.-
~~~~=~"~~~'~~~~~~'"'~'~-~~~~~~'~=~"--.-.'-.';--~'~~'~~~--='~~~~"~=~=~-' -c.
Mathematical
t'
Surveys
o
and Monographs
Volume 81
Tools for PDE Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials
Michael E. Taylor
i,l
$::::~'('«~
.. .. ~
rJi :
~ L
.
~
• 9,
~ r;; ..., "
American Mathematical Society
=o=~o~,~~="=~"'~~~"~~~===~·~=~"~=·~~~·=~=·==~=~~~=·~~"·~~~·"·==·~·=~=='·~='===='===~~l;:T.~~~~~~o~
Editorial B o a r d " " Georgia Benkart Peter Landweber
Michael Loss Tudor Ratiu, Chair
1991 Mathematics Subject Classification. Primary 35805, 35S50, 42B20.
Contents
The author was supported in part by NSF Grant #9877077. ABSTRACT. This book develops three related tools that are useful in the analysis of partial differ ential equations, arising from the classical study of singular integral operators: pseudodifferential operators, paradifferential operators, and layer potentials. A theme running throughout the work is the treatment of PDE in the presence of relatively little regularity. In the first chapter we study classes of pseudodifferential operators whose symbols have a limited degree of regularity. In the second chapter we show how paradifferential operators yield sharp estimates on various nonlinear operators on function spaces. In Chapter 3 we apply this material to an assortment of results in PDE, including regularity results for elliptic PDE with rough coefficients, planar fluid flows on rough domains, estimates on Riemannian manifolds given weak bounds on the Ricci tensor, div"cur! estimates, and results on propagation of singularities for wave equations with rough coefficients. Chapter 4 studies the method of layer potentials on Lipschitz domains, concentrating on applications to boundary problems for elliptic PDE with variable coefficients.
Library of Congress Cataloging-in-Publication Data Taylor, Michael Eugene, 1946 Tools for PDE : pseudodifferential operators, paradifferential operators, and layer potentials / Michael E. Taylor. p. em. - (Mathematical surveys and monographs, ISSN 0076"5376 j v. 81)
Includes bibliographical references and index.
ISBN 0-8218-2633-6 (alk. paper)
1. Differential equations, Partial. 1. Title. II. Mathematical surveys and monographs; nO. 8l. QA377.T37 2000 515'.353--dc21 00"036248
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10987654321
05 04 03 02 01 00
j
...f
I'
l',;;.. . . . • . :..
Preface
ix
Chapter 1. Pseudodifferential Operators with Mildly Regular Symbols §1. Spaces of continuous functions §2. Operator estimates on V', ~1, and bmo §3. Symbol classes and symbol smoothing §4. Operator estimates on Sobolev-like spaces §5. Operator estimates on spaces CO.. ) §6. Products §7. Commutator estimates §8. Operators with Sobolev coefficients §9. Operators with double symbols §10. The CRW commutator estimate §11. Operators with vmo coefficients §12. Estimates on a class of Besov spaces §13. Operators with coefficients in a function algebra §14. Some BKM-type estimates §15. Variations on an estimate of Tumanov §16. Estimates on Morrey-type spaces
3 17 31 37 44 54 58 61 63 75 78 82 86 88 92 94
Chapter 2. Paradifferential Operators and Nonlinear Estimates §1. A product estimate §2. A commutator estimate §3. Some handy estimates involving maximal functions §4. A composition estimate §5. More general composition estimate §6. Continuity of u f-t feu) on Hl,p §7. Estimates on F(u) - F(v) §8. A pseudodifferential operator estimate §9. Paradifferential operators on the spaces C CA ) §A. Paracomposition §B. Alinhac's lemma
101 105 106 108 110 112 113 116 118 120 125 132
Chapter 3. Applications to PDE §1. Interior elliptic regularity §2. Some natural first-order operators §3. Estimates for the Dirichlet problem §4. Layer potentials on Cl,w surfaces
135 137 148 155 159
:;',,':
vii
1
/If
~".~"".......= '::,,'
§5. §6. §7. §8. §9. §10. §11.
,..
""""""'"'-=,._
_0' ".
:--__ :~_:::=--_~>.-=-=--=."-=;"'"'L
.•
'''',''''_7';,"""",_j'''''_'''''''''''''''''""-""''''''I-'-'''''''''''''''''7'''',"'''''''-''''''''''''''''''£~'=-'''''''-~''~''''''-c_,c.".,,."''""-'",,
Parametrix estimates and trace asymptotics Euler flows on rough planar domains Persistence of solutions to semilinear wave equations Div-curl estimates Harmonic coordinates Riemannian manifolds with bounded Ricci tensor Propagation of singularities
"~.
173 178 183 186 194 202 205
Preface
Chapter 4. Layer Potentials on Lipschitz Surfaces §1. Cauchy kernels on Lipschitz curves §2. The method of rotations and extensions to higher dimensions §3. The variable-coefficient case §4. Boundary integral operators §5. The Dirichlet problem on Lipschitz domains §A. The Koebe-Bieberbach distortion theorem
217 218 228 230 235 241 246
Bibliography
249
List of Symbols
255
Index
257
Since the early part of the twentieth century, with the work of Fredholm, Hilbert, Riesz, et al., the use of singular integral operators has developed into a range of tools for the study of partial differential equations. This includes the use of single and double layer potentials on planar curves to treat classical boundary prob lems for the Laplace operator on a planar region and higher-dimensional extensions. It also includes the construction of parametrices for elliptic PDE with variable co efficients. Fourier integral representations of these operators have provided many useful insights, though this method has not entirely supplanted the singular integral representation. When the use of the Fourier integral representation is emphasized, the operators are often referred to as pseudodifferential operators. Paradifferential operators form a singular class of pseudodifferential operators, particularly suited for applications to nonlinear PDE. Treatments of pseudodifferential operators most frequently concentrate on op erators with smooth coefficients, but there has been a good bit of work on operators with symbols of minimal smoothness, with applications to diverse problems in PDE, from nonlinear problems to problems in nonsmooth domains. In this monograph we discuss a number of facets of the operator calculi that have arisen from the study of pseudodifferential operators, paradifferential operators, and layer potentials, with particular attention to the study of nonsmooth structures. In Chapter I we study pseudodifferential operators whose symbols have a lim ited degree of regularity. We consider various cases, including measures of regularity just barely better (or just barely worse) than merely continuous, measures either a little better or a little worse than Lipschitz, and others. Function spaces used to describe the degree of regularity of symbols include
e Here
e
w
w
eCA)
,~
LX n vmo ' pB,S l .
consists of functions with modulus of continuity w. The space e CA), with
A(j) = w(2- J ), is defined in terms of estimates on a Littlewood-Paley decomposition
j,,-~
.•
'
of a function. These spaces coincide for Holder-Zygmund classes of functions, but they diverge in other cases. The space vmo is the space of functions of vanishing mean oscillation, and B;,l are certain Besov spaces. The interplay between some of these function spaces is itself a significant object of study in this chapter. The class of paradifferential operators, introduced in [Bon], has had a substan tial impact on nonlinear analysis. In Chapter II we make use of paradifferential operator calculus to establish various nonlinear estimates, some of which have previ ously been established from other points of view. My interest in organizing some of this material, particularly in §§1-5, was stimulated by correspondence with T. Kato.
.
----~'. . . . . . .! .
• ,:'
..•...
':
ix
--.---,..----..-
..."""
-----~,
Other material in Chapter II includes investigations of paradifferential operators on the new function spaces C(>'). Chapter III gives a sample of applications of some of the results of Chapters I-II to topics in PDE. We treat some linear PDE with rough coefficients, includ ing some natural differential operators arising on Riemannian manifolds with non smooth metric tensors. We consider the method of layer potentials on domains that are not smooth (though not so rough as those considered in Chapter IV). We also treat a couple of topics in nonlinear PDE, including inviscid, incompressible fluid flow on rough planar domains and wave equations with quadratic nonlinear ities. We also discuss various div-curl estimates, including a number of estimates of [CLMS]. Some of the work in this section, especially variable-coefficient results, grew out of correspondence with P. Auscher, following up on our work in [AT]. Other topics studied in Chapter III include the construction of harmonic coordi nates on Riemannian manifolds with limited smoothness, regularity results for the metric tensor of a Riemannian manifold when one has estimates on the Ricci tensor, and propagation of singularities for PDE whose coefficients are more singular than C l • l , but which still have well defined null bicharacteristics by virtue of Osgood's theorem. Chapter IV deals with the method of layer potentials on Lipschitz domains. We establish the fundamental estimates of Cauchy integrals on Lipschitz curves of [Ca2] and [CMM] (via a method of [CJS]) and extensions to higher dimensions from [CDM]. We then discuss the Dirichlet problem for Laplace equations and variants on Lipschitz domains. We consider operators with variable coefficients, hence Lipschitz domains in Riemannian manifolds. Our treatment of this follows [MT], though here we restrict attention to the simpler case of smooth coefficients, whereas [MT] treats cases arising from C l metric tensors. This extends earlier work of [Ve] and others on the flat Laplacian on Lipschitz domains in Euclidean space. Prerequisites for this work include an acquaintance with basic results on pseu dodifferential operators and some methods from harmonic analysis, including the Littlewood-Paley theory. Sufficient material on these prerequisites could be ob tained from either [T2] or Chapters 7 and 13 of [T5]. Indeed, this present work can be viewed as a companion to [T2]. Michael Taylor
CHAPTER 1
Pseudodifferential Operators with Mildly Regular Symbols Introduction Studies of pseudodifferential operators whose symbols p(x,~) satisfy a HOlder condition in x have been found to be very useful in PDE. A number of their prop erties and applications are investigated in [Bon], [Bour], [KN], [Meyl]' [T2], and other places. There has also been an interest in symbols whose x-dependence is described by some other modulus of continuity; studies of this are made in [CM], [Ma2], and [Ma3], for example. Here we study related problems. We concentrate on measures of regularity just a bit better than mere continuity, with a secondary interest in measures of regularity either barely better or barely worse than Lipschitz. We measure such regularity in several different ways. To be explicit, we consider four types of function spaces, associated with a modulus of continuity w. First, (0.1)
u E CW
Second, with A(j) = w(2- j
),
+ y) -
u(x)1 ~ Cw(ly\)·
we say
u E C(>')
(0.2)
lu(x
{==}
{==}
II'lf'j (D)ull Loo
~ CA(j),
where {'If'j(~)} is a Littlewood-Paley partition of unity and 'If'j(D) the associated Fourier multiplier. Third, we say
(0.3)
U
E C[w]
{==}
\lu -
Wj(D)u\lLoo ~ Cw(Tj),
where wj(D) = 'L.f5,.j'lf'e(D), which is an approximate identity as j we say j (0.4) u E C{w} ~ II\7 x Wj(D)ullL"" ~ C2 w(Tj).
----t 00.
Fourth,
When w(h) = hr (so A(j) = 2- jT ) and r E (0,1), then all the spaces (0.1)-(0.4) coincide, due to well known results of Bernstein, Jackson, and Zygmund. For more general r E JR, (0.2) with A(j) = 2- jr defines the Zygmund space C:. For other interesting w, such as
(0.5)
1
w(h) = (log h
)-8
(for hE (0,1/2]) with s > 0, or (0.6)
w(h)
= h(log ~r,
......"..,..""~c~""'''-~--''-'-''=-~.c..... ''''''...,~~~,·_~~-.,.,''-~''~-''~'=''--=y''-.,....Sr]\Gi::i~r·''''!"""~~ffi'~Qv-u5""--.i'--i()~'I-·roi"il""6-'"""·"·~""""'~"-~~""-~·"""""""'~"'--""~"'''
not all of these spaces coincide. In §1 we explore various properties of spaces of the form (0.1)-(0.4), including containment relations among them. In §2 we establish LP-operator norm estimates for pseudodifferential operators
(0.7)
p(x, D)u
= (271-)-n/2
!
