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Consequently, when q(x, and
is
<
(8.20)
+
r)r2.
The dependence in (8.20) on e-derivatives is hidden, but implicit in the hypothesis that q(x, is an elementary symbol; the "constant" C in (8.20) depends on bounds on (e)knID?cck(e), for some finite range of a (as well as qj,rj, and other parameters). A general symbol can be written as a sum of elementary symbols (8.21)
p(x,e) =
= £
with (8.22)
k)O
2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
and Qkl(x) obtained from a Fourier analysis of for some a C Details on this are recalled in §2 of Chapter 1. Thus, we have:
on
PROPoSITION 8.1. AssumeO<s
For sufficiently large N, take
= sup sup t
(8.24)
and
p(x) =
(8.25)
t
IaI
Then (8.26) 9.
Paradifferential operators on the spaces
function on Let F be a regularity, we will analyze F(u)
as
If u is a function (with values in R") with a paradifferential operator. As in (O.1)—(O.4),
we can write
F(u) = F(uo) + {F(ui) — F(uo)]
(9.1) with
uk = Wk(D)u,
(9.2)
and
+ ...
+ [F(uk÷1) — F(uk)]
then we write
F(uk+1) — F(uk) = mk(x)?i'k+l(D)u,
where (9.3)
mk(x)
=
f
dt
F'(Wk(D)u +
Hence (9.4)
F(u) = Mp(u; x,
D)u + R(u),
where
(9.5)
(9.6)
R(u)
= F(tbo(D)u)
MF(u;x,e) =
We have
(9.7)
MF(u;x,e)
u
E
As we noted in the introduction to this chapter, if r > 0, (9.8)
u E
c'
MF(u;x,e)
E
+"m
9. PABADIFFEB.ENTIAL OPERATORS ON THE SPACES
121
We want to draw an analogous conclusion when we assume u E To do this, we need to estimate Imk Ic(Ah which is essentially equivalent to an where estimation of
Vk4!k(D)U+flbk+1(D)U,
(9.9)
07-<1.
Note that, parallel to (9.4), we have
F'(v) = MF'(v;x,D)v+R(v),
(9.10)
and
Now
Mc"(v;x,E)
v E LOC
(9.11)
applying Proposition 5.12 of Chapter I, we have
Cp((IvIIJ]oo)(1 +
(9.12)
provided
A2(i) \ and 5 CA(j).
(9.13)
ej (In
fact, (9.12) can often be seen to hold via a modulus of continuity argument.)
Now
(9.14) so
IVkIlcxA2
we have
(9.15)
ImkHccA)
+
Hence, if (9.13) holds, and E
Coo, so
ue
E
(9.16) We
c
L°°, we have K(2k)
recall from Proposition 5.2 of Chapter I that
(9.17)
E
==*
p(x,D) c(A)
provided C 0°,
(9.18)
i.e., in the setting of (9.16), provided A(k)
E A2(k) C
(9.19)
00.
Thus, if (9.13) and (9.19) both hold, we have (9.20)
U E
Mp(u;x,D) : c(A)
so
(9.21)
+
=
2. PARADIFFERENTIA L OPERATORS AND NON LiNEAR ESTIMATES
122
For example, we can take
(922)
Then (9.13) holds for s > 0, and (9.19) holds for s > 2. The classical case, with c(A)
is
A2(j) =
A(j)
(9.23)
1,
r>
0.
Of course, in this case, we already have OPS?1 acting on C, so the exponent of 2 in (9.21) is not needed. We want to extend the collection of A(j) for which it can be shown that (9.20)
holds. As a first step, assume we have p(j) \ such that Ap satisfies (9.13) and (9.19), i.e., (9.24)
A(k)1i(k)
EA(t)p(t)A2(t)
Then, parallel to (9.21), we have (9.25)
+
Hence we can replace (9.12)—(9.15) by
+ (9.26)
2
1
Thus, when (9.24) holds,
(9.27)
Mp(u;x.e) e
u E
This time, Proposition 5.2 of (9.28)
provided
=
Chapter 1 gives
p(x,D)
e
0(A)
(9.29)
This
condition fits together with (9.24) best when p(j)
=
We thus see
that if (9.30)
EA(4\2&)3/2
< Ic
then
(9.20) holds, and we have
(9.31)
IF(u)IL'A
We see that (9.30) holds for
(1+ as in (9.22), with s >
3/2.
9. PARAD1FFERENT1AL OPERATORS ON THE SPACES
123
We can iterate this argument. Suppose Ap satisfies (9.30), i.e.1 A(l?)p(t)A2(4?)3"2
> (9.32)
CA(j)p(j)A2(j)'12,
1/2
> A(k)p(k)A2(k)
Then we have (9.25), with the exponent 2 replaced by 3, and similarly for (9.26). Thus, when (9.32) holds, (9.33)
Mp(u;x,e) e c(AP)S?(y),
u E
_
Another invocation of Proposition 5.2 of Chapter 1 gives (9.28) in this case, provided
<00.
(9.34)
Taking
p(k)
=
we
see that (9.32) and (9.34) both hold provided
<00.
