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00. Assume fj ----> f in Loo-norm. Then
(11.35)
of q(x, ~):
lif - fj"IIbrno
which implies
f
s: Ilf -
fil/Loo
+ Ilfi
- 1i"llbmo,
E vmo.
Next, if f, 9 E £00 n vmo and B is some ball of radius
q(x,~) = Lqi(X,~), (11.28)
(11.31)
1,
PROPOSITION 11.2. Ifp(x,~) E LaoS~1 and 9 E bmo, then
(11.27)
j,k
00.
and, if 9 E vmo, this commutator is compact. Summing, we have the following
result of [CFL J:
(11.26)
+ L Mf; [aj(D), M9Jak(D).
The first sum in (11.30) clearly differs from a(x, D) by a compact operator. The second sum is equal to
Thus, by the fundamental estimate (10.1) on [Mg,aj(D)J,
(11.25)
L M!jaj(D).M9kak(D), j,k
Next, we consider the commutator [Mg,p(x, D)]. We have
(11.22)
+ p(x, D)qo(x, D)
j~O
qj(x,O =
qo(x,O
=
Lqe(x)1Pe(~), e
gi(x)ai(~)' j ~ 1,
(11.36)
1',
then
~ jlfg - fBgBI dx ~ IIfllL''' r~ Jig - gBI dx+ IlgllLoo ~r jlf - fBI dx, n
1'''
B
H
n
B
which implies f 9 E vmo.
with 1j!p as in (11.12)-(11.13). Note that under our current hypotheses we have
II/jIIL'>O, 119iliLoo ~ CN(j)-N.
As a consequence, under the hypotheses of Proposition 11.3 , if we also Suppose in (11.27).
p(x,~) E (£00 nvmo)S~/l then we have a(x,~) E (Loo nvmo)S~
_...,,,"" -'"="FoPE1tXfohg':Wi;r;H ~LDLy..:-~~~ti~A-~
S~M~i:S-';
_un ._":"::";",_:__._._==",,,"-'=.=c"=.~~="='=-'o IlgIIB;,l'
for s E JR, p E [1,00], by the analysis of (12.3)-(12.6). There is an obvious analogue for Tgf. Similarly the estimation of R(f, g) is a special case of the analysis of (12.7)-(12.11). In fact, with
are slowly varying. If (12.20) holds,
k+4 ;3(k) = L II7fj(D)gllu>o, j=k-4
ipk(D)f = 7fk(D)f,
we have precisely
p(x, D) : B~,l (JR n ) ~ B~,l (JR n ).
(12.32)
L
11 2i8 7fi(D)R(f,g)IILP ::; RES of (12.10),
i
Note that (12.20) is weaker than )"(k)~(2k) "'.,
Tfg = L Wk-4(D)f· 7fk(D)g,
(12.28)
(12.31)
PROPOSITION 12.2. Assume).. "'., ~ / then, for 1 ::; P ::; 00,
(12.22)
fg = Tfg
(12.27)
lJ-kl9
{j:j2:k}
(12.21)
PROOF. We use the paraproduct expansion as in (3.46)-(3.47):
(12.29)
the condition (12.6) holds, and furthermore the conditions (12.11) and (12.18) hold
provided
(12.20)
then B;,l (JRn) is an algebra under pointwise multiplication.
with
and we have
(12.18)
+ IlfIIB;)lgIIL~.
Consequently, when s > 0 and (12.26)
to
< 00.
We next show that certain spaces B;,l (JRn) are algebras.
(12.25)
If we look at the action of P on B~,l' we see that (12.10) needs to be modified
provided L)..(j)
so L)"(k)~(2k)
0 ==? IIR(f,g)IIB;,l ::; CllgIIL~llfIIB;,l'
86
J.
OPERATORS WITH MILDLY REOULAR SYMBOLS
This finishes the proof of Proposition 12.3.
y,,: ,,,,. :'.••
.',
REMARK. Proposition 12.3 is contained in results of [Run2]. The proof given above of this Proposition is essentially the same as the proof given by H. Triebel in [Tril], pp. 133-137. In fact, Triebel anticipates the use of paraproducts in establishing such results. The general result stated in Theorem 1 on p. 133 of [Tril] would also imply that B~,1 (~n) is an algebra under pointwise multiplication. However, the proof given there does not work for this space. The problem arises in the estimate (12.32) (or equivalently, in (9) on p. 136 of [Trill). This estimate works for s > 0 but fails for s = O. This correction has been noted in [SiT], where it is shown that B~,1(~n) is not an algebra.
13.
OPERATORS WITH COEFFICIENTS IN A FUNCTION ALGEBRA
'lji ,
PROPOSITION 13.1. Let B, B' be Banach spaces of functions on ~n with trans lation-invariant norms. Assume that B c Loo(~n) is an algebra 'under pointwise multiplication, and that B' is a B -module. Also assume
:.j~(
(13.4)
{'
"':' ..J
"':&[L ~I~f
P(~)
E
(13.5)
PROOF. Decompose p(x,~) as in (11.6), with m = O. An analysis parallel to (11.10)-(11.13) applies to Po(x,~), with IIPellB :s; C~(£)-N in place of (11.11). The behavior of B' as a B-module then gives
:s; C2 Jn / IIWj(D)fIILP'
fg
E B~.1 (~n).
In fact, the estimate
:s; CllfIIC(A) IlgIIB~,l
follows as in (12.17)-(12.18).
13. Operators with coefficients in a function algebra We recall that, whenever B
(13.1)
p(x,~) E BS?,o
c
Loo(~n) the symbol class BSP.o is defined by
-¢:::::}
IIDlp(-'~)I\B:S; Ca(~)m-Ial.
p(x, r~) = rmp(x, ~),
r ~ 1, I~I ~ 1,
= fj(x)aj(O,
(13.8)
Ilaj(D)uIIB'
for some M = M(B, B') < B-module, we have
00,
:s; C(j)MlluIIBI,
independent of j. Using again the fact that B' is a
IlpJ(x, D)uIIB'
:s; CN(j)-NlluIIBI,
which proves (13.5).
Let us denote by jj the closure of Co(~n) in B. The following result lacks the depth of Proposition 11.3, but it is still useful. PROPOSITION 13.2. In the setting of Proposition 13.1, assume that
p(x, 0, q(x,~) E jjs~I'
(13.10)
Furthermore, assume that (13.11)
r(x,O ==?
E
S~~ with compact x-support
r(x, D) : B'
-+
B' is compact.
Then
we say
(13.3)
B'.
IlfJIIB :s; CN(j)-N,
If we assume furthermore that (13.2)
---+
and, as a consequence of symbol estimates on aj (~) and the operator bounds from (13.4),
(13.9) IIR(J,g)IIB~,l
Pj(x,~)
(13.7)
A(k) = k-t, f E B~,1 (~n) n c(>.)(~n), 9 E B~,1 (~n) ==}
(12.37)
Po(x, D) : B'
with
p
and, as indicated above, the second inclusion in (12.34) is straightforward. Regard ing B~, 1 (~n), we can say that
(12.36)
B'.
p(x,~) E BS~I ===* p(x, D) : B' -+ B'.
(13.6)
The first inclusion can be established by showing
IIWj(D)fllv'O
-+
Next, for j ~ 1 we have, as in (11.8),
B;t,/~p(~n) C B~,1 (~n) C Loo(~n).
(12.35)
S?,o ===* P(D) : B'
Then
As an example of spaces satisfying (12.26), we mention the well known result that, for 1 :s; P < 00,
(12.34)
p(x,~) E
87
BS,'!['.
In this section we treat some results that are valid in great generality. At the end we indicate some interesting classes of examples to which these results apply.
(13.12)
p(x, D)q(x, D) = a(x, D)
+ K,
a(x, e) = p(x, Oq(x, e)
jjS~1
with
(13.13)
E
k"""'"
88
,"'
u
---_._-...,-_.--------_._----_
••
1.. OPERATORS WITH MILDLY REGULAR SYMBOLS
14. SOME BKM-TYPE ESTIMATES
and
_-_._89
which in turn follow from
(13.14)
K : B'
------t
B' compact.
Ilull£"o ::; C c Ilullc~ + C(log T
(14.3)
PROOF. We can construct Pj(x,~) E S~l' with compact x-support, such that ----+ p(x,~) in BS~I' with its natural topology. Hence pj(x,D) ----+ p(x,D) in £(B'), in operator norm. Similarly, qj(x, D) ----+ q(x, D) and, with aj(x,~) = Pj(x, ~)qj(x, ~), aj(x, D) -';t a(x, D). Now, for each j, aj(x, D) - Pj(x, D)qj(x, D) satisfies (13.11). Hence a(x, D) -p(x, D)q(x, D) is a norm limit in £(B') of compact operators on B'. Pj(x,~)
DIlullcz·
Here we note several extensions of these estimates, involving other function spaces and rougher operators. We start with some results in which C; is replaced by spaces of the form C[w] and C(>\). We retain our standard hypotheses on w(h) and on >'(k) = w(2- k ). With ~/Jk, II' k as in (1.2), we write k
There are, of course, many examples of function spaces Band B' to which the results above apply. We mention (13.15)
B
= B' = B;,I(~n),
s> 0,
(14.4)
U
=
L ~/Je(D)u + (I - Wk(D))u, £=0
and deduce that
sp:::: n,
k
as one interesting family of examples, studied in §12. Without going into details, we note that Propositions 13.1-13.2 have relatively straightforward analogues with ~n replaced by ']['n or some other compact manifold. Sometimes it is technically preferable to operate in this context. If one wants to apply Proposition 13.2 (or its analogue on a compact manifold M) to produce a Fredholm inverse of an operator q(x, D) which is elliptic, in the sense that (13.16)
q(x,O E 13S~I'
p(x,O
=
q(x, ~)-1(1- 'P(~)) E CS~I'
where 'P(~) is a convenient cut-off and C = C(M) is the space of continuous func tions on M, then it is desirable to know that (13.16) leads to p(x,~) E 13S~I' This works for function algebras 13 with the property that
I
(13.17)
E
ii, r
1
E C(M)
=? 1-1
14. Some BKM-type estimates
0
P E OPS1,0
log Ilulic IIPullu'oO ::; C Ilullv>o ( 1 + log Ilullu'" ' r
=?
for any r > 0. Such estimates are used in [BKM] , which has stimulated much further work. In [T2] these estimates and some extensions were derived as a con sequence of estimates of the form (14.2)
Ilullvx ::; C Ilullc o (1 + log T
•
IcW' Ilullr'o '
V k E Z+.
V k :::: 1,
or equivalently (14.7)
IlulIL= ::; Cw(c) Ilullclwj + C(IOg ~) Ilul!c~,
VeE (0,1].
This extends (14.3), in light of the inclusions recorded in (1.62)-(1.63), i.e., C w C C[w] C C a ,
u(h)
=
+h
w(h)
t w~t) dt. t
ih
To extend (14.1), we work with the spaces C(>,) , and apply estimates established in §5. We will also need the following inclusion, recorded in (1.63): (14.9)
C(A)
c
C[J.'J,
j.t(h)
=
l
h
w~t) dt,
t w(t)t dt
'(j) (14.11)
IlullL~
=
w(2- j
::; Cj.t(c)
),
Ilullc().) + (log ~) Ilullcz.
Replacing u by Pu we hence have (14.12)
IIPuIIL~
::; Cj.t(c)
IIPullc().) + C(log
D
IIPullcz.
-
~91"'j.....""""_.---
1. OPERATO-RS WITH MILDLY REOULAR
SYl>,~BOLS
"''='4
Hence
14, SOME BKM-TYPE ESTIMATES
91
and
P : C~
---+ C~,
P: CU.)
---+
CU.)
(14.13) =?
IIPullu>c :::; C J1,(c) Ilullc(A)
(14.21)
+ C(log ~) Ilullcz.
L
Ill}ik(D)uIIB~" :::; C
II'l/Je(D)uIIL=:::; C(k
+ 2)llullc~,
e~k+2
from which (14.18) follows. Then (14.19) follows from Proposition 12.2.
Note that
(14.14)
J1,(T
k
)
~ ,(k) =
L
In light of Proposition 14.2, we can replace the left side of (14.3) by IlullBD , =,1 and hence we can sharpen (14.2) to
,\(£),
£?:.k and we can also state the conclusion of (14.13) as (14.15)
(14.22)
+ Ckllullcz,
IIPullu>o :::; C,(k)llullc(A)
k ~ 1.
I;j
00,
and
PROOF. That P: CU.) -7 CP,) for such P is explicitly stated in Corollary 5.3. That P : C~ -7 C~ follows from Proposition 5.2, in light of (5.22).
'\(k)
= k- s -
1
when '\(k) is given by (14.16). One application of BKM-type estimates made in [T2] was to the following result on persistence of solutions to higher-order hyperbolic equations, of the form m-l
(14.24)
P E OPC()..) S?,u
=}
,
,(k) ~ k- s ,
.5
(14.25)
IIPuliL= :::; C1lull~?+s) Ilull~(\~+s),
PROPOSITION 14.2. In the setting of Proposition 14.1, we have
IlullBDoc:,l :::; C,(k)llullc(A)
+ CkllullcD, •
I;j
k ~ 1,
with ,(k) given by (14.14). Hence, for P E OPC()..)SP,u, (14.19)
IIPuIIB~., :::; C,(k)llullc(A) + Ckllullc~,
(14.20)
Aj(t, x, Dm-1u, DxWiu + C(t, x, D m - 1 u),
I;j
k ~ 1.
L
\[Jk(D))uIIB~.l :::; C
1I'l/J£(D)uIlL=
L e?:.k-2
aj,a(t,x,v)D~,
where the coefficients ajp (t, x, v) depend smoothly on their arguments. We take
t E~, x E M, dim M = n. The following is Theorem 5.3.A of [T2]. THEOREM. Assume the equation (14.24) is strictly hyperbolic. Let initial data be given: (14.26)
u(O) = fa, ... , a;n-lu(O) = fm-l,
!J
E Hs+m-1-j(M),
and assume s > n/2 + 1. Then there is a unique local solution
u E C(I, Hs+m-1(M)) n c m- 1(1, HS(M)). (14.27) This solution persists as long as Ilu(t)llc~ + ...
+ Ila;n-lu(t)llc~
::; K
k-2
:::; C{
L
Aj(t,x,v,D:IJ =
lal O.
IluIIH~,l ::; Cllull~?+s) Ilull~N)+s),
(14.23)
A particular case to consider is
(14.16)
::;
Similarly, by the arguments leading to (14.17), we have
Regarding the hypothesis of (14.13), we can apply Proposition 5.2 and Corollary
5.3 to obtain: PROPOSITION 14.1. Assume '\(k) '" 0 is slowly varying and L '\(k) < use (14.14) to define ,(k). Then (14.15) holds for all P E OPCP')S?'o'
o IlullB 00,1
,\(£) }lluIIC(A),
(14.30)
IlullciF :::; C s Ilullc; ( 1 +
log Ilu I1H ') 10Q' 1111.11", '
n
s>"2 + 1.
..
- - - -""--
c::::::.-=---=-..:::_~~-=::.---_._--~-_-=,.-=:===-:~"",;=--=--=-=----:::=.:....-~.,..._ ;:;...,~"..-
-
-,;.,",~~~~...,.";"",""""-~~-""",,,,,,
1. OPERATORS WITH MILDLY REGULAR SYMBOLS
92
At the time the author had not noticed that (14.22) is valid, and a somewhat ad hoc space C~ (M) was constructed. At this point we can state the following: PROPOSITION 14.3. The space B~,1 can be used in place ofC~ in the analysis
in §5.3 of[T2].
15. Variations on an estimate of Tumanov In the course of work on eR mappings, A. Thmanov [Tu] established the fol lowing estimate. Define an operator P+ on D'(SI) by
(15.1)
(l:::>keikll) = l:::akeikll.
P+
15. VARIATIONS ON AN ESTIMATE OF TUMANOV
for all r E R Also, setting Ruf = R(J, u), we have u E C~
rf. Z+, then ~ Cllfilerilullu>o.
11(1 -
(15.2)
P+)(fu)lle r
0, r
Note that (15.3)
where Mfu (15.4)
u E R(P+)
= fu, so
===}
(1 -
P+)(fu)
=
[Mf' P+]u,
(15.2) is equivalent to the commutator estimate
Cllfller Ilullu>o, \j u E R(P+). Holder spaces cr by Zygmund spaces C: and then these esti
I [Mf , P+]ulle
PROPOSITION 15.2. If 0
r
~
.(£).
k?£
Then, for f E CP')(Sl), P+ as in (15.1), (15.20)
II[Mt , P+]ullcc>.)
If, in addition, w(t)C OPS?,o(lR n ), (15.21)
S
"'-.
S;
Cllfllc(>.) (1Iulk"" + IIP+ullu>o + Ilullc(,,)).
for some s
then, g'iven f
E (0,1),
E
CP')(JR"), P
- n_-r~~~~~~~"""~~~~~~~~::~~~~~-~-~--~~~;"~~sTIMATEs- 6N-MORREY-TYPE SPACES
THEOREM 16.1. We have Vu E Mn,W(JR")
(16.4)
f
E
Mp,W(JR")
.) (1IuIIL= + IIPullLoo + Ilullcc,,)).
PROOF. This time, in the estimation of the terms in (15.8), we have
u E C{w}
Pr(D)lfl S; Cr-n/pw(r),
V r E (0,1).
On the other hand, the condition on the right side of (16.5) implies
IIPr(D)fIILOO S; Cr-,,/pw(r),
(16.6)
II[Mf,P]ullcc>.) S;
==}
Recall that C{w} is defined by (1.64) and satisfies the containment relation (1.65). In particular, we have continuity of u in (16.4) as long as J~ w(t)C l dt < 00. We prove Theorem 16.1 by a method similar to that used to prove Morrey's r2 theorem in [T2]. Consider the family of operators Pr(D) = e ,\ i.e., Pr(~) = e-lr~12. Now Pr(D) is the operator of convolution with (47Tr2)-n/2e-4Ix/rI2, so from the definition (16.3) we have (16.5)
E
95
which in turn implies
II\I!o(hD)fIILOO S; Ch-,,/pw(h).