In §9 we study operators arising from double symbols: (0.12)
p(x, ~)il(Oeix,~ d~,
where a(x,y,~) has limited regularity in x and y. For example, parallel to (0.8), we say
whose symbols p(x, 0 are of order 0 and satisfy various regularity conditions in x. We consider symbol classes of the form X Sf,'o, where (0.8)
(0.13)
p(x,O E XSr,'o ~ IIDfp(-'~)lIx ~ Co(~)m-lol,
p(x,~) E S~8 ~ ID~Dfp(x,~)1 ~ Col1(om-101+8/I1I.
We produce results in terms of CC>")-regularity, and compare these with results of [CM] and [Ma3], given in terms of CW-regularity. Neither set of results contains the other. We can slightly extend the scope of the latter class of estimates by recasting them in terms of C[wLregularity. In §3 we decompose a symbol, such as in (0.8), into a sum of two terms. One, p#(x,~), has better smoothness properties, of use in further results on operator calculus. The remainder, pb(x, ~), has no better smoothness, but does have lower order (by a degree depending on the smoothness of p(x, ~)). Such symbol smooth ings follow a number of earlier models, including [KN], [Bon], [Mey2], and [T2]. A particular case of fundamental importance gives rise to Bony's paraproduct. Constructions in §3 give rise to another class of symbol spaces, X Si,CK)' defined in (3.18)-(3.19) .
(0.14)
Ha,p(I~n) = {j
E
U(lR n) : A(D)j
E
U(lR n )},
where A(~) = (~)a(O is a symbol satisfying a certain "ellipticity" hypothesis. In §5 we treat operators p(x, D) on the spaces CC>"). These are the spaces from the list (0.1)-(0.4) that seem best suited for analysis involving pseudodiffer ential operators. Largely for this reason, we have emphasized symbol spaces of the form CC\) Sf,'o and CC>") Sf,'cK) over other symbol spaces. Indeed, the class of spaces CC>") is important even for analysis involving more usual sorts of pseudodifferential operators. We mention that, for slowly varying )...(j) '\. 0,
(0.11)
P E OPSP,8, 0 ~ J < 1 ====> P: CC>")
----+
j
E
n
B;,l (lR ) ~
I: 112js 1/'j(D)jIILP <
00.
with 1/'j (D) as in (0.2). In particular, the space B~,1 (lR n ) is of interest because it is contained in C(lR n ) but contains all spaces CC\) (lR Tl ) for which L)...(j) < 00. In §13 study a rather general class of operators with coefficients in a function algebra. One primary application of these results will be to cases involving Besov spaces that are algebras. In §14 we establish Loo estimates on Pu, when P is a pseudodifferential operator and u is barely more regular than continuous, of a sort pioneered in [BKM]. We also produce improvements, involving estimates in B~ l' and mention how such ' results relate to constructions in Chapter 5 of [T2]. In §15 we present some generalizations of an interesting inequality that arises in the study of CR manifolds in [Tu]. We interpret the estimates as commutator estimates. In §16 we look at some variants of Morrey spaces. We extend Morrey's imbed ding theorem to a class of Morrey-type spaces. We discuss advantages of expressing these results in terms of spaces C{w} rather than the spaces cw more commonly used for such imbedding theorems. We also study the action of various pseudodif ferential operators on these Morrey-type spaces.
In §4 we analyze operators p(x, D) on variants of Sobolev spaces, of the form
(0.10)
a(x, y,~) E XSr,'o ~ IIDfa(., ',~)llx ~ Co(~)m-lol,
where X is a Banach space of functions on lR n x lR Tl • There are similar definitions of symbol classes XSf,'8' XS~CK)' etc., for double symbols. While, as is well known, if a(x,y,~) E Sf,'8,J E [0,1), then A in (0.12) can be written in the form (0.7), with p(x,~) E Sr,'8' such a reduction need not hold for symbol classes with limited regularity, so a separate study of operators of the form (0.12) is called for. One way operators of this sort arise is in the use of layer potential techniques for elliptic PDE on domains whose boundaries have limited regularity (a little better than C 1 ). We will explore this in §4 of Chapter III; this can be contrasted with the more difficult study of such PDE on Lipschitz domains, taken up in Chapter IV. In §1O we establish a commutator estimate of [CRW]. The approach we take uses paraproducts, and follows a proof given in [AT]. This commutator estimate plays an important role in div-curllemmas, a topic that will be discussed in Chapter III. It also is a key tool in the analysis of operators with vmo coefficients, the subject of §11 of this chapter. In §12 we investigate operators on the Besov spaces B;,l(lR Tl ), defined by
and X is some function space, such as C W , C(>.. ), etc. We take (~) = (1 + 1~12)1/2 in all cases except when (~) occurs as an argument of the logarithm; then we take W = (4 + 1~12)1/2. We also define in (2.52) symbol classes XS~CK)' generalizing the H6rmander classes Sf,'8' We recall that
(0.9)
Au(x) = (21r)-n!! a(x,y,~)eiCx-YHu(y)dyd~,
CC\),
which is a consequence of Proposition 5.7. In §6 we study products of operators, particularly of the form p!(x, D) where p!(x,~) arises from p(x,O E XS~CK) via symbol smoothing. We apply this to commutator estimates in §7. In §8 we discuss results about operators whose symbols p(x,~) have Sobolev space regularity in x.
1. Spaces of continuous functions
j
Suppose w is a modulus of continuity, i.e., a continuous, increasing function on [0,00) such that w(O) = 0, and having the property that w does not vary too
1. S
rapidly, Le., that w(2h) ::::: Cw(h), V h E (O,IJ. We denote by bounded, continuous functions on lR n satisfying
(1.1)
lu(x
+ y) -
cw
the space of
(1.10)
u(x)/ ::::: Cw(lyl).
wd~) = WO(2-k~),
'l/Jk(~) = Wd~) - Wk-l(~),
7y ullux>
: : : lylllL V'I/'j (D)utc>o + II (I -
where 7yU(X) = u(x - y). Now
IIV'I/'j(D)ullu>c : : : C2jll'l/'j(D)ullu>o,
(1.11)
Ilu -
(1.12)
Tyullu'" :::::
C1yl L
j 2 ll'l/'j(D)uIILC>O
'lidO = 'l/'o(~) + ... + 'l/'k(O.
(1.4)
j>£
II'I/'j(D)ullu'" ::::: CA(j).
Generally, speaking, if w(h) is a modulus of continuity, the construction of a(h) via (1.8)-(1.9), using A(j) = w(2- j ), gives
>.(j) = w(2- j )
=}
cw
l'I/'j(D)u(x) I =
IJ
(1.6)
::::: C
(1.14)
'¢j(Y) [u(x - y) - u(x)] dyj
J
PROPOSITION 1.2. If A(j) is a positive decreasing sequence such that
where
(1.15)
inf (A (£) h + 0(£)),
£EZ+
with
j~£
0(£) =
Cl3w(t) '\.,
for t E (0,1],
f3
for some
E
(0,1).
In this case, we have
( 1.16)
L
A(j) <
(1.17)
(1.18)
L 2 A(j),
i
h
t w(t) dt < hw(h) (I t13hl3 i
ih
t2
-
2
dt < Cw(h)
h
-
,
(h w(t) dt 2: w(h) (h t13-1 dt = C'w(h).
io
L j>£
A(j).
io
h 13
t
Thus, in such a case, with >.(j) = w(2-
CC>') C C a ,
A(£) =
t
i
while
then
j
dt),
w is a modulus of continuity with the property that
Towards a converse, we have the following.
(1.9)
iro w(t) t
a(h);:::: (h w(t) dt + h w~) dt. t h o t In many cases, the last term in (1.14) can be absorbed. For example, assume
= 0,
since '¢j(Y) is concentrated on Iyl ~ ment is given in (1.42)-(1.43) below.)
=
dt +
J;
C2- j . (A more elaborate version of this argu
a(h)
t
T
provided w(t)C 1 dt < 00. Note that the 7-derivative of the function in paren theses vanishes precisely at 7 = h, if hE (0,1), so
l,¢j(y)1 w(lyl) dy
::::: C'w(2- j ),
(1.7)
(h
O
C CC>').
PROOF. Assume u E CWo Then, for j 2: 1, since '1/'(0)
1 w~) 1
a(h);:::: inf
(1.13)
PROPOSITION 1.1. If w is a modulus of continuity, then
(1.8)
W£(D))ullu>o'
Bounding the last term in (1.12) by 2 L II'I/'j(D)uIILC>O, we obtain (1.8).
We also make the standing assumption that A(j) is slowly varying, i.e., A(j) ::::: CA(j + 1).
00,
+ 211 (I -
j~£
Now we say u E CC>') if and only if
(1.5)
'1' £(D)) (u - TyU) 11£0>0,
so it follows that
for k ~ 1. Also set 'l/'o(~) = wo(~), so
(1.3)
Ilu -
j~£
Next, if A(j) is a positive decreasing sequence, we want to define CC>'). To do this, we use a Littlewood-Paley partition of unity. Pick '1'0 E COO (lR n ) , so that wo(~) = 1 for I~I : : : 1, 0 for I~I 2: 2. Set
(1.2)
PROOF. For any £ E Z+, write
CC>') C C a ,
j
),
we have
with a(h) = (h w(t) dt.
io
t
Let us consider some examples. First, consider w(h) = hT, 0 < r < 1. Then (1.15) applies, so (1.18) holds, with a(h) = Ch T, so we have the well known result:
(1.19)
w(h) = h
1' ,
A(j) = TjT
::=:}
C W = CC>'),
This identifies Holder spaces and Zygmund spaces, i.e.,
(1.20)
1'
C =C:,O
if 0
< r < 1.
IF L
1. 6 61
UI2
BitAielt5""eiIR M,ebef p~utftk~'~~~IFT"
Next, consider w(h) = h, so cw = Lip 1. Thus A(j) Zygmund space C;. Proposition 1.1 gives (1.21)
Li p 1
=
r
M
1
'
e_m
~11".Il!UillJmJl\IMm··~~·aw.M:&~~~~~,s'i!".4~~->'"'''~''''''''''''''''··<>''""l~~''SPACES\':Hr--vOP4.Lll~UUUb
1,
and C(A) is the
This extends (1.22). We mention that a closer analysis of (1.29) shows that it is ~ h(log(l/h)r as h \, 0, so for this class of examples the second term on the right side of (1.14) is dominant. We will return to this point shortly; see (1.54)-(1.57). It is desirable to estimate IIP(2~jD)ullu>Q for P(~) in a class somewhat larger than that figuring in the definition (1.4). We will take
C;.
C
In this case, (1.15) does not apply. Instead, the second term on the right side of
(1.14) is dominant, and we have (1.22)
C;
cC
er(h) = h log h (0 < h:::; 1/2).