EA(t)A2&)5/3 <
(9.35)
Then (9.20) holds and we have
+ IIuIIcA)).
(9.36)
We see that (9.35) holds for A, A2 as in (9.22), provided s Inductively,
PR0POsvJON 9.1. Let A(j) \ varying
A2(j) \
> 4/3.
we have the following. 0
be slowly varying. Suppose there exists slowly
that
such
< CA(j)A2(j)
(9.37)
EA(k)A2(k)
.c oo,
Ic
for somevEV. Then (9.38)
u
e
MF(u;x,D)
—>
and we have
+ IIlc(A)).
(9.39) We see that (9.37) holds for
(9.40)
A(y)=j
A2(j)=y
1
,
We now produce further information on the symbol M1.' (u; x, when u and A(j) satisfies the hypothesis (9.37). Then, parallel to (9.28), we have (9.41)
+
which we can use in place of (9.12) to obtain (9.42)
+ IkuIIC(A)),
e
2. PARADIPPERENTIAL OPERATORS AND NONLINEAR ESTIMATES
124
and hence, for )t(j) satisfying (9.37),
ue
(9.43)
Mp(u:x,e) E
substantial improvement over (9.16). We next obtain some more estimates on From the chain rule, we have a
and hence
(9.44)
the sum being over flt, > 0, if 0 > 0. We can deduce that, if )t(j) \ satisfies (9.37), then IIC(A)
(9.45)
.. .
>i:
This uses (9.39) plus the fact, established in Corollary 5.4 of Chapter I, that is a Banach algebra (as long as EA(j) coo), so that (9.46)
IIVWIIC(A)
Ii
Now, if v = vk has the form (9.9), we see that IlD0vkIIcA) = sup I?
C sup
(9.47)
I?
C SO
(9.45) yields
+
(9.48)
It follows that rnk(x) in (9.6) has such an estimate, and hence, for )t(j) satisfying (9.37), (9.49)
e
We are ready to record the following further properties of MF(u; x, PRoPOSITION
(9.50)
9.2. Under the hypotheses of Proposition 9.1, u e
MF(u;x,e)
E
Furthermore, we have the decomposition
(951)
MF(u;x,e) =M#(x,e)+Mb(x,e) = M#(x,e) + +
where the terms on the right have the following properties: (9.52)
M#(xC)
FD?M#(.,e)IICCA)
A. PARACOMPOSITIOM has support in
(9.53)
E
has
24JC(,
ku
125
support in IiuI
?
and
SjJ',
where (9.54)
with çh
= A(j),
(9.55)
&(h)
=
j
dt.
0
PROOF. We already have (9.50) and (9.52), from (9.42). To proceed, note that (9.48)
gives
+
(9.56) Hence (9.57)
k
5;
k
A
5;
=
These estimates yield (9.53)—(9.55).
We
note that the proof of Proposition 5.12 of Chapter I shows that
—s
(9.59)
P2(i) =
and (9.60)
We
P3(i) = i4i) >A(j).
M1(x,D)
have /22(j) 5; Cp3(j), and
typically /22 A. Paracomposition
F
In this appendix we discuss a construction of [Al], applied to a composition o u, and extend some of the estimates given there. The basic thrust of this
despite a similar appearance of material is somewhat different from that of the objects imder study. For one thing, it is necessary to assume here that u is a diffeomorphism. As we will see, that assumption will play a crucial role in Lemma
A.2.
Throughout this section, we make the standing hypothesis that all functions F have support in some fixed compact set. Then quantities like 'IuUca can be interpreted in that light. Also, subtracting off a smooth function (whose composition with u is estimated by previous techniques) we assume F(0) = 0.
2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMA'I'F)S
126
=
To begin, set
and write
Fou >[13(Wku) - F,(Wklu)]. j,k
and
We decompose the double sum into (A.2)
Note
that. due
to cancellation,
= EFk(Wku).
—
j>k
kO
Meanwhile,
(A.3)
—
Fj(Wk_lu)J =
—
j
(Wk_1F)(4'k
k?1
hence °
F a u(x) = 4F(u; x, D)u +
(A.4)
(Wku)(x),
k >0
where
= k?0
(A.ö)
1
+ (1 — = f 0(WkF)'(rWk÷lu With 4'k+r = 7-Wk÷1 + (1 — r)Wk, we can estimate Ak(x)
a
(AM)
dr.
= a
>
In particular, for /9 > 0, (A.7)
C
IWk÷luhcølI
>:i: 1
131
Of course, (A.8)
One consequence of (Al) is the following, in case u is Lipschitz. Note that, for (AS)
s
02k(I/3,j
Hence (A.7) implies C
(A.l0)
0
=
C2k3 E LcZlvl
127
A.
Hence
F,u
(A.12) p
Lip'
(1, oc), x, D)UIIHS,P
(A.13)
:5;
c
Now a variant of Alinhac's "paracomposition" operator is given by (A.14)
wk(C)= k
N
Here,
is
which
a number, determined by
will
be specified in Appendix
B. If we set (A.15)
then,
Fk 0
(Wkn)(x) —utF(x),
from (A.4), we have
Fou(x)
(AiM)
+utF(x)+
=
In
fact, the operator defined as
(A.17)
u*F =
paracomposition in [Al]
is
ou).