(16.7)
Thus (15.22)
II Tpufll c (>.) + IIPTuJllc(>.) S;
Gllfll c(>.) (1IuIILOO + IIPuIIL= ),
Vu E M",W(JR n ) =? IIV x\I!o(hD)uIILoo S;
(16.8)
CW~h),
by Proposition 5.7, and (15.23)
IIR(j,Pu)llc(>.)
+ IIPR(j,u)lb>.)
S;
Cllfllc(>.)llullcc,,),
by Proposition 5.8. It remains to estimate [Tf , P]u. Proposition 15.4 is enough to yield (15.20). On the other hand, if w(t)C S "'-. for some s E (0,1), we have from (7.22) that [Tf,P] E OPBS~t (mod OPS-X) when f E CP.), and this gives (15.21). Let us recall that examples when (15.19) holds are given in (5.52)--(5.54). 16. Estimates on Morrey-type spaces
For P E (1,00), the Morrey space MP(JR") is defined by
(16.1)
f
E
Mp(JR")
3/2. Thus, for such a family of examples,
j r, 2j +lr]} ,
and j ~ 1 in the sum. We want to estimate Tf on Br(z). Clearly
and the estimate (16.22) for Ix - yl ~ 1 implies
h
a(h) =
M n,w,l Sf,o(lR n ) ~ p(x, D) : LP(lRn ) ----+ LP(lRn ), 1 < p
0. If 1 < q :S p
1, than for the classical case A(j) = 2- rj , r > 0, because OPSP I does not map CC)..) to itself for such slowly decreasing sequences. We are able to succeed via a bootstrapping argument that treats progressively larger classes of sequences A(j). In Appendices A and B we discuss some results of [AI] on composition and "paracomposition." One of our original goals was to link this material with the estimates treated in §§4-5, but in the end it seems that our task is to report on their essential differences, both in perspective and in the flavor of the results. We complement Alinhac's results with some new estimates, both in LP-Sobolev norms for p f= 2 and in Besov norms. Estimations done here involve AS, with A = VI - ~. T. Kato has pointed out that one can make use of dilation arguments to obtain estimates involving instead DS, with D = J-~. A typical case of such an argument is described in [K2]. As preparation for material in the following sections, we briefly review the analysis of F(u), for smooth F, given in [Meyl], bringing in the paradifferential operator calculus. More detailed accounts can be found in Chapter 3 of [T2] and in Chapter 13 of [T5]. Take 'lI o E CO'(lR n), 'lI0(~) = 1 for I~l ~ 1, and set 'lId~) = 'lIo(2-k~), Uk = 'lI k (D )u. Then
F(u) = M(x, D)u + F(uo)
(0.1)
._._--:::::::-::~
~
._~
--_.~_u.
__
_
._~'
INTRODUCTION
2. PARAOiFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
103
To estimate M(x,~), given u E Loo(lR n ), we have, by the chain rule,
(0.5)
IID~mkIILoo ~
L
C{3
IID{11 Uk +1 1ILoo" ·IID{1uuk+lllu>o ·11F'llclul.
Islvls\{31
Also,
II D{3j uk+111 Loc
(0.6)
~
C{1J 2
kl {1j Illullv>o.
Since 2k£ ,. . ., (~)£ on the support of 'l/Jk+l (~), we see that
(0.7)
u
E
Loo(lR n ) ===* M(x,~)
E
SP,I'
Now, as proved in [SI] (see also [Bour]),
(0.8)
p(x,~)
E Sf,\
=}
p(x,D): Hs+m,p(lR n )
---7
HS,P(lR n ), p E (1,00),
as long as s > O. Proofs of this are also given in [T2] and in Chapter 13 of [T5]. One immediate consequence of this is that, for s > 0, P E (1,00),
(0.9)
IIF(u)IIH3,p ~ IIM(x, D)uIIH9,P ~
+ IIF(uo)IIH3,P
Csp(lluIILoo) (1IuIIH"'P + 1),
In fact, the estimates establishing (0.8) show that, for 0 < (0.10)
IIF 0 uIIH3,P
~ CspKN(F, u)(lluIlH"p
< N,
S
+ 1),
with
(0.11)
KN(F,u) =
IIF'IICN(O)(1 + lI ullt'oo).
The symbol class sp 1 obtained in (0.7) lacks many desirable properties. In order to have a symbol c~lculus available, one splits such a symbol as M(x,~) into two pieces, as discussed in §3 of Chapter I: M(x,~) = M#(x,~)
(0.12)
+ Mb(x,~),
where
M#(x,~) =
(0.13)
L Jkmk(X) 'l/Jk+I(~), k
where the formula
M(x, D)u =
(0.2)
and Jk are smoothing operators (in the x variable), forming an approximate iden tity. Possible choices of A are
L {F(Uk+d - F(Uk)} k;:::O
(0.14)
where 8 E (0,1). Given r > 0, we have u E C r
yields (0.3)
M(x,~) = Lmk(X)'l/Jk+l(o, k
ml.-(x)
=
t
io
F'('lIk+r(D)u) dT,
with (0.4)
J k = 'lIo(T k"D), or Jk
~}k+ dO = Wk+l(O - 'lI k (O,
Wk+r(D) = wk(D)
+ T'l/Jk+1 (D).
(0.15)
u E Cr
=}
M#(x,~) E Sr.",
= 'lIk-5(D),
=}
M(x,~) E
crs?,o n S?,I' and
Mb(x,~) E Sl,r"·
If we take 8 < 1, then the standard symbol calculus applies. If instead we take J k = 'lI k- 5(D), then there is a replacement operator calculus, given by [Bon] and [Meyl]. We have M#(x,~) in the symbol class BSP.I' where (0.16)
p(x,~) E BSrl
{==}
p(x,~) E srI' and supp p(TI,~) C
{ITiI
~ pl~I},
104
2. PARADtFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
for some p E (0,1). A more general operator calculus has been developed in [Bour2] and [H3]. If p(x,~) E 8S1,\' then (0.8) holds, for all s E R There is the following representation of a product (d. the discussion in §3 of Chapter I):
fg = Tfg
(0.17)
+ Tgf + R(f,g),
In this section we establish the following: PROPOSITION
(1.2)
k~5
This arises from the construction (0.1)-(0.14), applied to F(f,g) Jk given by the second formula in (0.14). Clearly
f
Also, if Rf9
E LOC>
===}
= fg, and with
Tf E OP8sf,l'
f
E LOC>
===}
R f E OPS?,l'
Hence (0.8) applies to Tf and Rf. There are also the following important estimates (used already in §10 of Chapter I): (0.21)
IITfglb
~
CpllfllLPllgllBMo,
IIR(f,g)lb ~
CpllfllBMollgllLP,
for p E (1,00), which follow from work of [CM]; proofs are also given in [T2]. From (0.19)-(0.20) and the operator estimate (0.8) we have, for s > 0,1 < P < 00, the Moser estimate (0.22)
IlfgllH8,p ~
M f(x) = sup r>O
vo
I
~r (
X
)
> 0, 1 < P < 00,
IlfgllHd,P ~ CllfllL Q11lgllHd,q2
1
1
1
1
1
P
Q1
Q2
T1
T2
- = - + - = - + -,
+ CllgllL r11lfllH8,r2
J
q2,T2 E (1,00), Q1,T1 E (1,00].
This result was established in [CW]. Note that the Moser estimate (0.22), Le.,
IlfgllHd,p ~ CllfllL~
fg = Tfg
(1.3)
IlgIIH8,P + CllfllHd,P IlgIIL~,
(1.5)
IITf gII H8,P ~ CllfllLQ1 IlgIIH8,Q2, IIR(f,g)IIH8,P ~ CllfIILQ11IgIIH8,Q2.
In fact, we have, for all
S
(1.4)
E
IITf91IHd,P
JR, ro.J
II (2: 22kslllfk_sfI21'I/JkgI2) 1/211LP k
(1.6)
~ CIIMf(2:22ksl'l/JkgI2) 1/2tp k
~ CII M fll LQ1 1 (2: 22kS \'l/Jk91 2) 1/2tQ2 k
If(y)1 dy,
~
Br(x)
and Littlewood-Paley theory. We mention two results that will be used frequently. First, if ide) are supported on shells (~) 2k , then, for p E (1,00), s E JR,
+ Tgf + R(f, g).
It suffices to show that, under the hypotheses of Proposition 1.1,
Cllfllu",llgIIH8,P + CllfIIH8,pllgllu".
Compare (0.10). In subsequent sections we will see some variants of (0.22), such as a result of [CW) in §1, and another variant in §6 of Chapter Ill. Material just discussed will provide some of the tools for the analysis in the following sections. Other tools include known estimates on the Hardy-Littlewood maximal function (0.23)
S
is the special case Q1 = T1 = 00 of (1.1). We give a proof that casts the analysis in terms of Bony's paraproduct. This will provide a warm-up for results in subsequent sections. As in the approach to (0.22) sketched in the introduction to this chapter, we begin by writing
= R(f,g), a simple symbol estimate yields
(0.20)
1.1. We have, fOT
pTovided
Tfg = I>Vk-5(D)f' 'l/Jk+1(D)g.
(0.19)
105
1. A product estimate
(1.1)
where Tf is Bony's "paraproduct," defined by (0.18)
1. A PRODUCT ESTIMATE
------
CllfllL01 IlgIIH8,Q2.
Here, M is the Hardy-Littlewood maximal operator. This proves (1.4). Next, for S > 0,
ro.J
(0.24)
112: iklIHd,P k~O
~ cil (2: 22kS lik1 2) 1/2tp' k~O
If fk = 'l/Jk(D)f, the converse estimate also holds. Second, if ik(~) are supported on balls lei ~ C2 k , then (0.24) holds, for p E (1,00), s > O. Recall that these tools have been used in Chapter I.
IIR(f,g)IIHd,P (1. 7)
cil ( 2: 22kSI'l/JjfI21'I/Jk912) 1/2tp ~ cIIM f (2: 22kS I'l/Jk91 2) 1/211 Lp ~
li-klS4
k
and, as in (1.6), this last quantity is ~
CIIM fIIL01llgIIHd,Q2,
so we have (1.5).
2. A COMMUTATOR ESTIMATE
2. A commutator estimate
PROOF.
(2.1)
IIAS(fu) - fAsuilLP ~
AS(fu) = ASTfu + ASTuf + AS R(f, u), fAsu = TfAsu + TAsuf + R(f,ASu).
u E L oo
===}
T u E OPBS?,l' R u E OPS?,1
where Ruf = R(f,u), and hence
(2.4)
CllullLoo IlfllHs,p, v s E JR, CllullLoo IlfllHs,p, V s > o.
(Note that these results are special cases of (1.4)-(1.5).) Next, (2.5)
uEL
oo
,
= L...[AS, Mh]'l/Jk u
k
where (2.12)
fk = Wk-sf.
Due to the spectral properties of 'l/Jku and of fk' we can write (modulo a negligible error) (2.13)
[AS, Mfk]'l/Jk u = 2ks['l/Jt,Mfk]'l/Jku
where 'l/Jt(D) = 'l/J#(2- kD) , and 'l/J#(f.) has compact support. Thus we have (2.14)
IIASTufllLP ~ liAS R(f, u)IILP ~
s> 0 ==} TAsu
[AS, Tf]U =
and
S II[A ,Tf]uIILP
(2.15)
'"
Cllull Loo IlfIIHs,p,
VS
II (I: 22ks I['l/Jt, Mh]'l/Jkun 1/21ILP' k
Now, with Vk
IITAsufilu ~
2: 2kS ['l/Jt,Mh]'l/Jk u k
E OPBSf,l
so
(2.6)
S
~
CllullLoollfllHs,P.
We will estimate four terms on the right side of (4.2) separately, and then estimate [AS, Tf]u. To begin, we have (2.3)
= 2:{A ((w k-sf)'l/Jk U) _ (w k-sf)'l/Jk ASu }
(2.11)
This result was given in [KPV]. As before, our main desire is to re-cast the proof in the language of Bony's paraproduct, and to motivate further arguments. As in treatments of commutator estimates in [AT] and [T2], we write (2.2)
We have
[A s, Tf]U
In this section we establish the following:
PROPOSITION 2.1. If S E (0,1), p E (1,00), then
> o.
(2.16)
= 'l/Jku,
I['l/Jt, Mfk]Vk(X)1
~ CllvkllLoo
We next establish that (2.7)
IIR(f,ASu)IILP
~
CllullLoollfllHs,p,
VS
> o.
In fact, by Proposition 3.5.B of [T2], (keeping in mind that R(f, v) = R(v,f)) we have
(2.8)
IIR(f, v)IILP ~
Cllvlix
s
IlfllHs,p,
as long as
(2.9)
P E OPSf,o
==}
P: X
---->
PROPOSITION
(2.10)
(2.17)
2.2. If 8
E
(0,1), P
E
fk(Y)I'I~t(x -
y)1 dy.
J
Ifk(X) -
fk(Y)I'I~t(x -
y)1 dy
~ C2: 2£-kM'l/Jef(x).
Plugging into (2.15), we get
II[AB, Tf]uIlLP ~ CIIullc~ II (2: 22kS II:2£-kM 'l/Jefn 1/21ILP' k
LEMMA 2.3. If s
(1,00), then
IIASTfu - TfAsullLP ~ CilullczllfllHs,v.
Ifk(X) -
£9
BMO.
If XS = A-S(LOO), then this criterion holds, and applies to give (2.7). It remains to estimate [AS, Tf ]u; for this we have the following.
J
Furthermore, a result we will establish in the next section (Lemma 3.3) implies
(2.18) S
107
(2.19)
£'5.k
< 1, then
2: 22ks l2: 2£-ka£1 2 ~ C2:22k8IakI2. k
£ 0,
n cc
Setting w = u - v and re-Iabeling u + v as u, we see that, if (7.6) is correct for F(u) = u 2, then, given B C HI (']['2) n Loo(']['2) bounded,
(7.8)
u, wEB
s::
IlaullHl
IluwllHl
+ IlaullLoo = 1,
CIIG(u, v)llu'" Ilu - vllHB,p
Hence (7.8), applied to au, bw, yields
+ C(llullu"',Ilvllv",) . (1+ IlullHB,P + IlvIIHs,p)llu-vllu"'.
(7.9)
PROOF. The well-known special case ql = rl = 00, q2 = r2 = P of Proposition 1.1, applied to the right side of (7.1), dominates the left side of (7.3) by the first term on the right plus
Then, the estimate (0.9) applied to G(u, v) yields (7.3). We remark that (7.3) is both sharper and more general than the estimate (3.8) in [Sog]. It also contains the estimate (3.8') of [Sog] when n is odd, but not when n is even. However, as we show below, (3.8') is false when n is even. The following provides an estimate on F(u) - F(v) when F is only C 2 • PROPOSITION 7.2. Assume F is C 2 , s E (0,1), p E (1,00), nee IR n , we have
IF' I s::
K 1 , and IF"I
s::
K 2 • Then, for
vlllf"p + CK2 (1 + IlullH"p +
(ab)lluwIlH'
IlbwllH'
+ IlbwllLDO = 1.
:s CbllwllHl.
Dividing by b, we deduce that, for all u,v E COO(']['2),
(7.10)
IluwllHl
(7.11)
:s C(llullHl + IlullLDO )IIwIlH"
IIwV'u11L2
:s C(\IV' u IIL2 + IlullLDO )llwllHl.
Let us set uc(x) = 1/J(x/c), given 1/J E Cij"(IR2), supported in Ixl < 1/2, equal to 1 for Ixl 1/4. Let w(x) = 'l,b(x) (log 1;1)1/3, so w E H 1(,][,2) but w ~ LOO(']['2). (Identify opposite sides of the unit square in 1R 2 to get ']['2.) Then
:s
(7.12)
IIF(u) - F(v)llu,',p(O)
s:: CKIilu -
:s C(B)llwIlHl.
Hence, this estimate must hold for all u E HI (']['2) n C (,[,2), W E HI (,][,2), We will show this is not possible. Since IIuwllL2 and IluV'wllu are both dominated by the right side of (7.10), we see that (7.10) holds if and only if
Cllu - vllu'" IIG(u, v)IIHB,P.
(7.4)
(7.5)
===?
Now, for arbitrary nonzero u, wE COO(,][,2), we can pick a, bE 1R+ such that
IIF(u) - F(v)IIH8,p(o) (7.3)
117
PROOF. As before, the product estimate dominates the left side of (7.5) by the first term on the right plus (7.4). Applying Proposition 4.1 to the estimation of G(u, v) in this case gives the rest of (7.5).
(7.6)
7. Estimates on F(u) - F(v)
(7.2)
F(u)- F(u)
ASSERTION. Let B be a bounded subset of H 1 (']['2)nLoo(']['2), Fa smoothfunc tion. Then, for u, v E B,
so (6.19) is proved.
(7.1)
ESTIMATES ON
IIV'ucllu
+ IlucllLoo
=
Co,
independent of c E (0,1]. Thus the right side of (7.11) (with u = u c ) is bounded
IlvIIH"p)llu - vlluX>.
for c E (0,1]. However, the left side of (7.11) blows up like (log ~) 1/3. This shows that (7.11) is false; hence (7.10) is false, so (7.6) is false.
118
8. A PSEUDODIFFERENTIAL OPERATOR ESTIMATE
2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
8. A pseudodifferential operator estimate Here we combine methods of [Bour] with methods of §1 to produce estimates on p(x, D)u. As with a number of estimates in Chapter I, we begin by considering an "elementary symbol:"
q(x,~) =
(8.1)
2: Qk(X)'Pk(~),
Hence (by a variant of (4.8»,
:s
I~I
+ Qk2(X),
(8.4)
cll :s cll
Ql(X) ~
sup {(k,~J:2k~I~I}
(8.15)
IIQlllLql
(2: 22ks l'Pk U 2f/2 tp'
(8.16)
Q2(X) ~
sup
(8.17)
Q2(X)
Ql(X) = sup IQkl(X)I·
q~~(x)
(8.18)
:s c
Mq~~(x),
= sup (O-T IIDxlaq(x,~)I. ~
Hence
:s GlIQlllLOl IluIIHS,q2,
Ilql(X,D)uIIHs,p
Hence, for Tl E (1,00],
s > 0, P E (1,00),
provided that 111 - = - + -, P ql q2
ql E (1,00], q2 E (1,00).