,
1)
w(h) = (log h
'
0< h:::;
1
We assume b 2: O. To say P(O) = 0 means P(~) is a sum of terms of the form ~vQv(~), with Qv(O E Slb-1 (lR n ). We may as well suppose P(~) = ~vQv(~)' Now we have
2'
This yields
(1.24)
A(j) =
r
P(Tj D)u = L 2- j DvQv(Tj D)'l/J£(D)u
8
£~o
in (1.5) (up to a constant factor). Proposition 1.1 applies here for any s > O. To apply Proposition 1.2, we need s > 1. In such a case, (1.15) holds for any (3 > 0, and we get
er(h) =
(1.25)
J(h( log t1)-8 tdt o
w(h)
= h(log~)
e~j
+ L P(2- j D)'l/J£(D)u. e>j
1)-8+1 .
(
= Clog h
If we assume u E C(A), then (1.34)
(1.35)
This yields
A(j) = FTj.
IIP(Tj D)'l/Je(D)uIILoo :::; CT(e-j)b A(£),
\;/ £ 2: j.
IIP(TjD)uIILoo:::; CLTU-e)A(£) +CLT(£-j)b A(£). e~j e>j j j Let us set A(j) = w(2- ), h = 2- , Then the two terms on the right side of (1.36) h are equivalent to h.f~ w(t)C 2 dt and to .fo w(t)C(l-b) dt, respectively. We have the (1.36)
Propositions 1.1 and 1.2 apply, for any s 2: O. We compute er(h) using (1.14). We have 1 2w / (t) t/2 1)8dt 1)8+1 (1.28) h h 72 dt =h (logt Ti=:::h(logh .
1
Jh
following.
Also,
l
h
-w(t) dt
=
l
h
1 8 dt = (log -)
tot
o
For h small, Le., for A
= log(1/h)
( 1)8+2100 l/h) T-
PROPOSITION 1.3. Assume P(~) and A(j) = w(2- j ), then
100
T8 e- T dT.
(1.37)
log(l/h)
large, we have
:::; h log h
2
T
8
+2 e- T
\,
( 1)8+1 .
(1.38)
dT = h log h
c
C(A),
C(A)
C
CU ,
Slb(lRn ), b 2: 0, and P(O) = O. If u
IIP(hD)ullL'XO :::;
C1lullcc;\) . erb(h),
erb(h) = h
1 1
h
l
h
w(t) w(t) dt + 1-b dt. tot
-2
If b > 0, we can take
Hence, with wand A given hy (1.26)-(1.27), we have C'"
E
where
on [A, 00), so this is
log(
(1.30)
\;/ j,£ 2: 0,
Hence
(1.27)
(1.29)
IIQv(Tj D)2-£D v'l/Je(D)ullu'O :::; CA(£),
and
O
8,
= LTU- e)Qv(2- j D)T eDv'l/Je(D)u
(1.33)
Note that er( h) > > w( h) as h \, 0 in this case, again illustrating that Proposition 1.2 is not exactly a converse of Proposition 1.1.
As another family of examples, let us take s > 0 and consider
(1.26)
\DfP(OI:::; CQ(~)-b-IQI.
(1.32)
-8
P(O) = 0,
where membership in the symbol class Slb(lRn ) means
For another class of examples, consider
(1.23)
P(~) E Slb(lRTI ) ,
(1.31)
1
U
.t'.Ul~vTlV1""
'~;f"
( h1)8+1 .
(1.39)
er(h) = h log
j
erb(h) = w(h)
+h
1 1
wS) dt,
E
C(A)
' ... _..
eo
I i &111
nUDOnI
1.
lCCiUOLRh' ijl1VlbUL-.::J
If b > 0 and t-iJw(t) "" for some f3 E (0,1), we can take
PROPOSITION
SPA-CES
O~(jtJS-FUNCTIONS
9
1.5. Let Wo(~) be as in (1.2). Then, for a modulus of continuity
w,
ab(h) = w(h).
(1.40)
Note that, if b = 0, we require L A(j) < 00 for the right side of (1.38) to be finite. If b > 0, this restriction is not required. Also note that all the estimates above work if we consider instead
(1.46)
Si"(JRn ),
(1.47)
(1.41)
p(e) = A(~)B(IW,
A
E
BE Slb-I-m(JR),
==}
P(O) = O.
IF(h-Iy)lw(/yl)dy
if Iy!
= t
=}
1
{)
IP(hD)u(x)!
~ Cw(h) + Ch
L ~~l
k
u EC ,
a(h) =
i to
w(t) t
dt.
if?v(~)
=
{)W 0 [)C
.
<"v
Since if?,/ (~) is supported on the shell 1 ~ I~I ~ 2, it follows that
IID,/
Hence the hypothesis in (1.47) implies that
Now, if
II :y Wo(YD)uIIL~ ~ Cw(y). y Xj E
JRn and IXI -
x21 = h, let us write
U(XI) - U(X2)
IF(y)! ~ F(t). If we assume that IF(y)j ~ Clyl-n-a, then
(1.43)
(T
==}
'"
(1.50)
F(h-It)w(t)t n- l dt,
a
w(h)
v
J\,~:·.l
~ Cw(h) + Ch- n
~ CW~h).
Loo(JRn ),
ay wo(yD)u = ~ Dvif?v(yD)u,
(1.49)
00
E
IIVxwo(hD)uIILoo
IIVxWo(hD)ullu'" ~ C
(1.48)
IP(hD)u(x)1 = h-nl! F(h-Iy)[u(x - y) - u(x)] dy!
!
==?
The result (1.46) follows from Proposition 1.4, applied to P(O ~,/wo(~). To get (1.47), we argue as follows. Note that
IIP(hD)u/lu", ~ Cllullcw . w(h).
~ Ch- n
CW
PROOF.
We will review that argument in more detail to treat hypotheses closer to those of Proposition 1.3. If FE LI(lR n ) and J F(x) dx = 0, then
(1.42)
E
On the other hand, given u
This will be useful later on. Note that the same argument as used in the proof of Proposition 1.1 shows that, if P E S(JRn) and P(O) = 0, then, when w is a modulus of continuity,
u E CW
u
(1.51)
XJ
=
[U(XI) - Wo(hD)u(XI)] + [Wo(hD)u(XI) -Wo(hD)u(X2)]
+ [Wo(hD)u(X2) - U(X2)],
dt.
and deduce that
Estimating the last term by such a device as used in (1.16), we have the following. PROPOSITION
1.4. Assume that P(O)
FE LI(lRn ),
(1.44)
IF(x)1 ~ C!xl- n- a.
If a > 0, then the estimate (1.43) holds. If also C f3 w(t) "" for some (3 then (1.45)
u
E
CW
==}
(1.52)
= 0 and
/IP(hD)ullu>o
~
IU(XI) - u(x2)1
~C t
io
w(y) dy y
+ Cw(h),
which gives (1.47), upon absorbing w(h). E
(O,a),
Cllu/lcw . w(h).
If P(~) E Slb(JRn), then F satisfies (1.44), for all a > 0, as long as b > 0. On the other hand, if P(O satisfies (1.41), with b > 0, then it satisfies (1.44), with a = 1. In such a case, the estimate (1.43) on IIP(hD)uliLoo has the same strength as the estimate (1.39), which is valid under the weaker hypothesis that u E CO\). We next give a relationship between the modulus of continuity and an Loo_ estimate, which is slightly closer to an equivalence than that given in Propositions 1.1-1.2.
The conclusion in (1.47) is stronger than the conclusion of Proposition 1.2, when the second term on the right side of (1.14) is dominant. This happens when w(h) is given by (1.26), i.e., (1.53)
( 1)8 '
w(h) = h log h
0
1
< h ~ 2'
The analysis of the last term in (1.14), in this case, is done in (1.28), while the analysis of the other term was begun in (1.29). It is time to complete that analysis, i.e., to evaluate asymptotically (1.54)
ioo
TSe- T dT,
1 A = log h
---> 00.
~:L::
-- ... ~ .........
-R£"'LiD~~~B'ij";'u"'T7~n
CI I .1vrD~0E~~-~··~--~~~-
- = " , •• "-:':"",,-
-
;.,-
1. SPACES OF CONTINUOUS FUNCTIONS
This is equal to
(1.55)
A"+1 /00 tOe-At dt = A"+1
/2
tOe-At dt
+ A"+1
1
00
tOe-At dt.
The first integral on the right is
(2
~ 2" Jl e- At dt = 2 8 e
(1.56)
-A
-. e
~ 2s+2e- 2A
(1.57)
100 c
">. on [2,(0) for A large
uE
1.6. If w(h) has the form (1.53), for some s ~ 0, then, given
Looc![~n),
(1.58)
u
E
C
w
~
II\7 x wo(hD)ullv>o
~
w(h)
CT'
We mention that the s = 1 case of this played a role in [Be]. Note that
(1.59)
u
E
C(;:.,)
(1.66)
(1.62)
c
c{a},
cw
c w c C[w] C C(;:.,) C CCT,
a(h) =
lh w~t)
dt
+h
[1 wS) dt.
When a(h) ~ w(h), all these spaces are equal. The identity of cw and C[w] when r w(h) = h (0 < T < 1) is the content of classical results of Jackson and Bernstein. It is useful to separate some reverse inclusions; we have
C(;:.,) C C[IL], (1.63)
C[W] ceo.,
p,(h) = fh w(t) dt Jo t
1
p,(h) =
a(h) = w(h)
-----+
C{ w}
-----+
c(;:.,)
----->
c{a}
1
Clw]
+h
lh w~t)
t w~) dt. t
Ca
CIL
-------t
CIIL]
-----+
CCT
1
1
1
The inclusions c{a} C CCT and C[IL] C CCT follow by substituting one modulus for another in (1.62)-(1.63) and manipulating some double integrals. We present the following examples in which the various function spaces ex amined above can be seen to coincide or differ, recalling several calculations done above: w(h) = (log ~) -8, s > ==* C[w] = c w =1= C(;:.,) = C{w},
w(h) = h(log
~r,
s
° ~°
==* C[w] = C(;:.,) =1= C W = C{w},
We make note of some desirable special properties one might require of a mod ulus of continuity. One is that w(h) be concave. All the examples given above have that property. Concavity implies slow variation; in fact, if w is concave, then w(2h) S 2w(h). The following result is useful. PROPOSITION
1.7. If w is a concave modulus of continuity, then
1
a(h)
=
w(h)
+h
wS) dt.
dt,
Jh
-------t
1
1
W
then, with AU) = w(2- j ), we have
II\7 x wo(hD)ullL'''' ~ CW~h).
In particular, again by (1.15)-(1.16), C(;:.,) = C{w} provided w(t)CI3 ">. for some (3 E (0,1). We collect the inclusions given above in the following diagram. Here, given w(h), we take A(j) = w(2- j ) and a(h), p,(h), and a(h) are as in (1.62)-(1.63).
C => Ilu - Wo(hD)uIILCX> ~ Cllullcw . w(h).
u E C[w] ~ Ilu - Wo(hD)uIILCX> ~ Cw(h),
{===?
while, by Proposition 1.3,
lu(x) - Wo(hD)u(x)1 = h-nl! ~O(h-ly)[u(X) - u(x - y)] dyl,
(Compare [eM], Lemma 3, p. 43.) We can summarize some of the results above as follows. If we denote by C[w] the space of functions satisfying
(1.61)
C{w}
C w c C{w} c C(;:.,) n CIL,
and this can be estimated in the same fashion as (1.42). Since Wo E S(JRn), the reasoning leading to (1.45) applies, so we have
(1.60)
E
Then (1.65)
when A is large enough. Hence (1.54) is ~ CAsCA for large A, so, when w(h) has h the form (1.53), we see that Jo C 1w(t) dt s Cw(h). COROLLARY
u
(1.64)
= 2"+l e -2A,
2 dt
The first inclusion in (1.63) follows by summing II'l/Jk(D)ullL'>O over k ;::: j, while the second follows from an examination of (1.12). Note that, in light of (1.15)-(1.16), we have C[w] = cw whenever w(t)t- 13 ">. for some (3 E (0,1). Another function space is suggested by the result of Proposition 1.5. We denote by C{w} the space of functions satisfying
-2A
To estimate the last integral in (1.55), note that t"+2 e -At enough, so this is
11
(1.67)
==;.