We estimate the difference of (A.14) and (A.17), given by
= u*F
(A.18)
—
utF = >Wk(D)gk
with gk(x) = Fk(u) (AdO)
—
Fk(4'ku) = bk(x)(I —
bk(x)= / FLfrU+(1—r)Wku)dr. Jo
LEMMA A.1. Given s > 0, we have
(A.20)
C
PROOF. We have
(A.21)
—
:5;
Now (A.22)
s > 0
ks —
so (A.20) follows.
We next estimate if F. The
following is due to [Al].
2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
128
LEMMA A.2. We have IIu*FIIc:
(A.23) PROOF. By (A.14),
i4'j(D)u*F=
(A.24)
j+N
C
(A.25)
CN2_Jr k=j-N
This gives (A.23).
We next estimate the "remainder" RFu in (A.16), folowing [Al]. LEMMA A.3. Assume u is a diffeomorphism of class CS, that u' is uniformly is uniformly bounded. Ifs> 1 and r + s > 0, then continuous, and that
(A.26)
IRp,uMc:+a
PROOF. Note
(A.27)
that
= Eu — Wk+N)(Fk °
+ EWk_N(Fk
We use the following two estimates of [Al], which will be discussed further in Appendix B: (A.28) for
2k(u-s+1)
°
j k + N, v > s —
1
0, and if also k K (depending on
and
IIDu1 (A.29)
2-k(s-1)
IWk.N(Fk o
will be seen in Appendix B, the hypothesis that u is a diffeomorphism is needed to establish (A.29). We can estimate in two parts. First, As
(A.30)
Eu - Wk+N)(Fk Ck<j—N E IkPj(Fk a 'Pku)lILoc Cu(MuIIcs) >
2jv2k(v—s-r)
k<j-N
by (A.28). Taking v> s + r, we dominate the last espression by (A.31)
r('+8)2.
A. PARACOMPOSITION
129
Next, we have, for k K, 0 WkuL)M
(A.32)
k>j-f-4 IIFIL;:+i,
by
(A.29). Combining this with (A.30)—(A.31), we have
2fr+sb,
(A.33)
which gives (A.26), upon taking note of the dependence on N in these estimates, and crudely estimating the last sum in (A.27) over k < K. We summarize what has been done above: is
PROPOSITION A.4. Assume it is a diffeornorphism of class C8, s> 1, that u' uniformly continuous, and (u')' is uniformly boundS. Assume F is Lipschitz.
Then (A.34)
Fou(x) = n*p(x) + 4F(u;x,D)u +
+
where the paracomposition u*F, given by (A.1 7), satisfies estimates (A.35) the second term
<
has
the property
F,u E Lip'
(A.36)
e OPS?1,
and the remainder terms satisfy the following estimates. (A.37)
and, for s 1. r (A.38)
s > 0, + 8
> 0,
IRF,tJIc:+s
[Al] there are estimates on L2-Sobolev norms of the various quantities in norm estimates. Such estimates on estimates x, D)u are already recorded in (A.13). We next give on the paracomposition. In
(A.34). Here, we produce some
PRoPosiTioN A.5. ff1
0, (A.48)
We now obtain Besov space estimates on the other terms in (A.34). PROPOSITION A.8. Under the hypotheses of Proposition A.4, if 1
0, (7.20)
186
3. APPLICATIONS TO PDE
Applying the Moser-type estimate given in (0.9) of Chapter II to W = B(x, it), we obtain the desired estimate (7.16).
We mention another known improvement on the straightforward results described in (7.2)-(7.4). Namely, one can relax the requirement s > rr/2. For example, when is the standard Laplacian on W', it is shown in [BB] that (7.l)—(7.2) has a local solution of the form (7.3) as long as s> fri — l)/2 if ri = 3, and as long as s (ii — l)/2, if ri 4. If, in addition, B(x, it, Vu) belongs to a certain class of "null forms" that includes ones arising in "wave maps," then it is shown in [KS] that (7.1)—(7.2) has a local solution as long as s > fri — 2)/2. We refer the reader to [BB], [KM], [KS], and references therein for more on this. While those results do not imply Proposition 7.1, they do lead one to wonder whether this Proposition might be improved.
8. Div-curl estimates The most basic div-curl lemma takes the following form. Suppose it and v are vector fields on JR3 satisfying
pE(loo),
(8.1)
Then (8.2)
where 551 denotes the hardy space. Equivalently, in view of the duality result of [FS], the conclusion in (8.2) is that it v can be paired with an element of BMO. Such a result arid many variants were presented in [CLMS]. One of the analytical techniques used in [CLMS] was the commutator estimate (8.3)
P E 0PS10,
IfPu —
of [CRW], which was established in §10 of Chapter I. Using the identity
f Pu - P(fu)]v dx =
(8.4)
f
f[(Pu)v - u(P'v)] dx,
one obtains (8.5)
I(Pu)v —
<
1