. y)
Ix
—
(20.32)
with the usual modifications iii case of n =
f(x) =
fE
I
or
(20.33)
in the usual interpretation according to (20.31). Let r be a compact d—set in with
n—2
(20.34)
Then. in the u.smial notation. G,
(Gf)(x)
=
f
G(x,
f(y) ;t(dy).
xE
(20.35)
III. Fractal elliptic operators
300
is called a single layer potential. If n 3 then we have the singularity (20.32). Since d> n — 2, it follows by (20.10) that the restriction of G to r, denoted by G",
(crf)(A)
1i(d'y),
G(A, 'y)
A E r,
(20.36)
= makes sense if, say, f D(r). Of interest arc the mapping properties of the operator G" between spaces H8(L') and the connection to the operator B, given by
B=
o
try',
with
= idr °
trr,
(20.37)
where trr is the above trace operator and idr is again the identification operator as mentioned in 20.2 (with a reference to (9.16) and (9.20)). As for B we have Theorem 19.7 and Theorem 19.17 (where n = 2). We collect what we need. By (19.69) and (20.14) the Weyl decomposition (19.176) is given by
=
=
As in Step 4 of the proof of Theorem 19.17 we denote the restriction of B to 'I —4 by B . In analogy to (19.179) we have
= If 1L2(r)II. Repeating the arguments given there,
(20.39)
can be extended by continuity
(using now the same letter) to an isomorphic map from L2(1') onto H' (r). Both on L2(r) and on H1 is positive, compact, and the operator (fl self-adjoint with the same eigenvalues and elgenfunctions. Now by (20.29) and the usual arguments, (identified with its dual) is also an isomorphic map from
onto L2(I').
(20.40)
Hence,
Br =
(20.41)
is an isomorphic map. As for the connection with the above Green's function we first remark that
idr
(20.42)
20. The fractal Dirichlet problem
This follows from (20.38) (or (20.14) with s = By (20.31) we have
1 —
301
) and (9.20), (20.17). (20.43)
Interpreting (20.33) as a dual pairing now with f replaced by idr h with h (F) we get
H—
xEQ Restricting this H'-function to r, hence x = A
(20.44)
F, we obtain
AEF,
(20.45)
and by (20.36). (20.41).
B" =
a
= G"
(20.46)
:
au isomorpluc map. After all these preparations we can now prove rather easily the following assertions. 20.7
Theorem
he a bounded domain in and let G(x, y) with x E y E be the classical Green's function for the Dirichiet Laplacian in Let F be a compact d-sel in with F C and n — 2 < d < n (interpreted as 0 < d < 1 in case of = 1). Let g E H' (F). Then the Dirichiet problem Lcl
mm
= 0 in
u E
tracmu
=0.
trru=g,
fl\F,
(20.47) (20.48)
has a unique solution u. Furthermore this solution u can be represented as a single layer potential (20.49)
u'ith a uniquely determined distribution ii €
III. Fractal elliptic operators
302
Proof Step 1 The existence of a solution u and the representation (20.49), including its interpretation, can be obtained from 20.6 as follows. By (20.46) we find for given
a distribution hE
gE
with
ojdj-h
u(x) =
E
!f'(Il)
trr'u=G"h=g.
(20.50)
(20.51)
c r, we have
Since
=
0
if
(classical harmonic function).
x
(20.52)
Hence. u satisfies (20.47), (20.48). Furthermore, (20.49) follows from (20.50) and (20.44). Conversely if u with (20.47), (20.48) is given by (20.49) with some h (F), then we have by (20.46) also (20.51) and hence h = is uniquely determined. Step 2 To prove the uniqueness of u with (20.47), (20.48), it is sufficient to show u = 0 ifg = 0. Hence, let u be a solution of (20.47), (20.48) with g = 0. U r. We apply Proposition 19.5 with p = 2 and Let w = Q\f'. Then Ow = I> s= 1 — > 0 and get u
{v
H'(w) : tr&, = o} =
(20.53)
where we used the notation introduced in 20.2. Then we have for any cc E D(w),
1>2 Since u 20.8
J°I'(w) it follows that
=
= 0.
(20.54)
= 0 with j = 1,... ,n and finally u = 0.
Comments and the variational approach
The above proof of uniqueness is based on Proposition 19.5, which, in turn, used the substantial Theorems 9.1.3 and 10.1.1 in [AdH96], pp. 234, 281 (here Theorem 9.1.3 dealing with Sobolev spaces is sufficient). We refer to 19.6 for further comments. The proof of the existence and the representation 20.8(i)
20. The fractal Dirichlet problem
303
of a solution u with (20.47)—(20.49) is based on the techniques developed in connection with Theorems 19.7 (with a reference to [Tri6], Sections 28 and 30) and 19.17. Basically one shifts the problem from fi to r and proves that the resulting operator B" is an isomorphic map as indicated in (20.41). The rest is interpretation of this observation. If r is a smooth or Lipschitz-contirnious (maybe the boundary of a respective domain), then d = n — 1. surface in Hence we have by (20.41)
B"
'—÷
H4(fl,
(20.55)
as indicated in (20.3). As will be seen below in Corollary 20.10 and its proof, the single layer potential on the right-hand side of (20.49) can be replaced by the single layer potential according to (20.1). Furthermore, after this modification (20.6) corresponds to (20.41). 20.8(11)
If one looks only for the existence of a solution u with (20.47),
(20.48), and not for its representation, one can use the very classical variational Then by definition there is a approach as follows. Let again g E function
ve Let again
'(&l)
with
v = g.
(20.56)
= 1l\F. Then we ask for a function h with h E J°f'(w)
and
—
=
E H'(c&).
(20.57)
This is the point where one needs Proposition 20.3. It follows that the righthand side of (20.58)
is a linear and continuous functional on H' (w) and, hence, can be represented by the left-hand side with some h E (ci.,). Then we have (20.57) and u = v+h is a solution of (20.47), (20.48).
We concentrate in this section as in Section 19 on the Dirichiet Laplacian —is. But it is quite clear that many of our considerations can be 20.8(iii)
generalized as indicated in 19.4 with a reference to [Trio], Sections 27—30. This
applies also to Theorem 20.7. Similarly the variational approach outlined in 20.8(u) can be used for wider classes of elliptic operators to prove the existence of weak solutions of the Dirichlet problem for elliptic equations in non-smooth domains. The corresponding uniqueness assertions can be based on density
III. Fractal efliptic operators
properties a.s mentioned at the beginning of 20.8(1). Both together (variational approach and density assertions) result in existence and uniqueness assertions for Dirichlet problems in non-smooth domains. This has a long history and goes back to [Sob5O] (and Sobolev's original papers in 1936- 1938). These problems have also been considered in some detail in [AIH96], Corollary 9.1.8, Theorem 9.1.9, PP. 236, 237, including Sobolev's original problems connected with the poly-harmonic operator. 20.9
The Dfrichlet problem in dom*mi with a fractal boundary
and let its boundary 1' = I be a d-set. Let Il be a bounded domain in Then we have necessarily d n — 1. We assume n — 1 < d < n (with d> 0 in case of n = 1; then 11 is a disconnected bounded open set). By (20.14) and in modification of Theorem 20.7 we ask for solutions u of
uEH'(IZ),
(20.59)
where g is given. Let K be an open ball with C K and let GK(x, y) be the Green's function for the Dirichlet Laplacian with respect to K. We apply Then one finds an element Theorem 20.7 to K in place of and to r = n— d hE H-'+r(r) such that
u(s) =
(20.60)
solves (20.47), (20.48), with K in place of ft We have in particular (20.59) as a solution. As for the uniqueness we recall first our point of view: By (20.12) all spaces on 11 are defined by restriction of the corresponding spaces on R". This avoids problems of extendability and traces on F = for intrinsically defined
spaces. As for traces we discussed this problem in some detail in 9.34(vi) (but this is hardly needed here from our point of view). Hence the uniqueness problem can be formulated as follows: Let u E and let
and trru=0.
(20.61)
Then one has to prove that u(s) = 0 in Ii By Proposition 19.5 it follows that D(R"\I') is dense in {v E
Let
E
H' (W') :
v = o}
.
(20.62)
Then by (20.61),
J
(20.63) )
20. The fractal Dirichlet problem
305
Since u belongs to the spaces in (20.62) it can be approximated in H' Then it follows from (20.63) by standard 0 in Il. Hence we have existence and uniqueness for the Dirichlet problem (20.59). But the representation (20.60) is unsatisfactory since it depends on K. However at least in case of n 3 this awkward description can be replaced by a more natural one. by functions belonging to
arguments, first u(s) =
20.10
in c and afterwards u(s) =
Corollary
where n E N, and let I' =
be a bounded domain in
Let n—
c
1
1
be a d-set with as described at the
beginning of 20.9). Let g E (i) Then the Dirichlet problem U E
au(s) = 0 mci, trr'u = g,
H'(Iz),
(20.64)
has a uniquely determined solution. (ii) Let, in addition, n 3. Then there is a uniquely determined distribution hEH (1') such that the solution u of (20.64) can be represented as
u(s)
=
x E ci.
J
(20.65)
Proof Part (i) is covered by 20.9. In addition, we have in any case the representation (20.60). It remains to prove (20.65). Let now n 3. Let K be a ball centred at the origin with radius R and ci C K. Then the representation (20.60) can be written as
u(s)
—
=
where c1 =
(n
x E K,
+ dK(X, tv))
is independent of K (here
—
is
(20.66)
the volume of the
unit sphere). Furthermore dK (x, 'y) is the harmonic correction of the singularity function such that = 0 ifs E OK. Recall dK(X,'y)
1
=
IR\"2
1
M -y
=
R2
(20.67)
is the inverse point of 'y 0 with respect to K. This may be found in any relevant textbook, for example [GiT77], p. 19. Denoting the operator with respect to the right-hand side of (20.60) by BK we obtain where
(BKh)(x) = (Ph)(x) + (DKh)(x), s e K,
(20.68)
III. Fractal elliptic operators
306
with
(Ph)(x)
=
= (n —
f1
(20.69)
and (20.70)
Let
p",
be the restrictions of BK, P, DK on r. We have (20.39) with
in place of B" (uniformly with respect to K). Then by the arguments given there, in particular (20.41), (20.71)
:
is for all K an isometric map. Let T, be a neighbourhood of r (maybe collecting
all points in R" with distance smaller than 1 to 1'). Let R be large. Now it follows from (20.70) in obvious notation (dual pairings) (x, y) h(y)
= =
(20.72)
),
1H'(I'i)II Ilidr h 1H'(l'i)II ,
We use (20.67) and also (20.42) adapted to our situation. The estimate is uniform with respect to x E r, and a, restricted to al = 0 and Iai = 1. In particular we have an estimate in H' (1',) and by application of trr as in (20.38) we obtain , j?n-2 where c is independent of K (and, hence, of R). By (20.68) we have
pr
=
—
=
o(id—
(20.73)
(20.74)
By (20.71) and (20.73) with R large it follows that the last operator on the right-hand side of (20.74) is invertible and hence P" is an isomorphic map between the spaces in (20.71). Then any g E can be uniquely represented as
g=P"h,
(20.75)
Now we are in the same position as in 20.6 and Theorem 20.7. In particular, u in (20.65) is a solution of (20.64).
20. The frac tat Dirichlet probLem 20.11
307
Spaces on d-sets, revisited
with 0 < d < n. We considered in
Let F be again a compact d-set in
20.5 the spaces (F). If s > 0 then we have the quarkontal description from Definition 9.29 and Theorem 9.33. But especially in the cases of interest for us in connection with Theorem 20.7 and Corollary 20.10 one has rather simple explicit descriptions. Let 0 < s < 1. Then by I.J0W84), p. 103.
lie
=
(I
+
(i (20.76)
This applies in particular to the spaces in is an equivalent norm in Theorem 20.7 and Corollary 20.10 where s = 1 — 20.12
Formalizations; the smooth case
Let F and
be as in Theorem 20.7. Then the assertioii that the Operator G" in (20.46) is an isomorphic map is a concise version of (20.47)—(20.49): existence and representation. This coincides with (20.7). The counterpart in case of Corollary 20.10(u) is given by
trroPoidr :
(20.77)
as an Lsomorphmc map, where P is given by (20.69). Let in addition F = domain in R71 with a 3). Then we have be smooth (say. is a bounded d = a — 1 1)0th (20.46) and (20.77), in particular
trroPoidi- :
(20.78)
as an isomorphic map. This is the precise version of (20.3). As mentioned there, one might consider F = as a compact. Riemannian manifold. Then can be defined directly for all s e R via local charts. Let the spaces be the related Laplace-Beltrarni operator. If p > 0 is sufficiently large then + pid
H8(I')
s E R.
(20.79)
is an isomorphic Imiap. Some information concerning spaces on Riemannian manifolds may be found in 7.2 (although in a soiriewhat different context); otherwise we refer to [Tri5]. 1.11 and Chapter 7. As in the eticlidean ease (or the n—torus) the fractional powers of the positive—definite self—adjoint operator
308
elliptic operators
III.
+ Qid with pure point spectrum have the expected mapping properties. In particular,
ER.
+ Qid :
(20.80)
combined with (20.78) one gets (20.81)
:
as an isomorphic map. This sheds some light on the relations between the Laplacians on Il and r = Oil. 20.13
with boundary r = [Xl. The quality Again let il be a bounded domain in of the boundary r has a strong influence on the Dirichiet problem as treated above. If F is C°° then one has a complete theory in all reasonable spaces even with p < 1, where the final results go back to [FrR95J. and which may also be found in [RuS96], 3.5. Looking for an Lu-theory for nonsmooth boundaries there is apparently a big difference between F E C' and F E Lip1. We refer to [Ken94], [JeK95], Theorem 5.1. p. 191, and iii particular to [FMM98]. The last paper deals also with mapping properties of single layer onto for some s potentials of type (20.69) on F E Lip' from 6 for some 6 > 0, where 6 depends oii with 0< s < 1 and p with — the Lipschitz constant of F, [FMM98I, Theorems 3.1 and 8.1. We refer also to [ZanOO]. This seems to suggest that in the fractal case as treated in Corollary
20.10 there might be little hope to step from £2 to
(or Br). Further
information and additional references concerning layer potentials, boundary integrals and Dirichlet problems for Laplacians (on Riemannian niauifolds) in Lipschitz domains may be found in (MeMOO] and in Chapter 4 of We refer in this context also to the survey [JoW97], where the authors summarize their contributions to boundary value problems of the above type. Let again P" be the operator P given by (20.69). restricted to the d-set F. We have the mapping properties as described at the end of 20.10. One may extend given by these considerations to Riesz potentials on d-sets F in
(P[h)(A) = where
/
A E F, IA
d> a > 0. Mapping properties of these operators and related spectral
problems have been studied recently by M. Zähle. LZahOO]. in the framework of an L2-theory.
20. The fractal Dirichlet problem 20.14
309
Classical solutions
be a bounded domain in where n 2 and let 1' = Oil be its boundary. be the space of all continuous functions on r, obviously formed, and let. be the space of all Lipschitz continuous functions on r. normed in analogy to (10.17). Let g E or, more restrictive. g E Lip(r). The classical Dirichlet problem for the Laplacian asks for harmonic functions u in il with Let Let
x—*',Er.
and
(20.82)
The first decisive step for arbitrary bounded domains Il goes back to Perron, [Per23J. and is known as the method of subharmonic and superharmonic functions. One gets always a harmonic function u in 11. The main problem is to clarify for which e r or for which boundaries F one ha.s the desired pointwise boundary behaviour as indicated in (20.82). The final solution goes back to Wiener and can be described as follows. Let !IC = R"\il be the complement of il in R" and let C1.2(K) be the (1,2)-capacity according to (19.44). this means with respect to the above classical Sobolev space Ht(Rn). Then one has (20.82) for a given point y e F and g E C(1') if, and only if,
/
C1,2
(B('y,r)
dr = 00,
(20.83)
r) is again a ball centred at E F and of radius r. In particular there is a natural connection to the theory of weak solutions. this means H'where
solutions. We do not go into detail. A description of the method of subharmonic functions may be found in [GiT77], 2.8. pp.23-27. As for the Wiener criterion (20.83) we refer to [Ad1196], Theorem 6.3.3. p. 165, and to [Ken94j, p. 5. In our context it is of interest under what additional conditions for F the solutions IL from Corollary 20.10 are classical solutions with respect to Perron's method. By [Ken94]. p.5 (restricted to our situation) one has the following assertion: The Wiener criterion is necessary and sufficient such that for every Lip(F) the (weak) solution u in (20.6.4) is continuous in Il (and hence a g classical solution).
We discuss (20.83) in connection with the domains in Corollary 20.10. First we remark that always
0
(20.84)
This can be checked easily: one can use the norm in (20.16). Secondly, by Remark 9.19 any d-set in with d < n satisfies the ball condition according to 9.16. This suggests the following modification of Definition 9.16:
III. Fractal elliptic operators A bounded domain Q in R'1 is said to satisfy the outer ball condition if there is
a number0
ir) with
B(y,
C
r).
fl
(20.85)
Now we get the following assertion:
be a domain as in Corollary 20.10 satisfying in addition the outer ball condition and let g Lip(I'). Then the solution u in (20.64) is a classical Let
solution.
As for the proof we first remark that
Lip(r) c H8(r),
0<
s
< 1,
(20.86)
as a consequence of (20.76). Secondly, by (20.84) we have (20.83) for all 'y E I'. Outer ball conditions and modifications of them have been used by several authors on different occasions (also in [Ken94), p.4, called class S). In [TrW96J we called such a domain exterior regular in connection with intrinsic atomic characterizations of spaces in domains. A short description and further references may also be found in [ET96], 2.5.1, p. 59.
21 21.1
Spectral theory on nrnnifolds Introduction
The aim of this section is twofold. First we continue our studies from Section 7 on function spaces on manifolds. Let M be the Riemannian manifold as introduced in 7.2 and let be the spaces defined in 7.8. We characterized in Proposition 7.17 the compact embeddings ptqa'
" '/ F32 'M,9
The first aim is to find out the degree of compactness expressed in terms of
entropy numbers as introduced in 19.16. The motivation comes from the close connection between entropy numbers and cigenvalues of compact operators described by (19.129). Under restriction to d-domains as introduced in 7.5 and discussed in 7.6 we get in Theorem 21.3 a definitive result. We rely on the quarkonial decomposition from Theorem 7.22 and the techniques developed in (Triöj and used so far in this book in connection with Theorem 19.17. The second (and main) aim of this section is to use this result in the indicated way as a starting point for a spectral theory of elliptic operators in d-domains. We developed in [ET96], Chapter 5, in a rather systematic way a spectral theory
21. Spectral theory on manifolds
311
for degenerate pseudodifferential operators based on estimates of entropy numbers. The fractal counterpart has been studied in [Thö1, Chapter 5. There are always two distinguished problems. The first is the question of the distribution of eigenvalues. Theorem 19.7, in particular (19.73), arid Theorem 19.17 with Definition 19.13, are typical examples in the present book. The second type of problems is connected with the so-called negative spectrum. Based on the techniques used in [Trio] and Theorem 21.3 one can develop a spectral theory of weighted (fractal) pseudodifferential operators in d-domains. But this will not be done here in detail. We restrict ourselves to an example, concentrating now on the negative spectrum for suitable operators. Let be the LaplaceBeltrami operator from (7.54). If9 is sufficiently large then we have the lifting property (7.57). In particular, with its domain of definition,
= H2(M) = F2(M), is sell-adjoint in L2(M) such that
(=
(21.2)
and bounded from below. Let Q E R be
C [1,00).