Ilq2(X,D)uIIHs,p:S CII(L:2
2jS
Let us assume that, for some a
)n
I L: ('l/JjQk2)('Pk U
j
k~j-4
> 0, T
~
l 2 / tp'
0
j
l'l/Jj(D)Qk2(X) I :s CT a+kTQ2(X),
Vj 2 k
+ 4.
(8.20)
)
2j
!L
(S-a J
k") rv
2k.
8.1. Assume 0 < s < a, T ::::: 0, 1 < P < 00, 1 1 1 1 1 - = - + - = - + -, Ql,rl E (1,00], Q2,r2 E (1,00).
PROPOSITION
(8.23)
p
ql
q2
Tl
(9.10)
p*(X) = sup sup (~)laIIDrp(x,~)I,
e
F'(v) = MF'(v; X, D)v v E L oo
(9.11)
+ R(v),
=? MF'(v;x,~) E S~,I'
Now applying Proposition 5.12 of Chapter I, we have
p~~(X) = sup sup (~)-'-+laIIIDxl'),2»)' x
provided A2 (j) "'" and
IIp(x, D)ull H8,P
.s Clip· II Ln Ilull Hs,Q2 + Cllp~~ IluI IluiIHs-("-1"),T2'
2: A(f)A2(f) .s CA(j).
(9.13)
C?j
9. Paradifferential operators on the spaces CU.) Let F be a Coo function on R K . If u is a function (with values in JRK) with C(>,L regularity, we will analyze F(u) as a paradifferential operator. As in (0.1)-(0.4), we can write
(9.1)
F(u) = F(uo)
+ [F(Ul)
- F(uo)]
+ ... + [F(Uk+d -
with Uk = wk(D)u, and then we write
(9.2)
F(Uk+l) - F(Uk) = mk(x)'l/Jk+l(D)u,
1 1
mk(x) =
+ ... ,
(9.14)
F'(Wk(D)u + T'l/Jk+l(D)u) dT.
C A2(k) Ilvkllc(A),
Ilmkllc(),) .s CFf(lluIILoo) (1
+ A2~k) Ilullc(),»).
L A(j) < 00, so CU.) c
Loo, we have
(9.16)
MF(U;X,~) E C(.X)S~.(I.. ",,) Sf,(T)'
T(2
k
=
)
===}
(9.37)
----+
> 4/3.
L 'x(P)'x2(p)2-1/v ::::; C,X(j)'x2(j)1-1/V, L 'x(k)'x2(k)-1/v < £~j
p(x, D) : C(,\)
'x(k) M(k)
2. The classical case, with C(,\) = C;', is (9.23)
C(A)
We can iterate this argument. Suppose 'xM satisfies (9.30), i.e.,
,X(j) =
(9.22)
9. PARADIFFERENTIAL OPERATORS ON THE SPACES
IIF(u)llc'Ai : : ; C(lluIILoo) (1 + IIUllcCA)
t
We see that (9.30) holds for .A, ,XZ as in (9.22), with s > 3/2.
00,
IIP(u)IIC(A) : : ; CF(lluliLoo )(1 + IluIICCA) )"'+1.
We see that (9.37) holds for (9.40)
'x(j)=F s ,
'x2(j)=Fl,
s>l+!. v
We now produce further information on the symbol MF(u; x, 0 when u E C(,\) and ,X(j) satisfies the hypothesis (9.37). Then, parallel to (9.28), we have (9.41)
IIP'(vk)llc(A) : : ; CF/(ll vkIILoo)(l + Ilvkllc(A))"'+\
which we can use in place of (9.12) to obtain (9.42)
IImk IIC(A) : : ; CF' (1IuIILoo) (1 + IluIIC(A») V+1,
124
2.
u E C(>')
(9.43)
===? Mp(u;x,~) E C(>')S~,o,
(9.53)
a substantial improvement over (9.16). We next obtain some more estimates on D~mk(x) and hence D~Mp(u;x,~). From the chain rule, we have
L
D~G(v) =
C(/h, ... , f3/-l)(Di31v) .,. (Di3,.v) (D~G) (v),
i31 +..+i3,.=i3
(9.45)
1 IIDi31 Vll c (A) .. , /IDi3"vllc(A) (1 + Ilvllc(A) r+ .
L
:::; Ci3 ,p (1IvllL'''')
i31 +"'+i3,,=i3 This uses (9.39) plus the fact, established in Corollary 5.4 of Chapter I, that C(>') is a Banach algebra (as long as ~ A(j) < 00), so that
(9.46)
IlvwIIC(A) :::; Cllvllc(A) Ilwllc(A).
(9.54)
(9.56)
£
= (h w(t) dt.
10
t
IID~mkllc(A) :::; Ci3,p(lluIILoo)(l + Ilullc(A))!i3I+v+!2kli31.
(9.57)
IID~'l,b£(D)mkIlLoo :::; C(llullLoo )2 k1i3I A(k),
k - 4:::; £ :::; k + 4,
IID~(I - Wk+2(D»mkIILoo :::; C(llullc(A»)2 k1i31 )'(k),
), =
L
A(£).
These estimates yield (9.53)-(9.55). We note that the proof of Proposition 5.12 of Chapter I shows that
(9.59)
M~(x, D) : C(/-l)
---+
C(/-l2) ,
P,2(j) =
L A(£)p,(£) , £?J
IID~F'(Vk)IIC(A) :::; Ci3 ,F(ll v kII Loo )2 k1i31 (1
+ Ilvkllc(A) )1i3I+ v+1.
u E C(>')
===?
IID~D, Mp(u; "~) Ilc(A) :::; Ci3 ,p (1Iullc(A)) (~)Ii3HQI.
We are ready to record the following further properties of
Mp(u;x,~).
PROPOSITION 9.2. Under the hypotheses of Proposition 9.1,
u E C(>')
===?
Mp(u;x,~) E S~1 , nc(>')s~o' ,
Furthermore, we have the decomposition
Mp(u; x,~)
=
=
+ Mb(x,~) M#(x,~) + M~(x,~) + M1(x,~), M#(x,~)
where the terms on the right have the following properties:
(9.52)
w(h)
= sup A(£)-III'l,b£(D)Di3vk II Loo
It follows that mk(x) in (9.6) has such an estimate, and hence, for A(j) satisfying (9.37),
(9.51)
= A(j),
£?k-2
so (9.45) yields
(9.50)
w(TJ)
Hence
£
(9.49)
(0-\10(0 = W((~)-I),
PROOF. We already have (9.50) and (9.52), from (9.42). To proceed, note that (9.48) gives
:::; C 2k1i3 !ll vkIlc(A),
(9.48)
(O-'NO = W((~)-I),
with
(9.58)
:::; C sup A(£)-12 k1i31 /1'l,b£(D)VkIILoo
(9.47)
M~(x,~) E S~r,
and
Now, if v = Vk has the form (9.9), we see that
IIDi3 Vk II C(A)
M~(x,~) E s~t,
where
(9.55)
the sum being over f3v > 0, if f3 > O. We can deduce that, if A(j) "" satisfies (9.37), then
IID~F'(V)IIC(A)
125
M~(",,~) has support in 2-41~1 :::; 1",1 :::; 241~1, Mg(",,~) has support in 1",1 ~ 21~1, and
and hence, for A(j) satisfying (9.37),
(9.44)
A. PARACOMPOSITION
PARADIFFErtENTIAL OPERATORS AND NONLINEAR ESTIMATES
M#(x,O E
ssP. 1 ,
liD, M#(-'~)lIc(A) :::; CQ(~)-IQI,
and (9.60)
M~(x, D) : C(/-l)
---+
C(/-l3) ,
P,3(j) = p,(j)
L A(j). £?J
We have P,2(j) :::; Cp,3(j), and typically
P,2 ::::::
P,3·
A. Paracomposition In this appendix we discuss a construction of [AI], applied to a composition F 0 u, and extend some of the estimates given there. The basic thrust of this material is somewhat different from that of §§1-5, despite a similar appearance of the objects under study. For one thing, it is necessary to assume here that u is a diffeomorphism. As we will see, that assumption will playa 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 lIullc- can be inter preted in that light. Also, subtracting off a smooth function (whose composition with u is estimated by previous techniques) we assume F(O) = O.
126
2. PARADIFFERENT1AL OPERATORS AND NONLINEAR ESTIMATES
To begin, set Fj = 'lj;j(D)F, and write
(A.l)
F
0
U=
L
We decompose the double sum into (A.2)
Hence
i.e.,
2: j k . Note that, due to cancellation,
:l]Fj(Wk U) - FJ (Wk-1 U)] j?>k
=
F,u E Lip 1
(A.12)
===}
(A.13)
L[Fj(Wk U) - FA Wk-1 U)] = '2)(Wk-1 F )(WkU) - (Wk-1 F )(Wk_1 U)]. jl
Csp(lluIILipl) IIF'IIL= IluIIH',p, II F (u; x, D)ull c: ::; Cs (1lullLipl ) IIF'II L= Ilullc:.
IIF(U; X, D)uIIHs,p ::;
Now a variant of Alinhac's "paracomposition" operator is given by (A.14)
Hence
u* F
L F(U;x,D)uIIBSp,q :s;
Cspq(l/uIILipl) IIF'llLoo IlullBs .
PROOF.
p,q
In view of the location of the spectrum of the various terms in (A.l 7),
we have
js jsq 112 VJj(D)(u* F)IIIp :s; C2
CIIF'I/c;·lluIIH8,P.
:s; C2
PROPOSITION
u~ FIIHs+r,p :s;
cil
I
:s; CIIF'lle: (L:: 22ks1M(1 - Wk)U I2 k
f/
2 tp'
II (L:: 22kSlL:: MVJjUn 1/2tp :s; cil (L:: 22kSIMVJkUI2f/2tp' k
A.9. If 1
j?k
k
and s
> 0, we have
IIVJj(D) (h(1 - Wk)U) II~p
L:: Ik-jISN
:s; CIIF'II~:
L::
js 1/2 (1
-wk)uIIIp·
Ik-jISN
To obtain (A.51) from this, it suffices to establish
L:: 112 ks (1 -wk)ulilp :s; Cs L:: 112 ks VJk(D)uIlIp,
(A. 53)
LP-Sobolev estimates on RF,u seem less accessable from the results of Appendix B than Zygmund space estimates. However, Lemma B.2 does readily yield Besov space estimates. Given 1 < p, q < 00, we define the Besov space B;,q (~n) to consist of v such that
which in turn follows from
Ilvll'hs = 2: 112 js VJj(D)vIIIp(lRn) < 00.
< p, q < 00
We have j 112 (s+r)VJj(D)R},ulllp :s; C2 j (s+r)q
provided s > 0, by (4.9). Making use of (2.24), we thus get (A.42).
(A.46)
L:: IlFklllp,
PROOF.
(A.52)
We dominate the last factor by (A.45)
jsq
1 lIu* F - u* FIIB;:qr :s; C(IIDullv"', IIDu- I/Loo) 11F'lle: IluIIB~,q'
(A.51)
(L:: 22k (s+r)loM(I - Wk)U(X),
we have
lIu* F -
IIVJj(D)(Fk 0
!k-jISN
l 0,
k
L::2 k
kSq
(L::la£lf:s; C.• L::2kSqlakIQ, £?k k
s
> 0.
This estimate is proved by the same argument as used for (4.9).
p,q
J
Then the estimates (B.15)··(B.16) yield the following analogue of Lemma A.3:
Thus we have Besov space estimates for all the terms in (A.34). One has (A.55)
00,
C(IIDullu>o, IIDu- 11Iu,o)IIFIIBs .
(A.50) PROOF.
P,q
A.8. Under the hypotheses of Proposition A.4, if 1 < p, q
0, we have
(1,00) and s
C(llulb, IIDu- 11Iu"o) 1IFIIBr+l.
The boundedness of operators in OPSP 1 on Besov spaces (for s > 0) is well known; see, e.g., [Maj. We recorded the pro~f for the case q = 1 in §12 of Chapter 1. We deduce from (A.12) that, for 1 < p, q < 00, s > 0,
PROPOSITION
We next obtain LP-Sobolev estimates on R},u.
A.6. If p E
00,
We now obtain Besov space estimates on the other terms in (A.34).
= C(IIDu-11Ivxo). This gives (A.39).
PROPOSITION
131
A.7. Under the hypotheses of Lemma A.3, if 1 < p, q
0, we have (B.1) II'l/Jj(D)(Fk 0 wku)IIL"" :::; Cv(llulles) Tjv 2k(v- s+l) IIHIIL"". LEMMA
IAkv(Y, 1],~) t
(B.lO)
:::;
C k2k(v-s+1) on 1~12 + 11]1 2= 1,
with similar estimates on all (1], ~)-derivatives, assuming u E cs. Next, on a box containing supp 'l/J(~)'l/J(2j-k1]) for all j 2 k, write
L
Akv(Y, 1],~) =
(B.ll)
akvo:/3(y)eiO:'~+i/3'TJ,
(o:,/3)EA
where A is an appropriate lattice. Note that Furthermore, if u is a diffeomorphism, u' is uniformly continuous, and (u')-1 uni formly bounded, then there exists K, depending on Ilulle. and IIDu-111 L"", such N lakvo:/3(Y) 1:::; CN2 k(v-s+l) (1 + lad + Il3Ir . (B.12) that, for k 2 K, We see that (B.8) (hence (B.4)) equals (B.2) Ilwk-N(Fk 0 wku)IIL"" :::; C(llullcs, IIDu- 1 1Iu>o) Tk(s-l) IlFkllLoo. (B.13) In view of the localization properties of 'l/Jj(D), it will suffice to establish such 2- jv ei(Vk '17-YHx,~) eiTj a.~+i2-j/3'TJ akv o:/3 (y )'ljJ(2- j ~)iA( 1]) d1] dy d1; estimates when we throw in a factor Ao(Y) E Coo, i.e., consider estimates on 0:,/3 'l/Jj(D) (Ao(y)(Fk 0 WkU)), etc. Also, to simplify notation, set = 2- jv Fk(Vk(Y) + Tj (3)ei(X-Y+Tjo:Hakvo:/3(Y)'l/J(Tj~)dy d1; (B.3) Vk = WkU.
L //
L/ a,/3
The proof of Lemma B.1, taken from [AI], makes use of some techniques one encounters in the theory of Fourier integral operators. To begin, write
= 2- jv
L/
Fk(Vk(Y)
+ 2- j (3)akvo:/3(Y) 2jn~(2j(x
- y)
+ 0:)
dy.
0:/3
'ljJj(D) (Ao(Fk OVk))(X) (B.4) = / /
= / ei(x-Y)'~'l/J(Tj~)Ao(y)Fk(Vk(Y))
ei(x-Y)'~'l/J(Tj ~)Ao(Y )Fk (1] )eivdY)'TJ
d1] dy d1;.
Set
d~
By (B.12), this is dominated by the right side of (B.1), so (B.l) is established. To establish (B.2), we analyze a quantity like (B.4), with 'l/J(2-j~) replaced by W(2- k +N ~). Our hypotheses on u and u- 1 imply that, for N large enough, and k large enough,
IV~(Y)1] - ~I 2
(B.14)
Sk = Sk(Y, 1],~) = Vk(Y)1] - y.~, 1 v~(y)t1]-~ L k = Lk(Y,1],~,{)y) = '1 '()t 1"12 • \J y . t V k Y 1]
(B.5)
Note that, if N is picked large enough, then (B.6)
IV~(Y)1] - ~I
provided j 2 k
+ N.
2 C(I1]1 + IW on supp 'l/J(Tj~)'l/J(2-k1]),
In fact, it suffices that
2
N
> sup Iv~(y)l.
Furthermore,
C(I1]1 + IW
k
on supp W(T +N~)'l/J(Tk1]),
so we have an argument parallel to (B.8)-(B.13), proving (B.2). Behind (B.14) is the fact that, at least for large k, DV"k1 is uniformly bounded. Then (B.13) also yields LP estimates. We record the variant of Lemma B.1 so produced.
B.2. If u is a diffeomorphism of class CS, s > 1, and Du- 1 is uni formly bounded, then there exist N, depending on IIDuIILoo, and K, depending on Ilulle and IIDu-11ILoo, such that, for k 2 K, j 2 k+N, 1/ 2 s-l, and 1:::; p:::; 00, LEMMA
B
(B.15)
II'l/Jj (D)(Fk 0
WkU) IILv
:s Cv (Ilulles, IIDu-11ILoo ) Tjv 2k(v-s+1) IlFk IILv,
and LkeiSk = eiSk .
(B.7)
Thus, we can integrate by parts many times, and write (B.4) as
(B.8)
dy
!!
ei(Sdxe) Akv(Y, 1], ~)'l/J(Tj ~)A(1]) d1] dy d~,
(B.16)
Ilwk-N(Fk
0
wku)IILV :::;
C(llullcs, IIDu-11ILoo) Tk(s-l) IlFkllLv,
t.m.Ei,:;:a:..;,;;""'~ii.,:.;;_~~~~~;~:~,:~~i;;,,~..;~~.:;;,.,T""'~
•.
,~::::;,:::~~;:;:,~;";~;::;;:';:~i::Z;;;~: :~.;;;:;:;;..,~~"",,,.·",
... ", ;.'_;.., .. '.......,""""","''''c"...".;"(j)7(2
j
)
< 00,
and
lim 7(0- 1 = 0,
(1.7)
I~I-+oo
IDf7(01 S CQ7(~)(~)-IQI.
We recall from Chapter I that these hypotheses imply operator bounds on various function spaces, for example OPC(A)sf,(T) : C(A) ---> C(A).
°
One example to keep in mind is (the T = case of) the example discussed in (3.28)-(3.32) of Chapter 1. Assuming a < 'Y < s - 1, we take
(1.8)
'>"(j) =
y-s,
(~)-ljJ(~) = (10g2(~))-(s-l-r),
7(~) = (10g2(~)r-l-r.