E
CW.
l'l
1.
UPt;;KA'l'URS WITH MILDLY REGULAR SYMBOLS
PROOF. If IXjl = t j , then It I - t21 S; IXI - x21, and
lip(xI) - ip(X2) I = Iw(t l ) - w(t2)1
S; w(ltl - t21)
S; W(IXI - X21),
the first inequality by concavity.
··l.SPACaS Of; cUN'iTNUOUS FUNCTIONS
13
is the Zygmund space. Also, C[w] might not be trivial. We will encounter a number of occasions to use such spaces. It is desirable to record the simple fact that the spaces defined above are inde pendent of the choice of the Littlewood-Paley partition of unity used in the defini tion. Thus, suppose {ipj} is another such partititon of unity, and set
j = I:e~j ipe. Then there exist M, N such that, whenever j ~ M,
j+N To state the next requirement we might impose, set Wo (h) = w( h) and, in ductively, set wj(h) = hwj_l (h), for j = 1,2,3, .... We will say that w is tame of order k if Wj is concave and wj(h) S; Cjwj_l(h), for all j S; k. Note that both w(h) = hr(O < r < 1) and w(h) = (log(l/h)) -8(8 > 0, 0 < h S; 1/2) have this property. The following is a useful complement to Proposition 1.7. PROPOSITION 1.8. If w is a modulus of continuity which is tame of order 1, then (1.68)
(X) = w(lxl)lxl
=}
\7 E CW.
L
e=j-N
VJe(~),
j(~)
= J(~)Wj-N(~)'
Hence, if u E C().), as in the definition (1.4), we have
j+N (1. 75)
Ilipj(D)ullux , S;
L
Ilipj(D)VJe(D)uIILoo.
e=j-N Note that {ipj(D) : j E Z+} is a uniformly bounded family of operators on each LP space, including Loo(IRn), so (1. 76)
\7(x) = w'(lxl)x
\7(XI) - \7(X2)
=
x/lxl. Our hypothesis implies that w is
= [W(IXII) -W(I X21)]Ho(XI) +W(I X21) [Ho(xI) - H O (X2)].
L
Ilipj(D)uIILoo S; C
+ w(lxl) 1:/ = w(lxl)Ho(x),
where w(h) = WI (h) + w(h) and Ho(x) concave and w(h) S; Cw(h). Now (1.70)
ipj(~)
=
j+N
PROOF. We have
(1.69)
ipj(~)
(1.74)
IIVJe(D)uIILoo.
e=j-n Similarly, {j(D) : j E Z+} is uniformly bounded on each LP space, including Loo(IRn), and hence (1.77)
11\7j(D)uIILOO S; CII\7wj-N(D)uIILoo.
Similarly, we have
11(1 -
C11(1 -
The first term on the right is estimated as in (1.67). It remains to show that
(1.78)
(1.71)
We can also establish the invariance of the spaces C().), C[w], and C{w} under a diffeomorphism F : IRn ---+ IRn that coincides with a rotation outside some compact set. Seeing this comes down to observing properties of the operators ipj (x, D) and j(x, D), obtained by conjugating VJj(D) and Wj(D) by the action of the diffeomorphism F. The symbols have the form
W(I X21) IHo(xI) - H O(X2)1 S; CW(IXI - X21).
To see this, note that
IXI - x21 IHo(xI) - H O(X2)1 S; C I I I r' Xl + X2 so it suffices to show that (1.72 )
IXI - x21 _ W(IX21) I I I I S; W(IXI - X21). Xl + X2
Say IXI - X2/ = h, IX21 = t. If t S; h, this follows because the left side of (1.72) is S; wet), which is then S; w(h). On the other hand, if t ~ h, the estimate (1.72) follows from the estimate
(1.73)
wet) S;
~w(h)
for
t
~
h,
which follows from the concavity of W. This proves (1.68). Note that, if w(h) = o(h), then the space cw is trivial, but C().) (with Y\(j) = j w(2- )) might not be trivial. For example, if w(h) = hr with r > 1, then C().) = C;
(1.79)
ipj(x,~)
I"V
j(D))uIILoo S;
VJj(DF(x)tO
+"',
wj+N(D))u/lLoo.
j(x,~)
I"V
Wj(DF(x)t~) + ...
,
where the remainder terms have relatively milder behavior. To estimate the quan tity Ilipj(x, D)uIILoo, we can write
j+N (1.80)
ipj(x, D)
I"V
Aj(x, D)
L
VJe(D)
+ ." ,
£=j-N where, for j (1.81 )
~
M, Aj(x,~) = A(x,rj~)
= VJI(DF(x)tr(j-I)~).
Note that A(x,~) is compactly supported in~, and we have, for the partial Fourier transform, (1.82)
~
Aj(x, z) S; CK(z)
-K
,
0.-
-
--,,-- -,..-- .•.•.•.- ---.---
.~~.'tT.in:1D'C:YCJ~
-
with C K independent of j. Since the integral kernel of Aj(x, D) is Aj(x, x - y), we see that {Aj(x, D) : j E Z+} is uniformly bounded on Loo(lR n ). Thus, making a similar analysis of the remainder terms in (1.80), we obtain
(1.83)
lI'Pj(x, D)ullL= ::; C
L
II7Pe(D)uIlLoo
+ C Ke 2- jK lluIIH-e.
l=j-N
The invariance of C(.X) (lR n ) under the action of such a diffeomorphism follows. In a similar fashion one establishes the invariance of C[w] (lR n ) and C{w}(lRn). It is then straightforward to define the spaces C()..) (M), C[w] (M), and C{w} (M), whenever M is a compact manifold. We need only remark that these spaces are easily shown to be invariant under multiplication by elements of Of course, (M) is also well defined in such a case. We next establish the following characterization of the spaces C()..) (M), etc.
Let M be a compact Riemannian manifold (with smooth metric tensor), with Laplace operator.6, and let A = V-.6 E OPSI(M). Let w and A be as before. Given u E D'(M), PROPOSITION 1.9.
(1.84)
u E C()..)(M) ~ II7Pj(A)uIILoo ::; CA(j),
(1.85)
u E C[wJ(M) ~ lIu - IJiAA)uIlL= ::; Cw(2- j ),
II7Pj(A)uIILoo ::; Cllullc(>.) . A(j),
HN (1.91)
7Pj(D)u = 7Pj(D)
with
IIRjullLOO ::; CTjllullLoo,
(1.92)
or, alternatively, given J < 1,
IIRjUllLoo ::; C2- j Ollullcz.
(1.93)
j+N
(1.94)
u E C{w}(M) ~ IIXwo(hA)uIILe<> ::; C(X) W~h),
7Pj(A) = 'Pj(x, D),
=
+ Tjrl (x, 2-j~) + ... ,
rl (x,~) (and subsequent terms in the expansion) having compact support vanishing for ~ near O.
in~, and
Given u E C()..) (M), to verify the right side of (1.84), we use a partition of unity to write u as a sum of elements of C()..) (M) with supports on coordinate patches. Thus it suffices to estimate II'Pj(x,D)ullu"', for 'Pj(x,~) satisfying (1.88). If we denote by 'Plj (x,~) the principal term on the right side of (1.88), we have, for some N, j+N
(1.89)
'PIJ(x,D)u = 'Plj(X,~)
L
l=j-N
II7Pe(A)uIILoo + C Ks 2- jK lluIIH-s.
p#(u(x)eikO ) = p(k-l, x, D)u(x) eikO ,
k E Z \ 0, j
/gjk(x)~j~k,
j 'Pj(x,~) '" 7Pj (2- 1~lx)
L
An alternative, and perhaps more transparent approach can be given via an analysis of families of operators such as 7PI(hA) and wo(hA), using techniques of [ST] and [T6]. There one associates to a family p(h, x, D) of operators on M, an operator on M x Sl:
(1.95)
(1.87)
IIRjuIILe<> ::; C
l=j-N
PROOF. These results follow from analyses of 7Pj(A) and Wj(A) as families of pseudodifferential operators. As shown in Chapter 12 of [Tl], we have, in local
coordinates,
(1.88)
7Pe(D)u + Rju,
These estimates establish the other half of (1.84), at least when A(j) tends to 0 no faster than 2- j . To treat more general A(j), one could take a closer look at the remainder terms in (1.88) and (1.91), and establish
for every smooth vector field X on M.
where, with 1~lx
L
l=j-N
and (1.86)
..
-.-=~~~-~ =~~~~.=~--.-i5-
hence giving half of (1.84). For one approach to the other half of (1.84), we could note that, for some N the principal part of the symbol of E~~f-N 7Pe(A) is equal to Ion a neighborhoo~ of the support of 7Pj(~), and that, by (1.87)-(1.88),
Co.
cw
C(jNTfNUous-FUN'-cTforr~f
From here, the same sort of arguments as in the proof of (1.83) (including an argument that {'Plj(X, D) : j E Z+} is uniformly bounded on Loo) show that
(1.90)
j+N
",- cin.-",-",,,,--i"lP
7Pe(D)u.
setting it equal to 0 when k = O. Thus, to 7Pj+1 (A) = 7PI (hA), h = 2- , we associate the operator 7Pl(D;1 A) E OPSO(M X Sl). If 'P(~) = E~-N 7PI (2-e~), then, for N large enough, B# = 'P(D;1 A) has total symbol (mod S-oo) equal to 1 on a conic neighborhood in T* (M x Sl) \ 0 of the essential support of the image of 7PI (hD) under this transformation. This in turn implies the identity (1.91), with the better estimate
(1.96)
'K
IIRjuilLoo ::; CKsTJ IluIIH-s,
Similar arguments apply to prove (1.85) and (1.86). We note parenthetically that this approach was used in §4 of [T6] specifically to study "excellent approximate identities," of which the family wj(A) is a paradigmatic example. As we have seen, elements of C()..) are continuous provided E A(j) < 00. If A(j) decays sufficiently slowly, elements of C()..) need not be locally integrable functions (or even locally finite measures) though they will be distributions. Here we make
~,
~w'~~~rm~~~
~~"";
2.
note of the following, which is closely related to a differentiability result of [WZ] , as improved by [IN]. PROPOSITION 1.10. We have 00 ===}-
CC).) C bmo.
PROOF. We bring in Besov spaces B~,2(JR.n) and Ttiebel-Lizorkin spaces
F~,2(.lRn), with norms defined by
= inf
::;
o2
1 k
17
I~I
::;
::; C0 2k , then
C
a(2- k ) Ilfkllu>o,
Le.,
a(lyl) IITy/k - fkllLoo ::; C a(2- k ) IlfkIIL=,
(1.104)
(1.105)
j
Ilfllh,2
II/kllc~
BMO
We need only treat Iyl ::; 2- k . We set y = 2- k z and write
Ilfll~~'2 = L II1/Jj(D)fllioo, (1.98)
,AND
PROOF. We are stating that, if Jk has support in C (1.103)
L .>..(j)2 <
(1.97)
ON
OPERAToR
{IlL liJ1 /ILOO : f = L tpj(D)fj},
Tyfk - fk
=
tpt z(D)/k
2
where
j
where 1/Jj is as in (1.2), and we allow any tpj(~) as long as there are definite bounds in Sro(JR.n) and tpj is supported in a shell I~I ~ 2j . Cf. [Tri2], Chapter 2. The hypothesis L .>..(j)2 < 00 yields (1.99)
O O (JR. n ) CC).) (JR. n ) C B 00,2 (JR. n ) C FCXJ,2,
(1.106)
tpt(~) = tp~(TkO,
Ilfl'~~'2 ::; cIIL /1/Jj(D)fI
(1.107)
tp E Cgo(lR n ),
tp(~)
F~,2(JR.n) = bmo(lR
I~I
::; Co.