(21.3)
Then we are interested in the negative spectrum of the relatively compact perturbation
> 0, 0, + 9id — = + gid. In other words, we ask for the behaviour of H,3
of
N0
= # {Spec(H0) fl (—oo,0)}
(21.4)
(21.5)
Problems of this type attracted a lot of attention in the euclidean setting (i.e.,with in place of M). They originate from (euclidean) quantum mechanics and the semi-classical limit h 0 (Planck's constant tending to zero) and /3 ?r2, considered here. In Theorem 21.7 we prove as /3 — oc.
N0 N0
"-j
if
0<x<
oo),
(/3
if
(/3
oo),
(21.6) (21.7)
where M is the above d-domain. This might be considered as the main assertion of this section. We complement these results in Corollary 21.11 and in 21.12 by a look at hydrogen-like operators of type
H =
+ Qid
—
flg
\
1+
— 1
'+J
(21.8)
in this hyperbolic world (M, g), where 1x19 is the Riemannian distance of x E M to a fixed off-point.
III. Fractal elliptic operators
312
21.2
Prellminsiries
We recall what we need in the sequel. We always assume that (1 is a d-domain in
according to 7.5. This means in particular that we have the covering
(7.23) with (7.24). Equivalences are used as in (7.10). We denote ci also by M or (M, g) when converted in a Riemannian manifold as in 7.2. Also, though
not really necessary, we assume n 2 and that ci is connected. Typical examples of d-domains, called thorny star-like d-domains, have been discussed in 7.6. In these distinguished but otherwise characteristic cases of infinite Riemannian manifolds of hyperbolic type we comment briefly on the relations between Riemannian distances, the generating function g given by (7.9) and the Riemannian volume of the slices ci, in (7.21). Let 1x19 be the Riemannian distance of x E M to, say, 0 E M. By (7.11), (7.21), the Riemannian width of each slice is approximately 1. If one starts from a point x E ci,, then g(x) 2', and one needs approximately j steps of Riemannian length 1 to reach 0. Hence, 2m(x)IxIg,
g(x)
E ci,,
(21.9)
with m(x)
1. Furthermore we have for the Riemannian volume vol9 ci, of ci, and of the balls ci' = cii, vol9IZ,
2dm(x)IxIg
vol9 ci3
XE ci,.
(21.10)
This reflects the well-known fact that the volume of balls in Riemannian manifolds of hyperbolic type grows exponentially with the radius. In the general case of the above Riemannian manifolds the argument of stepping down from slice to slice might be more complicated in dependence on the geometry of ci. But nevertheless (21.9), (21.10) in the above distinguished star-like d-domains reflect typical behaviour. and g"), where In Definition 7.8 we introduced the spaces
sEa, XER,
(21.11)
(with q = oc if p = oo ), where we now always assume that M is a d-domain For our later purposes we abbreviate
in
SE R, xE R,
=
H8(M) =
(21.12)
in particular,
L2(M) = H°(M) and
L2(M,gM) = HO(M,gM),
xE R.
(21.13)
Of special interest for us is the Laplace-Beltrami operator in M,
= ._g_fl
(gn_2
.
(21.14)
21. Spectral theory on manifolds
313
This operator has been studied in great detail on manifolds of the above type and more general Riemannian manifolds. with dom = D(M) is essentially self-adjoint and In particular, with dom = H2(M) self-adjoint and positive in L2(M). We refer to [Str83] [Dav89), 5.2.3. P. 151, [Shu92), [Skr98], (Tri88], ['fll'71, Chapter 7. Furthermore if p R is sufficiently large, then (21.15)
with (21.11) and, in particular,
+ pid :
H2(M) '—' L2(M)
(21.16)
are isomorphic mappings. We refer to Theorem 7.15. Recall that we describe the degree of compactness in terms of entropy numbers according to 19.16. 21.3
Theorem
Let n E N with n 2 and let n —
1
x1 E R, X2 e IR,
—00<82<81<00. (21.17)
(qi =
oc
if pi =
5=
oc,
= 00 if p2 = oc) with
and
/
n\
\
P11
n\ >0, I (821 \ P2/
—X2>0.
(21.18)
Then
id :
(21.19)
(M,gM1)
is compact and for the related entropy numbers ek = ek(zd) we have ek
k
,
kEN.
zf x>—,
kEN,
if x<—.
(21.20) (21.21)
Proof Step I First we wish to prove that tile entropy numbers ek in (21.20), (21.21) can be estimated from above by the respective right-hand sides. We
III. Fractal elliptic operators
314
begin with some preparation. By the lifting property (21.15) we may assume that si and are sufficiently large that Theorem 7.22 can be applied to the spaces in (21.19). By Theorem 7.15 we may also assume K2 = 0. Recall that the quasi-norms in (7.99) are defined by (7.87), (7.83). Using (2.10) we wish to replace the somewhat complicated sequence spaces by their simpler b-counterparts. We use the notation introduced in Sections 8 and 9 (slightly modified). Let (E,),ENO be a sequence of natural numbers, let
aER, O<poo, 0
we shall mean the linear space of all complex sequences A with (7.82) (and E, in place of L3), endowed with the quasi-norm I lao fE, I
)
<00
(21.22)
/ /
\J=O \i=1
J
with obvious modification if p = oo and/or q = 00. In case of a = 0 we write Let in addition R and let now A be given by (7.85), (7.86) (again with E, in place of L3). Then we put
=
IA Ieao,e [eq
sup
eq
(21.23)
as quasi-norms of corresponding spaces. In analogy to [Trio], pp. 163—165, we wish to reduce the estimate from above of the entropy numbers ek in (21.20), (21.21) to
ek
(21.24)
where
Id :
(21.25)
eao,e, [4'
is the identity. We always assume that
Q1>92>°
large,
(21.26)
such that Theorem 7.22 can be applied. Furthermore x = x1 > 0 (recall that = 0) has the above meaning. The numbers and are unimportant. We may assume P2 (the F-spaces are monotone with respect P1 and to the q-index). Crucial for the estimate of ek(Id) is the knowledge of Er. Afterwards one gets (21.24) factorizing id in (21.19) via Id in (21.25) in a
21. Spectral theory on manifolds
315
similar way as in Step 3 of the proof of Theorem 19.17. First we estimate Er. Let I E be given by (7.98), where ((3qu)3i(x) are (sj,pi,x)-flquarks according to (7.88). Comparing this expansion with a corresponding representation in (M) (recall = 0) one gets
1=
= j=0 1=1
j=O 1=1
(21.27)
where (/3qu)1(x) denote temporarily the corresponding (82, P2, Here 5 is given by (21.18) and k by (7.89). If r E N0 then we have to estimate the number Er of balls K3, given by (7.78) with dist(K31,i91Z)
for
m.=0,...,j+c
(21.28)
and
rK=mx+(j—m)t5. (21.29) (7.90). Forfixedmwithm =0,...,r+c'
Hereweusedm
2'
we have to estimate the number of balls K31 of radius —' which, say, intersect u1rn given by (7.21) with (7.24), and where j is calculated by (21.29). Hence
2md 2(r_m)?,
2—vn(n—d) 2jn
(VOlIlm)
(21.30)
where we used again (7.24). Summation over m results in (21.31) ,n=0 We
obtain
if d>
Er
if d <
Er
(21.32) (21.33)
Let Ar be the collection of those Er couples (j. 1) with the above property. Recall that the series (21.27) converge absolutely and can be rearranged by
f=
(f3qu)3,(x)
(21.34)
and
f=
(21.35) r=0
(ji)EA,
III. Fractal elliptic operators
316
For fixed r N0 the balls K31 with (j, 1) A,. have only a controlled overlapping such that we have an obvious counterpart of Proposition 2.3. Furthermore by 7.23 we may assume that the coefficients depend linearly on f. We construct the linear operator S,
S:
(21.36)
by
Sf =
=
fi E Nj}
:
=
:
rEN0, (j,1)EAr} ,
=
where f is given by (21.34) with
(21.37)
,
(21.38)
To justify that S is continuous
we claim 11Sf
[€qi
IIfI1'p81'q1(M,9M')II
C
II
(21.39)
.
The first equality comes from the construction (in obvious notation) and the notational agreement after (21.22). As mentioned above we can apply PropoTogether with (7.87) we obtain sition 2.3 now with p = Pi and q = the inequality in (21.39). The final equivalence follows from Theorem 7.22 and
the choice of 4(f) according to 7.23. Next we construct the linear operator T,
T:
(21.40)
[€q2
by
=
(21.41) r=O (j,1)EAr
where again
are the above (s2,p2,O)-fl-quarks and
rEN0, (j,1)EAr}
.
(21.42)
Obviously, T is linear. But it is also continuous, since we can again apply and afterwards Theorem Proposition 2.3 now with p = P2 and q = 7.22. Let id and Id be given by (21.19) and (21.25) with (21.26). Then by (21.34), (21.35),
id=ToIdoS.
(21.43)
21. Spectral theory on manifolds
317
Theorem 9.2, p.47 and (21.32) we have
By
ek(Id)
if d>
(21.44)
if
(21.45)
and by (21.33)
Together with (21.43) (and (19.133)) we get
ck1,
ek(id)
ckr,
ek(Zd)
kEN, if
kEN,
(21.46)
(21.47)
if
for some c> 0. Step 2
We prove the converse of (21.46). (21.47). In case of (21.20) we must
show that there is a number e> 0 such that ek (id
C
(21.48)
(and for all k N. By the lifting property (21.15) we may assume that hence also Si) iS large. Assume that there is no c > 0 with (21.48) for all
k E N. Then we find a sequence
with
—p
(id
—sO
if
v
00.
(21.49)
By Theorem 7.15(u), (21.46) and (19.133) one can even
1
=
=
where we used Theorem 7.10, obviously extended to .q2 = by (1.9), (1.10),
=
=
0,
(21.50)
1
Hence we must disprove (21.49) with the specification (21.50). As for the excluded case p2 = ocwe add a remark in 21.4 below. Similarly in case of (21.21). Let r E N0 and let Ar be a subset of the above set Ar, introduced after (21.33), such that corresponding balls K31 with (j, 1) E Ar are pairwise disjoint. By the above construction one may select Er balls with this property such that
Er"Er, rEN0,
(21.52)
III. Fractal elliptic operators
318
where Er is given by (21.32), (21.33). By (7.79) the corresponding quarks in (7.88) have disjoint supports. We put = (fiqu),, if fi = 0. Let
s>O, O
P2
P1
l4et
=
:
P2
(j, 1) E Ar} for fixed r
N0. Let A,
A:
(21.55)
be given by
=
(qu),,(x),
x
(21.56)
(j.l)EAr
where
= 2_oj_m)(3
)—mxl
(21.57)
are (s, P1, x1 )-O-quarks according to (7.88) and m = m(j, 1) has the same meaning as in (21.28). By Theorem 7.22 and the support properties of the balls K,, involved it follows that IF1q(M,g"2 )II
(21.58)
PtA
(where c is independent of r). By (21.29) and (21.54) we have
(qu),,(x) =
tPt(x), x
(21.59)
AA =
(21.60) (j,1)EA,.
Let again (j, 1)
Ar. We may assume that there are functions XjJ(X) e D(K,,,)
=
,
Ix,j(x)I c, (j,1)
(21.61)
Ar.
(21.62)
21. Spectral theory on manifolds
319
Let B,
B: be given by
Bf =
(j,L) E
(21.64)
By Holder's inequality we have IIBf
(3J)EAr
S
If(xW'2 dx
J
:S
Ill
(21.65)
IIBII
(21.66)
Hence,
Let id be given by (21.19) with the specification (21.50). Let id? be the identity from into Then we have by (21.60), (21.62), (21.64),
id' = B aid o A.
(21.67)
In particular one gets by (21.58), (21.66) for the related entropy numbers
kEN,
(21.68)
where c is independent of r E N0. By [Trio], Theorem 7.3, p. 37,
rENa,
(21.69)
c> 0. independent of r. In case of (21.52), (21.32) we have
r EN0. In case of (21.52), (21.33) we put a = ec2ro(id)
(21.70)
and get
=
,
r EN0,
(21.71)
where we used (21.54). Here c> 0, e' > 0, are independent of r E N. But this disproves (21.49) with (21.50). As said above, this is sufficient to prove the converse of (21.46), (21.47).
III. Fractal effiptic operators
320
21.4 Thecasep=oo In Step 2 of the above proof we excluded so far the case p2 = oo (and hence with 0 < s < 1 = oo and let C8(Q) = also = oo). Let now be the Holder-Zygmund spaces, normed by first differences as in (1.11)—(1.13)
and according to Definition 5.3. By Theorem 7.10 and direct arguments one gets
O
(21.72)
where by Proposition 7.12 and (5.103) these two spaces coincide if is smooth. By the same arguments as at the beginning of Step 2 it is sufficient to disprove (21.49) or the related counterpart based on (21.21) by the specification (21.50) with C32(Ifl in place of Combining this with the entropy numbers of
id
(21.73)
according to [Triö], Theorem 23.2, p. 186 (which works for any bounded doas a target space to disprove main), it turns out that one can take also (21.49). The rest is the same as in Step 2 which works (including the references) also for = 00. 21.5
Remark
Theorem 21.3 coincides with [Tri99cJ, Theorem 2.4.2, where we outlined also a proof. This theorem might be considered as the hyperbolic counterpart of the weighted euclidean situation with the replacement of by
= (1 ÷
(21.74)
in (21.20), (21.21) is then The outcome is similar. The splitting point = = 5. First results in this weighted euclidean case have been obtained in [HaT94a]. An improved version may be found in [ET96], 4.3.2, pp. 170—171. The state of art, including estimates for approximation numbers may be found in [Har95], [Har97], [Har98], [HarOOc]. In the papers by D. D. Haroske special attention is paid to limiting case = 5. We excluded here in the hyperbolic Furthermore in the euclidean situation the corresponding limiting case x = setting there is a rather satisfactory discussion of cases with p2
21. Spectral theory on manifolds
Spectral theory
21.6
Let A > 0, 0
1, and
b(.,D) be
321
a
pseudodifferential operator in
(21.75)
E
in the Hörmander class indicated. Let
1ri
+x2 > 0
x1
ER,
(in obvious notation according to (21.74)). Based
(21.76)
on [HaT94b]
we developed in [ET96). 5.4, a spectral theory of the degenerate pseudodifferential operators
B = b2b(.,D)b1
(21.77)
the weighted euclidean counterpart of Theorem 21.3 as indicated in 21.5 starting point. More recent results in this direction may be found in [Har98j. In [Trit5), Chapter 5, we discussed a corresponding spectral theory for fractal pseudodifferential operators. Armed with Theorem 21.3 one can try to extend this theory from the euclidean case to the above manifolds of hyperbolic type. Pseudodifferential operators on manifolds with bounded geometry and positive injectivity radius, especially the Laplace-Beltrami operator (21.14) have been considered with great intensity and may also be found in the references given in 21.2. In particular the mapping properties proved there can be taken as the starting point for a spectral theory of hyperbolic counterparts of (21.77). But this will not be done here. with as
As described in the Introduction 21.1 we concentrate ourselves on the special but interesting problem of the negative spectrum as explained in (21.3)(21.5). We commented there also on the physical background (at least in the euclidean case). As for the mathematical background we complement our pregiven by vious assertions. Recall that the Laplace-Beltrami operator (21.14) and considered as an operator in L2(M), is essentially self-adjoint on = H2(M). If E IR is suffiD(M) = D(fI). and self-adjoint with ciently large then not only (21.16), but also (21.15) are isomorphic maps. Of is symmetric and course for any x > 0 the multiplication operator f '—' bounded on L2(M). Furthermore, by Theorem 7.15,
I=
+ pid)_lg_)c
L2(M)
H2(M,9M),
(21.78)
is an isomorphic map and by Proposition 7.17 (or Theorem 21.3), B = (—A9 +
:
L2(M)
L2(M),
(21.79)
III. Fractal elliptic operators
322
is compact. Hence, H0, given by (21.4), is for any 0 a relatively compact perturbation of + id. By Kato's criterion, [Tri92], Theorem 3, p. 208, or [Dav95], Theorem 1.4.2, p.18 (attributed there to Rellich, 1939), which applies in particular to relatively compact perturbations, it follows that
x>0, dom(H0)=H2(M),
(21.80)
is a self-adjoint operator in L2(M). Furthermore, since the operator in (21.79) is compact one gets by [Dav95], Theorem 8.4.3, p. 167, or in a more general context of Banach spaces by [EdE87}, Theorem 2.1 on p.418, that the essential spectra of these two operators coincide,
EssSpec(H0) =
+ aid) C [1,00),
(21.81)
where the latter inclusion comes from our assumption (21.3) concerning the choice of p. Let Spec (H0) be the spectrum of the seif-adjoint operator H$ in L2(M), given by (21.80). Then N0 in (21.5) counts the finite number of negative eigenvalues of called the negative spectrum. As usual, #A denotes the number of elements of the finite set A. Recall again that equivalence, expressed by —.', is used as explained in (7.10) and (19.17), (19.18). 21.7
Theorem
Let n E N with n 2 and let n—i 0. Let N0 = # {Spec (H0) fl (—co, 0)}
(21.82)
be the number of the negative eigenvalues of H0. Then
N0
if
—.
(21.83)
if x>
N0
00.
/3
(21.84)
Proof
Step 1 Let the compact operator B and the isomorphic operator I be given by (21.79) and (21.78), respectively. Let
id
'—' L2(M)
:
(21.85)
be the identity. Then we have B = id o I
and
id = B
o
(21.86)
21. Spectral theory on manifolds
323
Let ek(B) and ek(id) be the corresponding entropy numbers according to 19.16. It follows by (21.86),
ek(id),
ek(B)
k E N.
(21.87)
In order to estimate from above we use the entropy version of the BirmanSchwinger principle as described in [Trio], 31.1, p. 243, with references to
[HaT94b] and [ET96], 5.4.1, p. 223. In the latter book one finds also further discussions about the Birman-Schwinger principle including the relevant literature. We have
Ns<#{kEN
i}
:
.
(21.88)
We specify id in (21.19) by (21.85) and obtain ek(id)
kEN,
ek(id)
k E N,
(21.89)
0<x<
(21.90)
Now (21.87)—(21.90) result in (21.91)
<
c/3-
if
2d 0<x< —,
(21.92)
for some c> 0 which is independent of /3. Step 2 We prove the converse inequalities. Recall that is self-adjoint in L2(M). We shift the problem to quadratic forms and the Max-Mm principle. Hence, by (21.80) we have (H0f, f)L2(M)
If IH'(M)112 —
= Ill IH'(M)112
1L2(M)
/3
— /3
Ill
112
(21.93)
I E H2(M).