Then w(2- j ) = y-s and 0-(2- j ) = j-(S-I). Another, more classical example is the case .>..(j) = 2- jr , T E jR+ \ Z+, in which case A; E CT Si,8 T 0, one has -(~u,v) (du, dv) + (d*u, d*v), which gives in local coordinates an expression of the form
~u = g-I/2{ &j(gjkgl/2&k U) + &j(blu) + bl&ju + cu}, where bi and c are matrix-valued functions, involving both the coefficients of the I. metric tensor and their first-order derivatives. Thus gjk E =} bi, c E In Chapter 7 of [Mor] there is a treatment of these operators, for manifolds with Riemannian metrics in Lip l(M), with data in Sobolev spaces H-I,P(M), p 2: 2. There is also a discussion of the related Hodge decomposition. We will not dwell on such problems further here. But see Proposition 2.5 in the next section for some results related to the Hodge decomposition. We now produce some Sobolev space analogues of the results above. Our first result requires very little regularity for the metric tensor.
C:
C:-
PROPOSITION 1. 7. Let M be a compact, connected Riemannian manifold with metric tensor of class LDO n vmo. Then ~
(1.28)
: H1,P(M)
H-1,P(M),
---+
1')S~,(T)
B(x,D): HS'lb,P
=}
---+
Hs!jJ,p
so we can readily verify that Fourier multiplication by 'l/'(~)-1 is compact on LP(1['n), for 1 < P < =, and this implies the desired compactness on HS1/!,p. Given that A is Fredholm in (1.39), the rest of Proposition 1.8 follows by reasoning similar to that used in Proposition 1.7. In particular we make use of the identity (H-l+s,p,P(M))' = H 1 - s'!J,P' (AI).
ip(()),
where ip E CO'(lR n ) is equal to 1 on a neighborhood of 0, then
(1.36)
E
ID"T(~)-11 :::; Cc«~)-Ic.) (M)
==}
==}
u
u E Hl+s,p,q(M),
E
C 1 ,(>.) (M).
PROOF. If the hypotheses of (1.46) hold, thenlet v E H1,q(M) be the unique solution to Lv = f. Hence u - v E H 1 ,P(M) solves L(u - v) = 0, so u - v = 0, and we get the conclusion of (1.46). The demonstrations of (1.47)-(1.48) are similar.
We claim that, given (1.6)-(1.7),
(1.41 )
=,
Fredholm of index zero. If we assume V has the property that L is injective on H1,2(M), then L in (1.45) is injective for 2 :::; p < 00, hence an isomorphism for such p, hence, by duality, an isomorphism for all p E (1, (0). An example of this is when V :::: 0 on M and V > 0 on a set of positive measure in M (which is assumed to be connected). For such L we record the following global regularity result.
The next result obtains somewhat stronger conclusions, granted more regularity for the metric tensor.
A: H1+ s1/J,P(M)
1
V
U E Clac
I ,().) ("'"')
u = E#(gl/2 f)
+ E#(//2Vu) -
L(ipu) = ipf + OJ (gjk gl/2(Oktp)U)
+ (Ojip)gjkgl/2(OkU).
Under the hypotheses of (1.49), the right side of (1.52) belongs to H-I'Pl (M), where PI = npl(n - p) if P < nand q ~ PI, with obvious modifications in other cases. Then Proposition 1.9 gives tpu E H 1 ,Pl(M), for all such ip, hence u E Hl~~l(O). A finite number of iterations of this argument gives the conclusion of (1.49). The demonstrations of (1.50)-(1.51) are similar, once we note that, by (1.49), if the hypotheses of (1.51) hold, then u E HI~~(O) for all q < 00, hence u E Cl~-:E(O) for all E: > 0, so the right side of (1.52) is in C-l,().) (M). REMARK. We mention that the implication (1.49) refines the r '\, a limit of Theorem 2.2.H of [T2]. A similar result is also established in [DiF]. A regularity result with a flavor similar to (1.49), but for operators in nondivergence form, was given in [CFL]. When
C().)
= cr, 0< r < 1, (1.50) gives the following result.
PROPOSITION 1.11. Assume the metric tensor on M is of class cr, 0 < r < 1. Then, given 1 < P:::: q < 00, -1:::: a < -1 + r,
(1.53)
1 u E H lac' ,P(0) Lu
=
f E Ha,q(O) lac
===}
2+a,q(0) u E Hl ac'
This result is established in [MT4], with the hypothesis on u weakened from u E Hl~~ to u E Hl~~' T > 1 - r. We now establish some estimates for metric tensors of class Hr,p, which will be of use in §1O. For simplicity of notation we use HI,q for H/';~(O), etc. PROPOSITION 1.12. Assume the metric tensor on M is of class HI,p, with p > n. Then
(1.54)
u E HI,q, q> 1, Lu
=
f E LP
==?
u E H 2,p.
L E#ojA~u,
mod Coo,
where, for some 8 E (0,1),
(1.57)
A b E OPH I ,Psl-a8
E# E OPS;),
)
1,6
n
'
a=l--. p
v.
PROOF. Pick ip E Co(O). We have
(1.52)
+ gl/2VU,
and, as in the proof of Proposition 1.1, we proceed as follows. We have
and I ,().)(,,",).
145
PROOF. Since £P c nr n, we see from (1.49) that u E HI,r for all r < 00. In particular we have u E HI,p. Now we write Lu = f as
(1.56)
S u E Hl+ 1/!·q(O) la c'
INTERIOR ELLIPTIC REGULARITY
Here we use the symbol class discussed in §8 of Chapter 1. As noted there, in (8.2), we need to pick 8 > a small enough that 1 - 8 > nip. The result on AJ given in (1.57) follows from Proposition 8.2 of Chapter 1. Now we have E#(gl/2 f) + E#(gl/2VU) E H 2,p. Furthermore, by Proposition 8.1 of Chapter I, AJ : Hr-a8,p ---. Hr-I,p for -(1 - 8) < T - 1 :::: 1, so
(1.58)
E#OjA~ : H r - a6 ,p ----. Hr,p,
8 < T:::: 2.
Applying this once to u E HI,p, we have u E Hl+a6,p. Iterating this argument a finite number of times, we obtain (1.54). The following can be proven in a similar fashion. PROPOSITION 1.13. Assume the metric tensor on M is of class Hr,p, r > 1, rp> n. Then
(1.59)
u E HI,q, q> 1,
~u =
f E Hr-I,p
===}
u E Hr+l,p.
REMARK. An alternative approach to proving Propositions 1.12-1.13, not us ing the results stated in §8 of Chapter I, can be given along the lines of Proposition 1.16 below; ef. also Proposition 9.4 in §9 of this Chapter. Next we look at a second order elliptic equation in nondivergence form:
(1.60)
L Ajk(x)OjOkU
=
f.
Let us denote the left side by Lu. We assume Ajk E smoothing, we write
(1.61)
L=L#+L b , L #( x,.,C) E S21,6' L b() X,~ E
C().).
Performing a symbol
C().)S2-,¢
I,(r)"
We take 'ljJ and T as in (1.4)-(1.5). Also, we &''lsume that (1.6)-(1.7) hold. The ellipticity hypothesis .on L implies L# E OPSr,8 is elliptic; let F# E OPS~~ be a parametrix. Using thiS, we establish the following regularity result.
146
3.
APPLICATIONS TO PDE
1.
C;
f
E
for some
CC5.) =;.. u E C 2 ,(5.).
(1.63)
As we know,
f
(1.64)
f, then, parallel to (1.11), we have u = F# f - F# Lbu mod CX].
E CO')
=}
F# f E C
2
1',
s
>0
and p E (1,00). Assume u solves
Lu=f,
with
PROOF. We have u E Hm+u,p with this by stages. Write (1.75)
Next, we establish a Fredholm result.
-;
C()..) (ll'n)
F#L=1+F#L b,
(1.66)
C()..)
to
C 2 ,()..).
(1.67)
-;
LF#=1+LbF#,
C()..)
compact.
F# L b : C 2 ,(),)
-;
C 2 ,()..)
compact,
and (1.69)
L#
= L#(x, D)
f"
of 0
EL
E:
>
O. We will improve
E OPBS';':I'
elliptic.
#
-r
=1+F,
FEOPS 1 ,1'
< l' < 1. Hence
(1.78)
u = Ef - Ell>(a, u) - Fu.
Now
f E HS'P
==}
Ef E Hm+s.P,
u E Hm+u,p
==}
Fu E Hm+u+r,p,
(1.79)
(1.80)
since we have m L b F#
: C()..)
- ; C()..)
compact.
Thus F# is a 2-sided Fredholm inverse of L. There are also analogues of Propositions 1.7-1.10, involving L : H 2 ,P(M) ---7 U(M), etc. The resulting analogue of (1.49) was established in [CFL]; see also §5 of [T4] for further discussion. We next consider operators of the form (1.70)
for any
-E:,
and
Hence we have (1.68)
=
2:: Tao< DO:u + (2:: R(ao:, D"u) + 2::TDo(a, u).
(1. 77)
Note that
1,(r)'
L b : C 2 ,()..)
(J
We can take E(x,~) E S~r', with E(x,O = L#(x,~)-l for 1~llarge, and, by Proposition 6.1 of Chapter I, since ao: E cr, E = E(x, D) satisfies
modulo operators that are infinitely smoothing, and hence clearly compact. Since Lb(x C) E C()..) S2-,p we have ,'"
Lu =
(1.76)
is Fredholm.
PROOF. The operator F# in (1.63) maps
E HS'p.
We have
PROPOSITION 1.15. If Ajk E C()..)(1l'n), L = LAjk(x)OjOk is elliptic, and the conditions (1.6)-(1.7) hold, then (1.65)
f
C m - 1,1,
u E Hm+s,p
(1.74)
where Ji, is as in (1.10). Now (1.63)~(1.64) yield u E C 2 ,(Mll, with Ji,1(k) _ ma:x(Ji,(k), ~(k)). Iterating this argument, as in the proof of Proposition 1.1, we obtain (1.62).
L: C 2 ,()..) (ll'n)
E
Then
E C 2 ,(M),
u E C; =;.. Lbu E C(M) =;.. F# Lbu
u
(1. 73)
FUrthermore, parallel to (1.14)-(1.15),
,(.\).
n HS'P,
ao: E C r
(1. 71)
(1.72)
PROOF. If Lu =
147
PROPOSITION 1.16. Assume L in (1.70) is elliptic. Assume
PROPOSITION 1.14. Assume u E satisfies the elliptic system (1.60), with Ajk E C()..). Let ~(k)/..\(k) / . Assume that the conditions (1.6)-(1.7) and (1.10) hold. Then (1.62)
tN'TERIOR ELLIPTIC REGULARITY
Lu =
2::
ao:(x) DO:u,
lol.:=m
when the coefficients have simultaneously some Holder continuity and certain Sobolev regularity,
(1.81) for
+ (J + r > O. To analyze Ell> (a, u), note that
u E Cm-1,1
==}
DO:u E LX
==}
TDo
C;+Tl5.
Thus EV b is compact on C;(M). Also, clearly B : C;(M) --t C2(M) is compact, since it can be factored through the compact inclusion C;(M) 0, VyFX
E £P
==}
00,
It is useful to complement these results with a regularity result.
P,W.,;;;;_,r.:
151
then
X E H
The interest in estimating X in the Zygmund class element of C; has the modulus of continuity:
Ilu - ullc; :::; Cllfllcz'
."ii_·,,;;;~_::
SOME NATURAL FIRST-ORDER OPERATORS
If the metric tensor belongs to Lip l (M), and 1 < P
1. Ifr > 1, and BE LX; in (2.6), then
generate uniquely defined flows. A proof of this classical result can be found at the end of Chapter 1 of [T5]. (2.16) u E L=, Vu E C? ==} u E As a consequence of these observations, Proposition 2.4 is related to studies of Ifr > 0, BE L= in (2.6), and 1 < P < 00, then
quasiconformal mappings. In fact, the special case of Proposition 2.4 in which AI = sn, with its standard smooth metric, has played an important role in such studies; (2.17) u E C;, Vu E £P ==} u E HI,p. see [McM], [Rei]. Proposition 2.4, specialized to M = STL, contains Theorem A.I0 of [McM]. PROOF. We have, modulo C=, Another geometrical example of interest is d + 8, acting on differential forms, u = EVu - EVbu - E13u.
(2.18) i.e., sections of A = tBAjT*(M). Here, d is the exterior derivative and 8 is its formal adjoint, with respect to a given Riemannian metric. In local coordinates, d + 8 has Of course, Vu E C? =} EVu E C;. Also, u E cs =} E13u E C;. It remains to
the form (2.5), where Aj is expressed in terms of the metric tensor gjk and B in analyze EVbu . Since Vb E OPCrst.6r 0,
DTFX E
02
==}
If the metric tensor belongs to C' (M) for some r > 1, then (2.23)
X E LX, DTFX E
02 ===? X
II
X E C;.
E 0;.
f'.:.'
j "
i' !
'\",
': '
N(d
+ 8) =
{u E C;(M,A): du
= 0,8u = O}.
PROOF. Only (2.29) remains to be established. When the metric tensor is Lip schitz, it and its volume element can be used to implement the duality of H1,2(M, A) and H- 1 ,2(M, A), and we have
u E H 1 ,2, wE L 2
==}
(8u,w) = (u,dw).
Hence (2.30)
u,v E H 1 ,2(M,A)
==}
(8u,dv)
=
(u,ddv)
=
0.
Now, if u E N(d + b), we know by (2.28) that u E HI,2, so (2.31)
If p;::: 2 we can take the inner product of both sides with ou/oz, obtaining
I (d + b)ulli2 = Ii du lli2 + Ilbulli2 + (du, bu) + (bu, dU) = Iidulli2 + Ilbulli2,
(2.40)
and this implies (2.29).
(2.32)
(2.41)
B
=
ou ou 8Z - A(x, y) oz'
IIAIILOO
(2.33)
S; k
< 1.
H- L q(1I'2)
PROPOSITION 2.7. Assume >"(j) "'" 0 is slowly varying and satisfies (2.42)
Iv(x, y)1 S; C(z)-1,
L >..(j)a < 00,
for some a < 1.
Let A E C(A) (1I'2) satisfy (2.32). Then
satisfying (2.35)
uz ,
Uz -
1 E LP(JR 2 ),
(2.43)
for some p> 2.
B : CI,(A) (1I'2)
---t
C(A) (1I'2)
is Fredholm, of index zero, and Ker B consists of constants.
The map u is a quasiconformal homeomorphism of C onto itself. We refer to [Ahl] or leG] for a proof of the result stated above. Here we estab lish further regularity results for solutions to Bu = 0, given some mild regularity hypotheses on A. First, we establish some Fredholm properties.
PROOF. If we take 7(2 j ) = >,,(j)a-l, then the hypothesis (2.42) implies that the conditions (1.6)-(1.7) hold. From here, the proof is exactly parallel to the proofs of Propositions 1.2-1.3. Here is a third Fredholm result.
PROPOSITION 2.6. Let A E L OO n vmo(1I'2) satisfy (2.33). Then, for all p E
PROPOSITION 2.8. Assume A E B~,l (1I'2) with s > 0, 1
(1,00),
B : H I ,P(1I'2)
(2.36)
---t
(2.44)
:
H I,P(1I'2)
---t
p(1I'2),
BT =
:z - :z' 7A
(2.45)
---t
S B p,l (1I'2)
q(x,D) = BA- I : B;,l(1I'2)
---t
B;,I(1['2),
with
0 S; 7 S; 1.
We see that these operators are all Fredholm, with the same index. The index of B o is clearly zero, and hence so is the index of B. Now suppose u E HI,P(1I'2) satisfies Bu = 0, Le.,
ou OU 8Z = A oz'
s (1I'2) B : B1+ p,l
PROOF. Under our hypotheses, B~,l (1I'2) is an algebra under pointwise mul tiplication, contained in C(1I'2). Furthermore, by Proposition 12.1 of Chapter I, OPS~,1 acts on B~,I(1['2), so paradifferential operators act on these spaces. Hence, under our hypotheses, whenever f E B~,l (1['2) and f- 1 E C(1I'2) , we can deduce that f -1 E B~, 1(1['2) . Now consider
This is an operator with symbol q(x,~) E (L oo nvmo)S2[, and the hypothesis (2.33) implies it is elliptic, Le., p(x,~) = q(x,~)-I(I - LP(1I'2) is a two-sided Fredholm inverse of BA -1. Thus B in (2.36) is Fredholm. To compute the index, consider the I-parameter family of operators
BT
S; 00, sp ;::: 2.
is Fredholm, of index zero, and Ker B consists of constants.
+ 1)1/2, consider q(x, D) = BA -1 : p(1I'2) ---t p(1I'2).
PROOF. Setting A = (-~ (2.37)
0, consider the family W s of moduli of continuity, given by
( 1)
ws(h) = log h
(3.10)
-s
'
0< h ::::;
1
2'
EXAMPLE 1. Let
0, and n :::: 2,
=
Iloglxll-s-yvCK gj(x),
AS+I(j)
where Rj is the jth Riesz transform. Since R j is a classical pseudodifferential operator of order and f has compact support, we have Rjojf E C(>'s+ll(JR. n ), by s (lR~+I). Corollary 5.3 of Chapter I. Then Proposition 3.1 gives OyU E
U(Y2, X2)] ,
IU(ZI) - u(z2)1 ::::; Ci(h)
OyU =
+ h, X2)]
(Yj, Xj), yields
(3.8)
'\7f E CWS+l(lR n ) C C(>'S+l)(lR n ),
Thus Proposition 3.1 directly implies that '\7 xU E tigate
Y
Then the standard method of setting h = IZI - z21 and writing (3.7)
u(zr) - U(Z2) = [U(YI' Xl) - U(YI + h, Xl)] + [U(YI + h, xr) - U(YI
+ R(x),
where R(x) is smoother than the principal term. Thus
'
(3.15)
(3.6)
s 1
'\7f(x) = Cilog Ix ll- - O.
\j r
fies
and consider its restriction to an, which we also denote f. The set of such restric tions to an is denoted C 1,W(an). We have the following.