#
Iltpk,z(D)fIIL= ::; C
a(2- k Izl) _In_1-\
IlfIIL=, Izl::; 1.
1IuK>'
On the other hand, it is a consequence of work of [FS2] that n
= 1 for COl ::;
Then (1.104) holds if we have (1.108)
J
(1.100)
= (e iz .'; - l)tp(~),
and
the latter inclusion by inspection of (1.98), noting that 2
tp~(~)
);
On the other hand it is clear from (1.106) that Ilrp~II£l (1.109)
::; Cizi
and hence
Iltpt(D)fIIL= ::; ClzlllfIIL=' Izl::; 1.
Now (1.109) implies (1.108) under the hypothesis (1.101).
see [Tri2], p. 93 for a proof. This establishes (1.97). We will deal with bmo in more detail in §2. We now consider an estimate, complementary to Proposition 1.2, which will be of use in the study of operators on bmo(lR n ) in §2. PROPOSITION 1.11. If a(h) is a modulus of continuity and '>"(j) "" 0, both slowly varying, and (1.101)
a(h)b::; a(bh),
0
< b,h::; 1,
then, with 1/Jk(~) as in (1.2), (1.102)
with C
= Cc; independent of k.
and bmo
We establish here some LP bounds for operators p(x, D) with symbols p(x,O which are well behaved in ~ but have limited regularity in x. Various specific symbol classes considered will be defined below. One tool in the analysis, used here as in other places, such as [eM], [Bow']' [Mal], and [T2], is the following result from Littlewood-Paley theory. Let Nd be as in (1.2). LEMMA
(2.1)
'>"(k) II1/JdD)fllc~ ::; C a(2- k ) Ilfllc(~),
ryl,
2. Operator estimates on LP,
2.1. Let fk E S'(JR. n ) be such that, for some B j
supp Jk C {~ : B o . 2k -
l
::;
I~I
::; B l
.
2kH },
Say Jo has compact support. Then, for p E (1, 00 ), we have DO
(2.2)
oc
III: /k!ILP ::; CI\{I: 1/k1 k=O
k=O
1/2
2 }
tp·
> 0, k
21.
2. OPERATOR
'i'Ci~'''''''''''''''''"''C~''''Cc=,y;"ovF£iDITUft'S~-wrriT'MltDty''ltEfeuLAR'''8'"Y"MB'OtS"~'"''''''''""''"~='''''<''''''''''''';o,",,,~,,,,,c;
I!ql (x, D)fIILP ::; CII{
o,n,-,
2j on the spectrum of
rv
(2.9)
2: 'l/Jj(~),
00
k-4
k=4
j=O
1/2
2:\2: Qkj!k\2} IILP 1/2
00
::; CII {2: IIQklll~lfkI2} IILP k=4
::; CAdlfIILP.
and with 00
p(x,O
(2.4)
00
To estimate q2(X, D)f, set
= 2:p(x,~)'l/Jj(~) = 2:Pj(x,~). j=O
Now, let us take a basis of L2(1~jl
k+3
j=O
(2.10)
< 7T) of the form eCt(~)
= eiCt .(
2:
k+3
Qkj(X),
2: {3(k) ::; A 2 <
00.
Ie
::; C2::{3(k)l!iple(D)fII LP
(2.12)
k
::; CA 21IfIILP. To estimate q3(X, D)f, we again apply Lemma 2.1, with fk replaced by gj
2k and bounded in S~, in fact ipk(~)
= ipl (2-k+l~),
(2.13)
j-4
j=4
Ie=O
2
1/2
j-4
00
::; CI\{ 2:(2: (3(k,j)likl) j=4 k=O
IQk(X)1 ::; AI, II'l/Jj(D)Qkllu>o ::; (3(k,j),
for j 2: k - 3. The array (3(k, j) will have properties which will be described below. In (2.7), as in (2.3), 'l/Jj(~) is the Littlewood-Paley partition of unity (1.2), so 'l/Jj(~) is supported on I~I rv 2j • Our next task is to estimate the V-operator norm of q(x, D) when q(x,~) is such an elementary symbol. Set Qkj(X) = 'l/Jj(D)Qk(X), and set
q(x,~) = 2: {I: Qkj(X) +
00
::; CII{ 2::\2:: Qlej!Jc!} IILP
Ilq3(X, D)fl\LP
k=O
where ipk is supported on (0 rv for k 2: 2, and Qk (x) satisfies
2:: IIQk'(X)ipk(D)fIILP
L:{~~ Qkj!Jc, to obtain
00
=
(3(j, k),
j=k-3
\\q2(X, D)fIILP ::;
q(X,~) = 2:Qk(X)ipk(~)'
k-4
2:
=
j=k-3
Then
where 'l/Jf(~) has support on 1/2 < I~I < 2 and is 1 on supp 'l/Jl, 'l/Jr(~) = 'l/Jf (2-j+I~), with an analogous decomposition for po(x, ~). Inserting these decom positions into (2.4) and summing over j, we obtain p(x,~) as a sum of a rapidly decreasing sequence of "elementary symbols." By definition, an elementary symbol is of the form
(2.7)
{3(k)
k~O
Ct
(2.6)
=
(2.11)
Pj(X,~) = 2:PjCt(x)eCt(2-j~)'l/Jr(~),
(2.5)
Qk'(x)
and assume
Then we write (for j 2: 1)
(2.8)
~-, ~'~1.J
QkJ'
The next ingredient (also used in [eM] and subsequent papers mentioned above) is a symbol decomposition. We begin with the Littlewood-Paley partition of unity (1.2):
1 =
UN L'.
First we estimate ql(X, D)j. By Lemma 2.1, since (~)
If fk = 'l/Jk(D)f, then the converse inequality also holds.
(2.3)
r;:::>nMA1~;';
k+3
L
00
Qkj(X) +
k j=O j=k-3 ql (x, 0 + q2(X,~) + q3(X, ~).
L
Qkj(X) }4?k(~)
j=k+4
We will perform separate estimates of these three pieces. In two of these cases, we will apply Lemma 2.1, with h: = ipk(D)f.
2
1/2
}
tp·
Now make the hypothesis that
j-4
T(3a(j)
(2.14)
=
2: (3(k,j)a(k)
k=O
defines a bounded linear operator on £2 :
(2.15)
2: IT(3a(j) 1
2
::;
j~O
A§
2: la(jW· j~O
We deduce from (2.13) that
f/ \ILP ::; 2
(2.16)
Ilq3(X,D)fIILP ::; CA3\\{2: Ifjl2 j
C A 31IfIILP.
=
-"''i-~----=--~====--=--=- -=--=-~L~ .....,-iU~n.-ruhi:j --y~-ii~--fi l.mLU~~=---R~l;aLAtrsy-=-MHO-CS-==-
Now
p(x,~)
----- ----
--------------,---c- - - -
;"Ai
can be written aB a sum
L", ~',
-------2:-0PERATOR ESTIMATES ON
ANU tlMU
Note that the bo=d (2;22) on Tp hold, P:ded
00
;;~"
p(x,~) = Lc"q,,(x,~) ,,=1
(2.17)
'L{3(k,j) ::; A 3 ,
(2.28)
k=O
where (c,,) is rapidly decreaBing and
L
(3(k,j)::; A 3 ·
j=k+l
In particular, (2.27) holds provided 00
K(2 k )
On the other hand, assuming A(j) "'" and K(j) / , we have j
PROPOSITION
We have the following conclusion.
2.2. Assume p(x,~) satisfies (2.20), where (3(k,j) satisfies L{3(k+i,k) <
(2.21)
-3::;
00,
i::; 3,
(2.32)
-+
£2.
1
00.
PROPOSITION
A(j)K(~),
j) =
K
IIDfp(-'~)lIc(~)::; CaK(~)(~)-lal.
A(j)K(2j
L
L{3(k + 3, k) <
(2.35)
AaW-1a l ,
do not vary too fast, the hypothesis (2.21) follows from
(2.26)
2.3. Assume p(x,~) satisfies (2.20). If (3(k,j) satisfies (2.32)
and
the hypothesis (2.20) is implied by
Then, if A and
)
then p(x, D) : U(lRn )
(2.37)
LP(JRn),
A(j) ",", K(j) /, and
1
AU) LK(2 k )a(k) k=O
L A(j)K(2 j
J
=
_
00.
In particular, this holds if p(x,~) satisfies (2.25), with
while (2.22) follows from Taa(j)
00,
k
(2.36)
< 00,
j
(2.27)
(3(j, j).
j=k+l
Thus we have the following.
(3(lOg2(~)'
IDfp(x,OI::;
{3(k, j)::; L
j=k+l
In the particular CaBe that
(2.25)
k=O
00
L
(2.34)
(2.24)
j
00
Then
U(lR n ),
in k, V j.
and
k=O
p(x, D) : LP(lR n ) _
),
L{3(k,j)::; L{3(k,k), k=O
T(3 : £2
j
j=k+l
(3(k, j) is "'" in j, V k and /
(2.33) =}
A(j)K(2
so (2.29) follows from (2.26). More generally, suppose
j
(2.23)
AU)::; L
j
(3(k, j)a(k)
),
Then
and T(3a(j) = L
k
00
K(2 k ) L
k
(2.22)
A(k)K(2
k=O
k+l
IDf7/lj(Dx)p(x,~)1 ::; Ca{3(log2(~),j)(~)-lal,
::; 2j +4.
L
00
(2.31)
ID~p(x,~)1 ::; Aa(~)-Ial,
for all a ? 0 and I~I
::;
and
for all a ? 0 and j ? k - 3. For this, it suffices to have (2.20)
j
A(j) L K(2 k ) k=O
[Df7/lj(D." )Pk(X, ~)l ::; Ca {3(k, j)Tklal,
A(j)::; A 3 K(2 k )-1.
j=k+l
(2,30)
I ::; Aa2-klal,
L
A 3 A(j)-I,
::;
k=O
where Qk all satisfy (2.7), with fixed Al and (3, and
L
(2.29)
k=O
IDfPk (x,~)
00
j
q,,(x,~) = L Qk(X)
(2.18)
=?
T(3: £2
-+
£2.
provided K(2j) ::; CK(j).
j
)
< 00,
_.
a.~
__ ~""'" ....L..luv.u.cl..I."lo
d- • .lY~DVi.Ji:J
.---.---------~
Let us look at some examples. First, picking AU) as in (1.13), fix r > 0 and let (2.38)
AU)
= 2- jr ,
~(~)
=
(~)ro.
p(x,~) E C:S~,o(JRn),
in this case. Now (2.37) holds if and only if 6 ~ 0 and 2: j 2-jr2jro and only if 0 :::; r5 < 1. We recover the well known result that
(2.40)
<
00,
Le., if
p(x,~) E C: S~,.5(IRn) => p(x, D) : £P(JR n ) ~ £P(IR n ),
for 1 < P < 00, provided r > 0, 6 E [0,1). On the other hand, if we pick (2.41)
AU)
=
Tjr,
then the hypothesis (2.25) implies that
PROPOSITION
2.4. If A(j) '\. and
L. >"(j) < 00,
2~
then
n
n
p(x,~) E CC>')S~,o ==? p(x,D): U(IR ) - U(IR ),
(2.45)
The hypothesis (2.25) is equivalent to (2.39)
2. OPEltATOR ESTIMATES ON LP, ~1, AND BMO
1
< P < 00.