Let r E N and let A,. be the same set of couples (j, 1) as in Step 2 of the proof of Theorem 21.3. Recall that balls K3, with (j, 1) e A,. are pairwise disjoint. Otherwise they are located by (21.28) with (21.29). We apply this construction to
H1(M) and In particular we have 0 = to
1
and x must be replaced
(21.94) Hence, (21.29) specifies
(21.95)
III. Fractal elliptic operators
324
Furthermore, Er, the number of balls involved, can be estimated by (21.52) with (21.32), (21.33) as 0<
Er if
Er
<
(21.96)
)(>
(21.97)
Now one could follow the quarkomal arguments in Step 2 of the proof of Theorem 21.3 Starting with (21.55), (21.57). But maybe it is more transparent to do the calculations directly. Recall that the radius of K,1 is 'P-' 2'. Let l',j (x) be the same functions as in (21.57), given by (7.79), (7.80). In particular they have disjoint supports. To estimate the H'(M)-norms we use Theorem 7.10. By (21.28) and the necessary notational modifications in (7.38) we may assume
IH'(M)II
(21.98)
On the other hand, again by (7.38), (21.99)
Comparing (21.98) with (21.99) it follows from (21.95) that there is an orthonormal system
: t=1,...,Er} , rEN, in H'(M) which is orthogonal in
(21.100)
and
r
N,
(21.101)
where the equivalence constants are independent of r. We choose = c with c> 0 small (but independent of r E N) and insert finite linear combinations of these functions xr in (21.93). We find that for these functions the quadratic form is always negative. Hence, with fi = c2TM
(21.102)
is negative on a subspace of H' (M) of dimension Er. By the Max-Mm principle, which may be found in [EdE87J, pp. 489—492, it follows that
NflEr with
(21.103)
By (21.96), (21.97) we get
if Nfl
(21.104) if
(21.105)
21. Spectral theory on manifolds
325
for some positive numbers c1 and C2. This is the desired converse of (21.91). (21.92). 21.8
Local singularities and hydrogen-like operators
As said in the introduction 21.1, in a euclidean setting problems as treated in Theorem 21.7 originate from semi-classical limits h 0 (Planck's constant tending to zero) and ,3 IF2. Related potentials, with the Coulomb potential as an outstanding example. have natural local singularities. But this is not the case with the potential in (21.80). However there are no serious problems in incorporating local singularities in our context. This will be done in the remaining subsections of this Section 21. To provide a better understanding we have first a glance at the hydrogen atom in R3. The corresponding Hamiltonian operator 1IH is given by dorn(flhc) = H2(R3),
=
(21.106)
—
where e is the charge and m is the mass of the electron (spinning around the origin in 1R3. where the nucleus is located). It is a self-adjoint operator in L2(R3). its essential spectrum coincides with [0, oc), and the pairwise different eigenvalues E2 are given by
=
2h2j2
with multiplicityj2,
(21.107)
N. These assertions may be found in many books, for example where j [Tri92], 7.3.3, 7.3.4. pp. 434. 438. We adapt to (21.80), (21.81) and put with =
dom(H8)=H2(R3), (21.108)
where a > 0 is the resulting constant. We have
EssSpec(Ha)= [1.x), eigenvalues
A3 =
(21.109)
with multiplicity j2 for some c> 0 and j E N. Hence we get in this case for given by (21.82). in obvious interpretation, (21.110) j=1
elliptic operators
III.
326
We preferred a direct approach to obtain (21.110). But this has little to do with the explicit knowledge of the eigenvalues. it follows also from qualitative arguments using the technique developed in 1ET96}, 5.4.8, 5.4.9, pp. 238-242. Now we switch from the euclidean situation to the hyperbolic one. Hence we replace 1R3 by the d-domain in R" according to 7.5 and denote again the related non-compact Riemannian manifold by M. We have Theorem 21.7. On
the other hand, the Coulomb potential in (21.108) and its first derivatives describing the resulting forces are comparable with the volume of spheres in R3 centred at the origin. The situation in spaces M of the above type is different. We discussed this point in 21.2. In particular, (21.9), (21.10) suggest that something like Coulomb potentials in M should have an exponential decay when measured in terms of the Riemanman distance from, say, the origin. Hence the far-distance behaviour might be reasonably expressed by with
x > 0 as in (21.80), which coincides with (21.4). On the other hand locally, near the nucleus, the Coulomb potential in M (say with n = 3) should be Hence something like Izi;' looks reasonable. Then one gets, similar as in with the additional specification x = 1, the operator in (21.8). We call such operators hydrogen-like. But we could not find in the literature anything like a quantum mechanics in hyperbolic spaces providing a better justification of operators like in (21.8) than the above vague arguments. We look at operators of type (21.8) from a slightly more general point of view expressing local singularities by b L,. ( M). Then we have first to complement
Theorem 21.3 in the operator version of (21.87). This will be done in 21.9. Afterwards we extend Theorem 21.7 to potentials with local singularities. We introduced the weighted Sobolev spaces H(M,gM) in (7.42) with the weighted Lebesgue spaces
1
(21.111)
as special cases. The latter follows from (7.33), (7.39), (7.34) with .s = 0, stands for the Laplace-Beltrarni operator (21.14) and q = 2. As always, we assume again that
e
R is chosen so large that, in specification of (21.15),
+ gid
p—'
1
(21.112)
is an isomorphic map. 21.9
Corollary (to Theorem 21.3)
Let Il be the same d-domain as in Theorem 21.3 and let M be the related Riemannian manifold. Let
l0,
(21.113)
21. Spectral theory on manifolds
327
and
E Lr(M)
1)
q pr n
with
—
=
(21.114)
Then the operator B6
+ Qid)1
=
(21.115)
'—b
is compact. There is a number c> 0 such that for the related entropy numbers ek = ek
(2_n)
kEN, if
<
(2_!!i)
kEN, if Proof
(21.116) (21.117)
We factorize B6 by (21.118)
with (21.119) (21.120)
b
id
+
Lq(M,g") H12(M)
:
aid)'
:
(21.121)
is interpreted as a multiplication operator; the assertion itself folHere lows from (21.111) and Holder's inequality based on (21.114). By (21.112) we have (21.121). As for id we can apply Theorem 21.3 with
= 2+
n —
p
n
— — =2— q
n —
r
>0.
(21.122)
Then (21.116), (21.117) follow from (21.20), (21.21), respectively. 21.10
Discussion
We are not so much interested in the operator itself. Our aim is to perturb the operator in (21.80) and in Theorem 21.7 so that local singularities are included. Using again the Birman-Schwinger principle as explained in Step 1 of the proof of Theorem 21.7, we must estimate the entropy numbers ek(Bb) of the operator B,,, B,, = B + B6 =
+
êid)'
(1 + b),
(21.123)
III. Fractal elliptic operators
where B is given by (21.79) and Bb by (21.114), (21.115). We claim that k e N.
+ ek(Bb) cek(B),
e2k(Bb) <
(21.124)
The first inequality comes from (19.134). As for the second inequality we consider B in We have again (21.87), where id in (21.85) is modified in an obvious way, and by (21.89), (21.90), equivalence in (21.116), (21.117) with
r=oo.Thiscoversalsothecaser=ooin(21.124).Nowletr
Corollary (to Theorem 21.7)
Let
be the same d-domain as in Theorem 21.7 and let Al be the related
Riemannian manifold. Let 00
r> max (2, suppb
real,
b
(21.125)
,
compact in
(21.126)
Let
x>0, dom(Hj)=H2(M). (21.127)
Then for fi >
0
is self-adjoint in L2(M) with
and p E R large,
+gid) C [1,oo).
=
(21.128)
(21.129)
= be the number of negative eigenvalues of
if if
Then ,t3—*oo,
(21.130)
(21.131)
Proof By Corollary 21.9 with p = 2 the remarks in 21.6 apply also to in place of H0. Hence is self-adjoint, + b) is a relatively compact perturbation of —A9 + pid and we have (21.128). By Step 1 of the proof
22. Isotropic fractals and related function spaces
329
of Theorem 21.7, the estimates (21.124), combined with (21.87) and (21.89), (21.90), we get the counterpart of (21.91), (21.92). Since we assumed that supp b is compact in Q we can apply the arguments from Step 2 of the proof of Theorem 21.7, restricted to those values of m such that the balls with (21.28) have an empty intersection with suppb. Then we obtain the counterpart of (21.104), (21.105). 21.12
Hydrogen-like operators
Let
2
(21.132)
and let H3 be the operator from (21.8), H3 =
+gid—flg' (i + in (21.127) with x =
—
i))
,
(21.133)
i).
— Recall and b(x) = we n that a get (21.128). Now it follows by Corollary 21.11 and in particular by (21.130)
in specification of
1
that N3 =
(21.134)
In 21.8 we called operators of type (21.133) hydrogen-like. At least in case of n = 3 this might he justified as far as the Coulomb potential b(x) lxi;' near the off-point is concerned. Whether there is an argument in favour of x = 1 or any other value of 0 is not so clear. Let us take this for granted. Then
the physicists iii such a hypothetical hyperbolic world are in a remarkable position. They claim that they can find out by experiments in a laboratory, calculating the exponent d in (21.134), how fractal the invisible boundary of their infinite world might be (explaining to their astonished country(wo)men that this originates from tiny irregularities in the first millionth of a second after the Big Bang). Recall that we have (21.110) in our eudidean world R3. 22 22.1
Isotropic fractals and related function spaces Analysis versus (fractal) geometry
Entropy numbers and eigenvalues may be considered as two sides of the same coin. Entropy numbers are the geometrical twins of eigenvalues. This opinion
HI. Fractal elliptic operators
is based on Carl's observation (19.129) at an abstract level. It is also supported by the equivalences in (19.73), Theorem 19.17 combined with (19.109), and the assertions about the negative spectrum in Theorem 21.7' and Corollary 21.11
where estimates for entropy numbers play an important role. In 19.18(vii) we remarked that in case of Hilbert spaces, entropy numbers and eigenvalues might be even equivalent. The decisive link between the abstract inequality (19.129) on the one hand, and the more concrete sharp estimates from above of related eigenvalue distributions on the other hand, is the possibility to calculate the entropy numbers of compact enibeddings between function spaces and the consequences for respective operators. Theorem 21.3 and Corollary 21.9 may serve as typical examples. We followed this path in [ET96j, Chapter
5, based on the earlier papers mentioned there, for regular and degenerate elliptic pseudodifferential operators in R" and on (lomains fi in R'1. At that time estimates of entropy numbers of compact embeddings between fimction (W') and spaces were based on the original Fourier-analytical definitions of as described in (2.37), (2.38), and Theorem 2.9. These tools7including and atomic decompositions) proved to be inadequate when switching from domains fI in R" to fractals, preferably d-scts according to (9.148). Extending a d-set with this definition to d = 0 and d = vi we call a compact set F in 0
0< r <1,
F,
(22.1)
where again B(y, r) is a ball in K'1 centred at 'y E F and of radius r. In par-
ticular, if Il is a smooth bounded domain in R'1, then F =
is
a (rather
special) n-set. To handle related fractal elliptic (pseudo)differentiat operators we developed in [Trio) the theory of quarkonial (subatomic) decompositions of anti Fq(R'1). This has been extended in Chapter I of the the spaces present book in several directions. Seen from the point of view of applications to fractal elliptic operators, quarkonial decompositions compared with, say, atomic representations have two major advantages. First the building blocks, called fl-quarks, are given in a constructive way as in Definition 2.4, and secondly they are highly flexible as indicated iii 2.14. This paved the way for the Chapter W, of a theory of function spaces on fractals, development in which in case of d-sets, looks satisfactory. On this basis we dealt in [TriO), Theorem 30.2, p. 234, with isotropic fractal drums, described by operators B,
B = (—s)—'
(22.2)
where 1' is a compact d-set in R'1 with n — 2 < d < n. We repeated the assertions obtained there at the beginning of Theorem 19.7, where one finds also the necessary explanations. in particular, if Qk are the positive eigenvalues
22. Isotropic fractals and related function spaces
of B, then according to Theorem 19.7(i),
kEN. As said above, this part of Theorem 19.7 is not new, and (22.3) coincides with [Triö}, Theorem 30.2, (30.7), p. 234. One can take the equivalence (22.3) as a strong hint that also in case of d-sets F, the method outlined above is natural and well adapted and characterized by the key words: Spaces of type special cases on
(22.4)
their relations to compact d-sets 1', estimates of entropy numbers of
corresponding compact embeddings. and assertions of type (22.3) for respective
operators B in (22.2). The outcome is remarkable (in our opinion): The spaces in (22.4), including are naluall their special cases and their restrictions to domains Il in ml, effective and well adapted not only when dealing with partial differential operators in R". and on domains fl, but also in connection with d-sets. One may ask for indicators providing an understanding of this success. The trace operator trr and the identification operator idr have the same meaning as in 9.2. Furthermore by 9.19 any compact d-set in with d < n satisfies the ball condition. Let 1' be a compact d-set with 0 < d < n. Then one gets: (i)
([Trio], Theorem 18.2, p. 136)
idr
=
(R"),
I
1
-+
1
= 1,
(22.5)
where we used the notation introduced in 9.34(viii). (ii) ([TriOI, Corollary 18.12, p. 142, combined with 9.19 above)
trrB;T(Rh1)=Lp(r), 0
(22.6)
On the other hand we discussed in Section 9 in some detail traces of spaces of type (22.4) on compact sets F which are supports of finite Radon measures. We refer to Theorem 9.3 and. as far as spaces on F are concerned, to Definition
9.29 and Theorem 9.33. The outcome is more or less satisfactory. But it is apparently a rather different question to ask for tailored spaces to meet the needs of operators B of type (22.2) with the desired outcome of type (22.3) for the distribution of the positive eigenvalues of B. We dealt several times with operators of type B in (22.2) and their spectral theory when r is not a d-set. As for anisotropic and nonisotropic fractals in
332
III. Fractal elliptic operators
the plane R2. we refer to the discussion in 19.4 and the literature mentioned there. Special but very interesting cases are the Weyl measures from Definition 19.13 with Theorem 19.17 as the (isotropic) outcome. The rather direct iroof of Theorem 19.17 might be considered as an interesting example of the intense interplay between fractal geometry and analysis in the above franiework and in a particuliar situation. We refer also to the discussion in 19.18. and the literature mentioned there. Whereas the special multifractal measures ji in with n = 2 mentioned there fit in the above scheme since they are Weyl measures, this is presumably iiot the case when n $ 2. Hence it would be of interest to deal in greater detail with these multifractals, to find tailored function spaces, or arguments with geometry versus analysis of the above type. We return to the original problem: analysis versus geometry. In Section 9 spaces
of type (22.4) are given (analysis) and we ask for traces on given conipact sets IT (geometry). Conversely if r is given (geometry) and B in (22.2) is considered, then one may ask for tailored spaces (analysis) producing (22.3) (if possible). In case of d-sets these two ways of looking at function spaces coincide and we took (22.5), (22.6) as indicators of this fortunate outcome. In general this cannot he expected, although one has (in our opinion) the striking interplay between analysis and fractal geometry described in 19.18(vii). We discussed this situation in 9.34(i) and (ii) and we indicated there and also at the beginning of 9.1 and in Introduction 18 what intentions we have now: We replace d-sets by more general but isotropic (d, 'P)-sets IT, construct tailored
spaces according to B in (22.2) and ask for counterparts of (22.3). Section 22 deals with (d, W)-sets and related tailored spaces where the counterparts of (22.5). (22.6) serve as criteria. The spectral theory of the corresponding operators will be described in Section 23. As for the (d, W)-fractals we give proofs. Otherwise we restrict ourselves to a report written in the style of a survey.
To justify this procedure we have a look at the state of the art and at the literature. Our inspiration to extend what has been done in Chapters IV and V, for d-sets and related operators B of type (22.2) to the more general (d, '1')sets comes from the note [Leo98] by H.-G. Leopold. who studied eznbeddmg properties of spaces of type Bg' —b) (Rn) as briefly mentioned in 17.2 in delicate limiting situations, including estimates for entropy numbers. We summarized our own results in [EdT98] and in an extended version [EdT99a]. Obviously in {EdT98J, but also in IEdT99aI. we gave only outlines of proofs. We restricted all parameters as much as possible and claimed afterwards that everything is parallel to the respective arguments in [Trio]. The situation is now much better. There are several (forthcoming) papers, preprints, PhD..theses dealing with all relevant aspects in detail. First one must ask for a generalization of [TriO], Sections 8 and 9, dealing with entropy numbers of compact embeddings
22. Isotropic fractals and related function spaces
333
between weighted sequence spaces of a similar type as has been used before in Step 3 of the proof of Theorem 19.17 and in Step 1 of the proof of Theorem 21.3. All that one needs (and more) may be now found in [Le000a], [LeoOOb], [MouOlb], [LeoOl] and [KuSQO]. As for the related function spaces of type and
(22.7)
all details are given in [Mou99], [MouOla] and [MouOlb]. This applies also to the corresponding spaces on (d, W)-sets. We refer in this context also to [Bri99], [BriOO], [BriOl].
22.2
Admissible functions
We outlined our intentions in Section 18: We wish to generalize compact d-sets, given by (18.1), to (d, W)-sets with (18.6). Here r"W(r) might be considered as a perturbation of rd. Hence first we must clarify which type of functions 'I' we have in mind. As for notation we recall first our use of "i, explained in (19.17), (19.18). Furthermore we adopt the following convention. A real function 'I' on the interval (0, 1] is said to be monotone if it is either monotonically decreasing
or increasing, where monotone decreasing (increasing) means monotone not increasing (not decreasing). As always log is taken with respect to base 2. A real function 'I' on the interval (0, 1] i.s called admissible if it is positive and monotone on (0, 1], and if
W(22i)
j E N0.
(22.8)
0
(22.9)
Let 0
Proposition
Let 'I' be an admissible function on the interval (0, 1]. (i) Let x E 1R. Then is also admissible. (ii)
(iii)
LetO
jEN0.
(22.10)
0<x1.
(22.11)
Furthermore,
III. Fractal elliptic operators
334
There are positive numbers c1, c2, b and c with 0 < c <
(iv)
1
0<x<1
such that (22.12)
Proof Step 1 Part (i) is obvious. In particular. to prove (ii) and (iii) we may assume that 'P is decreasing. First we remark that for given x with 0 < x < 1. there is a number d > 0 such that
W(xx)
for all 0 < x < 1.
(22.13)
Now (22.11) follows from (22.8) (it is the continuous version of (22.8)). Furthermore it is sufficient to prove (22.10) for 4 and 4 in place of r1 and r2. respectively, where 1 N. In particular we assume that both Ti and r2 are small.
<TI
where
and
k1 E N0
k2 E N0.
(22.14)
Then it follows by iterative application of (22.8).
< '1'(r?)
<
<
'P
j E N0.
(22.15)
for some b> 0. This proves (22.10). Step 2 It remains to prove the right-hand side of (22.12) for an admissible decreasing function 'I'. Let
<x <
2_2k
some kEN.
(22.16)
As in (22.15) we obtain by iterative application of (22.8), 'I'(x) <2bk '1l(2_2)
(22.17)
for
This proves (22.12). 22.4
Definition
Let F be a compact set in where n E N. Let 0 < d < n and let 'P be an admissible function according to 22.2. (i) such that for Then r is called a (d, 'P)-set if there is a Radon measure jz in
any ball B('y.r) in R" centred at'y Er and of radius r wit/tO < r <1. suppp = r. (ii) oc
rdIJ!(r).
(22.18)
Let 'I' be a decreasing admissible function according to 22.2 with '11(x) —÷ if x 0. Then F is called an (n, '11)-set if there is a Radon measure p in with the above properties and d = n in (22.18).
22. Isotropic fractals and related function spaces
335
Discussion
22.5
If 'I' =
1 in (22.18) then we have (22.1) and hence I' is a d-set. So far d-sets played a (lecisive role at several places in this book. Some discussions may be found in 22.1. By the above definition and by (22.12) it is clear that (d, '11)-sets might be considered as isotropic perturbations of the (isotropic) d-sets. As for basic notation of fractal geometry we refer again to [Fa185j, [Fa190j, [Mat95] and the brief summary in Sections 1-3. Note that the requirement that p in (22.18) be Radon is not an additional restriction but a natural outcome, see [Trig], Section 2. pp. 2—4, with the references given there to [Fa185j and [Mat95J. In particular, [Mat95], p.55, is now of relevance. Recall the following basic facts of fractal geometry: For any d with 0 d n there are compact dIf dimH F denotes the Hausdorif dimension ofF, then dirnj, F = d. sets I' in If in addition d < n, then Fl = 0, where IFI is the Lebesgue measure of F. Furthermore for any d with 0 d n there are self-similar d-sets defined by IFS (Iterated Function Systems). Details may be found in the above references. We recall that any d-set F with d < n satisfies the ball condition. We refer to 9.19 and 9.16. By Theorem 9.21 this property is of great interest in connection with traces of functions on F. The question arises whether for any admitted couple (d, 'I') there is a (d, W)-set according to Definition 22.4 and whether the above properties of d-sets can be extended to (d, '11)-sets. We describe the outcome in 22.6 and 22.8 and define in 22.7 what we wish to call pseudo self-similar sets.