(3.20)
Lip(n),
PROPOSITION 3.5. Let 0. be a bounded convex domain in IR n
I
C 1 (JR n ),
---t
The specific result we need for §5 is the following:
where here PI is the solution operator (3.1), with n+ 1 replaced by n, and y replaced by Xn. Now let f be a function on IR n satisfying (3.19)
PI: cl+T(an)
(3.23)
k(
H1,P(r),
1
(4.59)
with a as in (4.44). If an is of class Cl,w, with w satisfying (4.9), then (4.60)
II(Df)*IILP(an) :::; CpllfIILP(an),
for 1 < p < 00. We now establish solvability of some boundary integral equations that will apply to the study of the Dirichlet and Neumann problems. Some of these results will be extended to other spaces further on, but the following proposition will provide a useful start. PROPOSITION 4.5. If an is of class (4·9), then we have isomorphisms
v + K,
(4.61)
Cl,w
V + K* : V(Bn)
---t
and w satisfies the Dini condition
LP(Bn),
1 < p < 00.
If an is of class C1+r, r E (0, I)! then we have isomorphisms (4.62)
~I
+ K,
~I
+ K* : HS,p(an)
---t
HS,p(an),
-r < s < r, 1 < p < 00,
and (4.63)
v + K,
~I
+ K* : c:(aD,)
---t
c:(an),
-r < s < r.
(4.64)
f E Ker (~I + K*) ===> f E Hrf,q(an),
167
Va < r, q < 00.
Now given such f in the null space, set u = Sf, so (~ - V)u = 0 on M \ an. According to (4.48), (aulav)_ = 0 on an. If f E HCT,q(an) with aq > n, then f E C 8 (an) for some 0> O. One has for u = Sf that ul!L E C 1 (0), by Proposition 4.2. Then Zaremba's principle implies thdt aulav is nonzero at a point Xo E an where u is maximal, unless u is constant on each connected component of 0,_. Since we are assuming that V = 0 somewhere on each such component, this requires u = 0 on 0,_. Then Sf = 0 on an. Thus, by the maximum principle, u = 0 on n. Since, by (4.48), f is equal to the jump of avu across an, we have f = 0, so we have the isomorphism in (4.62) in the case that an is of class C1+r, r E (0,1). We now establish that the maps in (4.63) are isomorphisms. Again we need only prove injectivity, but this time we cannot appeal to duality, so we must treat both operators in (4.63), First, suppose f E Ker ( ~ I + K), f E (an) for some s E (-r,r). This implies that f E HCT,p(an) for each a E (-r,s), p < 00. But the isomorphism just established in (4.62) then implies f = 0, so ?I 1+ K is injective in (4.63). Injectivity of ~I + K* on c:(an) follows similarly. Now we tackle the case where an is of class C 1 ,w, w satisfying (4.9), and establish injectivity of ~I + K* on V(an), for all p E [1,00]. We first show that, for any f E V(an),
C:
(4.65)
f E Ker (~I + K*)
===?
f E Lq(an),
Vq
0, let B8 = {x EM: dist(x, TO) < o}. Pick 'P,1/J E C(f(B) such that 'P = 1 for dist(x, xo) :::; 013, 0 for dist(x,xo) 2: 2613, while 1/J = 1 on supP'P. Note that, if f = -2K* f, then 'Pf = -2[M'P' K*]j - 2K*('Pf), so
(4.66)
(I
+ 2M,pK* M1/J)('Pf)
= g,
9 = -21/J[M"", K*]j.
Examination of the integral kernel of [M"", K*] shows that (4.67)
Ig(x)1 :::; C
J
If(y)1 dist(x y)n-1 da(y). an '
Hence f E V(an), lip - lin = 1/q =* 9 E Lq(an), while lip - lin:::; 0 =? 9 E Lq(an) for all q < 00. If 0 is small enough, the operator norm of 2M,pK* M'Ij! will be < 1 on Lq(an), and we can deduce from (4.66) that 'Pf E Lq(an). A covering argument yields f E Lq(an). Iterating this argument then gives (4.65).
· .__
_._.__. __..__.
~~O
~::=
lti8
:
. __.....
""1.
.
__
.._.::_.. "..."'.
-
-M"'"?
-~.-
APPLICATI6N~ TO" POE
. ..
.~"'"':
F
We complete the proof by establishing that + K* is injective on £2(an). Suppose f E L 2 (an) and + K*)f = o. As above, set u = Sf, so (~- V)u = 0 on M \ an. The estimate (4.47) allows the use of Green's formula, to write
(F
(4.68)
f,L
Ju(~~)
2
{I Vu I + Vlun dv(x) = -
_ da(x),
an
and by (4.48) the right side of (4.68) vanishes when f E Ker(~I + K*). Thus u is constant on each connected component of n_ and u = 0 on supp V, so u = 0 on n_. Thus Sf = 0 on an so, again by Green's formula (justified as before) we have
(4.69)
f
{I Vu I2 + Vlul 2 } dv(x) =
n
f u(~~)
+ da(x) = O.
Hence u is constant on n, so avu+ = 0 on an. Since f is equal to the jump of a v across an, we have f = 0, so the proof is complete. With Proposition 4.5 in hand, we can produce solutions to the Dirichlet problem
Lu = 0 on
n, ulan
=
f,
(4.71)
u
=
PIf
=
V((~I + K)-l f).
The fact that, when u is given by (4.71), then (4.70) holds, follows from (4.55). We also have from (4.60)-(4.61) that, for 1 < p < 00,
II(PIf)*IILP(an) ::::: CpllfIILP(an).
(4.72)
This holds for an of class Cl,w, w satisfying (4.9). If an is of class Cl+r, r E (0,1), then, by (4.58) and (4.63),
PI: CS(an)
----t
C S (f2),
0
< s < r.
Uniqueness of solutions satisfying (4.73) follows from the classical maximum principle. For solutions satisfying (4.72), uniqueness is a special case of results that we will establish in Chapter IV (ef. Proposition 5.5 there) when p ~ 2. We omit a treatment of uniqueness for p E (1, 2); ef. [F JR] for a uniqueness result of such a nature, when an is merely C l . We next establish solvability results for Sf = g, which complement Proposition 4.5 as tools to treat the Dirichlet problem. PROPOSITION 4.6. If an is of class Cl,w and w satisfies the Dini condition (4·9), then we have isomorphisms
(4.74)
S : LP(an)
----t
Hl,p(an),
2Sp
< 00.
If an is of class Cl+r, r E (0,1), then we have isomorphisms (4.75)
S : Hs·p(an)
----t
Hl+s,p(an),
-r
< s < r,
1o hypersurface in M. Flowing along V produces a C 1 ,w diffeomorphism between 80, and E, and this induces a refinement of the differentiable structure of 80" useful for the proof of Proposition 4.6. Using Proposition 4.6, we can obtain solutions to the Dirichlet problem (4.70) in the form
PROOF. As in the proof of Proposition 4.6, we can show that T: C(A) (80) ---. C(A) (80) is Fredholm, by combining the arguments used in that proof with results of §5 in Chapter I, particularly Proposition 5.2 there. It follows that Sin (4.88) is Fredholm, of index zero. Injectivity is a consequence of results of Proposition 4.6, so the proof is complete.
u = 5(S-11).
An important operator for the study of boundary problems for L is the Neu mann operator N, defined by
(4.81)
We can also denote this by u = PI f, in view of uniqueness results. We have (4.82)
11(\7 PIf)*IILP(ol1) :::; Cp llfIIHl,p(ol1),
2:::; p
< 00,
when 80, is of class C 1 ,w, w satisfying (4.9), by (4.74) and (4.47). In fact, this can be extended to p E (1,2), but we will not carry this out here. If 80 is of class C1+ 1' , r E (0,1), we do have the extension of (4.82) to all p E (1,00), by (4.75). We also have (4.83)
PI: C1+ 8 (80)
--+
C1+ 8 (0),
0
PI: C 8 (80)
--+
C 8 (0),
0
< s < 1 + r, s
=1=
We have the following mapping properties. PROPOSITION 4.9. If 80 is of class C 1 ,w and w satisfies (4.9), then (4.90)
N: H 1 ,P(80)
-----+ LP(80),
2:::; p < 00.
If 80 is of class C1+I', r E (0,1), then
< s < r,
by (4,76) and (4.29). Furth~rmore, we can interpolate between (4.83) and (4.43), to obtain (4.84)
N f = 8" (PI I) 1011'
(4.89)
(4.91)
We next obtain variants of the property (3.20) for domains with boundary of class C1+ 1' • PROPOSITION 4.7. Assume 80 is of class C1+ ', r E (0,1). Let w be a modulus of continuity satisfying w(t) 2: t P , t E (0,1), for some p E (0, r). Then
-----+
H 5 ,P(80),
0:::; s
< r,
1
< P < 00,
and (4.92)
1.
N: H1+ 8 ,P(80)
N: C1+ 8 (80)
-----+
C 8 (80),
0
< s < r.
Furthermore, if w is as in Proposition 4,7,
N: C 1 ,W(80)
(4.93)
-----+
C(((80),
7
(4,85)
PI: C 1 ,W(80)
--+
PROOF. The result (4.90) follows from (4.82), (4.92) follows from (4.83), and (4.93) follows from (4.85). Note that if we write PIf = 5(S-1 I) and apply (4.48), we have
C 1 ,(((0),
with (4.86)
PROOF. Set )"(k) (4,87)
with (J given by (4.86).
N
(4.94)
(J(h) = ('(log !!.)w(y) dy. io y y
= (-!I + K*)S-l
on the various spaces listed in Proposition 4.9. This also establishes (4.91).
= w(2- k ). We claim that PIf = 5(S-1 f) has the property f E C 1 ,(A)(80) = } S-l f E C(A)(80).
(Here we refine the differentiable structure of 80 to a smooth one, in order to define C(A) (80) and Cl,(A) (80), as in Chapter I.) Granted (4.87), since C(A) c C' h with ,(h) = I O w(t)C 1 dt, by Proposition 1.2 (together with the remarks (1.15) (1.16)) of Chapter I, we have PIf = 5g, g E C', and the result (4.85) follows from h Proposition 4.3 (plus the same remarks from Chapter I), with u(h) = ,(t)C 1 dt. The relations just stated between w, " and (J then lead to (4.86). Thus Proposition 4.7 is proven once we establish the following.
Io
We can produce another formula relating Nand S, as follows. On M \ 80, consider (4.95)
Df(x) - 5N f(x)
=
J
{f(Y) ;~ (x, y) - N f(y) E(x, y)} d(J(Y)·
011
Applying Green's formula, we see that (4.96)
Df(x) - 5N f(x) = u(x), 0,
Approaching 80, we obtain PROPOSITION 4.8. Let 80 and w be as in Proposition 4.7, and )"(k) Then we have an isomorphism (4.88)
S: c(Al(an)
--+
Cl,(A)(80).
=
w(2- k ).
(4.97)
SN= -!I +K.
Comparison with (4.94), which we can rewrite as (4.98)
NS
=
-F + K*,
x
E
x E
0, 0_.
172
3.
APPLICATIONS TO Pj)E
K
= S K* S-l.
In particular, we deduce that K: H l ,p(8n)
(4.100)
--->
Hl,p(an)
PROPOSITION 4.10. Assume V = 0 on n (which, recall, is assumed to be connected). In the cases (4.90)-(4.92) considered in Proposition 4.9, KerN is the one-dimensional space of constant functions on 8n. Furthermore, given f in ani! of the target spaces in (4.90)-(4.92), f E R(N) {::::::::}
j
f drr
173
(-~I+K*)g=f,
(4.104)
is compact, for"2::; p < 00 if 8n is of class Gl,w, w satisfying (4.9), and if an is of class G!+r, r E (0,1), then we can combine (4.53) and Proposition 4.6 to deduce further mapping properties of K. Since S is an isomorphism, we deduce from (4.94) that N is Fredholm of index zero, in (4.90)-(4.92). In fact, we have the following.
(4.101)
PARAMETRIX ESTIMATES AND TRACE ASYMPTOTICS
PROOF. The only part not a consequence of Propositions 4.9-4.10 is the ex istence for f E LP(an), p E (1,2), when an is of class Gl,w. This follows from solving
shows that (4.99)
5.
= O.
and writing u = 8g. In turn, solvability of (4.104) on LP(an) , for p E (1, (0), granted Jan f drr = 0, follows by arguments similar to those used to prove Propo sition 4.5. In particular, if 9 E Ker (-~I + K*), one shows that 9 E Lq(an) for all q < 00. Then consider u = 8g. Formula (4.69) applies, so u is constant on n; say u = Co there. Then, using (4.68), we have, (4.105)
j {1'\7uI + Vlun 2
dv(x) = -
n_
j u(~~) _drr
= -Co
an
j
gdrr,
an
where the last identity uses the fact that 9 is equal to the jump of al/U across an. If Jan 9 drr = 0 it follows that u = 0 on n_. Hence u = 0 on n, so 9 = O. This shows that - ~ I + K* is injective on (4.106)
Lb(an)
=
{g E LP(an) :
an
j gdrr = O}.
Thus its range on LP(an) has codimension one, and the rest of the proof follows. PROOF. In case (4.92), the fact that Ker N consists of constants follows from Zaremba's principle. In case (4.90) this fact follows from Green's formula (4.69), justified by the estimate (4.47). In case (4.91), we have f E Ker N if and only if 9 = S-l f (which is in HS,p(8n)) belongs to Ker (-~I + K*). The same regularity argument used in the proof of Proposition 4.5 then implies that 9 E Ha,q(an) for all rr < r, q < 00, so f E H!+a,q(8n), and the previous arguments apply. Knowing that N has one-dimensional null space and is Fredholm of index zero, we now know that R(N) has codimension one in each case (4.90)-(4.92). Now the implication =} in (4.101) follows from Green's formula if f is sufficiently regular, which includes all cases in (4.90) and (4.92). For cases in (4.91) that lack sufficient regularity, the Green's formula argument still shows that the implication holds on a dense subspace, hence, by continuity, everywhere. This is enough to complete the proof. Using these results, we can treat the Neumann boundary problem (4.102)
Lu
= 0 on n,
au 8v =
f on 8n.
PROPOSITION 4.11. Assume V = 0 on n and assume Jan f drr = O. Assume an is of class Gl,w, w satisfying (4.9). Then, given f E LP(8n), 1 < p < (0) (4. 102) has a solution u satisfying
11('\7u)*llv'(iJn)
(4.103)
~ GllfIILP(an).
If an is of class G1+ 1', r E (0, 1), then, given f E GS(an), s E (0, r), (4.102) has a solution u Eel J , uniqup up to an additive constant.
en).
Note that, in the case (4.103), uniqueness of u (modulo an additive constant) follows from Green's formula, when p 2: 2. Following our previous pattern, we omit a treatment of uniqueness for the case p E (1,2). One can use the Neumann operator to investigate numerous other regular el liptic boundary problems. Compare treatments of the smooth case, discussed, e.g., in [TI].
5. Parametrix estimates and trace asymptotics Let Ai be a smooth, compact, n-dimensional manifold, equipped with a Rie mannian metric tensor of class Gk+ r , in local coordinates. We assume k E Z+ and r E (0,1). Let 6. be the associated Laplace operator, and consider L = -6. + V, with V E C k -2+ r (M), if k 2: 2, or V E C(M) if k = 1. Assume L is positive definite (adding a constant to V if necessary). One of our goals here is to prove the following. THEOREM 5.1. If 1 ::; k (5.1)
Tre-s-..!L
f'V
< n,
then there is an asymptotic expansion
s-n(A o + Als
+ ... + Aks k + o(sk)),
as s "" O.
Before proving this, we first demonstrate the following result, on the integral kernel of L -1, given by (5.2)
L~lf(x) =
J
E(x,y)f(y)dv(y).
At
~~17r~~----~~=~~-~-~~=-~-~~-~-=-~-r-~APPLiCkriQNS---;O-PDE~~=-~=---~-~--~--~~~~
5. PARAMETRIX ESTIMATES AND TRACE ASYMPTOTICS
THEOREM 5.2. If 1 ::; k < n - 2, then there is an asymptotic expansion of E(x, y), of the following sort, in local coordinates:
(5.3)
E(x, y)Jg(y)
rv
EAy, x - y)
where E 2H (y, z) is continuous on z in z, and
i=
+ ... + E k+2(y, x -
y)
+ Rk(X, y),
-a;
Fu(x) = (27r)-n
II F(y,~)ei(X-Y)'~u(y) dyd~,
F(y,~) = F2(Y'~)
+ F3(Y,~) + '" + Fk+2(y, ~),
with F H2 (y, ~) homogeneous of degree - 2 - f in ~. We will construct these symbols so that F 2+£ ( y, n/q 2: n - k - r - 2, with a similar implication for 1- n/ql' Given such bounds on up in crude norms, and given the bound (5.29), the fact that up satisfies the elliptic equation (5.26) and the regularity estimates on this PDE give
Ilu pIIH2.q(Bl/ 2 ):S C(q,c:)p-(n-k-r-2+o),
Hence, for any c:
If q
> 0,
Ilupll L~ (B Ll2 ) :s Co p-(n-k-r-2+o)
< 00,
c:
s
< 0.
We see that
8 S2u
Lu + 2Jb(s)r.p(x).
=
lsi
---t
u = -2(8;
00, we have
+ L)-18sJo(s)r.p(x),
> 0, e-sVir.p(x)
f p(s,x,y)r.p(y)dv(y),
=
M
with
p(s,x,y) = -28sE(s,x,0,y),
(5.39)
where E(s,x,t,y) is the integral kernel of (-8; + L)-I. There is an extra 8 s here, compared to the quantity analyzed in Theorem 5.2, but an entirely parallel argument yields the following. PROPOSITION 5.3. Assume 1 :s k < n, and that M has a metric tensor of class Ck+ r . If L is given a.s in Theorem 5.1, then the integml kernel p(s,x,y) of e- sVi has the following asymptotic behavior on the dia.gonal x = y, as s "" 0:
< n _ k ~ r _ 2'
and
(5.33)
u(s,x) = e-sVir.p(x) , s > 0,
(5.35)
jk 1 2 v g-I/28·g p Jp gp/ 8k,
The equation (5.26) holds on B 1 = {x : Ixl 1}. On this set, the family of coefficients gt k are of class Ck+ r , and t1 p is elliptic, uniformly in p E (0, Pol (for some Po > 0). Also, by (5.11),
(5.32)
This completes the proof of Theorem 5.2. The analysis just described has some points in common with estimates on E(x,y)Jg(y) - E 2(y,x - y) made in [MT], in the case where M has a Cl metric tensor, though the analysis done there had a different purpose. To study the integral kernel of e- sVi , we use the following device. Given r.p E L 2(M), set
(5.38)
Vp(x) = V(px),
:s Co Ix _ yl-(n-2-k-r+o).