In [CM] it is established that, if p(x,~) E cw Sf,o, then p(x, D) : LP _ LP as long as L. >..(j)2 < 00, where >"(j) = w(2- j ). We recall the proof of their result here. To emphasize the mechanism of their proof we replace CW by C[w], though by the remark made after (1.63) we actually have cw = C[w] in cases of greatest interest for the application of this result. PROPOSITION
2.5. Let AU) = w(2- j
),
and assume
p(x,~) E C[w]S~.o => p(x, D) : U(IR n )
(2.46)
_
L. A(j)2 < 00.
LP(IR n ),
1
Then
< P < 00.
PROOF. The argument here is essentially the same as in [CM]. It suffices to estimate q(x, D)f when q(x,~) is an elementary symbol (2.6). We decompose q(x,~) into two pieces; instead of (2.8) we write
(~)r
~(O =
(log2(6jS'
(2.4 7)
p(x,~)
is just a little better behaved than
where
q(x,~) = ql(X,~)
ql (x,~)
+ ih(x, ~),
is as in (2.8), and 00
p(x,~) E C:S~,l(IRn),
(2.42)
provided 8 > O. In fact, to check (2.37), note that 2: AU)~(2j) :::::: L. F- s in this case; this is finite if and only if 8 > 1. Consequently we have LP operator bounds on p(x, D) if p(x, 0 satisfies (2.25), with A, ~ given by (2.41), provided r > 0 and 8> 1. Next, picking A(j) as in (1.15), fix s, r > 0 and let (2.43)
= r",
= (log2 (~) is equivalent to s > r + 1.
AU)
K(~)
e
II D P(-,Ollx :::; c,,(~)m-IQI.
q(x,O
E XSr,'o
=> IIQkllx :::; C2 km .
In particular, q(x,~) E CC>') S~.o
=> II1/Jj (D)Qk IIL= :::; C>"(j),
Le., we have (3( k, j) = >"(j) in (2.7). Hence, if we apply Proposition 2.3 to the case (3(k,j) = A(j), we obtain the following.
(I - Wk-4(D))Qdx).
The estimate (2.9) still applies to ql(x,D)f. To estimate ih(x,D)f, note that
q(x,O
(2.49)
Now, with
!k
E
C[w]S~,o
-¢::=}
IIQkllc[w\ :::; C =>
IIBkliL= :::; CA(k).
= 'Pk(D)f, write 2
00
(2: IBk(x)I·lfdx)l) k=O : :; (2: B k(X)1 (2: Ifk(XW) :::; C A 2 2: Ifk(X)1 2,
!Q2(X, D)f(x)1 2 :::; (2.50)
I
2
)
2
where -,12 = L. >"(k? < 00. Raising both sides of (2.50) to the power p/2, integrating and using the converse inequality to (2.2), we obtain (2.51)
Spaces X of particular interest include C; (r > 0), cw, C(>'), and C[wl. If such a symbol is decomposed into simple symbols, and such a simple symbol q(x,~) is written as in (2.6), we have
Bdx) =
k=O
r.
Then the hypothesis (2.37) The estimates involved in establishing Proposition 2.3 suggest that we pay special attention to certain spaces of symbols, of the following sort. Given a Banach space X of continuous functions on IRn, define X Slo to consist of symbols p(x,~) such that ' (2.44)
ih(x,~) = 2:Bdx)'Pk(~)'
(2.48)
IIq2(x,D)fIlLP :::; CAllflb·
Comparing Propositions 2.4 and 2.5, we see that neither result contains the other. Some examples for which Proposition 2.4 has an advantage over Proposition 2.5 arise in §16. It is also useful to consider the following more general type of symbol classes. If X is a Banach space of continuous functions on IRn, and ~ is slowly increasing (so K(2r) :::; CK(r)), we say (2.52)
p(x,~) E XSi','CII:)
-¢::=}
IDeP(x,~)1 :::; CQ(~)Tn-IQI,
and
IIDeP(-'~)llx ~ CQ~(IW(~)Tn-I"I.
~'-
...
~_
..
~IORb WI~YMBOLS
. _...,
Then the part of Proposition 2.3 to which (2.37) applies can be phrased as follows. PROPOSITION 2.6. We have (2.53) p(X,~) E C P.) SP,(K) (IR n ) ===? p(x, D) : U(lR n ) - t U(lR n ),
provided that ,\ '\"
K /
1
00,
L: '\(j)K(2 j ) < 00. j
PROPOSITION 2.7. We have
p(x,~) E C[wJSP,(K)
===?
1
~1(IRTI)
= {f E L 1(IR
TI
)
:
25
(ib f E L 1(IR TI ) } .
a E C"(lR n ), f E ~1(lRn)
To see this, note that for all I~t
* (af)(x) -
~ E
af E ~1(lRn),
===?
provided
F,
a(x)~t
J
a(t) * f(x)1 :S C~
t
Jo
a(t) dt <
If(y)1 dy,
so it suffices to verify the membership in £1 (IR TI ) of
L:[w(T k )K(2 k )] 2 < 00.
(2.56)
(2.65)
k
q(x,~)
E
a(t) v(x) = sup - n O
PROOF. Set '\(j) = w(2- j ). As in the proof of Proposition 2.5, we consider an elementary symbol q(x, ~), make the decomposition (2.47)-(2.48), and note that the bound on ql (x, D) is already done. As for the bound on ih(x, D), note that
C[w1Sp,(K) ~ ===?
IIQkllux, :S C, and IIQkllc[wj:S CK(2 k ) IIBk IIL= :S CW(2- k )K(2 k ),
J
If(y)1 dy
:S
JI
X(x - y) I a(lx - yl) If(y)1 dy, x yn
Bt(x)
where X(x) is the characteristic function of {Ixl :S I}, which is clear given the Dini condition on a(h). (We may as well assume a(t)jt Tl ' \ , . ) An alternative characterization of 1)1 (IR n ), which will be of use here, is the following: 1/2
00
Ilfll~l ~ II{L: l1Pk(D)fI
(2.66)
2
Ilv'
}
k=O
- 2 and then (2.50) holds, with A 2 = E[w(2- k )K(2 k )] . The rest of the proof follows as in the proof of Proposition 2.5.
with 1Pk as in (1.2). Furthermore, in the setting of Lemma 2.1, with (2.1),
We remark that you can also improve the estimate on q2(X, D)f in (2.12), by making an argument like that above, but there is no analogous improvement of the estimate on q3(x,D)f in (2.31)-(2.16), so one does not obtain by such reasoning an improvement of Proposition 2.2. Having analyzed a number of classes of operators on LP(lR n ), we now consider their action on Hardy spaces. There are several approaches to the definition of Hardy spaces. Here is one; set
(2.67)
(9f)(x) = sup {I~t
(2.58) where ~t(x)
= t-n~(xjt)
(2.60)
F =
{~
n E C(j"'(lR ) : ~(x)
n1(lR n ) = {f E L 1(lR n ) : (if E L 1(lRn )}.
(2.61)
(9bf)(x) = sup{l~t *f(x)l: ~
E
F,O <
t:S
I},
00
k=O
k=O
!k
satisfying
1/2
Ilv'
The results (2.66)-(2.67) are established in §2.5 of [Tri2]. Using these results, we can check that (2.9), (2.12), and (2.16) have analogues of the form IIql(X,D)fll~l:S Cllfll~l,
(2.68)
IIq2(x,D)fllv IIq3(x, D)fll ~1
:S CllfIIL" :S Cllfll ~1,
under the hypotheses (2.11) and (2.15). Consequently the propositions above ex tend, as follows: PROPOSITION 2.8. Under the hypotheses of Propositions 2.2, 2.3, 2.4, or 2.6, we also have p(x,D) : 1)1(lRn ) - t L 1(lR n ). In particular
(2.69)
A related but slightly larger space is ~ 1 (IR n ), defined as follows. Set
00
2 11L:!k11~1:S CII{L:lfkI }
* f(x)1 : ~ E F, t > OJ,
and
= 0 for Ixl ~ 1, 11\7~IIL= :S I}. This is called the grand maximal function of f. Then we define (2.59)
00.
t
B,(x)
00,
provided
(2.57)
AND BMO
A great deal of information about Hardy spaces is given in [S3]. Brief introductions can be found in [Sem] and in Chapter 13 of [T5]. Here we concentrate on ~1(lRn), which has the desirable localization property:
(2.64)
p(x, D) : U(lR n ) - t U(lR n ),
~',
and define
(2.63)
If we use X = c[w] , we can establish the following extension of Proposition 2.5. A similar result, but with X = CW, was given in [Ma3].
(2.55)
2. OPERATOR ESTIMATES ON LP,
(2.62)
are slowly varying and
(2.54)
%---,
p(x,~) E CP·)SP,o
provided I: '\(j) < (2.70)
00,
==}
p(x, D): 1)1 (IR TI )
-t
L 1(lR n ),
and more generally
p(x,~) E CP')SP,(K)
===}
p(x,D): 1)1 (IR n ) - t L 1(lR n ),
;:
"---r;~"7l"- cnxn:JK1> -Wrrin\7IIL-DLTriEGULA~a-crLfj
provided'\ "-", /\, /' are slowly varying and
PROPOSITION
L '\(j)/\'(2
(2.71 )
j
)
p(x,~) E BS?1 ==? p(x,D): ~1(JRn)
(2.81)
~1(JRn).
---+
Here 13SP 1 consists of symbols p(x,~) whose partial Fourier transforms p(T/,~) are supported in IT/I ::; pl~1 for some p < 1. See (3.36)-(3.39) for further discussion of this symbol class (which will play an increasingly important role later in this work). The result (2.72) follows because when an element of BS?,1 is expanded into elementary symbols, only symbols of the type q1 (x,~) arise. From (2.72) one obtains
p(x,O E S?,iS, <5 < 1 ==? p(x, D) : 1)1 (JR n )
---+
27
p(x,D): 1)1 (JR n )
->
~1(JRn),
PROPOSITION
2.9. If 2:w(2- k )2
L 1 (JRn).
E
C[wlS~,(I<)
==?
p(x,D): 1)1 (JR n )
---+
p(x, e) E C()..) S~,(I<)'
(2.83)
< 00,
I:['\(j)/\'(2J)]1/2 <
00.
IlfllBMO = sup
vtQ)
J
If - fQI dx,
Q
the sup taken over all n-dimensional cubes Q in JRn, where V(Q) is the volume of Q and fQ is the mean value of f over Q. Actually, we concentrate on the space bmo(JR n ), with norm
(2.85)
L 1 (JRn).
,\(j)1/2
Now we bring in the John-Nirenberg space BMO(JRn), introduced in [IN], with semi-norm given by
Ilfllbmo = IlfllBMO + II'lJo(D)fIIL=,
where 'lJo E CO'(JR n ) satisfies 'lJo(O) The fundamental duality
More generally, if (2.56) holds, then
p(x,~)
(2.82)
(2.84) ---+
L
C()..) SO1.0'
Q
< 00, then
p(x,~) E C[wJSP,o ==? p(x,D): 1)1 (JR n )
p(x,~) E
or more generally provided
1)1 (JR n ).