22.6
Proposition
Let F be a compact (d. '11)-set according to Definition the related measure. (i)
in
and let p be
Then Fl
=0 and dimHl'=d.
(22.19)
(ii) Let, in addition, n = 2. Then p is a Weyl measure according to Definition 19.18.
be a second Radon measure on I', satisfying (22.18) with Let place of jz. Then p and are equivalent. (iii)
in
Furthermore, F satisfies the ball condition according to Definition 9.16 if, and only if. d < n.
(iv)
Proof Step 1 We prove (i)—(iii). We again use covering arguments. Let centred at such e F and of radius j E There are N, balls
III. Fractal elliptic operators
336
that
r c tJ
1.
(22.20)
Hence, by (22.18),
jEN,
(22.21)
where all equivalence constants are independent of j. Then we have (22.22)
Ifl
If j —i oo then we obtain the first assertion in (22.19) in all cases. The second assertion in (22.19) is well known for d-seLs. This can be extended to (d, '11)-sets
by using standard arguments which can be found, for example, in [Tri5], pp. 3/4, and (22.12). Details and further information may be found in [MouOlb], 2.1. This completes the proof of (i). As for (ii) we note that satisfies the doubling condition and is strongly diffuse according to (19.124), (19.125). We obtain by Theorem 19.17 that iz is a Weyl measure (in the plane). Let and ji he two measures as in (iii). To compare and one can apply the same covering arguments as for d-sets as in, for example, [Trio], proof of Theorem 3.4, pp. 5/6. It follows that j.t and are equivalent. Details may be found in [MouOlb], Proposition 2.4. (9.83) we must check whether Step 2 We prove (iv). By Proposition 9.18 there are numbers A > 0 and c> 0 such that for all j E N0 and all x N0, '1,(2_i_X)
(22.23)
If d < n and if'1' is decreasing then one may choose A = n — let 'I' be increasing. If j = 0 and
0
E
N and x
d.
Let d < n and (22.24)
2j. Then (22.23) with
N and with k EN.
(22.25)
By iterative application of (22.8) and (22.25) we obtain (22.26)
22. Isotropic fractaLs and related function spaces
337
where c and b are suitable positive numbers. \Vc again get (22.23) with (22.24). Hence r satisfies the ball condition if d < ii. Let d = ii. Assume that r satisfies the ball condition. Then we have (22.23) with d = n for some C> U and .\ > 0 and all j E N0 and E N0. By (22.12) we obtain (22.27)
(j +
for some c> 0 and b> 0. For fixed j E N0 and oc we get a contradiction. Hence if d = n then r does not satisfy the ball condition. 22.7 Pseudo self-similar sets and t/,IFS
So far it is not clear whether there are (d,
for each admitted couple
(d. W). As mentioned in 22.5 in case of d-sets we have distinguished related selfsimilar fractals created by IFS (Iterated Function Systems). The corresponding procedure may be found in the standard references quoted in 22.5 or in [Triöl, Section 4. Using the notation introduced there we describe now what we wish to call pseudo iterated function systems, abbreviated by But we do not for their own sake here. We outline the typical procedure of varying study
contractions and restrict otherwise ourselves to a simple. but for our purpose sufficient case. We follow be the closed unit cube in Let Q = [0.
R". Let N E N with N 2. Let A,1
be
j E N,
x
contractions in R'1 with
0<
1
ik < N,
and
E
SUPêj < 1.
(22.29)
j
J
(22.28)
The cubes
= have side-length tion
We
.1, fl Q,ni •rn, = 0,
at each level j
E
(22.30)
o A,1...,3_1
assume here that the strong separation condi(Ii. .. ,l,)
(mi
.
.
(22.31)
N. and the inclusion property
c
I
=
1
N.
(22.32)
at each level j E N0 hold. Of course, j = 0 in (22.32) means C Q. If are chosen = g for some 0 < g < 1 and all j E N and if the points
III. Fractal elliptic operators
appropriately, then one has the well-known procedure of IFS, creating selfp. 11, we put also in the above similar (isotropic) fractals. As there, more general case
j EN,
=
(22.33)
where the union is taken for fixed j over all cubes in (22.30). By (22.32) the resulting sequence of sets is decreasing and its attractor
I' = (AQ)°° = fl (AQ)' = llrn(AQ)'
(22.34)
jEN
is called a pseudo self-similar set. The strong separation condition (22.31) is not really necessary. It would be 8ufficient to have (22.31), (22.32) for the respective open interiors. But as said we do not study and resulting
pseudo self-similar sets for their own sake here. 22.8
Proposition
For any couple d and W according to Definition 22.4 there ts a pseudo selfsimilar (d, W)-set. Proof Step 1 First we construct a sequence of cubes with the properties described in 22.7. Let m E N be large,
N=
=
and
)(j)
(22.35)
3EN.
(22.36)
=
with
)
,
where
By (22.11) there are two positive numbers c and C such that
for jEN. Let first d =
(22.37)
Then 'F is decreasing and we may choose C = 1 in (22.37). < 1 for j EN in this case where d = n. If not, one on the right-hand side of (22.36) by with some k = k(j) such that we have 0(j) < 1 after this replacement. This does not influence the following constructions (after appropriate modifications). By (22.37) we have (22.29). Let j = 1. Then we divide Q naturally in N cubes with 8ide length =
One may even assume can replace p3 (but not
with (22.34).
22. Isotropic fractals and related function spaces
339
Let now d < n. Let j = 1. Then we again divide Q naturally in N subcubes By (22.35) and (22.37) we have of side-length =
m is large.
(22.38)
and a Hence (22.31) with j = 1 can be obtained. Iteration gives again a related pseudo self-similar set r according to (22.34). Step 2 We construct a measure jL on 1' with (22.18). We use the wellknown mass distribution procedure, see [Fa190], pp. 13-14, and Proposition 1.7 there, distributing evenly the unit mass during the procedure described above. The resulting Radon measure /L has, by construction, the support property in (22.18). As for the second property in (22.18) we note that the cubes in (22.30) have side-length
= c
=
>
0.
By the mass distribution procedure and (22.35) we
obtain
=
=
j EN.
=
(22.40)
Using (22.12) and (22.11) we have C1
where ci >
0
C2
and
'11(r3),
(22.41)
and C2 > 0. Hence,
j EN. By (22.39) and (22.11) we have
--' r3.
(22.42)
Then (22.18) is a consequence of
(22.42). 22.9
Function spaces: Preliminaries
As indicated in Section 18 and in greater detail in 22.1 we are now interested in spaces of type (22.7) tailored to (d, '11)-sets according to Definition 22.4. Since (d, '11)-sets might be considered as perturbations of d-sets, one can expect someand thing similar for the spaces in (22.7) compared with This is largely the case and one could follow the Weierstrassian approach to the latter spaces developed in Sections 2 and 3. But this will not be done here. We restrict ourselves to a summary. Then it might be better to begin with the Fourier-analytical definition and to say afterwards how atoms, /3-quarks etc. appear. The first steps have been carried out in [EdT98] and [EdT99a) under
III. Fractal elliptic operators
340
the restriction to the B-scale and the range of the admitted parameters. By the recent work of S. D. de Moura, we refer in particular to [MouOlb], the full theory for B-spaces, F-spaces and all parameters is now available. As a compromise we follow mainly [EdT99a}, complemented by some assertions taken from [MouOlbI, avoiding full generality. As said above, no are given. Brief outlines may be found in [EdT99al (under the indicated restrictions),
full details are presented in [MouOlb] and [MouOlaJ. We use the notation introduced in 2.8. In particular, the functions have the same meaning as in (2.33)—(2.35). First we define the counterparts of (2.37) and (2.38). 22.10
Definition
Let
sER,
(22.43)
and let 'I' be an admissible function according to 22.2.
is the collection of all f E
Then
(i)
such that 1
=
Ill
(22.44)
11(w21)"
(with the usual modification if q = oo) is finite. (ii) Let in addition S'(RTh) 8uCh that
p < oo. Then
is the collection of all I E I
=
hf
(22.45)
(with the usual modification if q = 22.11
00
) is finite.
Comments
If 'I' = 1 then
has the
usual Fourier-analytical definition of the spaces On this basis we developed in [Tri/3] and [Th'y] the theory of these spaces in detail. Many properties obtained there can be carried over to the above spaces and In particular, they are one
and
independent of the
chosen
resolution of unity {cok}. Furthermore they are
22. Isotropic fractals and related function spaces
341
quasi-Banach spaces (Banach spaces if p 1 and q 1). We refer to [MouOlbJ, 1.2 and 1.3. where one finds detailed proofs. In particular characterizations in terms of local means as developed in [Tk17j, 2.4.6, 2.5.3, for the spaces Fq (IR?') and can be extended to the above spaces; Theorems 1.10 and 1.12 in [MouOlb) include also some improvements even in case of 'I' = 1. By (22.12) it is quite clear that a remains the main smoothness and W stands for an
additional finer tuning. 22.12
Atomsandf3-quarks
In [Triö], Sections 13 and 14, and in Sections 2 and 3 of this book we developed
the theory of atomic and quarkonial decompositions for the spaces and (R"). There one finds also the necessary references to the literature. By [MouOlbJ, 1.4, there is a full counterpart of this theory for the spaces introduced in Definition 22.10. In particular the simultaneous proof of Theorems 1.18 and 1.23 in [MouOlb], covering atomic and quarkonial decompositions. fits in the scheme of the above Sections 2 and 3. We do not describe the full
theory here. We wish to provide an understanding of the necessary modifications needed now, compared with the above Sections 2 and 3, where it is sufficient for our later purpose to restrict the considerations to those spaces where no moment conditions for atoms and [3-quarks are necessary. Otherwise we use the notation introduced in Section 2. Let i[' and with i3 e be the same functions as in Definition 2.4. Let
a E R and 0
cc. Let 4' be an admissible function according to 22.2. Then, in generalization of (2.16). (fiqu)pm(x) =
—
in),
x E IR",
(22.46)
is called an (s.p, 'P)-Ø-quark. Again ii E N0 and in E Z". We describe briefly the atomic counterpart of (flqu)vm. The cubes and CQvm with c > 0 have the same meaning as at the beginning of 2.2. Let c> 1. Then a K times differentiable complex-valued function a(x) in R" is called an (s.p, if for some v E N0.
suppa C cQ,,,,,
for some
in E Z", for
j x'
a(x) dx = 0 if
(aI
L.
(22.47)
K.
(22.48)
(22.49)
elliptic operators
IlL
342
Explanations may be found in [Th8], pp. 73—74, and in [MouOlb], Definition 1.14. Let and be the sequence spaces introduced in (2.7) and (2.8), respectively. Let 0
o-,=nI ——ii\ \P 1+ (1
u,,=n 11 —ii \mln(p,q)
and
.
.
(22.50)
a E R. We shaM use the abbreviation (2.22). Let r be the same number as in (2.14). Then we have the following generalization of Definition 2.6 and Theorem 2.9 (now in the reverse order). Here
22.13
Let (I)
Theorem
be an admissible function according to 22.2. Let
s>o,,,
O<poo, r. Then / E and let repre8ented as
f
if, and only if, it can be
belongs to
(22.52)
t'O
unconditional convergence in hA
(22.51)
where
are (s,p, W)-/3-quarks and
= sup
(22.53)
Purthermore, the infimum in (22.53) over all admissible representations (22.52) is an equivalent quasi-norm in (ii)
Let
8>o,q,
O
(22.54)
if, and only if, it can be and let r. Then f Sl(R?z) belongs to where represented by (22.52), unconditional convergence in are (8,p, W)-13 -quarks and hA
= sup
IA'9
(22.55)
Furthermore, the infimum in (22.55) over all admissible representations (22.52) is an equivalent quasi-norm in
22. Isotropic fractals and related function spaces 22.14 Remark
The above theorem coincides essentially with [MouO!b], Theorem 1.23. First
steps, especially concerning the correct normalizing factors in (22.46) and (22.48), may be found in [EdT99aI, 3.3, p. 97. The discussions in 2.5 and 2.7 can be taken over without substantial changes. This explains the role of r. Furthermore (22.52) converges unconditionally in with p = max(p, 1). The generalization of Definition 3.4 combined with Theorem 3.6 to spaces for all a E R (again in reverse order) may be found and in [MouOlbJ, Corollary 1.27. Tailored spaces: preliminiuies
22.15
are optimally adapted not only to R' and domains in R' but also to compact d-sets. Apparently, (22.5) and (22.6) are the decisive criteria. Hence we ask for counterparts for the spaces (Rn) with respect to (d, '11)-sets. The trace operator trr. and the identification operator idr have the same nieaning as before. We refer to 9.2 and 9.32, obviously extended to with p < 1 in case of trr, and also to [Thö], By our discussions in 22.1 the spaces
18.5, p. 138. Furthermore we extend (9.168) by 37(s,'V), £Jpq
=
{i E
=
:
0
E
= o}
and
(22.56)
30, and where F is a compact set in R". where a E R, 0
is a subset of In particular 0. Recall that, as usual, +
if
1 p
cc and a >
As before.
is trivial if
=
if 1
00. p and its quasi-norm must always be understood according to 1
(9.13). 22.16
Theorem
Let!
Let F be a compact (d, '11)-set according to Definition 22..4 (i). Then
=
"-a (22.57)
III. Fractal elliptic operators
344
Let r be a compact (n, 41)-set according to Definition 22.4 (ii) and let in addition (ii)
(22.58)
Then
= 22.17
(22.59)
Remark
This theorem coincides with Theorem 2.16 in [EdT99a]. It generalizes Theorem 18.2 in [Trio], p. 136, from d-sets to (d, 41)-sets. In [EdT99a] we outlined also a proof claiming that one can follow the arguments in [TriO] since the main ingredients have appropriate counterparts: local means and atoms. This is now
available in detail. We refer to the comments in 22.11 and in 22.12. On this basis a detailed proof has been given in [BriOO], Theorem 3.12. Some arguments (in particular the application of local means) may also be found in [MouOlbJ,
Proposition 2.12. Obviously, (22.57) extends (22.5). This, together with an appropriate counterpart of (22.6), is our criterion for whether spaces of type are tailored with respect to (d, 41)-sets. The restrictionsp> 1 and, in case of d = n the additional assumption (22.58), are needed to prove the sharp equalities in (22.57), (22.59). The inclusion (R")
idrLp(1') c
holds for all p with 1
oc
(22.60)
and all (d, 41)-sets r according to Definition
22.4. The special case
idrLi(1') c
(22.61)
is an immediate consequence of
c where
(22.62)
is the space of all complex-valued finite Radon measures in
normed in an obvious way. We refer to [TriO), 18.3, p. 138. A direct proof has
also been given in 9.2, formula (9.9).
22. Isotropic fractals and related function spaces 22.18 (1)
345
Theorem Let r be a compact (d, '11)-set according to Definition 22.4(i). Let
O
(22.63)
Then
trr (ii)
..—d
= L9(r).
(22.64)
Let r be a compact (n, '11)-set according to Definition 22.4 (ii). Let
l
(22.65)
Then
= 22.19
(22.66)
Remark
If 1
The proof outlined there is based on the duality of trr and id1 as described in 9.2, Theorem 22.16, and (22.67)
the latter in generalization of [Trio], 2.11.2, PP. 178—179. As for p 1 in part (i) we refer to [BriOOJ, Theorem 4.1. Here one needs that r satisfies the ball condition according to Proposition 22.6 (this makes also clear that at least the proof in case of p 1 cannot be extended to d = n). If = 1 then (i) coincides witii (22.6) and also with [Tho1, Corollary 18.12(i) on p. 142.
Next we use the above theorem as the starting point for the introduction of B-spaces on F. First we recall what has been done so far in this context. In case of arbitrary finite compactly supported Radon measures in W' we defined in 9.29 sonic spaces on r = suppj.i which, according to Theorem 9.33 are traces of suitable spaces Now we r as a compact (d, '11)-set eqwpped with the respective Radon measure. One can follow Definition 9.29 and introduce spaces via (9.132), (9.133), where now are of type (22.46). Afterwards one asks for counterparts of (9.143). \Ve I)r('f('r here the converse procedure taking the appropriate modification of
III. Fractal elliptic operators
346
(9.143) as starting point. This approach has the advantage that it works for all values of p with 0 < p oo (since, at least in this definition, there is no need to bother about moment conditions for respective atoms and 13-quarks). according to both This applies in particular to p = oo and to all (d, parts of Definition 22.4 because we have always (fl_d
O
'
(22.68)
We refer to [MouOlb], Proposition 2.14, where p = oo is covered by (19.21). As for quarkonial characterizations of spaces of type (F) we refer to [Bri991 and [BriOl]. There one finds also a detailed discussion of further properties of
these spaces, especially in connection with the approach by A. Jonsson and H. Wallin. Further references and comments have been given in 9.34(i) and 9.34(iii). 22.20
Definition
Let F be a compact (d, W)-set in R" according to (both parts of) Definition Let
0O and aER.
(22.69)
Then ,,—a
=
)(R?2)
(22.70)
equipped with the quasi-norm ,
where the infimum is taken over all g E 22.21
(22.71)
with trrg =
f.
Remark
The definition is justified by Theorem 22.18, complemented by (22.68) in com-
bination with Definition 22.10(i) and Proposition 22.3(iv). In particular. the spaces on the right-hand side of (22.70) are continuously embedded in the spaces on the left-hand sides of (22.64), (22.68). If '1' 1 and 0 < d < ii. thei we have compact d-sets and the above definition coincides with Definition 20.2, p. 159. There we preferred the letter B in place of B for reasons
22. Isotropic fract.als and related function spaces
347
explained in [ThoJ, 20.3, pp. 160-161. But it seems to be reasonable now to use the same notation as in Definition 9.29 and Theorem 9.33 and to denote now for all the resulting spaces by
Oczpoo,
s>O.
(22.72)
It was one of the main aims of [Triö], Chapter IV, to study compact embeddings on compact ti-sets in RTh and to apply these results between the spaces Chapter V. Now to the spectral theory of fractal elliptic operators in we describe the extension of this theory from d-sets to (d, 41)-sets (where we
shift the spectral theory to the next section). To avoid awkward formulations we agree on
0<poo,
a=O,
(22.73)
simply as a notation. As always we measure compactness in terms of entropy = max(c,O) if cE It. numbers ek(id) as introduced in 19.16. Recall 22.22
Theorem
Let r be a compact (d, 41)-set in 2L4. Let
according to (both parts of) Definition
O
(22.74)
IR (with a2 =
0
ifs2 = 0),
\Pi
(22.75)
Then the embedding
id :
(22.76)
is compact and for the related entropy numbers, €k(id) 22.23
W(kl)0201, kEN.
(22.77)
Remark
By our notational agreement (22.73) if = 0 (and hence a2 = 0) the target In this version and restricted to 1 < space in (22.76) coincides with
III. Fractal elliptic operators
348
that the respective proof is similar to that in [Trio], Section 20. In case of d-sets with 0 < d < n and, hence, 'I' = 1 the above theorem coincides with [TriO], Theorem 20.6 on p. 166. Then we have
kEN.
(22.78)
The proof in [TriO] is based on two ingredients: Quarkonial decompositions for the spaces which are considered here in the above Sections 2 and 3,
on the one hand, and entropy numbers for compact embeddings Id between sequence spaces of type (22.79)
as used here in the proof of Theorem 21.3 above, on the other hand. We refer in particular to (21.22)—(21.26). Ignoring technical details, the proofs of (22.78)
in [TriO] on the one hand and of Theorem 21.3 in this book on the other hand and also of the above theorem follow the same scheme. First one needs quarkonial representations. In the above case this is covered by Theorem 22.13 and fl-quarks given by (22.46). This reduces, roughly speaking, the embedding
(22.76) to an embedding Id of type (21.25), where one has to ask what is meant by Er and what is the appropriate substitute of It comes out that one ha.s to choose
rEN0, and
must be replaced in case of
by
O=81_s2+d(!__L'L \P'
(22.80)
P2/
Pi
P2
(22.81)
This requires an extension of the results obtained in [TriO], Sections 8 and 9, which we described in (21.44), (21.45). This has been done in [Leo98] and [LeoOl] covering also more general cases. A direct proof restricted to (22.80), (22.81) as substitutes in (22.79) has been givemi in [MouOlbI, Proposition 3.9. hi this paper one finds also a complete detailed proof of the above theorem.