IRk(x,y)1
(5.34)
(5.37)
and
177
In other words, we have the following more precise version of (5.4):
so this gives, for s
t1 pv
(5.27)
(5.31)
5. PARAMETRIX ESTIMATES AND TRACE ASYMPTOTICS
In view of the decay of u(s, x) as
where
(5.28)
'..:
0.
(5.40)
p(s,x,x)
"-J
s-n(Bo(x)
+ B 1 (x)s+'" + Bk(x)sk + o(sk)),
with (5.41)
Bi
E
Ck+r-i(M).
We are now ready to prove Theorem 5.1. From (5.39) we easily have p(s, x, y) continuous on (0,00) x M x M. Of course, an operator with a continuous integral kernel might not be of trace class. However, since M is compact, such continuity certainly implies that e- sVi is Hilbert-Schmidt, for each s > 0. Since e- sVi = e-(s/2)Vr;e-(s/2)v'L, we deduce that e-sv'L is in fact of trace class.
::
'!iI_
ZE2_~:_J!!!E
~!
,..,.,.,....
178
l!,i!ll--~iiIIiu:
~J!Io--E2-
&~~'
Em
A well known argument (d. Appendix A of [T5]) shows that, whenever a trace-class operator K on £2(M) has a continuous integral kernel k(x, y), then Tr K is equal to the integral of k(x,x), so we have Tre-sy'[ =
J
(6.6)
Uj(t)
= *dfj(t),
where fj E COO(1R x OJ) satisfies, for each t,
p(s,x,x)dv(x),
(6.7)
for each s > O. Given this and the behavior (5.40) ofp(s,x,x), we have (5.1), and Theorem 5.1 is proved.
6. Euler flows on rough planar domains Let 0 be a bounded open set in 1R 2. The Euler equation for ideal incompressible flow in 0 is OtU
+ 'Vuu =
-'Vp,
div U = O.
Here U is the fluid velocity and p the pressure. The boundary condition is that U be tangent to the boundary. There is a good existence theory when 00 is smooth, which can be found for example in Chapter 17 of [T5]. Here we want to allow 00 to be rough. Our basic strategy will be to use an approximation procedure. Let OJ have smooth boundary and satisfy
0 1 CC O2 CC ... CC OJ / O. For convenience, we assume each domain has connected boundary. Assume Vj E COO(Oj) are vector fields, tangent to oOj, satisfying div Vj = 0 and (6.2)
IIVjIICl(TIj )
::::
K
o,
180
181
6. EULER FLOWS ON ROUGH PLANAR DOMAINS
3. APPLICATIONS TO pbE
satisfies
We have IIUj(t) ® Uj(t)IIH1(flj ) ~ C'IIUj(t)IILoo IIUj(t)IIH' ~ C;
(6.18) hence
(6.33)
IIVPjllp(n j ) $ C,
as a consequence of (6.19). Normalizing Pj so that In Pj(t,x) dx Ildiv(Uj(t) ® Uj(t)) IIp(flj) ~
(6.19)
J
c,
= 0,
we have
Ejpj bounded in L OO (JR,H 1(0)),
(6.34)
so
IIPj
(6.20)
so (perhaps passing to a subsequence), we have
div(Uj(t) ® Uj(t)) IIL2(flj) ~ C.
(6.36)
Interpolation with the bound (6.15) yields Ejuj compact in C([-T, T], H 8 (0)),
\;/ T
+ C)No(s) ds.
= B(x,u) is a
IIB(x, u)v· vllHs,p ~ CIIB(x, u)v/lux> Ilvll Hs,p + CllvllLoo IIB(x, u)vIIHs,p ::::: C(lluIILoo) IlvllLoo IlvllHs,p + CIIvlluxo IIB(x, u)vIIHs,p.
It remains to estimate the last factor in the last term, Le., IIWvIIHs,p. For this, we use
Wv = Twv + TvW + RvW.
(7.18)
(7.9)
(7.11)
c
Gronwall's inequality applied to (7.14) yields a bound on No(t) for all tEl, and the proof is complete.
. liAs Jc;B(x, u, V'U)/I£2,
where the second identity in (7.7) uses (Ot - D.)AsJe = ASJc;(o; - D.). Now, in Proposition 7.2 we will establish an estimate that implies:
and, letting
1
00
(7.15)
/lB(x, u, v)IIHs,p(x) ::::: Csp,,(lluIILOO(X)) IlvIILoo(x)
~ (IIV' xAs Jeulli2 +
(7.8)
No(t) ::::: C + C i t (C +'log+ No(s))No(s) ds.
Since
Applying this to AS Jeu, we have
(7.7)
/lOtu/lHS]
/lotul/Loo ~ C/lOtu/lc~ [1 + log Ilo.ullr n/2 in (7.9),
the solution extends to an open interval J =:J [-TI , T 2 ]. A proof can be found in
[T2]. We aim to prove the following sharper result. As usual, C;(X) are Zygmund
spaces.
(7.5)
185
We estimate Twv by using the implication W E Loo =} T w E OPSP l' as stated in
(0.19) of Chapter II. As for the other terms, we use the following implication, valid for a > 0: (7.19)
v
E
C;"
==}
Tv, Rv
E
OPSf,I,
which follows from a straightforward symbol estimate; compare (3.5.7) and (3,5.11) in [T2]. Hence, for s > 0,1 < p < 00, and a > 0,
(7.20)
IIWvlIHs,p ~ CllWllLoo IlvllHs,p + CIIvll c :-" IIWIIH'+",P'
186
3. APPLICATIONS TO PDE
8. DIv·CURL ESTIMATES
Applying the Moser-type estimate given in (0.9) of Chapter II to W obtain the desired estimate (7.16).
=
B(x, u), we
We mention another known improvement on the straightforward results de scribed in (7.2)-(7.4). Namely, one can relax the requirement s > n/2. For exam ple, when 6. is the standard Laplacian on ]Rn, it is shown in [BB] that (7.1 )-(7.2) has a local solution of the form (7.3) as long as s > (n - 1)/2 if n = 3, and as long as s .:::: (n - 1)/2, if n,:::: 4. If, in addition, B(x, u, 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 > (n - 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.
The most basic div-curl lemma takes the following form. Suppose u and v are vector fields on ]R3 satisfying u E LP,
v E V/,
P E (1, (0),
Ilfllbmo = Ilfllm.lO + IIWa(D)fIIL=,
(8.6)
where wa E Co(]Rn), wa(O = 1 for I~I ~ 1. The two types of estimates have the same implications for local analysis. Most div-curl type estimates have been established in the context of constant coefficient PDE. The end of this section deals with a variable-coefficient div-curl type estimate. We begin with the following "abstract div-curl lemma," whose statement in this form arose in the course of correspondence of the author and P. Auscher: PROPOSITION 8.1. Let u and v be defined on ]Rn and take values in ]Rk and respectively. Let P, Q E OPSP a (or more generally in OPBSPtJ be a k x N ' and an £ x N matrix of operators, ~nd consider
1
N
(8.7)
1
Pu· Qv
-+=1 p p' '
(8.2)
j=1
divu
= 0, curl v =
°
==}
u· v
E Sjl,
(8.8)
where Sjl denotes the Hardy space. Equivalently, in view of the duality result of [FS], the conclusion in (8.2) is that u . v can be paired with an element of BMO. Such a result and many variants were presented in [CLMS]. One of the analytical techniques used in [CLMS] was the commutator estimate
Ilf Pu -
P(fu)lb ~
Cpllfllm.wlluliLp,
1;.i4 "~"".""':~::.~ ~
3. APPLICATIONS TO POE
8. DIV·CURL ESTIMATES
PROOF. The constant rank hypothesis on al implies that the orthogonal pro jection 11"1 (x,~) of ffi.k onto Ker at{x,~) is Coo on T*ffi.n \ OJ it is also homogeneous of degree 0 in~. Similarly, the orthogonal projection Pl(X,~) of R£' onto R(al(x,~)) is Coo on T*~n \ O. Now, for each (x,~) E T*ffi.n \ 0 we have an isomorphism
COROLLARY 8.4. Assume u and v are j-forms on a Riemannian manifold. Take p E (1,00). Then
(8.54)
Pl(x,~)al(x,~): (Ker at{x,~))l. ---... R(al(x,~)).
imply
(8.46)
Denote this isomorphism by a(x, ~), and set (8.47)
bl(X,~)
=
q(x,~)a(x,O-lpt{x,~).
bl(x,~)al(x,~)V = q(x,~)v
We next apply Proposition 8.2 to an extension of (the N = 2 case of) an estimate given on pp. 276-277 of [eLMS]. In this case, u and v are defined on ffi.2 and we take
Al = D 1 ,
(8.57)
where D j
q(x, ~)vv(x,~) = a2(x, ~)twv(x, ~).
b2(x, ~)tvv(x,~) = wv(x, ~),
1:::; v
with E = (Dr
:::; r.
a2(x,~)tb2(X,~)tv
= q(x,~)v
o
o for
v ..1
Q = E(Dr
+ D~) + R =
+ D~ + 1)-1 =
given (x,~) E r. Putting together (8.48) and (8.51), we have (8.34), but so far only for (x,~) E r. However, we can take a locally finite covering of T*ffi.n \ 0 by cones on which such a construction works, and use a partition of unity to obtain (8.34) globally.
COROLLARY 8.5. Assume that 1 < P < 00 and u E U(ffi. 2), V E U ' (ffi.2). Then
(8.59)
DIU E H- 1 ,T(ffi. 2),
(8.52)
Al = d,
acting on differential forms. We take Q (8.38) is
(8.53)
Q = E(t5d + dt5)
=1=
A 2, we derive a result containing
= I, and the relevant identity of the form
+R =
The following natural generalization of Corollary 8.5 is also a simple corollary of Proposition 8.2.
pi,
with A j E OPSo. Then (8.62)
Char Al n Char A 2 = 0 ===} uv E fJtoc(ffi. n ).
(Et5)d + d(Et5)
where we take E = (1 - Ll)l E OPS-2, Ll = Laplacian. Then Proposition 8.2 directly gives:
+ R, -(dt5 + t5d)
r > p, P > pi
uv E fJtoc(~2).
(8.60)
(8.61)
A 2 = 15,
D 2v E H- 1,P(ffi.2),
implies
COROLLARY 8.6. Assume 1 As an example of Proposition 8.2, when Al the div-curl lemma, upon taking
(ED1)D 1 + D 2(ED 2) + R,
(1 - Ll)-1 E OPS- 2. Then Proposition 8.2 gives:
for v E Ker al(x,~), for v..l Ker al(x,~),
A 2 = D 2,
= i8/8xj. Again we take Q = I, and
(8.58)
Extend b2(X,~)t by linearity on Ker al(x,~), and set b2(X,~)tv Ker al (x, ~). then we have (8.51)
IJ f du /\ dvl :::; CpllfllbmolluIIH"pllvIIH"pl.
(8.56)
Then, set (8.50)
r > p, P > pi
u· v E fJtoc'
for v E Ker al (x, ~).
We will next define b2(X,~)t for (x,~) in a conic neighborhood r of any given (xo,~o) E T*ffi.n \ O. To begin, take vv(x,~), 1 :::; v :::; r = Dim Ker al(x,O, smooth on r, forming for each (x,~) Era basis of Ker al(x,~). Now 'Pv(x,~) = q(x,~)vv(x,~) belongs to the range of a2(x,~)t, by hypothesis (8.45). Since a2(x,~) has constant rank, we can find wv(x,~) E ffi.£2, smooth on r, such that (8.49)
t5v E H- 1 ,p,
When j = 1, this is equivalent to the standard div-curl lemma. Also, via the Hodge star operator, one deduces from Corollary 8.4 that
for v..l Ker al(x,~),
o
du E H- 1 ,T,
v E LPI,
(8.55)
Thus (8.48)
U E LP,
193
being the Hodge
PROOF. The hypothesis of (8.62) implies that (8.34) holds with q(x,~) 1, bj(x,~) E So.
194
9. HARMONIC COORDINATES
3. APPLICATIONS TO PDE
and
9. Harmonic coordinates
The use of harmonic coordinates is an important tool in differential geometry. Here we produce harmonic coordinates when the metric tensor has limited regu larity. We first consider the classical case of Holder continuous metric tensors. We then extend the results to a class of metric tensors with less regularity. To begin, let M be an n-dimensional C l manifold, with a continuous metric tensor. Then the Laplace-Beltrami operator ~ : Hl~c(M) ----t Hl~~(M) is well defined, as is the notion of a harmonic function on an open subset of M. We now assume M has a finer structure. Namely, we assume there exist K o, K l E (0,00) and s E (0,1) and, for each z E M a C l diffeomorphism t.pz : B l ----t Uz C M (B l denoting the unit ball in IRn centered at the origin), such that t.pz(O) = z and the metric tensor pulled back to B l via t.pz belongs to CS(B l ) and satisfies
(9.1)
gjk(O) = Jjb
0
< Kr;l I::;: (gjk(X)) ::;: KoI,
Ilgjklb(Bd::;: K l .
We take up the task of constructing harmonic coordinates, centered at a given point
zEM.
8jajk(x)8kU~
(9.2)
(9.3)
=
= 0,
u~laB
p
= Xl,
1::;: f.::;: n,
g(X)1/2 g j k(x). Note that 0 < K:;l I::;: (ajk(x)) ::;: K 2I,
'k
Ila J Ib(Bd ::;: K 3 •
-p 8japjk( x )8kUl = 0, -PI u l BB , = Xl,
with a~k(x) = ajk(px). Note that (9.5)
0< K:;l I::;: (abk(x)) ::;: K 2 I,
k
Ila~ Ilcs(Bd ::;: K 4.
We estimate how close are u~ and Xl. Note that (9.6)
'k
8ja~ (x)8kXl
=
'k
Jkl8ja~
= h lp .
(v,8ja~k) =
j[ajk(O) - a jk (px)]8 j v(x)dx, B,
and hence (9.8)
118ja~kIIH-lq(Bd ::;: K 5 •qps,
\f
q
< 00.
Hence
(9.9)
(9.11)
q
< 00.
llu~
-
Xl I H1.2(B,) ::;: K 6p s.
To derive further bounds on u~ -Xl, note that the bound in (9.5) on a~k implies IIh~IIH-lh,q(Btl
(9.12)
::;: K 7 ,a,q,
\f
(J'
< s,
q
< 00.
Interpolation with (9.10) gives
Ilh~IIH-l+s/2,q(Bd ::;: K S ,qps/4,
(9.13)
\f q
< 00.
Then interior regularity results established in §1 yield, for all q (9.14)
llu~
- xlIIHl+s/2,q(B,/2) ::;:
K9, qp S/4,
Ilu~lb+s(BI/2)
(9.15)
Ilu~
< 00,
- XellC'+S/4(B ,/
2
)
::;:
K lOp s/4.
aJa~k(x)8du~ - Xl) = h~,
u~ - xll aBl = 0,
::;: K n .
t
lt follows that there exists p = p(n, K lO ) > 0 such that (x) = (ui(x), ... ,u~(x)) is a diffeomorphism of B l / 2 onto a region containing B l / 3 , such that
"t(B l / 4) C B l / 3 C l,P(B l / 2),
having inverse map
""t satisfying
-p
(9.17)
B l / 4 C ( (B l / 3 ) C B l / 2
and
-p
II( Ib+
(9.18)
s
(B 1 / 3 l
::;: K 12 ·
This argument establishes the following. PROPOSITION 9.1. Assume the existence of an atlas t.pz : B l ----t Uz c M such that (9.1) holds. Then there exists Po = po(n,s,Ko,K l ) > 0 and for each z E M a C l diffeomorphism 'ljJz : B po ----t U~ such that 'ljJz(O) = z and such that the inverse 'IjJ-;1 : U~ ----t B po C IR n is harmonic. Furthermore, the metric tensor pulled back to B po via 'ljJz belongs to CS(B po ) and satisfies
(9.19)
Given v E CO(B l ), we have (9.7)
\f
An elementary consequence is that
(9.16)
Here and below, K" will denote a constant, which might depend on n, s, and KJ.' for J1, < v, but not on other quantities. To analyze the solution to (9.2), it is convenient to dilate coordinates. Consider u~ (x) = p-1U~ (px), defined on B 1. This solves (9.4)
Ilh~IIH-l.q(B,) ::;: K 5 ,qps,
(9.10)
. Also interior regularity results applied to (9.4) yield
To begin, for 0 < P < 1, using the coordinate system t.pz to translate the PDE a neighborhood of the origin in IRn, solve on the ball B p where xi + ... + x;' < p 2 the Dirichlet problem
where ajk(x)
195
0
< K l31 I::;: (9jk(X)) ::;: K 13 I,
Ilgjklb(B po )::;: K 14.
Note that on overlaps U~l nU~2 the harmonic functions are of class C1+ s in both coordinate systems, so we have a C 1+s-coordinate system. We can hence deduce that the old coordinate atlas {t.pz : z E M} must have been of class C1+ s and the two coordinate atlases are C1+s-compatible. Except perhaps for the bounds recorded above, particularly on Po, Proposition 9.1 is classical, the n = 2 case going back to Korn and Lichtenstein; d. [Mor], §9.3. See also [ehe]. As shown in [HW], this cannot be extended to arbitrary continuous metric tensors. Here we obtain harmonic coordinates when 9jk belongs to a class of spaces CP.).
, 1 • .~.
196
3. APPLICATIONS Tc>PDE
Let w be a slowly varying modulus of continuity and assume A(j) = w(2- j ) satisfies the following condition, which is slightly stronger than the Dini condition:
L
(9.20)
A(j)a < 00,
1 0 < K G I:::; (gjdx )) :::; KoI,
gjk(O) = 6jk ,
Ilwgjkllcp.):::; K l ,
where we fix W E C O(B 1) such that w(x) = 1 for x :::; 9/10. We solve the DiricWet problem (9.2) for u~, provided P :::; 1/4. This time, making use of Proposition 9.1 of Chapter II, we have (9.22)
0
.)