Examining the proofs of Propositions 2.5 and 2.7, we have the following, the first part of which was established in [eM] (with 1)1 replaced by 5)1).
(2.75)
oN~;~-~T,-A;D HMO
provided that
(2.72)
(2.74)
ESTIMATES
2.10. We have
(2.80)
< 00.
Furthermore,
(2.73)
~P;~ATOR
i-
O.
5)1(JRn)* = BMO(JRn )
was established in [FS2]. The parallel result We want to obtain more general conditions under which P : I) 1(JRn) This will involve a modification of (2.12). We have
---+
I) 1(JRn).
k
(2.76)
(2.87)
::; C{I:J3#(k) }Ilfll~l,
where J3#(k) is the 1)1-mult iplier norm of Qr:
IIQrgll~l
::; J3#(k)llgll~l.
As shown in (2.63), we have (2.78)
J3#(k) ::;
J3#(k) ::;
LEMMA 2.11. Assume 1 ::; p < Mkk(e) = Mk(2k~), and assume
CIIQrllorr,
given any u(h) satisfying the Dini condition. Taking p,(j) = u(2- j (1.102), we have (assuming (1.101) holds)
P E OPSP,iSl <5 < 1 ==? P : bmo(JR n )
---+
bmo(JR n ).
However most of the operator classes dealt with in Propositions 2.8-2.9 are not closed under the operation of passing to the adjoint. To get bounds on bmo for such operators we follow an idea used in [Mal], in a related setting, and obtain estimates on the adjoints of these operators, acting on I) 1(JRn). To do this, we will use the p = 1, q = 2 case of the following vector Fourier multiplier result, established in §2.4.9 of [Tri2].
k
(2.77)
1)1(JRn)* = bmo(JRn)
was proven in [Gol]. Exploiting this duality, we can draw an immediate conclusion from (2.73):
Ilq2(X, D)fll~l ::; I: IIQk''Pk(D)fll~l
(2.79)
(2.86)
)
and using
CIIQrllo(,,) ::; p,~) IIQ/;'IIL'''o.
This gives the following result, proven by taking p,(k) = ,\(k)1/2 and p,(k) ['\(k),..(2 k )jl/2, respectively:
(2.88)
1 ::; q
00,
IIMkkIIH'(IRn) ::; A,
<
00.
Let Mk(e) be given, set
n
s> 2'
Then (2.89)
II(Mk(D)!k)IILP(IRn,i q )
::;
CA
IIUk)lb(IRn,i
provided (2.90)
suppf~ c {~: I~I
::; C2 2k }.
q ),
28
29
L OPERATORS WITH MILDLY REGULAR SYMBOLS
We look at qj(x, D)* f, j
= 1,2,3. To begin, we look at k-4 Q k = LQkj'
ql(x,D)*f = L'Pk(D)(Q:J),
(2.91)
k~O
Since
Qk(O
is supported in I~I ~ 2k -
(2.92)
,
we can write
k+5 !k = L 1/Ji(D)j.
£=k-5
We then have
IIql(x,D)*fll~' ~CII{LI'Pk(D)(QkfkW} 00
-8
k=O 00
~ CII{L IQ:fk
1/2
2 I
}
k=O
00
~CAlll{LlfkI2}
1/2
k=O
1/2
IlL'
Ilq2(X,D)*fll~, ~
'" _
~ C { LJI3(k)2
00
Ilfll~',
(2.101)
00
q3(x,D)*f=L L 'PdD)(QkjJ)· k=Oj=kH
Using the support properties of write
the first inequality by (2.65), the second by Lemma 2.11, the third under the hy pothesis
IIQkIIL=
}1/2
the last inequality being essentially a consequence of (2.61)-(2.62), since F k ~k+5 * f with ~o E S(lR n ). We now look at
IlL'
t,
" _ }1/2 c { 'LJI3(k)2 Ils~p IFklllL'
(2.99)
(2.100)
~ C' Alllfll~l,
Qkj
and reversing the order of summation, we can
00 j-4 q3(X, D)* f = L L 'Pk(D)(QkjiJ), j=4 k=O
j+5 fj = L 1/Ji(D)f· £=j-5
A crude approach to (2.100) yields, with
Qk = L Qkj = (I - Wk+3(D))Qk, j2:k+4
~ AI,
the estimates
as in (2.7), and the fourth by (2.64). Next we look at
(2.95)
{L;3(k)2}S~p IFk1 2 ,
we have 3
k~O
(2.94)
2 L;3(k)2IFkI ~
(2.98)
j=O
ql (x, D)* f = L 'Pk(D)(Q~!k),
(2.93)
the first inequality by (2.65), the second by Lemma 2.11, and the third by the definition of ;3(k) in (2.10). Using
00
k+3 Qk'= L Qkj. j=k-3
q2(x,D)*f=L'Pk(D)(q;J), k~O
Ilq3(X, D)* filL'
(2.102)
~ C{L IIQkIIL= } IlfllL' k=O
and 00
This time, since (2.96)
Qicn(O
is supported in I~I ~ 2kH , we can write
q2(X, D)* f = L 'Pk(D)(q; Fk), k2:0
Fk = Wk+5(D)j.
We then have
~ cll{I: l'Pk(D)(Q~lFkW} 1/2t, k=O 00
~ C!I{L Iq;Fk 2}
1/2
I
k=O 00
~ CII{L;3(k)2IFk 2} I
k=O
t,
1/2
~ C{L IIQklic }llfll~', u
k=O
given any a(h) satisfying the Dini condition. From (2.102) plus the estimates on ql(x,D)*f and q2(x,D)*f, we have: PROPOSITION
Ilq2(X, D)* fll~' (2.97)
Ilq3(x, D)* fill)'
(2.103)
2.12. Assume p(x,~) E C[wJS~.o,
(2.104)
Lw(Tj) <
00,
or more generally (2.105)
p(x,O E C[w]S~'(")'
LW(Tj)I\;(2j) <
00.
Then
IlL"
(2.106)
p(x,D)* : ~l(lRn)
--+
Ll(JRn ),
p(x,D): Loo(JR n )
--+
bmo(lR n ).
:>O---~-===---'~-l;-l)P):<.:I:tA'{dItS"WITH
PROOF.
(2.107)
MILDLYR.EGULAR SYMBOLS
Note that the hypothesis in (2.105) implies for q(x, D)
II (I -
\{f£(D) )Qk II L''''
:s; CW(2-£)K(2 k) ===}
IIQ£ IIL~
:s; CW(2- k)K(2 k),
--~---------~ ;:-SYMBOi,-CLASS-ES AND
SYMBOL SMOOTHING
Ojl
3. Symbol classes and symbol smoothing As in (2.44), if L:.A(j) < 00 we define C(A)Sf:O(JRn ) to consist of functions p(x,~) on JRn x R n such that .
so the sum in (2.102) is finite.
(3.1)
From (2.103) and the aforementioned estimates on q1(X, D)* 1 and q2(X, D)* 1, we obtain:
More generally, as in (2.52), if K(~) is slowly increasing, we define C(A) S~(",) (ffi- n ) to consist of functions p(x,~) such that
Assume
PROPOSITION 2.13.
2:w(2-i)1/2 < 00,
or more generally (2.109)
p(x,~)
E
ClwJ sp,(I<) ,
2:[w(2- i )K(2 j )F/2 < 00.
IID~p(·,~)llc(>.):S; CaK(O(~)m-lal.
p(x,~) = p#(x,~)
+ pb(x,~),
with
p#(x,O = 2:\{fo(ejDx)P(X,O~i(~)'
(3.4)
j
p(x, Dr : fJ1(JRn) ----; fJ1(JRn),
p(x, D) : bmo(JR n ) ----; bmo(JRn).
FUrthermore, (2.111)
IIDfp(-'~)llc(>'):S; Cawm-Ial.
In particular, p(x, 0 E C(A) Sp,(I<) if and only if the hypothesis (2.25) holds. We will find it convenient to split p(x,~) into two pieces: (3.3)
Then
(2.110)
:s; A.(~)m-Ial,
ID€p(x,~)I:S; Aa(~)m-Ial,
(3.2)
p(x,~) E c1w],
(2.108)
IDfP(x, ~)I
p(x,~) E BSP,l
=}
p(x, D)* : fJ1(JRn) ----; fJ1(ffi-n)
=}
p(x, D) : bmo(JRn) ----; bmo(ffi-n).
To use (2.103), we recall from the comment following (1.63) that C" = f3 C[o-) whenever u(t)t- '\. for some (3 E (0,1), which we may as well assume here. It is readily verified that, if a( h) is another modulus of continuity, PROOF.
Here, {~j} is the partition of unity (1.2), and \{fo E C8"(ffi- n ) is assumed to be equal to 1 for I~\ :s; 1. We take ej '\. 0 as j ----; 00. The properties of the decomposition (3.3) depend on the choice of ej, as we will see below. Let us set Je; = \{fo(eDx )' It is easy to verify the following. LEMMA
3.1. For e E (0,1/2]'
IID~Je;Illc(>.) :s; Cf3 e-\ f3I IIIllc(>.)·
(3.5)
il p,(j)
Also,
~ .AU),
and .AU) / p,(j) '\., then
111 -
(3.6)
Jdllc(l')
:s; C,(e)IIIllc(>.),
with (2.112)
II(I -\{fk+3(D))QllcI0"] :s; Ca(2-k)IIQllc1aO"].
_ A(log2 ~) ( ) ,e(log2 g- 1)'
(3.7)
Taking a(h) = u(h) = w(h)1/2 shows that, under hypothesis (2.108), the sum in (2.103) is finite. To treat the hypothesis (2.109), note that this implies IIQkllc[w\ :s; CK(2 k ), and hence, by (2.112), (2.113)
IIQilielO"]
:s; Ca(2- k)K(2 k) ~ a(h)u(h)
= w(h).
Now, if p(x,~) satisfies (3.1), it follows that
IDr D~p#(x, ~)I
(3.8)
If we pick 6 E (0,1] and
C
:s; Caf3 ejlf31 (~)m-Ial
so ej
ej
a(h) _ [ w(h) ] 1/2 - K(I/h) ,u(h) = [w(h)K(I/h)F/2.
Finally, the result (2.111) follows from the analysis of Q1(X, D)* fin (2.93).
rv
(3.10)
on supp ~j(~)'
E (0,00) and set
(3.9)
So take (2.114)
P,
=
c2- j8 ,
(~)-li on supp ~j, this implies
ID~ D~p#(x, ~)I
:s;
Caf3(~)m-lal+JIf3I,
i.e.,
p#(x,~) E S~8'
Furthermore, if p(x,~) satisfies (3.1), then, by (3.6),
(3.11)
IIDrpb(-'~)llc(l')
:s;
Ca,(ej)(~)m-Ial
on supp ~j(~).
- -- -_. '-
-------------------_._._-_._- VVI.ln NUJ.,UJ.,Y l{~UULAK
'-'C"'Ll.ruv",~
- - - - - - - - - - - - - - - - , c ' - -----. -_..
ISYMBOLS
If Cj is given by (3.9), then
PROPOSITION
I Dfpb C e)llc(!')