23
Isotropic fractal drums
23.1
Preliminpries
Let
be a bounded C°° domain in R'1 where n E N. Let. F' be a compact set in We described in 19.1 and in 19.2 what (in our context) is meant,
R" with I' c
23. Isotropic fractal drums
349
by a fractal drum where the influence of the membrane is characterized by a Radon measure ji with r = An adequate mathematical description has been given 19.3 in terms of the operator B in (19.25), B = (—s)—' otr1'.
(23.1)
-' lia.s always the same meaning: the inverse of the Dirichiet Laplawith respect to fI. In case of d-sets we gave detailed explanations under circumstances B in (23.1) makes sense and what is meant by tr1'. We refer to (19.26) on the one hand and (19.23), (19.24) on the other hand. We now generalize d-sets by (d, W)-sets according to Definition 22.4. Then we have the counterparts (22.64), (22.57) of (19.20), (19.22), respectively. Hence, if d < n, Here (—
then
tr r
n—d (—.'I'P)
(Cl)
(.......,.,W
(Il),
1
(23.2)
is he generalization of (19.24). If p = oo and/or d = n, then one can use (22.68) and 1)0th parts of Theorem 22.16 in order to extend (23.2) to these cases. In Theorem 19.7 we collected what we know about the properties of B when I' is a d-set. Now we wish to extend a few assertions to (d, 111)_sets, concentrating on the generalization of the distribution of eigenvalues given in case of d-sets hr (19.73). This part was taken over from [Trio] with a reference to [TriO], Theorems 28.6 and 30.2, pp. 226, 234. However the decisive ingredients for the proofs. entropy numbers for the estimates of the eigenvalues from above and approxilliation nunibers for the estimates from below, have also been used in this book several times, especially in connection with Theorem 19.17. So we refer to these points when only technical adaptions are needed. This applies especially to the estimates of the eigenvalues from below. In the theorem below we clip together Theorem 2.28, Corollary 2.30 (rusty drum), Theorem 2.33
(sintererl drum) in [EdT99a] (with [EdT98] as a forerunner). We outline a new and niore transparent (so we hope) proof concerning the estimates of the eigenvalues from al)ove. Beside the extension from d-sets to (d, '71)-sets, we incorporate also au additional multiplication by a function b on 1' and replace this purpose trr in (23.1) and (19.23) by
=idrobotrr where bE Lr(I').
(23.3)
Of c(mrse. and frr have the previous meaning. Otherwise we use the nota1011 introduced in connection with Theorem 19.7 without further explanations. 23.2
Theorem
Let !l he a hounded C°° domain in R" (where n E N) and let r be a compact (d. '11)-set according to Definition 22.4 such that r C Cl and n — 2 < d n
III. Fractal elliptic operators
350
(with 0 < d 1 when n = re8pect to
1).
Let b be a positive function on r (a.e. with
such that <1 — n —2
b E Lr(r) for some r with r> 1 arul o <
(234)
and for some c> 0, b('y)
c
if
'y E
r0,
(23.5)
whereroisa(d,'I')-setwzthr0cr. ThenB, B=
(23.6)
o
with (23.3), is a non-negative compact seif-adjoint operator in
with
null-space
N(B) =
(23.7)
Furthermore, B is generated by the quadratic form
=
fE
gE
(23.8)
with (19.29) as the scalar product in (1k). Let Qk be the positive eigenvalue8 of B, repeated according to multiplicity and ordered by their magnitude, (23.9)
Then
kEN.
,
Proof in
(outline) and
Step 1
(23.10)
First we check that B, given by (23.6), is compact
that ek(B)
ck'
,
k E N,
(23.11)
where again ek(B) are the respective entropy numbers according to 19.16. For this purpose we need only (23.4) where b might be even complex. Similarly as before we abbreviate
=
if
p=
v
= 2.
(23.12)
23. Isotropic fractal drums
351
By Definition 22.20 we have
trr
(23.13)
:
By (23.4) we obtain 1
1
n—2
1
22>2d
(with > 0 if n = (22.73) it follows that id
1).
and
d+2—n
1\
11
<
(23.14)
2
Hence by Theorem 22.22 with the interpretation
(23.15)
:
is compact and
=
Ck(Zd)
We split b by b = b1
.
b2
.
(23.16)
with b1 E L2r(r), b2 E L2r(r). Let T5' be the
multiplication operator, Tbhf = b1 1;
Tb!
:
H(1
ii!
L2(r).
(23.17)
To justify (23.17) we decompose TbI by id in (23.15) and the multiplication by b1 according to Holder's inequality based on (23.14). In particular, T5' is compact and by (23.16), ek(Tb1)
,
k E N.
(23.18)
In 20.5 we introduced spaces H8(F) with s <0 in case of d-sets. This can he with .s < 0 extended, again by duality, to (d, and spaces and a E R. In particular. Tb2
(23.19)
can be obtained as the dual of (23.17) (with b2 in place of b1). In case of mappings between Hilbert spaces the entropy numbers of a compact operator and of its dual coincide. We refer to [ETO6], Theorem 1.3.1. p. 9. where one finds also a short proof. As a consequence we have also ek(Tb2)
ck4
kEN.
(23.20)
III. Fractal elliptic operators
352
Clipping together (23.17)—(23.20) we get
Tbf=bf;
T6 :
(23.21)
ek(Tb)
,
k eN.
(23.22)
The operator B given by (23.6), (23.3), can be factorized as (23.23)
By (9.20) the identification operator idr is the dual of trr. Then we get by (23.13) and the same arguments as in connection with (20.42),
idr :
(23.24)
Finally we apply again (20.31). This justifies the factorization of B in (23.23). Now (23.11) follows from (23.22) and the indicated mapping properties of the other factors in (23.23). Step 2 Now the situation is very much the same as in the relevant parts of Theorem 19.7, covered by Step 1 of its proof with the respective references
to [ThoJ, Sections 28 and 30. This applies also to (23.7) since we assumed > 0 in r a.e. and Proposition 19.5 (appropriately adapted) can be used. In particular, the operator B is non-negative, compact, seif-adjoint and generated by the quadratic form in (23.8). As before,
ck'
,
k E N,
(23.25)
follows now from (23.11) and (19.129). For the estimates of from below one may rely on approximation numbers. One can follow tile scheme used in Step 5 of the proof of Theorem 19.17. Furthermore, in [Trio], Step 4 of the proof of Theorem 28.6 on pp. 228-229, we have given a detailed proof in case of d-sets, resulting in the desired estimate from below in (19.73). This can be taken over, where one has to replace by
jEN0,
(23.26)
according to (22.80). This is the point where one needs (23.5). 23.3
Discussion
The above theorem covers essentially corresponding assertions iii [EdT99aJ with [EdT98J as a forerunner. The main point of the proof in [EdT99a) and
23. Isotropic fractal drums
353
also here is the inequality (23.11). The new argument here is the use of duality resulting in (23.20) and (23.22). In particular in this part of the proof one needs only (23.4), where b might be even complex. Then B remains to be compact but is not necessarily seif-adjoint. Then one gets okI
ck' (kW(k1)) n_d2
,
k
N,
(23.27)
where now the non-zero eigenvalues are repeated according to algebraic multi-
plicity and ordered as in (19.128). Otherwise the role of the additional assumptions for b are quite clear by the proof. If b 0, but not necessarily positive
on r. then N(13) might be larger than H'(c\F). The assumption (23.5) is needed only for the estimate of the eigenvalues L'k in (23.10) from below by tim right-hand side. In [EdT99a] we distinguished between d < n, b = 1 as the ittain case and d = n, b = 1 and b Lr(r) come in as additional cases. Vvhereas d < n. b = 1 is largely parallel to the relevant parts of Theorem 19.7 (covered by the quoted assertions form [TriJ]), the two other cases might express sonic new phenomena. If d = n then we have by (23.10), Qk
Recall that.
W(k_1)I_*,
k E N.
(23.28)
is the expected classical Weyl behaviour. Hence (23.28)
might be considered as a tiny distortion. Maybe the originally evenly dist.rilmted niass of the membrane r = is crumbling or the membrane becomes rusty but remains otherwise in shape. If this process is getting worse and the membrane is sintering unevenly, then one could try to describe this effect by an arldit.ioiial unevenly distributed function b. However in case of n = 2 of (lirect physical relevance we have always
k',
where
k E
N,
(23.29)
hence all the above measures are Weyl measures according to Definition 19.13. But one can construct functions b such that the measure bj.i does not
and
satisfy the lkypotheses of Theorem 19.17. We refer to 19.18(iii) for a discussion.
Chapter IV Truncations and Semi-linear Equations 24
Introduction
The aim of this chapter is twofold. We assume that a. p, q are given such that be the real part of is a subspace of Let Then we say that has the truncation property if
= max(f(x),O),
1(x) '—p
x E R",
(24.1)
We call a) a truncation couple if all spaces is a bounded map in be the collection (R") with 0 < q < oc have the truncation property. Let of all truncation couples. Then (24.2)
(the shaded region in Fig. 24.1): which might be considered as one of the main with results of this chapter. There are similar assertions for the spaces
(but a curious special case remains open). With T+ one Ol)taiflS a) E also the boundedness of operators with the same non-linearity behaviour, for example,
f(x)
max(f(x),g(x)) = (1— g)÷(x) + g(x), and, in particular, where, say, g E
xE
=2f÷(x)—f(x), xER".
T:
(24.3)
(24.4)
Problems of this type have attracted some attention. References will be given later. We only mention here the distinguished well-known property,
IT!
= If
IE
(24.5)
IV. Truncations and semi-linear equations
356
1
u(x)
=
j K(x — y)
+ h(x),
XE
(24.6)
x E R",
(24.7)
or
+ id) u(x) =
c
ju(x)I + h(x).
with in spaces having the truncation property. Here, say, h E a) E R7, is given and one asks for the maximal smoothness of solutions u(x). This
means u E
for the semi-linear integral equation (24.6). In case of with (24.7) one might think of bootstrapping arguments starting from 1
boundary). Furthermore, as will be indicated at the end of this chapter in 27.10, bootstrapping arguments are of very limited use in connection with the problems treated here. It is the second aim of this chapter to circumvent this difficulty with the help of the Q-method. Let f E with a) E Then we have by (2.31) and Corollary 2.12 the quarkonial decomposition
I=
(I3qu)vm(x).
X E
R",
(24.8)
with the optimal coefficients which depend linearly on f. One = may even assume that all functions ([3qu)pm(x) 0. Then the operator Q is defined by
f(x)
(Qf)(x) =
X E
R'2.
(24.9)
It turns out that the operator Q is not only bounded in those spaces ]3q(R") where and (and hence T) are bounded but it is continuous (in sharp contrast to T+ and T). Hence if one replaces, for example, (y) in (24.6) by (Qu)(y), then one can try to apply Banach's contraction method. One obtains a solution u°(x) of this modified equation (24.6). If, in addition,
IV. Truncations and semi-linear equations
if the related context as the original spaces. Sometimes we write (under the natural restrictions for the and applies both to parameters s, p, q, which here always means p < oo for the F-spaces). Then The Sobolev spaces H(R") and the it is clear what is meant by have the same meaning as in (1.9) and in classical Sobolev spaces (1.10), respectively. Let 1
H(R") with s 0
with s
and
(25.3)
N0
are the corresponding real parts. Again we recall that a+ = max(a, 0) if a E lit We are interested in the mapping properties of the truncation operators T and given
by
(Tf)(x) = 1/1(x) = lf(x)l, =
=
x
(25.4)
x E R".
(25.5)
is restricted with (25.1)1 whereas we have (24.4), which can In
Here T makes sense for all spaces to the corresponding real parts
also be written as
=T+
(25.6)
id.
have in real spaces the same mapping propThis makes it clear that T and erties (boundedness, continuity). But first we have a look at T in the (complex) For given n N we again use our standard abbreviations, spaces
= n (!
\P
—
i'I
and
=
1+
(
1
—
j+
,
(25.7)
Asbefore,
= f(x + h)
—
f(x),
x
R", h
(25.8)
cr,<s
(25.9)
are the first differences. Example 1 Let n N,
O<poo, By ITriflI, Theorem 2.5.12, p. 110,
I
Ill
= Ill +
(j (25.10)
25. Truncations
359
(modification if q = oc) is an equivalent quasi-norm in
= If(x + h)I — If(x)I
If(x + h) — f(x)I =
We have that (25.11)
Inserting this in (25.10) one obtains that
I
IJTf
(25.12)
There remains a technical point, that of ensuring that Tf E There are two possibilities. First, if for some f E fl the right-hand side of (25.10) is finite, then I E This assertion is covered by ITrii3l, Remark 2.5.12/3, p. 113. This means, applied to our situation, Tf E Secondly, one uses the so-called Fatou property which will be described at the end of this subsection. Hence T is a bounded map in with (25.9). Example 2 Let
O
Tpq<S
(25.13)
By [Trifl], Corollary 2.5.11, p. 108 (extended to q = oc),
If
/
1
If
f
dh)
(25.14)
IhII
0
(modification if q = oc) is an equivalent quasi-norm in
By (25.11) it
follows that
c
IE
(25.15)
Using again the Fatou property. explained below, we have Tf Hence T is a bounded map in with (25.13). Example 3 Let 1 < v < oc. Recall that f belongs to the classical Sobolev space if, and only if, the norm in (17.23) is finite (and then it is an equivalent norm). This is covered by the references given there, in particular (Ste7OJ, Proposition 3, p. 139. Then it follows by (25.11) that IITf
and Tf E
c Hf
,
f
(25.16)
Hence T is a bounded map in Example 4 (Real spaces) Let be the real part of one of the (complex) spaces covered by the Examples 1-3. Then T, and by (25.6) also
IV. Truncations and semi-linear equations
In case of are bounded operators in equality (24.5). be one of the spaces Fatou property Again let without any restrictions for the parameters, which means
with 1
one has now even the
O<poo,
8EJR,
(with p < co in the F-case). Let with in
g3—+g
or
be
(25.17)
a bounded sequence in
if j—+oo.
S'(W')
(25.18)
and there is a positive constant c (depending only on the given and the chosen quasi-norm) such that
Then g E space
(25.19)
C SUP I19j 3
This follows from the Fourier-analytical characterization of the spaces
given by (2.37), (2.38), Theorem 3.6, and Fourier multiplier assertions, resulting in the constant c in (25.19). This notation was introduced in [Fra86] in a wider context. We refer also to [RuS96I, p. 15. The examples revisited We apply the Fatou property to the above examples. with (25.13). Let Let I E
xE
=
j E N0,
(25.20)
is given by (2.33). Let
where
jEN0.
and
(25.21)
C L,.(W') for some 1
f
f,
and 9, —.
g
= Tf in Lr(R") when j : 00.
(25.22)
are the Holderfor any a, 0 < a < 1, where g, E Zygmund spaces from (1.11), (1.13). Since every 9j has compact support it follows that g, By (25.15), pointwise in particular g, E and Fourier multipliers, one has that 1g3
S c2 Il/
(25.23)
25. Truncations
361
By (25.18). (25.19) one obtains g = Tf E and (25.15). This seals the left opeii so far in Example 2 and in a similar way in Example 1. The main aim of the above examples is the following. Ignoring technical cornplirat ions (apphicat ion of the Fatou property), the boundedness of T and in the indicated spaces with s < 1 is a direct consequence of known equivalent (Iliasi-norms. The situation is different ifs 1. This will be discussed in detail in what follows, where we restrict ourselves to real spaces. 25.2 Let
Definition
be the or for
part of
iii
stands either for is a bounded map
).
is said to be uniformly continuous in < I there is a ó = 5(e) > 0 such that e
11+ — g+
for al/fE 25.3
where
with (25.1). is said to have the truncation property if
(i) (ii) (I <
neal
when
Ill — g
if for any e with 5
(25.24)
andg E
Remark
Recall that
is given by (25.5). By (25.6) one can replace in the above
(lefinition 7'
by T. given by (25.4). This applies also to the following simple
ol)servat iou. 25.4 (1)
Proposition has the truncation property if, and only if, there is a constant
(
e > U sue/i that
broil
111÷
(ii)
is amformly continuous if, and only if, it is Lipschitz continuous:
there is a e(mstant
C> 111+
for a/If E
(25.25)
0 such that 9+
c
—g
(25.26)
andg E Proof This follows immediately from Definition 25.2 and the special strucore of T (positive 1-homogeneous).
IV. Truncations and semi-linear equations
362
25.5
Proposition
(Necessary condition, n = 1) If has the truncation property, then s < 1 + Proof Step 1 If (R) has the truncation property, then
=
with
(R),
(25.27)
= max(x, 0). Taking the first derivative it
Recall that belongs to follows easily that XE
S(R), x E R,
x characteristic function of [0, 1].
It is well known under what conditions x belongs to embeddings and [RuS96], p. 53, it follows that s — 1 <
(25.28)
By elementary in all cases in which
has the truncation property with the possible exceptions of (R), where 0 < q < oo. In the latter case one where 0 < p oo, and can argue as follows. Taking the second derivative of (25.27), one obtains that for the delta-distribution. Since q < oo, one has a contradiction. öE
(R) with 0
Oifx >0, and w'(O) >
0.
ifyE R,
(25.29)
According to [Trio], Theorem 13.7 on p. 75, xE
1(x) =
(25.30)
j=O is
(R). In particular,
an atomic decomposition in
fE
1+1
(IR)
for all
0
00.
(25.31)
By (25.30) we have for k E N that
f(2_k)
and, hence,
f(x)
x<
(25.32)
On the other hand, by Theorem 5.10 and (5.18) it follows that x)O
X
clIgIC'(R)II,
gEC'(R),
8UppgC [0,11.
(25.33)
In particular, f÷ does not belong to C'(R) and, hence, by embedding, also not
to (IR) with 0
oo.
This proves that also
(R) does not have
25. Truncations
363
Let n E N. Let 0< p oo ands ER. Then
is called a truncation has the truncation property.
25.6
Definition
if for any q. 0 < q oo, the space The colleciton of all truncation couples is denoted by 25.7
Proposition
Let n E N. Then (25.34) (the shaded region in Fig. 24.1).
Proof Step 1 Let , s) E By Definition 25.2, (25.1) and Theorem 11.2(u) it follows that s > where is given by (25.7). If n = 1, then < I + is a consequence of Proposition 25.5. Step 2 It remains to prove a < 1 + if and n 2. Since E we have in addition a > a,, (we again refer to Fig. 24.1), the spaces with s) E have the Fubini property according to Theorem 4.4. Let = (x1 x') with x' E and let .
0,
E
he a fixed function. Let g E
> 0,
= 0 if
1,
(25.35)
Then
1(x) =
(25.36)
This follows (formally) from Theorem 4.4. But it can also be proved directly. One starts with an (optimal) one-dimensional atomic decomposition for g(xi). Au atom at level j is multiplied with which in turn is decomposed by a natural resolution of unity related to the lattice resulting for fixed j, after normalization, in atoms in R". Checking the coefficients one obtains (25.36) and, by Theorem 4.4, fig
If
Ill
(25.37)
has the truncation property, then Ig(xi)I
= If(x)I E B,,(R"),
(25.38)
IV. Truncations and semi-linear equations
364
and again by Theorem 4.4 we obtain that Tg = gj
with
E
(25.39)
lIT9
Hence,
25.5 that a <
1
has the truncation property. Then it follows by Proposition +
Theorem
25.8
Let n E N and let and be given by (25.7,). be the collection of all truncation couples according to Definition (i) Let 25.6. Then
0
(25.40)
('Fig. 24.1). (ii)
Let either
l
(25.41)
or
O
(25.42)
has the truncation property according to Definition 25.