C
C a,
r w(t) dt. 1 t h
a(h) =
0
Using (9.7) we see that, for v E CO'(Bd, (9.25)
I(v, h~)1
:::; Ksa(p)
IIVvllu(Bd'
This implies (9.26)
Ilh~IIH-l.P(BJl
:::; K 6 ,pa(p),
. .~~~.
~.
Ilwh~llc;l:::; K 7 a(p),
HAR~~;; COORDINATES
197
Proposition 1.10 also yields from (9.27) the estimate Ilw3/4U~lb,CA) ~ K 1Z •
(9.32)
for some a < 1.
For example, we could take AU) = j-S for any s > 1. (For such sequences, Proposition 1.10 can be applied.) Let us begin afresh, assuming we have a C1_ diffeomorphism 'Pz : B 1 ~ Uz eM, and that the metric tensor pulled back to B l satisfies, in place of (9.1), the following:
(9.21)
. '.
These estimates take the place of (9.14)-(9.15). It follows that there exists p = p(n,Kll ) > 0 such that t(x) = (ui(x), ... ,u~(x)) is a diffeomorphism of B l / Z onto a region containing B 1 / 3 , with inverse map 7/, such that (9.16)-(9.17) hold, and -P
(9.33)
II( Ib'''(B
l / 3)
;5.
K 13 ,
with a given by (9.24). Hence we have: PROPOSITION 9.2. Assume the existence of an atlas 'Pz : B 1 ~ Uz c M such that (9.21) holds, with AU) "'" 0 a slowly varying sequence satisfying (9.20). Then there exists Po = po(n, A, K o, Kd > 0 and for each Z E M a C1 diffeomorphism 7/lz : B po ~ U~ such that 7/lz(O) = z and such that the inverse 7/1-;1 : U~ ~ B po C JRn is harmonic. Furthermore, the metric tensor pulled back to B po via 7/lz belongs to ca and satisfies
(9.34)
0< Ki41I :::; (gjk(X)) :::; K l4 I,
Ilgjkllc"(B po ):::; K lS .
It would be of interest to see if we can replace ca by C(>.) in these conclusions. In the original coordinate atlas, harmonic functions belong to C l ,(>.) C cl,a. In view of (9.32)-(9.33), in the new coordinate atlas harmonic functions are still in Cl,a. Hence on overlaps U~l n U~2 the harmonic functions are still of class C1,a in both coordinate systems, so we have a Cl,a-coordinate system. We can hence deduce that the old coordinate atlas must have been of class Cl,a and the two coordinate atlases are cl,a -compatible. We note how the two-dimensional version of this analysis leads to the existence of isothermal coordinates.
for p E (1,00), and with Was in (9.21). We also have Ilwh~llc-l.()')
(9.27)
:::; Kg.
From (9.26)-(9.27) we obtain, for each () E [0,1],
(9.28)
II7/lj(D)(wh~)IIL'x, :::; Kga(p)IJ 2j AU)1-8,
where Nj} is the usual Littlewood-Paley partition of unity. Pick () = (1 - a)/2 with a as in (9.20), and set p,(j) = A(j)I-8. Then (9.29)
Ilwh~llc-1,(,,)
:::; Kga(pt.
Parallel to (9.11), we have the elementary estimate (9.30)
Ilu~
-
xeIIHl.2(Bd :::; KlOa(p).
Then we can apply the regularity result (1.51) in Proposition 1.10 (with A replaced by p,), to obtain (9.31)
IIw:I/4(U;.' -
xe)lb,(,,) :::; Klla(p)lJ.
PROPOSITION 9.3. In the setting of Proposition 9.2, assume n = 2, and assume M is oriented. Then there exist PI > 0 and junctions (VI, vz) on B Pl satisfying
8jajk(x)8kve
(9.35)
=
0 on B pll
Ve E C l ,(>.),
such that, for all x E BPI' (9.36)
(where
dvz(x) = *dVl(X),
* is the Hodge
star-operator given by the metric tensor (gjk)), and such that
~(x) = (VI (x), vz(x)) is a diffeomorphism of B pdz onto a region containing B pd3'
Hence there exists Po > 0 and a conformal Cl,a -diffeomorphism 7/lz : B po ~ U~ such that 7/lz (0) = z, and such that the metric tensor pulled back to B po via 7/lz satisfies (9.37)
gjk(X)
=
f(x)6 j k,
0
A(X)-l is of class cs n HI,p, so it suffices to show that D~ 0 ( E HI,p. This is a consequence of the following. LEMMA 9.6.
£b : C1,(A) (']['n) n H 2,p(r)
----t
C- 1 ,(A)(']['n) n £p(,][,n) is compact.
As for £#, we have
D((x) = (D~(((x))rl.
(9.51)
(9.57)
If U E HI,p and
0 on f \ 0,
Furthermore, we assume
Hp,g:::; 0
on f.
(11.18)
H P1
c; - 2M c; =
2dE(Hpl dE)e2>.f
+ d;e2>.f (2)'Hp,J -
2M),
where dE = d(cg)-I and HpldE = (cg)-I[Hpld-c2(cg)-I(Hplg)d].
(11.20)
(CE{cE,pd Mc;)(x,~)
2:: rpE(X,~)2 - E(x,~)2,
with
for some open conic fer c assuming that u E H;::cta-I(O) for some open conic set 0 C (The closure is taken in T'n \ 0.) The key estimates, following [H4], use the following basic commutator identity: 1m (CP#u, Cu) = Re ({ -iC'[C, A]
Hp,j
Hp1d
Hence, if (11.14)-(11.17) hold, we can pick>' large enough that, for all c E (0,1]'
We look for conditions under which we can deduce that
(11.10)
r,
Later we will give geometrical conditions under which we can produce d, j, 9 sat isfying (11.14)-(11.17); for now we take their existence as an hypothesis. Then, if CE(X,~) is given by (11.13), we have
(11.19)
r.
g(x,O E S~l'
g(x,~) 2:: C1~1 > 0 on
2:: 0,
where 0 is some conic open subset of
-(1 - 8)r < a < r.
(11.9)
S~l'
and, with PI denoting the principal symbol of p(x, D), homogeneous in ~ of degree 1, we assume
We will assume 8 E (0,1), r/j> 1, and
(11. 7)
E
m =
n c lR,Tl, with
(11.6)
j(x,~)
J1..
p(x,D)u = j,
(11.5)
+ c2g(x, ~)2) - 1/2 ,
where>. > 0 will be taken large (fixed) and c > 0 small (tending to 0). We assume
(11.15)
(11.4)
CE(X,~) = d(x, ~)e>'f(x,O (1
(ILl3)
all homogeneous for I~ I large. We will say more about J1.. below, but we do take > O. We assume
B E OPAO-IS~6-I.
Let us assume that
on a region
and to get from (11.10) to (lLl1) we have used I(CP#u,Cu)[ :::; IICP#uI1 2 + IICuI1 2 /4. In (11.12) we use the convention that ReT = (T+T")/2, for an operator T. For C we will actually define a family of operators C E = cE(x, D), with
(11.14)
B = B*,
207
(11.21)
rpE(X,~) = rp(x,~)(cg)-I,
rp, E
E S~,
and with rp elliptic on f and supp E C O. Now to relate this to (11.11) we need to replace PI (x,~) by a(x, ~), the real part of ther complete symbol of A, given by (11.3), and we need to record the difference between symbols of products and commutators and the leading expansions of these symbols. A check of symbol expansions shows that, if (11.22)
r
=
1 + ro,
0 < ro :::; 1,
then (11.23)
-ReiC:[CE,A](x,~) - cE{cE,a}(x,~) E S~~-IH(I-rol,
bounded for 0 < c :::; 1, and W = ReC"[B,C],
(11.24)
PI (x,~) - a(x, 0 E C r st,-;SP,
p = min (1, d),
-_--------- •
3.
20
APPLICATIONS TO
-,.
_
~."',""
=?-';==:c:;;:;~~~~ _=_=_'".===_=~==,~==,,..=~_==~
11.
POE
PROPAGATION
OF
SINGULARITIES
209
We can carry out parallel estimates with p# replaced by
so
(11.36)
l{cE, a}(x,O - {CE,pd(x,~)1 ::; C(OIJ.-P+f>.
(11.25)
QE(X,O
(11.26)
(11.37)
= CE{CE,a}(x,~),
as well as
-Re iC; [CE, A] - QE(X, D)
with natural bounds for 0
< E ::;
::;
IICEsP#vllks
+ IIEs(x, D)vll~8 + Kllvllks,
+ E(X,~)2 2:
_K(~)21J.-p+f>,
SHARP GARDING INEQUALITY. Letq(x,~) E CSS['o be scalarandsatisfyq(x,~) 2: -Co· Then, for all u E Co,
Re (q(x, D)u, u) 2:
(11.30)
-C11IuII12,
provided
s > 0,
(11.31)
ro J-L ::;
2 + ro '
and
J-L
(l-1'o). The inequality (11.33) hence implies an L2-operator bound on WE' a~d we have 2 II'PE(X' D)vll[l:S
1
i\
r,
= r.
r
p#u E H:ncl(r).
Then
u
(11.40)
E
H:ncl(f).
PROOF. Shrinking 0, we can assume u E HS(O) for some s E R If J-L > 0 is small enough that (11.32)-(11.33) hold, and if s + J-L ::; t, we can apply (11.37) and let E ----t 0, to get u E H~~i (f 1)' Then we can construct f3} (x,~) E S~I' supported in f 1 , equal to 1 on 2 , and then U1 = (31 (x, D)u belongs to HS+IJ.(D.) while p#u coincides with p#u microlocally on 2. The proof is completed by a straightforward iteration.
r
This gives the following result on p(x, D).
PROPOSITION 11.2. Let p(x,~) E C1'S~1 satisfy the hypotheses of Proposition
< r ::; 2.) Assume
11.1. (So 1
p(x,D)u =
f,
with
(11.42)
(11.43)
-
C 1\ c r
,
::; 1, and let p# and satisfying
u E Hl~~(7' 1, and
ro).
When the corresponding consequence of the sharp Garding inequality is combined with (11.11), we obtain
II'PE(X, D)v1112
u E H;ncl(O),
(11.39)
(11.41 )
2'
where ro is as in (11.22), and we can replace QE(x,D) by -ReiC;[CE,A], since (11.32) implies (11.33)
1
< ro
Make the hypothesis that, for each j, there exist d = d j , f = fj, and g = gj replaced by f j and j . Pick t E JR and satisfying (11.14)-(11.17), with f and assume u E D'(O) satisfies
r
2s m::;2+s'
This result is Proposition 2.4.A of [T2]. The proof given there makes use of a symbol smoothing, q = q# + qb, and an application of the Fefferman-Phong inequality to q#(x, D). We can apply this sharp Garding inequality when q(x,~) = qE(X,~) is given by the left side of (11.29), provided (11.32)
... C r 2 C f
r
1. Furthermore,
QE(X,~) - MCE(X,~)2 - 'PE(x,~)2
rc
(11.38)
E OPS~~-l+f>(l-ro),
with p as in (11.24). This puts us in a position to exploit the following:
(11.35)
II'PE(x, D)vllk s
PROPOSITION 11.1. Assume p(x,O E Cl+roS~I' with 0 be given by (11.2). Suppose we have open conic sets 0, f, f j
QE(X,~) E S;~, ncrosi~, ,
(11.27)
(11.34)
= (1 _ ~)s/2.
with CES = ASCEA-s, etc. We are now ready to establish the following result.
we have
(11.29)
= AS p# A -s = As + iE s, AS
In such a fashion we obtain estimates of the form
Thus, if we set
(11.28)
p!
2 11('/op # V 1/ 2£2 + IIE(x, D)vIl 2L + Kllvll£2' 2
-(1 - o)r < 0' < r.
Then (11.44)
u E H~cl (0)
===?
u E Hl~lcl (f).
PROOF. By (11.7)-(11.8), we see that u satisfies (11.38), with t =
0'.
In case r = 2, one can take p(x,~) E C1,lS~I' Furthermore, by Proposition 4.9 of Chapter I, we can allow the endpoint case of (11.6), 0' = r = 2, and still have pbu E Ht;,c(D.), given that u satisfies (11.5). This leads to the following:
~~;,......~ ....~==- ._=.=:--"3. APPLIC~T~S TO.PD;---~ "",~-~~~~~~-~"''''"~''"OC''''-_·~;:;;~G~-~F·~;~-~-~~ARITIES PROPOSITION
11.3. Let p( x,~) E C1,1 S;1 satisfy the hypotheses of Proposition
11.1. Assume p(x,D)u = f with
a-(2J-1) ( u E H IDe 0),
(11.45 )
f
E H~el(r).
Assume 8 E (1/2,1) and -2(1- 8) < a :S 2.
(11.46)
Then the implication (11.44) holds. In case p(x, D) has order m, by examining p(x, D)A 1-m we see that if (11.4) (11.6) hold and if PI (x,~) = Pm(x, OI~II-m satisfies the hypotheses of Proposition 11.1, then, for a satisfying (11.43) we have
u
(11.47)
m + a - 1 (0) m a 1 H mel ====r. u E H mel+ - (r) .
E
Let us also say something about equations in divergence form. We consider PDE of order two:
p(x, D)u = ojAjk(x)ihu.
(11.48)
In such a case, it is natural to replace (11.2) by
p(x, D) = ajAj
(11.49)
+ ajA; =
p#
+ pb,
with
Aj
(11.50)
given Ajk E
cr.
(11.51)
E
OP.AFJSI,J,
A;
We see that, if -(1 - 8)1'
-1 : Wnr ----> (-a,e) x ~ is Holder continuous of class Hence (11.64) defines a function hI E CS(Wnr), and h 2 = h 1 - Kj2 satisfies (11.71). Of course, if X P1 ELL, then (11.72) holds for all 8' < 1. Hence we can apply Propositions 11.2-11.4 to obtain propagation of singularities results along null bicharacteristics, for operators with coefficients having one derivative in LL. As noted in (2.25), this happens if the coefficients belong to the Zygmund space C;. In particular it happens if the coefficients have two derivatives in bmo. For example, Proposition 11.4 can be applied to obtain propagation of singu larities results along null bicharacteristics for solutions to the wave equation (11.90)
[MaJ , 4?E(D)]ojh 2 ----> 0,
X P1 ELL
cs.
•
We deduce that, under the hypotheses (11.71)-(11.72), for 0 < (11.82)
'l9(a, t) = aCXp(-Kt).
(11.88)
so we have
(11.79)
215
1
< -.
- e
Utt -
~u =
0,
where ~ is the Laplace-Beltrami operator on a Riemannian manifold with bounded Ricci tensor, as we see by writing (11.90) in local coordinates as (11.91)
Otg(X)1/2 0tU - Ojg(X)1/2gjk(x)OkU
=
0,
and use Proposition 10.2 to see that the coefficients have one derivative in LL.
-_...._---------=======:;;:;::;;====::;;.;:;;,..
..--_._-_.-:_._. -====;:;;
~:'--:--
CHAPTER 4
Layer Potentials on Lipschitz Surfaces Introduction In this chapter we discuss results on layer potentials on Lipschitz surfaces and applications to the Dirichlet problem on Lipschitz domains. When a surface lacks moderate regularity beyond the class 0 1 , it becomes difficult to establish the basic operator norm estimates on single and double layer potentials. The first break through on this was initiated by A. P. Calderon [Ca2], and completed by R. Coif man, A. McIntosh, and Y. Meyer [CMM], estimating the Cauchy kernel on Lip schitz curves. Estimates were also established for an appropriate class of potentials on higher-dimensional Lipschitz surfaces in [CMM] and [CDM]. In §§1-2 we treat these estimates, in one and higher dimensions, respectively. Our treatment of the basic estimate of the Cauchy integral on Lipschitz curves follows a proof given in [CJS]. Other proofs have been produced; we mention particularly [GM] and
[MeV]. These estimates on layer potentials allow one to apply Fredholm theory to the study of regular elliptic boundary problems in 0 1 domains. This was carried out in [F JR]. However, for Lipschitz domains that are not 0 1 one can lose such properties as compactness of double-layer potentials, and further effort is required. This was accomplished in [Ve], for the Dirichlet and Neumann problems. Among other things, an identity of Rellich was brought to bear, to establish unique solvability of appropriate boundary integral equations. A number of other boundary problems on Lipschitz domains have subsequently been treated via layer potential techniques; we mention the works [DKV], [FKV], [EFV], and [MMP]. All these works confine their attention to constant-coefficient equations on Lipschitz regions in jRn. Along with this restriction comes a topological restriction on the domain; only domains with connected boundary are treated. It was not expected that such a restriction should be necessary for the basic results to hold. Recently, [MiD] developed a technique to treat the Dirichlet problem for the Laplace operator on domains in jRn whose boundaries were not required to be connected. In [MT], tools were developed to apply the method of layer potentials to equa tions with variable coefficients on Lipschitz domains. There the authors studied operators of the form L = Li. - V where Li. is the Laplace operator on a compact Riemannian manifold M and V E LCXJ(M). The metric tensor was assumed to be of class 0 1 (an assumption that was relaxed to Lipschitz in [MT2] and relaxed further in [MT4]). The authors treated the Dirichlet and Neumann problems, and oblique derivative problems on Lipschitz domains in M. In [MMT] the scope of this work was extended to other boundary problems, including natural boundary
-"~
218
4. LAYER POTENTIALS ON LIPSCHITZ SURFACES
problems for the Hodge Laplacian on Lipschitz domains in Riemannian manifolds. It is worth mentioning that, once one moves to the variable-coefficient setting, the need for topological restrictions evaporates; one can treat arbitrary compact Lip schitz domains in a smooth manifold. In §§3-5 we present some of the material developed in [MT], but here it is specialized to the case of smooth metric tensors, for simplicity of exposition. Section 3 extends the layer potential estimates of §2 from potentials of "convolution type" to "variable-coefficient" generalizations. Section 4 investigates solvability of boundary integral equations arising in the layer potential approach to the Dirichlet problem. Section 5 then appies these results to the Dirichlet problem. 'While we restrict attention to the case of smooth coefficients, we mention that some of the techniques used in this monograph, particularly in §§1-2 of Chapter III, were brought to bear in the more general cases treated in [MT], [MMT], and [MT2]-[MT4]. The key estimate on Cauchy integrals on Lipschitz curves in §1 makes use of the Koebe-Bieberbach distortion theorem. As this is outside the circle of results we have described as prerequisites, we present a proof of it in Appendix A, at the end of this chapter. Our treatment draws from those in [Porn] and [Mil]. Taking a cue from [Mil], we present an endgame to the proof that is somewhat more geometrical, and less computational, than usual.