Then 24.1).
(Fig.
Proof We break the rather long proof into seven steps and shift a technical adaption to 25.9. We give a guide. Steps 1-4 deal with the one-dimensional = with case n = 1, p = q, and a We prove that the spaces
n=1,
\P
1+
p
p
(25.43)
have the truncation property (Fig. 24.1. n = 1). Afterwards we prove part (i) in Step 5. In Step 6 we return to the one-dimensional case and prove that the spaces
(]R) with
n=1, O
(1
1
1
(25.44) (25.45)
have the truncation property. Finally Step 7 contains the proof of part (ii).
25. Truncations
Step 1
365
Under the assumption (25.43) we prove in Steps 1—4 that
IE
B(R) and that there is a constant c> 0 with
cIIfJB(R)II
11111
(25.46)
= By Example 1 in 25.1 we may assume p < oo. We for all I E S(IR) be a real even function with begin with some preparation. Let =
suppço C (—2, 2),
when
1
1,
(25.47)
where J E N. Then
and let WJ(x) =
fj(x) = (cpjf)"(x) E
if
f
(25.48)
and, in particular, fj(x) is a real analytic function. As in (25.22) we have B(R) C Lr(R) for some r, 1
fj—if and,hence, IfjI—'IfI
when
J—÷oo,
(25.49)
in Lr(R). Let us assume that we have (25.46) for all real analytic functions of the above type. Then by the Fourier multiplier properties of B(R),
1B(R)II,
1B(R)II
II
(25.50)
where Cl and c2 are independent of J E N. Now we can apply the Fatou property described in 25.1. As in (25.23) we obtain fl E B(R) and (25.46). Hence it is sufficient to prove (25.46) for real analytic functions
f
with
C {y
]R
:
<2'}
(25.51)
for some J N. Step 2 In Step 3, Step 4 and in the technical modification described in 25.9 below, we construct optimal real atomic decompositions for functions f given by (25.51) which respect the zeroes of f. For this purpose we need to prepare by constructing a hierarchy of resolutions of unity on the interval (0,1). Let with tPo,k(x) with k N0 be non-negative C°° functions on
c
= Qk,
1
if XE (0,1), (25.52)
and
tEN0,
kEN0, XER,
(25.53)
IV. Truncations and semi-linear equations
366
for some q > 0 which are independent of x and k. Let x R with C (—1,1),
a C°° function on
be
x(x) 0,
x
where
E
E
E
R,
(25.55)
where only, say, N + 1 functions are different from zero. Put
=
1,bO.k(x)
if k 1
with k =
and
0, —1,. ..
,
—N, (25.56)
for the non-vanishing functions on the right-hand side of (25.55). Now we apply this procedure to with k = 1,0, —1,.. . , —N with x2,L(x) in place of xi,z(x), whereas the functions with k 2 remain unchanged. = Iteration yields a resolution of unity by non-negative functions where
we may assume (after a slight shifting of indices) k E Z with k =1 ?I'j,k(X)
if XE (0,1),
(25.57)
k j,
(25.58)
if
=
and if
k<j. (25.59)
Step S
We prove (25.46) with (25.43) for
IE
{y ER :
2J+1}
(25.60)
where J E N. With J = 0 the arguments cover also the case
suppfc {y E R
< 2} .
(In this step we do not need the restriction s
= {y E R : for some suitable c> 0, and
if
—
With
2'II IEZ,
XE
(25.61)
(25.62)
(25.63)
25. Truncations
367
we have that
If
E
I
fS7)
(25.64)
(Here and in what follows we assume p < 00. If p = oo then one has to modify the calculations in the usual way.) The first equivalence in (25.64) comes from (2.37) and (25.60). The second equivalence is covered by the scalar case of [Trifi], Theorem 1.6.2, P. 30 (or (Tri/3], Theorem 1.4.1, P. 22, combined with a dilation argument which includes also p = oo). The equivalence constants in (25.64) are independent of J E N0. Let 1(0) = 0 and, say, f(x) > 0 in the interval (0,2). We wish to construct an optimal atomic decomposition of f(x) in the interval (0,1). We use without further comments atomic decompositions as described in [Trio], Section 13, especially Theorem 13.8 on of and with k be the resolution of unity in (0, 1) according to p. 75. Let (25.57). Let, in analogy to (25.63), fJ,k =
sup
k
k
(25.65)
x E (0, 1),
(25.66)
e Z,
'J'J.k
and
1(x)
bJ,k(x),
= with bJ,k(X) = 2_(8*) max(k,J)
(25.67)
and QJ,k = 2(8_p max(k,J)
(25.68)
We assume that the bJ,k(x) are (essentially normalized) atoms in B(R) according to [TriO], Theorem 13.8, p. 75 (no moment conditions are necessary). The functions bJ,k(X) themselves are correctly normalized. This is not so clear as far as the needed derivatives of bJ,k(X) up to order 1+ [a] are concerned. We shift this technical question to 25.9 below where we modify (25.66) slightly. But this does not influence our reasoning and so we take (25.65)—(25.68) as a model atomic decomposition. We estimate the coefficients 2J,k and assume by (25.58) with (25.52), (25.53). Since first k J + 1. Then = 1(0) = 0 we have for x E
If(x)I=
sup 0
If'(y)I.
(25.69)
IV. Truncations and semi-Linear equations
368
Hence,
sup
c
If'(y)I
O
k=J+1
where we used s < 1 +
k=J+1 1f'(y)1P, sup
By (25.64) with f' and s —
1
(25.70)
in place off and s,
respectively, we have that
>
sup
B(R)II".
C'
S c
(25.71)
In particular, the right-hand side of (25.70) is one summand of the left-hand side of (25.71). If k S J, then coincides essentially with related terms on the right-hand side of (25.64). This construction can be made on both sides of any zero of 1. Summing up all these (appropriately modified) local representations (25.66) and using (25.64), (25.71) we obtain an optimal atomic decomposition
f(x) =
(25.72) v=O mEZ
according to [Trio], Theorem 13.8, p. 75, where the atoms and the real coincide locally with bJ,k(x) and PJ,k1 respectively, including the numbers indicated modifications and summations. In particular by (25.64) and (25.71), it follows that
C
(25.73)
vOmcZ
where c> 0 is independent of J and f. Since f is real and since the real atoms respect the zeroes of f we have that If(x)I =
(sgn 1(x)) Aiim
X
JR.
(25.74)
v0 mEZ
Again by (TriO], p. 75, and 8>
—
i)
it follows that (25.74) is an admis8ible
atomic decomposition of 1(x) I. In particular, 11111
< C' Ill
,
(25.75)
25. Thincations
369
where we used (25.73). Here c and c' are independent of I with (25.60) and J E N0. Step 4 We prove (25.46) with (25.43) for functions I given by (25.51). Again let p < oo; when p = 00 one has to modify the calculations in the usual way. = We may assume that
I
with
i,
=
()V,
x E R,
(25.76)
according to (2.37). We ask for the counterpart of (25.66) again under the assumption f(0) = 0 and 1(x) > 0 in the interval (0,2). Now we use the special structure of the hierarchy of resolutions of unity {'J.'j.k } as constructed in Step 2. Since
= f(0) = 0 we have for x
(25.77)
(0,1) that
f(s) = 2=0 k=0
=
(25.78)
j=0 k=—2J J
J
i
= j=Ok=—2
and Oj.k have the same meaning as in (25.67), (25.68), respectively. if k = in place of f). If k j + 1, then b,,k(X) and . ,j (with are modified by where
=
fTh
= 2(8_n
with
fj**k =
sup yEsupp
11'j,k(X),
—
(25.79)
x E (0,1).
(25.80)
1f3(y)
in place of f(s), which We have a counterpart of (25.69) with f,(x) — results in a counterpart of (25.70). The terms Oj,k b2.k(x) with k j also fit
IV. Truncations and semi-linear equations
we have for the remaining
in the above scheme. By the construction of terms on the right-hand side of (25.78) that J
j
j0 krr—2J
J
j
j=O
k=O
= =
(25.81)
=
f3 (0)
= This is an atomic decomposition in B(R) with the coefficients dk. First let <s < 1 + Then (25.82)
c
which again fits in the above scheme. Secondly, let
—
+
<.9 <
In the
middle term in (25.81) one can replace (25.83)
by
This follows from (25.77). With a similar replacement in dk one again obtains (25.82). [This does not work if a = excluded for this reason in (25.43).1 We have by (25.78) in the interval (0, 1), that
1(x) =
+ 12(x)
= f'(x) +f2(x),
(25.84)
j=O k=j+1
where f2(x) collects all the terms in (25.78) with k <j. The atoms in f2(x) with k j resulting from the first sum on the right-hand side of (25.78) belong
25.
371
22-levels and, hence, do not interfere with each other in atomic representations. As for the second sum we have (25.81). Hence by the above estimates for the coefficients 9j,k and dk in (25.82) one obtains in a similar way as in Step 3, resulting there in (25.72), (25.73), that to
XER,
(25.85)
v=OmEZ
with
cIIfpB;(R)II",
(25.86)
wrrOmEZ
respect the zeroes of 1. As for f1 the situation is
where the atoms
different. To the interval Qk in (25.52) the atoms bj,k(x) with, say, j = 0,. . contribute on the atomic level Hence we get a resulting atom
1k bk(X)
=
(
\j=O
. ,
k,
k
Qj,k
9j,k )
/
(25.87)
bj,k(x)
j=O
with the coefficients
=
where
(25.88)
Qj,k,
9k
is given by (25.79). We modify the estimates in (25.69), (25.70) and
obtain that 2(k—J)(s—*—1)
sup
Pj,k
(25.89)
and, since .s < + 1 00
sup
SC k=O
p
00
j=O
O
.
(25.90)
that the atoms bk in (25.87) belong to different atomic levels and here is no overlapping with the exception of direct neighbours (the last observation is not nee(led at this moment but of some use in connection with the F;q*spa•eS in Step 6 and the modifications described in 25.9 and Recall
IV. Truncations and semi-linear equations
372
25.10 below). Now the counterpart of (25.85), (25.86) is given by the atomic decomposition
XER,
(25.91)
v=Om€Z
with sup
c
vOmEZ
j=O IEZ
(25.92)
c"
where we again used the above-indicated modification of (25.64). The atoms also respect the zeroes of f. Hence, (25.84), (25.85), (25.91) is the desired atomic decomposition. Now we are in the same position as at the end of Step 3 and arrive finally at (25.46) with (25.43) for functions f given by (25.51). By Step I this proves (25.46) with (25.43) for all f Step 5 We prove part (i) of the theorem. Let
n2,
(25.93)
According to Theorem 4.4 the spaces property and, in particular,
=
have the Fubini
(25.94)
1B(R)II
Ill
are equivalent quasi-norms (in the notation used there). We add a technical is so smooth that Ill E remark. First one may assume that f E
where now a is additionally restricted by a < 1 +1 and a We add a remark about this technical point in 25.9 below. 'fhen one can extend the considerations afterwards to arbitrary f in the same way as at the end of 25.1 using the Fatou property. Hence the question of whether we have c
11111
If
f
(25.95)
under the restriction (25.93) can be reduced to the one-dimensional case (25.46). This proves (25.95) with
\P
1+
p
p
(25.96)
25. Truncations
373
It remains to extend (25.95) from (Rn) to with 0 < q oo and to remove the restriction s We use non-linear real interpolation. First we describe the underlying abstract assertions. Let A0 and A1 be two (real or complex) quasi-Banach spaces with A1 C A0 (continuous embedding). Let T be a (non-linear) operator with a E A1,
IITaIA1II
(25.97)
(boundedness) and IlTa' — Ta2
IA0II
cila' — a2 IA0II,
(Lipschitz continuity) for some c>
0.
LITa (Ao, Ai)9,qII
a1 E A0,
a2 E A0,
(25.98)
Then, whenever 0 < 9 < 1, 0
(25.99)
(boundedness). Here, (Ao,Aj)o,q are the usual real interpolation spaces. The general background may be found in Chapter 1, or in [BeL76]. The above specific (nonlinear) assertion goes back to [Pee7Oj and [Tar72a], [Tar72b] (we refer also to the more recent paper IMa184D. The above formulation coincides essentially with [RuS96]. Proposition 1. p. 88.
First let
0<9<1,
s>0.
(25.100)
Then
=
(25.101)
with an obvious counterpart if the spaces involved are restricted to their real parts. Of course, Tf = is Lipschitz continuous in Furthermore by (25.95). the operator T is bounded in (Rn) with (25.96). Application of (25.97)—(25.99) with A0 (tile real part of and A, = results in
I
11111
(25.102)
with
n1,
(25.103)
It remains to remove the restriction p 1. First we recall that
ill +
j (
IhI
(25.104)
IV. Truncations and semi-linear equations
374
(with the usual modification if q = oo) is an equivalent quasi-norm in the space
with O o,. This follows from [DeS93], Theorem 6.3, which, in turn, is a special case within a far-reaching non-linear approximation theory for Lu-spaces with 0
0
nEN,
(25.105)
This covers the right-hand side of (25.40), and completes, together with Proposition 25.7, the proof of part (i) of the theorem. Step 6 We prove part (ii) of the theorem in two steps. First we modify Step 4 and prove (25.106)
11111
with (25.44), (25.45) for real analytic functions
Ie
sizpplc {y
JR
:
2J},
(25. 107)
for some J N. This is the counterpart of (25.46), (25.43), (25.51). We follow the arguments in Step 4, resulting in (25.76)—(25.82) with the splitting (25.84) of I in f' and f2 and the atomic representation (25.85). But now we must replace 4 in (25.86) by the sequence spaces introduced in (2.8), onedimensional case n = 1. We need the counterpart of (25.62)—(25.64), now with I given by (25.107) instead of (25.60). Let 13,1 =
{y E JR
Iv
—
c23}
where
j
No
and I E Z, (25.108)
(again c> 0 is independent of
1,
and, of course, f). Let / be decomposed
by (25.76) and
f;1= sup
xEI,,
jENo, I€Z.
(25.109)
25. Truncations
375
As an immediate consequence of [Tri'y], 2.3.2, p. 93, we have (in obvious no-
tation) that
If Here
the equivalence constants are independent of f
(25.110) and
J. Now we refine
(25.84) by
1(x) =
f'(x) + f2(x) +
(25.111)
has the above meaning, I 2(x) collects all terms with k j resulting by the above procedure from the first sum on the right-hand side of (25.78) and f3(x) collects all the second sums on the right-hand side of (25.78) resulting by the above procedure. Again we recall that the atoms (x) of ft (x) in (25.91) have the property mentioned after (25.90) (controlled overlapping of in (25.91) (most of the respective supports). Let be the collection of all originate from (25.88), (25.89), them are zero). The (non-zero) coefficients which can be re-written as
where f'(x)
Qk—i,k
(with Qm,k = 0 if m <0),
(25.112)
= and sup
Qk—Lk
(25.113)
Ifk—L(Y)I
This convolution structure, the influence of index-shifting in the spaces fpq * with as discussed in (2.103), and the convergence-generating factors s — — 1 <0 now prove the counterpart of (25.92), IA'
IfpqII
fT}
If'
C' (25. 114)
in place of a and f, respectively. where we used (25.110) with a — I and Hence, (25.91) is again an optimal atomic decomposition which respects the zeroes of f. The same argument applies to f3(x) which originates from (25.81) since the coefficients dk have a similar convolution structure as in (25.113). Again one has the splitting in
\ <s<-. -<s<1+and (1 (——ii p p 1
1
1
IV. Truncations and semi-linear equations
376
There are no problems as far as /2
is
concerned. This is the usual atomic
decomposition. In analogy to (25.85), (25.86) one gets the respective counterparts where one now has to use (25.110). The rest is the same as at the end of Step 4 and Step 3. In particular, since 8> — i) + no moment conditions in the atomic representations for are required, [Th5J, Theorem 13.8, p. 75. Then the counterpart of (25.74) is an atomic decomposition for This proves (25.106) with (25.44), (25.45).
Step 7 We prove part (ii) of the theorem. First we remark that (25.106) holds for all / E with (25.44), (25.45). This is again a matter of the Fatou property as indicated in Step 1 with a reference to 25.1, now applied to Let
n1, 0
(25.115)
By Theorem 4.4 the spaces with n 2 have the Fubini property. Then we can argue as in Step 5. This proves part (ii) with exception of the case 8 = with 1
Points left open
We left open two points so far. (i) We used as a model case that (25.66) with (25.67), (25.68) is an atomic decomposition for f given by (25.60). As mentioned after (25.68) one needs not only a control of the size of bJ,k (x) itself but also of its derivatives with
m=0,...,1+Isl,
(25.116)
according to (Tri5l, Theorem 13.8, p. 75 (no moment conditions are required). The shortest way to circumvent this problem is to modify (25.65)-(25.68) as follows. For brevity let M = 1 + [s]. Let be given by (25.65) with respect to the derivatives f(L)(y), where I = 0,1,... , M. Then we have (25.66) with the modifications of bJ,k(x) and QJ,k in (25.67), (25.68), respectively, given by -1
M
max(k,J) (>
bJ,k (x) =
QJ,k =
2—" max(k,J)
f(m)s)
f(m)*
f(x) 7I)J,k(X) (25.117)
(25.118)
25. Thincations
377
Now it is clear that all derivatives with m = 0,. . . , M, are correctly normalized and that, hence, are atoms. The additional terms in (25.118). compared with (25.68), fit in the above scheme and do not influence the chain of arguments significantly. The right-hand side of (25.70) is now given by (25.119) m=1
Then the middle term and the last term in (25.71) must be modified by
E 111(m) IB;_moR)pIP S c'
IB;(R)Ir.
(25.120)
The same adaption can be made in Step 4. In particular we have in (25.89) and. using p
for
m=1,...,M,
also in (25,90), derivatives up to order M in the same way as in (25.119). This results on the right-hand side of (25.92) in the same terms as in (25.120).
Corresponding adaptions can be made in Step 6. We add a comment. In a posit ivity interval of f(z) there might be small local minima. But this does not influence the above arguments since constant factors in the atoms in (25.67),
or (25.117) simply cancel out. In other words, only the behaviour near the zeroes must be considered. (ii) At the beginning of Step 5 of the above proof it was convenient for us to assume if / E is smooth III E under the restrictions of 8 and p indicated there. For justification we recall that the spaces in question can be characterized in terms of differences, [Tn/il. Remark 3 on p. 113. Together with [Th/3j, Theorem 2.5.13, p. 115, and formula (11) on p. 116, combined with the one-dimensional case as treated in Step 4 of the above proof one gets the desired assertion. 25.10
Comments and references
Mapping properties of nonlinear operators in function spaces have attracted a lot of attention, including applications to nonlinear partial differential equations. A far-reaching treatment, especially in connection with (complex and real) spaces of type and in and in domains has been given in [RuS96}. It is quite clear that among the large bulk of nonlinear operators
IV.
378
and semi-linear equations
worth treating in function spaces, the truncation operators T and T+, givea by (25.4) and (25.5), respectively, are distinguished examples. First discussions of mapping properties of T and of more general operators of power type
I
and
in
spaces
were given in [Tri84]. Restricted to the case treated here, which means = 1, Proposition 1 in [Tri84] coincides essentially with the Examples 1 and 2 and equivalent quasiin 25.1. If one has in the respective spaces then the boundedness of T is a more or norms based on first differences less immediate consequence. Then one has necessarily s < 1. In other words, or becomes more complicated if s 1 whether T is bounded in (ignoring the classical Sobolev spaces considered in Example 3 in 25.1 and in (24.5)). We refer to [BoM91], [Bod93], [0sw92], and [RuS96], 5.4.1, pp. 350—359. In the last book one finds further references, especially in the Note
Section on pp. 391—392, and a description of the state of the art. Roughly speaking, at that time, and covered by the above-mentioned references, the
boundedness of T was known in the F-case if 1
Of special interest are the Sobolev spaces
sER, 0
(25.121)
We extended here the notation introduced in (1.9) to 0
bmo(R") =
(25.122)
as considered in 13.7, fits also in the above scheme. 25.11 (i)
Corollary
Let either
l
(25.123)
or (25.124)
25. Truncations
379
has the truncation property according to Definition 25.2. Fur-
Then
thermore, the real part of bmo (Rn) has the truncation property.
(ii)
Let
O
s1+!
and
(25.125)
does not have the truncation property. Proof Of course with 1
Ill Ibmo
sup IQI1
inf
cEC IQI
111(x)
—
dx +
sup —k-iqi>i IQI J
If(x)I dx. (25.126)
This is well known. It follows also immediately from (13.82). The first term on
the right-hand side of (13.82) remains unchanged if one replaces f(s) in each cube by f(s)—eQ with CQ E C, and correspondingly by (f—cQ)Q = fQ—eQ. Then the left-hand side of (25.126), given by (13.82), can be estimated from above by its right-hand side multiplied with 2. Since the opposite estimate is obvious we obtain (25.126). Now it follows easily that has the truncation property (real and complex, the latter restricted to T). The other assertions of part (i) are covered by Theorem 25.8(u). If n = 1 then part (ii) have the follows from Proposition 25.5. If n 2, then the spaces Fubini property from Theorem 4.4. If one assumes that such a space H(Rvz) has the truncation property, then the corresponding space also has the truncation property. This is a contradiction. 25.12
Remark
This corollary clarifies the question of whether or not cation property with exception of the disturbing restriction s
and of the limiting case s = and for the F-scale). 25.13
with .s
n
—
i)
has the trunin (25.124)
if p 1. Similar limiting cases for
and 0
oc. remain also open (p < oc
Continuity
(Rn) with 1 < p < oc any bounded composition operator in is also continuous. This applies in particular to T+ and T. This assertion was By [MaM
IV. Truncations and semi-linear equations
380
extended in [RuS96J, Theorem 3 on p. 377, to
0
with
(25.127)
However even for the comparatively simple nonlinear operator T+ the situation is different if one asks for uniform continuity according to Definition 25.2(u). By Proposition 25.4(u) this question is equivalent to the problem of whether is Lipschitz continuous. By (25.6) in real spaces the nonlinearity behaviour is Lipschitz is the same. Obviously, T, and hence in real spaces ofT and continuous in with 1 p 00. But otherwise the situation is less favourable. 25.14
Theorem be a truncation couple according to (25.40) and or be either oo. Let
s) E Let n E N and let Definition 25.6. Let 0 < q
(and hence also T) is not a Lipschitz (with p < oo in the F-case). Then continuous map in Proof We break the proof into three steps and justify an interpolation formula used in between, afterwards in 25.15. By (25.6) it is sufficient to prove that T is not Lipschitz continuous. Step 1 The first (and crucial) step deals with n = 1 and E(R) =
s.
where
O
be
f(x) =
given by (25.54) and let j
E
(25.128)
N. We put
2'' —
2m)
and g(x) =
—
—
2m
—
1).
(25.129)
Then 1(x), g(x), and If(x)I
—
= 1(x) +g(x),
XE IR,
(25.130)
are smoothed zigzag functions with supports in [_2—i, 1 + 2-3k'], whereas 1(x) — g(x)
=
1
if x E [0, 1].
(25.131)
The aim of this step is to prove that 111—
91B(IR)II
max
(25.132)
25. Truncations
1B(R)II
11111 —
2's,
(25.133)
where the equivalence constants are independent of j E N. First we prove be the same functions as in (25.52), (25.53) and let (25.132). Let Then the ip3(x) = and f —g are of the same type at least as far as (25.132) is concerned. We have that Ill
—
B(R)II"
lIv'
(25.134)
The first equivalence has been explained above. The second and the third equivalences follow from Theorem 5.14 and Corollary 5.16, respectively. Here one needs s> = F,;,. In the last equivalence we used s — i) and
This proves (25.132). Next we prove (25.133). By (25.129), (25.130) and the localization property from [ET961, Theorem 2.3.2, pp. 35-36, we obtain that .1
11111-
1)
1B(R)II
=
(Because of the required support properties in [ET96]
(25.135) we
do not obtain the
desired equivalence immediately.) But (25.135) is also a consequence of atomic
decompositions. Hence it remains to prove that there is a constant c> 0 such that 11111
for
—191 B;(]R)II
First let, in addition, we have that
—
i) <s <
1
j EN.
(25.136)
and let H(x) = If(x)I — Ig(x)I. Then
IIH
IIH
r2jsP
+ sup
0
J 0
sup IIH( O
+ r) — H(.)
+ r) — H(.)
t
(25.137)
382
and semi-linear equations
IV.
where c> 0 and c' > 0 are independent of j E N. The equivalent quasi-norm is covered by [TIi/3], p. 110. The first estimate is obvious, whereas the last estimate is a consequence of the oscillating nature of H. This proves (25.136) under the indicated restriction for a. Finally, let
O
choose
(25.138)
1
1—0 0 so=(1—0)s<1 with1 —=—+—, p P0
1—0 0 1 —=——-—+—. 2 p
(25.139)
By the complex interpolation formula in [Trif3l, Theorem 2.4.7, P. 69, we have
that =
C
(25.140)
This formula is also covered by the more recent complex interpolation method developed in IMeMOOI. By p. 110, there is a counterpart of (25.137) with respect to Then there are positive constants c1 > 0, c2 > 0, C3 > 0 such that for all j N, < IIH 1B(R)II'—° IIH
(R)jI° (25.141)
This proves (25.136) for all admitted cases.
Step 2 Let n E N. We prove the theorem for the spaces under the additional restrictions p < 00 and a With f and g given by (25.129) we put
F(x)=f(xi)x(x2)
and
(25.142)
if n = I
f
F = and G = g.) By Theorem 4.4 the spaces have the Fubini property. Then it follows from (25.132) and (25.133)
(Of course
then
that
C
—
II
1Ff — id
max
(i,
(25.143)
(25. 144)
25. Truncations
383
where again the equivalence constants are independent of j E N. However (25.143) and (25.144) prove that T is not Lipschitz continuous in provided that, so far, p < oo and a Step S Now we prove the theorem under the restriction p < co. We rely on the real interpolation formula (25.101) in the extended version as indicated at the end of Step 5 of the proof of Theorem 25.8. Together with the reiteration theorem of interpolation theory it follows that
= where again
is either
(25.145)
with
or
0
O
(25.146)
Assume that T is also LipOf course, T is Lipschitz continuous in Then it follows by real interpolation of Lipschitz schitz continuous in operators, [Tar72b], Theorem 4, p. 476, that T is also Lipschitz continuous in We may assume that sO We get a contradiction to Step 2. This Finally, let p = cc. Let proves the theorem if p < cc.
0<s<1, 1
(25.147)
and q
qo
so=(1—O)s.
(25.148)
Then we prove in 25.15 below the real interpolation formula
=
(25.149)
.
Taking this for granted, we are in the same position as above. Assuming that T (Rn) we obtain that T is also Lipschitz continuous is Lipschitz continuous in in from
But we know that this is not the case. To extend this assertion to with 0 < q < cc we follow the above scheme of
arguments, now based on the well-known interpolation formula
=
where0<s< land0<9<1.
,
(25.150)
IV. Truncations and semi-linear equations
384
25.15
An interpolation formula
+ = + = 1. Then we have by (Tria], Theorem 2.4.2(a), P. 185 (obviously extended) that We prove (25.149) with (25.148). As usual,
B7°(R'2) =
(25.151)
with (25.152)
qq'0
By [Tri/3), Theorem 2.11.2, p. 178, complemented as in [RuS96], p. 20, the dual of is Hence by the duality theory of real interpolation, 1.11.2, p. 69, we have that
=
=
.
(25.153)
This proves (25.149). 25.16
Holder continuity
T as described in Theorem 25.14 can also be demonstrated by looking for best Holder exponents, when these operators are considered as mappings between different source and target spaces. We restrict ourselves to an example. Again we put B = By (25.6) the operators T and always have the same continuity properties. Hence we may restrict the formulation to, say, T. The bad continuity properties of
25.17
Corollary
Let
0<8<1+! Let for some
a with0
and
a
and somec> 0,
+
Ill
0<9<1.
(25.154)
IITf — Tg
ThenOaO.
(25.155)
26. The Q-operator
385
Proof Step 1 First we remark that (25.155) with a = Theorem 25.8 and the well-known interpolation formula
= This proves (25.155) for 0 < a
8
follows from (25.156)
.
8.
It remains to prove that a = 8 is the best possible exponent in (25.155). We use the functions F and C in (25.142). Then we have (25.144) and by the same arguments as there that Step 2
hF
110
2's,
j
E
N.
(25.157)
This follows from Step 1 and Step 2 of the proof of Theorem 25.14, where the restriction 8 is not needed for this purpose. Inserting these equivalences, and (25.144) with s(1 — 8) in place of s, in (25.155) we obtain that 238(1_0)
c1
IITF — TO
j
+ E N,
IF — GILp(Rn)IIa
where the positive constants c1, c2, C3 are independent of j
26 26.1
(25.158) E
N. Hence, a
8.
The Q-operator Preliminaries
As described in Section 24 it is our intention to develop a smoothness theory for some semi-linear integral equations and differential equations with prototypes (24.6), (24.7). It is natural to interpret the right-hand side of (24.6) as an operator and to ask for a fixed point in a suitable (real) function space. As a minimal requirement, the operators T and given by (25.4) and (25.5), should be bounded in such function spaces. By Theorem 25.8 we have a rather satisfactory answer. Let, for example, h E in (24.6) with 8) E according to (25.40). Then it is natural to ask for solutions u with (24.6). However at least at first glance there are no fixed point theorems which can be applied. Since we are in there is no compactness such that Schauder or Leray-Schauder arguments can be used. Banach's fixed point theorem (contraction mappings) in does not work if R,-, as above, since by Theorem 25.14 the operators T have poor continuity properties. This is the point where the Q-operator, roughly described so far in (24.8), (24.9), comes in. It has much better properties than T and Under some posi-
N. Thincations and semi-linear equations
386
tivity assumptions for the kernels K it produces supersolutions of (24.6) and its generalizations. Together with the supersolution technique which will be described in Section 27 one gets finally solutions of (24.6), and also of (24.7), with optimal smoothness properties. In the present section we study the operator Q. Based on qua.rkonial decompositions as proved in Section 2 one obtains rather easily the described assertions. First we collect what we need. Let n E N and let
y = (yi,.
= {y
.
with y2 > O}.
.
In specification of Definition 2.4 we assume that
is
(26.1)
a non-negative C°° func-
tion in R" with C {y E (R÷)" :
(26.2)
liii
= have the same for some r > C) and with (2.15). Otherwise the By (26.2) we have now that meaning as there, where /3 E
0
for all
xE
and
/3 E
(26.3)
were introduced in Definition 2.4. In The related (8, p) — fl-quarks be the spaces particular they are also non-negative. Let and according to Definition 2.6. We are not interested in the most general cases. and complement the results So we concentrate mainly on the spaces obtained by a look at the Sobolev spaces
11(R") =
(26.4)
In other words, in what follows we always assume that n E N and that
O
(26.5)
co in case of the Sobolev spaces in (26.4). By the discussion in 2.7 we have in all these spaces the quarkonial decomposition
with p <
f(x) =
(flqu)um(x),
xE
(26.6)
fJ,v,m
with = which converges absolutely and (hence) unconditionally in max(p, 1). By Theorem 2.9 these spaces coincide with the usual ones. We refer also to 2.11-2.13. In particular by Corollary 2.12 there are optimal coefficients
=
(26.7)
which depend linearly on f. Furthermore we may assume that the generating and in Corollary 2.12 are real. We justify this remark. Let functions
26. The Q-operator
hence cok, in 2.8 and xk in (2.58) be even (with respect to the origin). Then the inverse Fourier transforms of these functions are real, and, by (2.86) the functions coefficients view.
are
also real. In particular, if f is a real function then its in (26.6), (26.7) are also real. We summarize our point of
The even functions the (real) functions
and Xk are fixed according to 2.8 and (2.58); and are calculated as in Corollary 2.12, in particular by
(2.86). Let p, q, s be given by (26.5), with p < oo in case of the Sobolev spaces (26.4).
and H(R") are
Then we have (25.1). As in (25.2) the real parts of denoted by and
respectively.
(26.8)
We have optimal quarkonial decompositions (26.6) in all these spaces
If, in addition, f belongs to the real parts according to (26.8), then the representation (26.6) is also real. Recall what is meant by optimaL be given Let be the sequence spaces introduced in (2.7) and let (f) be calculated as indicated above. Then by (2.23). Let the coefficients and
If
IIA(f)IbpqiIg,
fE Bq(W'),
(26.9)
where the equivalence constants are independent off. We use the abbreviations introduced in Definition 2.6; in particular .\(f) refers to (2.22) with (26.7), and
p> r has the same meaning as there. In case of one can argue in a similar way. where must be replaced by fp.2. Hence optimal coefficients means that we have (26.9). and H(R"). We introduce the operator Q for the full spaces 26.2
Definition
Let p. q. s be given by (26.5) (with p < oc in case of the H-spaces). Let
orfEH(R.") and let f(x) =
X E W',
(26.10)
be the above distinguished quarkonial decomposition (26.6), (26.7), (26.9). where (/3qu)um are (s, p) — fl-quarks according to Definition 2.4, now based on (26.2). Then the operator Q is given by
(Qf)(x)
xER".
(26.11)
IV. Truncations and semi-linear equations
388
26.3
Remark
First we remark that
fE
if
Qf
(26.12)
(similarly for This is covered by Definition 2.6, the discussion in 2.7 and the detailed arguments in Step 1 of the proof of Theorem 2.9, in particular, on the role of p> r. Hence, the right-hand side of (26.11) converges absolutely By (26.9) with = max(p, 1) and, hence, unconditionally in in
it follows that there is a constant c> 0 such that
fE
(26.13)
in place of Hence, Q is (and an obvious counterpart with a bounded operator in all these spaces. But it has much better continuity properties than the operators T and (m real spaces) considered in the previous section. Obviously, we have that If(x)I 26.4
(Qf)(x),
x E Re',
fE
(26.14)
Theorem
Let n E N and let p, q, s be given bij (26.5). Let Q be the operator according to Definition 26.2. (1)
Thereisaconstantc>Osuchthat IIQf
for all f (ii)
c
— Qui
and all g
—g
(26.15)
(Lipschitz continuity).
Let, in addition, p < oo. Then there is a constant c> 0 such that DQI
c
—
(26.16)
for all f E H(R') and g E H(RTh). Proof After the above preparations the proof is simple. We prove part (i). By (26.10), (26.11), and the obvious counterparts with g in place off, we have
that Qf — Q9 =
—
(26.17)
27. Semi-linear equations; the Q-method
389
This is an admitted quarkonial decomposition of Qf — Qg in accordiiig to Definition 2.6 and the above considerations. Using the linearity of the (listinguishe(l coefficients we have that
I
I
=
—
By (26.9) and the discussion there the sequence )t(f
—
—
g)
g)I
.
(26.18)
is optimal. Then it
follows that IIQf for
c
—
some c > 0 and all f
(26.19)
—g
and g
This proves (26.15).
The proof of (26.16) is the same. 26.5
Remark
be a truncation couple according to Definition 25.6 and
s) E
Let
Then (26.14) can be complemented by
(25.40). Let 0 < q
< (Tf)(x) < (Qf)(x), x E R'2, f E
(26.20)
By the above theorem and Theorem 25.14 the continuity properties of Q are much better than the continwty properties of T and T+. This will be the decisive observation in the applications given in the following section. We followed [TriOlJ.
27 27.1
Semi-linear equations; the Q-method Semi-linear integral equations
Let K E L1(R") be real and let h be real, where 1 p oo. Then one can apply Banach's contraction theorem to find a real (unique) solution u
f
K(y)u+(x — y)dy + h(x),
x E W',
(27.1)
provided that IlK is sufficiently small. This follows from the obvious fact t hat the truncation operator T+, given by (25.5), is Lipschitz continuous in But what can be said if h belongs to some real spaces as, for example. in (26.8)? Since a) is is involved the assumption that a truncation couple accorchng to Definition 25.6 and Theorem 25.8 seems to be natural, or at least reasonable. If one has no further information for the
27. Semi-linear equations; the Q-method 27.2
Definition
Ltr/
> 0. Then
is the collection of all (complex-valued) functions
f(.r. y) with x E
and y E
such that
=j 27.3
391
dy
flf(•,y)
<00.
(27.4)
Remark
Of course. (27.4) is applied to x au(l have
the Ce-norm in isaBanach space. For any flxedy then we put K(x,y) = K(y) f(x,y). If K(y) E
for any o>0. Recall that the real spaces
in (26.8) are specifications of
and
(25.1). (25.2). 27.4
Theorem
Lt is E N.
0<8<1+!,
(27.5)
(the shaded region in Fig. 27.1). Let a> s and let
K(x. y) 0
in
x
with K
Then there is a positive number e with the following property. If e, then for any h has a uniquely determined solution u (i)
the equation (27.2)
(ii)
(1, 1). If flK 1L1(Ca)(R2n)U Let, in addztwn, p < 00 and s) then for any Is H1(RTh) the equation (27.2,) has a uniquely determined
Solution it
E
).
In the first (and main) step we prove part (i) for the model equation (27.1). Here we present what we wish to call the Q.method. In Steps 2 and 3 we extend this assertion from B-spaces to H-spaces and from (27.1) to (27.2). Proof
Step I We prove part (i) for the model equation (27.1). Let h (27.5). By the interpretation given in 27.3 we must prove that (27.1) has a unique solution a E under the assumptions that IlK is
IV. Truncations and semi-linear equations
392
small and K(y) 0. Let
and Q be the operators introduced in (25.5) and in Definition 26.2, respectively. Let B and J3Q be given by
(Bu)(x)
=
j K(y)
—
y) dy + h(x),
x E W1,
(27.6)
y) dy + h(x),
xE
(27.7)
and
(B'?u)(x)
=
j K(y) (Qu)(x
—
By (27.5) we can apply Theorem 25.8(i) to and Theorem 26.4 to Q. Using the triangle inequality for Bausch spaces we obtain that IlK
I
C2, c3
(27.9)
flu — v
(27.10)
IlK
—
where Cl,
q(llr)Il +
are positive constants. Let C3
IlK IL1(RTh)II <1
is a contraction in in (27.10). Then and hence also in We use Banach's contraction mapping theorem. It goes back to S. Banach, EBan22J, and may be found in any book on non-linear functional analysis or fixed-point theorems. We refer, for example, to [Sma74J, p.2. Hence, under the a uniquely assumption (27.11) the equation (27.7) has for given h E determined solution u°(x)
= j K(y) Let j E N and let, by iteration,
(Qu°)(x — y) dy + h(x),
u°(x)
=
=
j
—
Again by Theoreni 25.8 we have E by (26.20) that u°(x) is a supersolution for B,
u'(x)
=
j
—
x E R".
y)dy + h(x).
(27.12)
(27.13)
Since K(y) 0 it follows
y)dy + h(x) u°(x).
(27.14)
27. Semi-linear equations; the Q-method
393
We obtain by iteration a monotonically decreasing sequence,
XERTh, jEN.