1. CAUCHY KERNELS ON LIPSCHITZ CURVES
219
A. We discuss later in this section how to pass from this case to the case of general Lipschitz A. The exponent 2 in (1.5) is better than obtained in [CMM]; the optimal expo nent 3/2 can be found in [Mur]. The proof we give here is taken from [CJS]. It exploits the behavior of d
(1.6)
£r f(z) = dz K r f(z) = -
J
f(C,)
(z _ ()2 d(
r
on f!±. The key analysis is contained in the next three lemmas. Let 'H+ denote the space of functions on f!+ satisfying
Ilfll~+
(1.7)
=
J
2
If(z)1 d(z) dx dy
O. Hence
jG(xWI'(x)! dx ::;
cII:
A < C(B -
+ e11' B 1/ 2A 1/ 2) < !A + CB + ~e211'C2 B - 2 2 '
which in turn implies A 5: CB (with a different C). The proof of Lemma 1.2 is complete. LEMMA
I:
2
+
The identity (1.12) follows from Green's theorem. To obtain (1.13), use (1.12) to write
(1.14)
r/
12 12 < - e11' B / A / ,
PROOF.
IH(x)IZdx
JR~
5: B 1/ 2e11'/2 ( / IG( ,)1/ZI 2\D'I,zy dx dy
+
I:
JR~
(1.19)
1.4. Let f E H+ and define
Tf()
=/
f(z)d(z) (z _ ()" dxdy,
(E
r.
11+
Then IG(xWiI>'(x) d4
(1.20)
IITfIIU(r) 5: Cz (l + L)llfllx+,
for some absolute constant C z .
so by Green's theorem
A 5:
ci/ ~(GG')YdXdyl 4cl/ (IG'IZ' + GO'")ydxdy! 1R 2
PROOF.
(1.21)
By Lemma 1.2,
II TfllL2(r) 5: CII(Tf)'llx_,
+
=
(1.15)
since Tf is holomorphic in fL. Here C
1R 2
+
5: CB + C
JIGO'"ly
I(Tf)'(w)\ = dx dy.
(1.22)
]K2
+
Now set Hence (1.16)
' = eV ,
so
" = V'e F = V''.
= C1 (1 + L).
12/
Now
f(z)d(z) dxdyl (z - w)3
11+
(z) is given by (1.38), then
(1.67)
and f E V(f), we have
00
0 in Co,
(1.74)
(1. 75) CT->O
----t
LEMMA 1.9. Suppose A : IR
Then (1.66)
- ia)I}.
pointwise a. e. and in LP -norm, where
Fix s E IR. Assume that A (hence () is differentiable at s, and that
(1.65)
Also, as a
+ ia)1 + IKr f(z
II(Krf)*IILP(r):S CpllfIILP(r).
(1. 73)
C
227
Co = {a E C : 0 < Re a :S 1, 11m a I :S J Re a},
lrovided J is sufficiently small that iC~ does not overlap with OL. Details are an
,xercise.
(1. 76)
IIEAfllp :S C(1
+ L)31Ifllp·
PROOF. Let n be the region in C consisting of points of distance :S 1 from the interval [-2L,2L] in the real axis, and denote its boundary by,. Now, for ( E " set
(1. 77)
Ae,(t)
=
ke,(s, t)
(t - A(t),
=.
1
Then Cauchy's formula gives
(1. 78)
eA(s,t)
=
~Jeie,k(,(s,t)d(. 2m I
To prove the lemma, it suffices to show that the operator Ke, with kernel P.V. ke,(s, t) satisfies the estimate
(1. 79)
IIKdllp :S C(1
+ L)21Ifllp,
V ( E ,.
Writing ( = ~+i'TJ, we have two cases to consider. First, suppose -2L :S ~ :S 2L. Then'TJ = ±1 and Ae,(t) = ±i(~ ± iBe,(t)), where Be,(t) = A(t) - ~t. Hence, up to a factor ±i, ke, (s, t) is in this case precisely the Cauchy kernel (1.33) associated to the Lipschitz graph of Be" whose Lipschitz constant is :S 3L, so the desired estimate on IIKc.ll.c(L2) follows from Theorem 1.1. Next we assume ~ 2' 2L (the case ~ :S -2L being similar). Then we can write Ae,(t) = ((t + Ce,(t)) with IC((t)1 :S 1/2, a.e. Again by Theorem 1.1, the kernel
228
4.
LAYER POTENTIALS ON LIPSCItITZ StJRFACES
(s - t - CdS) - Cdt))-I defines a bounded operator on L 2 (1R), with uniformly bounded norm. Hence IIKd.C(L2) ~ C/ L for such (. PROPOSITION 1.10. Assume 'P : IR N> n + 3, and consider the kernel
---+
IRn is Lipschitz. Let F E C N (IRn) with
7(S,t) = _1_ F('P(s) - 'P(t)) s-t s-t .
(1.80)
Then P. V. 7(S, t) is the kernel of a bounded operator on L2(IR). PROOF. If 'P has Lipschitz constant L, then the argument of F is contained in the ball {z E IR n- k : Izl ~ L}, and we can alter F at will outside this ball without affecting 7(S, t). If we alter F to a function 1>, smooth of class C N and periodic, so that 1>(z + 27rL",) = 1>(z) for all '" E zn-k, then we can expand 1> in a Fourier series
1>(z) =
(1.81)
I: a" eil 0 a.e. on 80.
I
+ C {l h l2 + Ihul} dv(x) 0
+ cll udal2.
ao We are now ready to prove Proposition 4.2. Given f E £2(80), let u = Sf; first restrict u to O. Since ~u = Vu on 0, we can apply (4.15) and use (3.29) to obtain
0
o
0
while (4.11) yields
= 2 1(\7TWU)(811u)da(x) - 2/(\7wU)hdV(X)
I
I
+ C {l h l2 + Ihul} dv(x)
ao
ao
+
ao
+cll udal2,
+ (\7 Tw u)(811 u),
2 {(div w)l\7uI + 2(L:wg) (\7u, \7u)} dv(x),
uh dv(x).
0
2 da SCi I\7TuI da + cll udal2.
1(II,W){I\7Tu I2 - (811 u)2}da(x)
ao
I
Hence (4.10) yields an estimate
where Tw is the component of w tangent to 80. Hence we can rewrite the (limiting case of) identity (4.7) as
(4.8)
u ~~ da(x) -
ao
ao
{(div w)l\7uI 2 - 2(L: wg)(\7u, \7u)} dv(x),
(II, w)(811 u)2
I lul
(4.13)
0
whenever 0 is smoothly bounded and ceO. To prove this identity, you just compute div ((\7u, \7u)w) and 2 div (\7 wU . \7u), and apply the divergence theorem to the difference. If, in addition, (\7u)* E £2(80), we can take OJ / 0 with bounded Lipschitz constants, and pass to the limit, replacing 0 by 0 in (4.7). In the last integral in (4.7), L:wg denotes the Lie derivative with respect to w of the metric tensor g. Regarding the first integral on the right side of (4.7), note that (a.e. on 80) =
I
dv(x) =
Also, there is the Poincare estimate:
o
(\7wu)(811u)
2
o
2 1(\7wU)(811U)da(x) - 2/(\7wU)hdV(x)
ao
0
= h on 0, Green's formula gives
I l\7ul
(4.12)
As in [Ve], we use a Rellich-type identity, of the following sort. Suppose U E C 2 (0), and ~u = h E £2(0). Let w be a smooth vector field on M. Then we have the identity
I
+ C {l h l2 + l\7un dv(x).
ao
Furthermore, if
ao
2
2
IIfllL2 s CII(±~I + K*)fll£2 + CllsfIIHl(M)'
=
0
1811 ul da(x) SCi I\7TuI da(x)
ao
Hence
1(II,w)l\7uI 2da(x)
I
+ C {l h l2 + l\7uI 2} dv(x),
and also the inequality
ao
(4.7)
I 00
= (V 2 + V)1/2. Also,
(4.6)
237
l' j¥
II(~I - K*)fIII2(aO)
1
(4.16)
S CII\7T SfIII2(ao)
+ cll Sf dal2 + cllwsfIII2(O)' ao
Next, we use (4.14) to estimate II\7TSfIII2(an)' except we replace 0 by 0 = M\O. Then the first integral on the right side of (4.14) is equal to CII(~I + K*)fIII2(aO)'
238
4.
LAYER POTENTIALS ON LIPSCHl'I'Z SURFACES
by (3.29). Hence, with u = Sf on 0, so again ~u
(4.17)
=
Vu, we have
j l 'il T U 12 d(J"(x) an : 0 somewhere on n. The invertibility on L 2 (on) then follows from Proposition 4.6. On the other hand, if V = 0 on n, then Green's formula implies (-~I + K')f belongs to L5(on) for all f E L 2(on), so (4.30) is well defined, and one deduces from Proposition 4.6 that this operator is also Fredholm, of index zero. We show this operator is injective. Indeed, if f E L5(On) belongs to its kernel, then the arguments involving (4.31) again hold, and again (4.32) vanishes, so again we have f = O.
5. The Dirichlet problem on Lipschitz domains We now apply the invertibility results of §4 to the Dirichlet problem. As in §4, we assume M is a compact, connected, smooth manifold, with a smooth Riemannian metric tensor, a domain in M with nonempty Lipschitz boundary, and L of the form (4.1), with smooth V ~ 0, and V > 0 somewhere on each connected component of 0 = M \ n. We begin with the following existence result.
n
=
J
u:; d17(x) = 0,
&0.
Lu = 0 on 0.,
u· E L 2 (on),
L 2(on)
We complement this with the following result on -~I +K·. Let L5(on) denote the subspace of L2(on) orthogonal to constants.
(4.31)
!{I'VU I2 + VluI2}dv(x) = - ! u(~~)_ d17(x) = -co! fd17(x), o
(5.1)
are invertible.
(4.30)
241
PROPOSITION 5.1. Given f E L 2 (on), there exists u E CDO(n) such that
From the injectivity (4.1) we deduce:
(4.29)
THE DIRICHLET PROBLEM ON LIPSCHITZ DOMAINS
so u is a constant (say CO) on 0. (which we are assuming is connected). If V > 0 somewhere in 0., then CO = 0; in any case, Sf = Co a.e. on on. Hence
(4.32)
8n
(4.27)
5.
ul 8n = f
a.e.
PROOF. By Corollary 4.7, there exists a unique 9 E L 2 (on) such that (~I + K)g = f. Then u = 'Dg satisfies (5.1), by (3.27) and (3.30). The interior regularity stated above is standard. Note that the solution to (5.1) constructed above is given by
u = 'D((~I + K)-l f).
(5.2)
We wish to establish uniqueness of u satisfying (5.1). For this, it will be useful to have some elementary results on solutions to the Dirichlet problem in Sobolev spaces. PROPOSITION 5.2. Given f E H 1 / 2 (on), there exists a unique u satisfying
(5.3)
Lu
=
0,
u E H1(n),
ul 8n =
f.
----------.---.-.
242
4.
._u_.. _u."
LAYER POTENTIALS ON LIPSCHITZ SURFACES
PROOF. Since the relevant Sobolev spaces are invariant under composition by bi-Lipschitz maps, one can locally flatten the boundary and produce
0, (5.29)
f E H1,p(aD), 1 < P < 2 + e = } (Vu)* E V(aD).
As with other results treated here, these results were treated in [MT] , [MMT], and [MT2], for equations whose coefficients have minimal regularity. Here we have studied the homogeneous Dirichlet problem (5.1). There are further results on the inhomogeneous problem (5.30)
Lu = 9 on D,
ulao
=
f
on Lipschitz domains, for the flat Laplacian on Euclidean space in [JK] , and for e Lipschitz domains in Riemannian manifolds with metric tensor in Cl+ in [MT3].
---------------=====
-,,'fo-~~"
4.
LAYt;tt PUTENTIALS ON LlPSCHITZSURFACI'5S
A. The Koebe-Bieberbach distortion theorem The Koebe-Bieberbach distortion theorem, used in §1, is a result about uni
valent (Le., one-to-one) holomorphic functions defined in the upper half-plane, or
equivalently functions defined on the unit disk in Co It fits within a small collection
of results about univalent functions, which we present here.
Let 5 denote the set of univalent holomorphic functions, defined on the disk
7) = {z E C : Izi < I}, with the additional property that f(O) = 0 and 1'(0) = 1,
so, for f E 5,
The transformation f 1---+ g, given by g(() = f(I/0- l , takes 5 to E, consisting of
univalent holomorphic functions on 7)* = {( E C : 1(1 > I}, having the form
g(() = (+ bo + blC I + ....
(A.2)
A calculation connects the coefficients in these expansions. In particular, one ob
tains
bo = -a2·
(A.3)
There has been an intensive study of the coefficients ak in (A.l), leading to
the proof of the Bieberbach conjecture, that lakl ~ k for all k. For the distortion
theorem, the case k = 2 is relevant. The first tool for this is the following "area
theorem," due to Gronwall:
PROPOSITION A.1. For gEE! the area of K
=C\
g(7)*) is given by
oc
A(K) =
(A.4)
7r(I- I:klbkI2). k=l
PROOF. For any p > 1, the image of the circle Izi = p under 9 is a simple
closed curve 1'(p) in C, enclosing a region of area A(p). Green's theorem gives
A(p)
J
=
'Y(p)
Setting z
J
xdy = -
ydx =
-~ I:
'Y(p)
kbjbkP-j-k
j''' e(k-j)iOdO
=
-7r I:
PROOF. Let gEE be given as above, Le., g(() = f(I/()-I. Note that 0 ~ g(7)*). Now
h(() = g((2)1/2 = (+ bo C 1 + ...
(A.8)
2
The next result is the Koebe-Bieberbach quarter theorem. THEOREM A.3. Suppose f : and set ro = dist( wo, an). Then
n is
a biholomorphic map. Let Wo = f(O)
11'(0)1 ~ 4ro·
PROOF. Thanslating and rescaling, we can assume that Wo = 0 and 1'(0) = 1, so f E 5. Then the claim is that 1/4 ~ ro ~ 1. Let Zo E have minimal distance from the origin, so Izo\ = roo Consider
an
(A.I0)
n
dO' = 1'(z) Idzl,
I
r~\\ = ~(1- [zI2)II'(z)l.
There are biholomorphic maps of 7) to itself given by linear fractional transfor mations:
p"" 1 yields (A.4).
(A.13)
As a corollary, we have gEE
7) ---->
ro ~
(A.9)
(A.12)
klb k I p-2k.
k;:::-I
Taking the limit
la21 ~ 2.
(A.7)
-71"
2
247
PROPOSITION A.2. For f E 5, we have
zdz.
'Y(p)
j,k;:::-]
(A.5)
(A.6)
;i J
= g(pe iO ) = 2:k;:::-1 bkp-ke-ikO (with L l = 1), we obtain A(p) =
A. THE KOEBE·BIEBERBACH DISTORTION THEOREM
is a single-valued (odd) function of ( on 7)*, and in fact is seen to belong to E. Applying Proposition A.l, we have Ibol ~ 2, and then (A.3) gives la2/ ~ 2.
f(z) = z + a2z2 + a3 z3 + ....
(A.l)
•
===?
Ibit
~ 1.
Using this, we can prove Bieberbach's theorem:
r(z)
=
az + b
1 + baz'
Ibj < 1, lal
=
1.
It is an exercise to show that each such map r preserves the Poincare metric, i.e., (1 -lzI2)jr'(z)1 = 1 -lr(zW,
z
E 7).
248
4.
LAYER POTENTIALS ON LIPSCHITZ SURFACES
It is also an exercise to show that the set of linear fractional transformations of the form (A.13) forms a group that acts transitively on V. We can now re-cast Theorem A.3 in terms of the Poincare metric induced on a simply connected domain in C.
PROPOSITION A.4. Let 0 be a proper, simply connected domain in C. Let "Y(z) )dzl be the Poincare metric induced on 0, via a biholomorphic map f : V ~ O. Then, for z E 0, (A.14)
~ dist(z, (0) S "Y(~)
PROOF. Consider a point Zl En. Composing f with a linear fractional trans formation, we can assume without loss of generality that f(O) = Zl. In such a case, we have by (A.12) that _1_ = "Y(Zl)
~lf'(O)I, 2
and the conclusion (A.14) follows directly from Theorem A.3. to
NOTE. an.
In (A.14), dist(z, an) of course refers to the Euclidean distance from z
Applying (A.12) and (A.14) again, for general zED, we have the distortion theorem: THEOREM A.5. If f : V ~ 0 is a biholomorphic map, then, for all ZED, (A.15)
dist(J(z),aO) S (1-lzj2)IJ'(z)1 S 4dist(J(z),an).
Via a linear fractional transformation taking the upper half plane U in C onto V, we readily translate this to the estimate (1.10), noting that the Poincare metric induced on U is given by ds = (I/Y)ldzl. REMARK. The estimate (A.16)
Bibliography
S 2 dist(z, (0).
~dist(J(z),aO) S (l-jz/)IJ'(z)1 S 4dist(J(z),aO)
follows directly from (A.9) and a scaling argument. This is only slightly weaker than (A.15), and it will suffice for application to (1.10), with nonoptimal 0: and {3, namely 0: = 1/4, {3 = 4.
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List of Symbols Various function spaces and other objects occur in this monograph, labeled by the following special symbols. We give the chapter and formula number where each one is introduced.
AoSr,'o
BMO bmo BSn Bp Sr,'l B;,l B;,q CU.) c(>,)sm
1,0
C(>\)S1fJ
1,1
Cw C[w] C{w}
Cr
* Cr,C),)
C:Sr,'o 'D TF Ha,p Hs,l Hs,oo 5jl(~n)
~l(~n)
(£00
n vmo) Sci
MP(~n) MP,w(~n)
0PXSro R(f, u)'
Sro Tf
XSro X Sr,'(I