(27.15)
it follows by (27.15) and Lebesgue's
Since both h and u0 bounded convergence theorem that u(x)
(27.16)
in
and that u is a solution of (27.1). By
We wish to prove that u E
(27.8), (27.9) and (27.12), (27.13) we have for sufficiently small OK that
2
Otto
(27.17)
111z
j E N,
+ IIh lB;q(R")II,
lu'
(27.18)
and by iteration
2
lu'
Since {U3} is uniformly bounded in
apply the Fatou property of Obviously,
j E N.
IIh
(27.19)
and u' —' u in as described in 25.1. Hence, u E
u is real. Then it follows that u E
we can
By (27.15) and the
monotone pointwise convergence in (27.13) (which can be assumed) we obtain
that u(x)
=
j K(y)
—
y) dy
+ h(x),
x
R'7,
(27.20)
follows from is the solution we are looking for. The uniqueness in where Banach's contraction mapping theorem is the uniqueness in
available. Step 2 The proof of part (ii) for the model equation (27.1) is the same. Now we rely on Theorem 25.8(u) and on Theorem 26.4. This explains also the curious restriction s) (1, 1) which we could not remove.
Step 3
Now we prove the theorem for the general equation (27.2). We replace
K(y) in (27.6), (27.7) by K(x, y). Then we obtain that IlBu
f
IIK(.,y)
—
y)
dy
+
Oh
(27.21)
IV. Truncations and semi-linear equations
394
the function x '—' K(x,y) is a pointwise multiplier in and we have by [Th'y], Corollary on p. 205, that
For fixed y E y)
—
y) C
IIK(, I!)
(27.22)
I
c'
Inserting this estimate in (27.21) we obtain the counterpart of (27.8),
cIIK!Lj(C0)(R2n)II In
+
(27.23)
Obviously there are similar counterparts of (27.9), (27.10). Afterwards one can
follow the arguments given in Step 1 and Step 2. 27.5
Semi-linear differential equations
Again let —
Let e >
0
be
=—
and. say, h E
the Laplacian in and let id be the identity. We ask for solutions u(s) of
+ h(s), x E
+ id)u(x) =
(27.24)
Now we 1)refer (27.24) instead of (24.7). But this is unimportant. By (24.4) one can reduce (24.7) essentially to (27.24). Then we have + Aid with some A > 0 on the left-hand side of (27.24). But this does not influence the arguments. Let C(s) be the Green's function of + id. Then (27.24) can be reformulated as
u(s) = f
—
y)dy + H(s),
xe
(27.25)
where
H(x) =
+
id)'h E
(27.26)
By the well-known properties of G(y) which may be found in 17.1, we have that.
G(y) >
0
with
y
and C
L1 (W').
(27.27)
Hence (27.25) is a special case of (24.6) and one can apply Theorem 27.4. But
the situation is now much better. In particular under the restrictions (27.5) there is no need to apply the Q-method. To discuss this point we assume temporarily that.
iO small,
(27.28)
N. Truncations and semi-linear equations
396
with 1 < r 00. The situation in R" is different. By Theorem 25.8 and, by Theorem 26.4, and Fig. 24.1 we have the natural region R,, where also Q can be applied. Furthermore there is the indicated lifting property of + id)' improving the smoothness by 2. Hence we get the natural region of admusibiity
+ id)
in Fig. 27.2 for the problem (27.24) given by
=O
(!,s+.A)
:
E
.
(27.30)
To avoid a wrong impression we add the following remark. If one replaces + id in (27.24) by a somewhat more complicated operator, for example, a(x) (—is + id), where a(x) is a positive function with some singularities, then bootstrapping arguments do not work any longer. We return to this point in the later considerations and refer in particular to 27.10. In other words, for semi-linear differential equations of slightly more complicated type than (27.24) bootstrapping is of no use. Theorem
27.6
Let n E N and 1 11 \ <s+A<1+—, n(——1)
O<poo, for some \
\P
p
1+
(27.31)
[0,21 (the shaded region in Fig. 27.2). Then there is a number
Lo > 0 with the following property. (i)
110 <
then
for any h E
uniquely determined solution u E (ii)
the equation (27.24) has a
(RTh).
Let, in addition, p < 00. 110 < E EO,
then
for any h E
the
equation (27.24) has a uniquely determined solution
Proof
Step 1 We prove part (i). We modify Step 1 of the proof of Theorem 27.4. The functions h and H have the same meaning as in (27.24)—(27.26). The in (27.6) and (27.7) are given by counterparts of B and
(Bu)(x)
=
e
=
E
f
+
id)' o
+ H(x)
(27.32)
XERIZ,
27. Semi-linear equations; the Q-method
and
=
+ id)' o Qu(s) + H(x)
e
(27.33)
zER".
=
Recall that G(y) > 0 if yE R" and C e Li(R"). Let E [0,2] be chosen such that (27.31) is satisfied. This corresponds to o on the line segment in Fig. 27.2. Then both We consider B and in and Q can be applied in and obtain the counterpart of (27.10),
cc
—
(27.34)
lu — v
and corresponding counterparts of (27.8), (27.9) with H in place of h. Now one can proceed as in Step 1 of the proof of Theorem 27.4 with the following modifications. First we remark that (R") is a complete metric space with the metric
p(f,g)=
K=min(1,p,q).
(27.35)
Hence Banach's contraction mapping theorem can be applied, [Sma74], p. 2. Secondly, using well-known embedding theorems, one must modify (27.16) in case of p < 1 by with
in
>
(27.36)
With the modifications indicated one can follow Step 1 of the proof of Theorem 27.4. Hence there is a (uniquely determined) solution u E 13w' (R") of
u(x) =
c
+ id)'
o
+ H(x).
(27.37)
The first term on the right-hand side of (27.37) belongs even to Since H E we get finally u E 2 The proof of part (ii) is the same. Now we can choose E [0,2] in such a way that s +.\ avoiding the exceptional cases in Theorem 25.8 concerning the boundedness of T+.
27.7 Two comments We are more interested in presenting the Q-method than in the most general applications. This applies both to Theorem 27.4 and, to an even larger extent. to Theorem 27.6. In connection with (27.24) we relied on the properties (27.27)
398
IV. Truncations and semi-linear equations
for its Green's function. Here C E L1 (R'2) follows from the exponential decay of C(y) if oc. But these properties of Green's functions, positivity and exponential decay, are well known for a large class of second order elliptic differential operators in R'1. All that one needs may be found in [Dav89]. This
applies also to some functions of these operators, in particular to fractional powers. In other words there is a good chance to replace + id in (27.24) by some classes of elliptic pseudodifferential operators.
Secondly, we add a comment on the supersolution technique. It goes back to 0. Perron, [Per231, in connection with boundary value problems for the Laplacian in bounded domains in R". This method was used later on, especially in the 1970s and 1980s, in connection with nonlinear equations of type
Lu = f(x, u)
and
Lu = f(x, u, Vu),
(27.38)
where L is a second-order linear elliptic differential operator. The underlying function spaces are preferably Holder spaces, but occasionally also classical Sobolev spaces. In contrast to our very special nonlinearities 1(u) = lul and the admitted nonlinearities in (27.38) are described in qualitative 1(u) = terms, by smoothness and growth conditions. Details of the use of the supersolution and subsolution technique in the just-outlined context may be found in [Ama76], [AmC 781, [KaK78], [FKY87], to mention only a few typical papers. At this moment it is not so clear whether the techniques developed in these and related papers can be combined with the methods presented in this chapter. But it might be worth consideration. As an indispensable ingredient one needs sharp mapping properties of composition operators of type u f(z, u) in function spaces of type We refer to [RuS96] where one finds and far-reaching assertions in this direction. 27.8
Local singularities
As just said there might be many possibilities to apply the Q-method. We have a closer look at how to incorporate local singularities in the right-hand side of (27.24), which then looks like
+ id)u(x) = E(1 + b(x) 0. To find out which type of singularities can be admitted one may consult Fig. 27.2. In the proof of Theorem 27.6 we used suitable points on the vertical line segments of length 2 characterized by o, say the point C representing for example H (R"). However, instead of going down from A to C on the vertical line, one may use sharp embeddings from A to D and afterward Holder inequalities at level a to step from D to C. This is the point where b
27. Semi-linear equations; the Q-method
399
I___ Fig. 27.3
in. The underlying theory has been developed in [SiT95] and may also he found in [ET96J. 2.4 and [RuS96], 4.8.2, pp. 238—239. We describe a special case given in [ET96], p. 54, formula (17). Let
1
U
U1
U2
s>0,
118
1 1 8 8 —=—+-, -=-+-, —=—+-, Ui V2 fl U fl Vi 1
1
(27.40)
(27.41)
a.' indicated iii Fig. 27.3. Then we have that
c
(27.42)
Iieiice. the classical Holder inequality
(R") .
c
is shifted along lines of slope n at the level s. In our case we have the triangle ADC iii Fig. 27.2 with the base line DC of length Let n 3. Then <1 and it makes sense to assume that the triangle ADC is located as seen in Fig. 27.2 lwtween the two lines s = and s = n — i), which appear also in Fig. 27.3. Let
then
and
(27.43)
IV. Truncations and semi-linear equations
400
with = — To be sure that not only C but also A and D are points within the distinguished strip, we must assume in addition that
nfl
q
n
(27.44)
p
By (27.42) and (27.44) we have that
c
.
(27.45)
Hence, bE
with
q
=
+
(27.46)
n n is a reasonable condition for b we are looking for. To get a feeling for what type of singularities can be included we remark that
c
q
(WL).
(27.47)
This is well known and also covered by Theorem 11.4. Let be a non-negative C°° function in with compact support and > 0. Then we have that
0<s
b(x)=1x12
.!
(27.48)
if, and only if, x> We give a short proof. Using (27.44) we have by Theorem 15.2 for the corresponding growth envelope that
=
,q)
(27.49)
,
and, hence by (15.9), .1
(/ One may assume that follows that b*(t)
c lb
.
(27.50)
in (27.48) is identically 1 near the origin. Then it
t*
0< t <e.
(27.51)
If x <
then the integral on the left-hand side of (27.50) diverges and b does not belong to Let x> Let B3,, with j = J, J + 1,..., and
forsomec>Oand B,,, C {y :
vi
2—i+1}
.
(27.52)
27. Semi-linear equations; the Q-method
401
If properly located and if {1,l'fl } is a suitable subordinated resolution of unity
with the usual properties, then
b(s) =
I log IxI
j=J is
IxI
<E.
(27.53)
1=1
an atomic decomposition in (WL) of b(s) near the origin. This follows = —2. Theorem 13.8 in p. 75, and, with =
from s —
I <00.
IIAIfq.211
(27.54)
This makes clear which type of singularities of b(s) in (27.39) can be expected. 27.9
Corollary
Letn3. O
11
\
1
n
and
(27.55)
(27.56)
then Then there is a number > 0 with the following property. 110 < e the equation (27.39) has a uniquely determined solution for any h E
Proof
Since b(s) 0. equation (27.39) can be rewritten as
+ b)u) (x) +h(x), xE R".
+ id)u(x) =
(27.57)
Furtherniore by the discussion in 27.8, in particular by (27.43), (27.44), the triangle ADC is located as indicated in Fig 27.2 and as discussed there. The counterpart of the operator B in (27.32) is now given by
(Bu)(x) = (Biu)(x) + (Bbu)(s) + H(s),
5 E R'1,
(27.58)
with
(Biu)(s) = (B5u)(x) =
(27.59) e
+ id)' o
(27.60)
402
IV. Truncations and semi-linear equations
One has similar counterparts for the operator in (27.33), where one must replace in (27.59), (27.60) by Q. The operators B1 and B,, (and their counterparts B? and ) are treated differently. They are decomposed by B1
oid1
and
oboidb
B,,
(27.61)
with (27.62)
id1
(27.63)
'-+
:
(27.64)
and
(2765)
idb
ilhl(R")
b
(27 66)
(27.67)
:
'—p
(27.68)
The necessary explanations have been given in 27.8. Recall that (27.65) is a well-known embedding with constant differential dimension. We refer to [ET96], p. 45, formula (8). Otherwise (27.62)—(27.64) correspond to tile line segment AC and (27.65)—(27.68) to the triangle ADC. In particular, (27.66) is covered by (27.45). One deals in a similar way with the operator Be?. Then one obtains the counterparts of (27.8)—(27.10) now with respect to the space Afterwards the rest is the same as before. 27.10
Comments; the Q-method revisited
We followed [TriOll. The aim to present the above corollary is twofold. First we wanted to discuss which type of (local and global) singularities, expressed by the function b, can be admitted. Secondly, the corollary sheds new light on the usefulness of the Q-method. Let p and s be given by (27.28). We discussed in 27.5 that under these restrictions the Q-method is not needed to get maximal regularity for solutions u of (27.24). In this particular case one can use bootstrapping arguments. But the latter method does not work if one replaces (27.24) by (27.39). The mapping properties of and
are different. In other words, in contrast to the Q—method bootstrapping arguments are restricted to rather special situations.
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Taylor, M. E., Partial differential equations, I. New York, Springer, 1996
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Symbols Sets
44
44
C, in IR(n),
K,.51 Kt, 54
119
MR(n),
102
N0, if)
Th8
241
(M,g),
2fi11
149
R, 3411
R'1, 2,
343
51
357, 387
UB,278 7Z,iO
zn,iQ B(1l), 113 trp
Spaces 44 4LL 358, 3fil 359•
44. 113
53
Symbols
420
4-'
155
113
340
106
357
11
106
113 113
156
LL3
LI
115
br
IIAIfpgIIQ, 1.3
1.48
149
4.
296
C(R?z), 168
C(r), 309 C'(lir), iLlS
H8(M), 312
H;(M),a1
396 90
254
227
296
H,(1Z), 254 H(s,*), 3511
47
.0(r), 298
1.57
D'(r),
358,
298
387
2611.
249 85 119
255,
132 126 SE
4
L(A, B), L(A), 278 10
Symbols
421
322
167
Lr(t'), 123 L1(cz,gr), 99
31L 329 328 27
43 1.74
'81
12.1
114
312
249
Lru(logL)a(Ic), 115 391
255
(—s)—',
idr, 123
1.64
227 248
Lip(r), 309
48
trr, 121. 123 tr1',
256
349
tracz,
5.fl
T,
358 358
I I
1.0
Functions, numbers, relations
S(Ifl, 112 [a],212 ak(T), 279
lB W(R"), a
341
358
115
W;(M),a2
1.Q5
AN
260
i
t,uJ
4.'
54 41.
Operators .6,
aas
258, 299
ek(T),
14 15
279
125
422
Symbols
f(t), 161, 181
f"(t), 161
wspIL,
j.ij(t),
181 1.62. 1.90 4
192 164
Xv,n,
1-93 ,
216
21,
13,v,m
1+a
198
P), 1+" 'fcApq ", 195
Ess Spec, 322 256, 290
133k
Ma, 56 128 1.36
145 161
311. 322 £
164. 196
1.64.196 247
Spec, 322 35
#A, 322
30, 76, 149
Index Animal Farm, 188 approach, Weierstrassian,
continuous, uniformly, 381 convergence, absolute, 21 convergence, unconditional,
atom, 81, 205 atom, hydrogen, 325 atom, (s,p,W)K,L, 341
Banach, contraction mapping theorem, 392, 397 Big Bang, capacity, 261 capacity, 309 case, borderline, case, critical,
d-domain, 811
d-domain, thorny starlike, d-set, (d, W)-set, 173.. 1.724
237
case, degenerate, 291 case, sub-critical,
covering, locally finite, 88 couple, truncation, 363. criterion, D. R. Adams, 1.32 criterion, Wiener, 309
1.73.
case, super-critical, 110., 173. 1Th2Th coefficient, Cauchy, 74. 33 coefficient, optimal, 77, 73. 724 Zfi, 11124 386
condition, ball, 138 condition, doubling, 277 condition, interior ball, 158 condition, moment, 28 condition, outer ball, 310 condition, strong separation, 33.7 constant, Planck's, 325 continuity, Lipschitz, 3734 388 continuous, Lipschitz, 381
312 330
334
decomposition, Weyl, 157 differences, 4 dimension, differential, 112 dimension, Hausdorif, 33.5 dimension, interior Minkowski, 87 Dirichiet Laplacian, 255 distance, 43. 235 distribution function, 181. 1113.. 181, 183 distribution, regular, 167 distribution, tempered, 104 distribution, tempered, in domains, 117 domain, minimally regular, 155 domain, interior regular, 155 drum, fractal, 274 embedding, 94 embedding, continuous, 167 embedding, limiting, 173
Index
424
envelope, continuity, 198 envelope, growth, Th4. 2111 equation, semi-linear integral,
equation, semi-linear differential,
Lebesgue point, 2f1L1
limit, semi-classical, 325 Lipschitz function, 87 localization, 59 localization, refined, 68
394
euclidean n-space, 2 extension, 48 formula, Cauchy, 9., 24 function, admissible, 333 function, continuity envelope, 198
function, Green's, 244 function, growth envelope, function, maximal, 136 function, monotone, 333 function, smooth, 226 function system, pseudo iterated, 337
function, tame, 289 geometry, bounded,
83
Habilitationsschrift, Riemann, 226 Hainiltonian, 325
IFS,
337 337
manifold, Riemannian, 312 means, ball, M. 66 means, cone, 54 means, local, 125 measure, diffuse, 278 measure, directionally strongly diffuse, 278 measure, Hausdorif, 256, measure, multifractal, 289 measure, Radon, 122 measure, strongly diffuse, 277 measure, Weyl, 274, 33,5 membrane, rusty, 353 41)2 method, Q, 357, modulus of continuity, 196. 247 modulus of continuity, divided, monomial, 1(1 multi-index, 11)
multiplier, pointwise, 67 music, of the ferns, 288
inequality, Carl, 279 inequality, Hardy, 236 inequality, Hardy-Littlewood maximal, 64 injectivity radius, 83 interpolation, non-linear, 373 interpolation, real, 178
number, approximation, 279 number, entropy, 279 Nullstellenfreiheit, 271
K-functional, 262
operator, pseudodifferential, 321 operator, Q, 356. 387 operator, senii-effiptic, 259 operator, trace, 349 operator, truncation, 358
lattice, if) lattice, approximate, lattice, irregular, 71
144
operator, hydrogen-like, 326, 329 operator, identification, 124 operator, Laplace-Beltrami,
425
Index
potential, Bessel, 121 potential, Hedberg-Wolff, 134 potential, Riesz, 308 300, potential, single layer, 301
primus inter pares, 186 principle, Birinan-Schwinger, 323. principle, Max-Mm, 323 procedure, constructive, 22 property, Fatou, 360 36 property, Fubini, property, Markov, 158 361 property, truncation, quark, generalized, quark, (s, p) — j3,
72 147
quark,
quark, (s,p,x) —
98
quark, quark, (s,p, 'I') — /3, 341 quasi-everywhere, (cr, p), 261 quasi-metric,
rearrangement, non-increasing, region of admissibility, 396 regular, interior, 155 regular, minimally, resolution of unity, 5., 15.,
us
82
space, classical target, 168 space, Hardy, 4 space, Holder-Zygmund, 4. 254 space, homogeneous type, 159 space, Lebesgue, 10 space, Lipschitz type, 248 space, logarithmic Sobolev, 2511 space, Lorentz, 174 space, Lorentz-Zygmund, 115 space, quasi-metric. space, Schwartz, 103 9& 148 space. sequence, space, sequence, modified, 26 space. Sobolev, 3.. 82, 379. 387 space, trace, 1SL 152 space, very classical Sobolev, 255 space, Zygmund, 174 264 spectral synthesis, spectrum, essential, 322 328 spectrum, negative, 3fl, supersolution, 392, 3.98
Taylor expansion, LQL ilL 1J1 transform, Fourier, 16 transform, Fourier, inverse, 15 weight, 85
resolution of unity, approximate,
resolution of unity, approximate, family,
space, classical Sobolev, 3. 34.
97
338 set, pseudo self-similar, solution, classical, 309 102 space, space, 88 space, bounded mean oscillation, 4