(3.12)
(3.3)-(3.4), with
~ C,/Yo(e)(e)m-1a l,
(3.20)
where
More generally, if p(x, e) satisfies (3.2) and p#(x, e), and for pb(x, e) we have
Cj
3.2. Assume p(x,e) E C(>')Si,(t<)" Then, in the decomposition given by (3.9), we have
Cj
p#(x,e)
E
Si,o, i.e., IDtD~p#(x,e)1 ~ Cal3(e)'P(~)-lal+Oll3l,
pb(x, e)
(3.21)
= c2- jO , then we have (3.10) for
C(/,) S'P-'" 1, ('T) ,
E
where we can take various functions '¢(e) and r(e), satisfying (3.22)
IIDfpbCe)/lC(!') ~ Ca70(e) ",(e) (e)m-1a l .
(3.14)
33
and
A(8 log2 (e)) .
70(e) = p,(8 log2(e))
(3.13)
_._------,----_.---
~=,==--===
3. SYMBOL CLASSES AND SYMBOL SMOOTHING
(e)-"'(~) ~ Ao(e)",(e),
r(e) = 70(e)",(e)(e)"'(O.
For example, we could take
If LP,(j) < 00, then IDfpb(x,e)1 is bounded by a constant times the left side of (3.14). For a better estimate, note that
(e)-"'(~)
= Ao(O",(e) = ",(e)w( (e)-O),
(3.23)
Ilf - Jdllu'" ~ I : II'¢j(D)fllux> ~ I:A(j)llfllc().), j~f
if £
'"
2- f , so, with fl(£)
=
or equivalently, if A(j) (3.15)
/If -
j~f
Jdllux>
~ fl(log2 ;)llfllc<).),
= w(2- j ),
J,J/luX> ~ w(c)llfllc().), w(h) = fh w(t) dt.
io
t
Hence
IDfpb(x,e)1 ~ CaAo(e) ",(e) (e)m-1a l.
(3.16)
70 (e) Ao(e)
=
=
A(j) = Tjr,
(3.24)
",(e) = (oor,
with 8 E [0,1). Then C(>.·)Si,(t<)(lR n ) = C:Si,o(lR n ). If Cj = c2-,j in (3.4), then (3.20) holds (with 8 replaced by ,/,), Le., p#(x,e) E S'f". Meanwhile, let us take
p,(j)=TjS,
(3.25)
O<s~r.
Since fl(j) '" 2- jr when A(j) is given by (3.24), we have
where (3.17)
Ao(e)
=
w( (e)-O).
These estimates suggest making a further generalization of the symbol classes defined by (3.1)-(3.2). Namely, if rp(e) is a positive, slowly varying function, and X is one of our favorite function spaces, we say
(3.26)
(e)-"'(~) = (e)or(e)-rf, o
(3.27)
p(x, e)
E
C:S'f,o(lRn ) ===> p#(x, e) pb(x,e)
if and only if
/DfP(X, e)1 ~ Aa(e)'P(~Hal,
IIDfp(·, e)llx ~ Ca"'(O(e)'P(~Hal.
Estimates parallel to those done above prove the following.
1 = (e)S,. r(e) ~ p,('Y log2(e))
Thus, pb(X, e) E C:SG rb - )(IR n ), provided 0 < S ~ r. We may as well take s = r in (3.25), and we have a satisfactory symbol decomposition provided 0 ~ 8 < '/' < 1. In conclusion, if r > 0 and 0 ~ 8 < '/' < 1,
p(x, e) EX Si,(t<) (IR n )
(3.18)
(3.19)
w( (e) -0)
w((e)-0)p,(8 log2(e))'
with A(j) = w(2- j ) and wgiven by (3.15). For this to be useful, we want (e)-"'(~) -+ o as lei -+ 00. Note that, if we take p, = A, so 7(e) = 1, then r(e) = ",(e)(e)"'W 1 in (3.22), and r(e) = w((e)-Or in (3.23). We will see below some examples for which it is not desirable to use (3.23). Let us consider some examples. First, as in (2.38), fix r > 0 and take
Lj~t A(j), as in (1.9), we have
Ilf -
r(e)
This is a known result; cf. (1.3.21) of [T2]. For the next example, as in (2.43), fix r, S (3.28)
AU) = r
s
,
E
Si,,(lR n ),
E C:Si,~rb-o)(lRn).
> 0 and let
",(e) = (log2(e)f·
"""""'-ii"moY"S- ',nii rvuetiC"l"ftEl:fufuilTsfM6\j15?
Take €J = 2- lij in (3.4), so again we have (3.20), i.e., p#(x,~) if s > 1, flU)
J.L(j)
=
r
8
rv
so fl (t5 logz (~))
r(s-l),
>'(j), we have pb(x,~)
=
W-,pro
(3.29)
=
-'"'_.-
Sj' '" Note that,
E
~ (logz(~)) -(8-1). Hen~e, if we
E e(>o·) S'i,(.r~,
(logz(~) r(8-1-r),
pick
with
T(~) = (logz(~)
r-
I
BSri
(3.37)
(3.38)
p(x,~)
)
<
00,
in order to apply Proposition 2.6, which states that q(x, D) is bounded on IJ'(JRn), for 1 < p < 00, provided that q(x,~) E e(A)S~,(T) and (3.31) holds. As will be shown in Proposition 5.2, this condition also implies q(x, D) : erA) ~ erA). But if T(~) is given by (3.29), then >'(j)T(2J ) ~ j-I, so the condition (3.31) fails. We can fix this by modifying 'l/J(~) and T(O as follows_ Take 'Y so that
o < 'Y < s -
1-
(~) -,pre)
= (logz (~)) -(s-I-r-,),
T(~)
p(x,~) E erSrno ,
==?
=
(log2 (~)
r-
-,.
p#(x,~) E A~Srno' ,
E
Sr,'lr,
n A~Sf,\,
BrSf,\
= U B;Sr,l' p
tj
s E JR.
A more general result will be established in Proposition 4.7. Also, the action on
erA) is considered in Proposition 5.7. There are also useful results on compositions of such operators, which will be discussed and extended in §6. The same analysis used to obtain Proposition 3.2 also yields: 3.3. Suppose that >.(j) satisfies erA)
c e r , and you use the
p(x,~)
E e(A)sr,o ===;.
p#(x,~)
E
l(x,~) E e(A)s~(~j,
BrSf\,
(~)-,p(e) ~ W((~)-I),
(3.42)
T(~) = (~)1/Jm,
with >.(j) = w(2- J ) and w given by (3.15). Suppose
2j >'U) '\. 0,
(3.43)
L: 2 ).,(j) < j
00,
so erA) eel. Then fl(R):S 2-£· LJ>£2J ).,(j):S e2-£, so we can take 'l/J(~) = 1 in
q(x,~) E A(;Srli
{=}
II Dfq(-, ~) IIcr+8
:S ect8(~)m-lctl+08,
s
~
O.
(3.41), to get, with T(~) = (~),
(3.44)
q(x,~) E AQSrno , ===;. D~q(x,~) E Sro ,
for 113\ :S r
Sm+li(Ii3I- r ) 1,0
for 113\ ~ r.
This is significant for operator calculus. In examples considered above, we have taken <5 < 1 in (3.9). As first shown in [Bon], it can be very useful to take t5 = 1, provided we take c sufficiently large that (3.36)
pb(x,~)
where, as in (3.22), we can take various junctions 'l/J(O and T(O, satisfying
As noted in (3.1.30) of [T2], (3.35)
p#(x,~) E BrSr,I'
p(x, D) : Hs+m,p(JR n ) ----t H 8 ,p(JRn),
(3.40)
(3.41) I
where we say (3.34)
==?
paradifferential symbol smoothing. Then
Now (3.30) holds also with these new 'l/J(~), T(~), and this symbol decomposition can be useful. It is often useful to record more information on the term p# (x, ~). For example, as shown in Proposition 3.1.C of [T2], if Cj = 2- oJ in (3.4), then, for r > 0, (3.33)
ersr,o
One desirable property of these symbol classes is that operators with symbols in these classes operate on all the proper Sobolev spaces. As shown by [Meyl], if p(x,~) E BSr,\ and 1 < P < 00, then
PROPOSITION
T,
assuming as above that s - 1 - r > O. Then replace (3.29) by (3.32)
E
B;Sr,l = BpSr,J
(3.39)
where (~)-,p(O and T(~) are given by (3.29). For this to be satisfactory, we need not only 0 :S T < S - 1, which implies (~) -,p(O ~ 0 as I~I ~ 00 and L >'(j)K(2J ) < 00; we also want j
U BpSri'
p
where
pb(x,~) E e(A)si,(Tj(l~n),
L: >.(j)T(2
=
Applying such a symbol smoothing we have, for r > 0,
p(x,~) E e(>') S'i, (K) (lR. n ) ===;. p#(x,~) E Si,,,(lR. n ),
(3.31)
35
for some p < 1. We will call this the paradifferential symbol smoothing. We define BpSr\ to consist of symbols in S1,\ whose partial Fourier transforms satisfy such a support condition, and we set
.
Thus, if 0 < t5 < 1, and >.(j), "'(0 are given by (3.28), then (3.30)
"'~l~""""~~='-'-=~-~~'---"--'---i-SYMBOL C LASSES AND SYMBOL SMOOTHING
p#(71,O is supported in 1711 <
pl~l,
p(x,~)
E e(A)sf,'o ===;.
p#(x,~)
E B1Sr,1'
pb(x,E) E e(A)s~(~~,
provided >'U) satisfies (3.43). Note that, by Proposition 2.3 and (2.37), if (3.43) holds, and T(~) = (0, then (3.45)
q(x,~) E erA) S?,(T) ==? q(x, D) : LP(JRn)
-+
LP(JR n ),
1
< P<
00.
An important example of paradifferential symbol smoothing occurs in Bony's paraproduct (3.46)
Tl u
= '2,:{Wk-5(D)f) 'l/Jk+l(D)u,
;..,
?"'1'Pi't'·~;-':'~~~~~"i'J.l~
_n
...""
.......
x"....'
·,~~~·~
_ _>_ _
·_·c.
.
_
4. OPERATOR ESTIMATES ON SOBOLEV·LIKE SPACES
which arises in a decomposition of the product: (3.47)
The same arguments used to establish Propositions 3.2-3.3 yield:
+ Tul + R(f, u),
lu = Tju
PROPOSITION
with
(3.58)
(3.48)
L
= Rju =
R(f, u)
37
'l/Jj(D)/' '1h(D)u.
p(x,e)
3.4. II you use paradifferential symbol smoothing, then E
C[wlSi,(I<)
=?
p#(x,O
13Si,l'
E
pb(x,e)
E
CIWJSi,(Tj.
Here, you can choose a slowly varying B(le[) ~ 1, and set
jj-kl::::S
(3.59)
Clearly
I E L oo =? Tj E OP13S~,l'
(3.49)
IE C(>,)
=?
IIDtF(·,e)llo(A) ~ Ca(e>-Ia l.
This observation is useful in analyzing products of the form a(x, D)Tj, as will be seen in Proposition 6.1. As for estimating the remainder, we have easily from (3.48) the following:
I
(3.51)
C()..)
E
=?
Rj E OPCw s~t)
for a pair 'I/J(e), T(e) described as follows. You can choose a slowly varying B(IW ~ 1 and set
(e> -,p(~) = B(le\)w( (e> -1),
(3.52)
T(O =
"."
1, .., .. ,
Also we have IE C()..)
(3.53)
==:}
Rj E OPSl,f,
(e>-l/J(~) = w((O-l).
aU) = (log2 (j) r,
'Y
> 0.
Then E.j '" (e>-1(log2(e>r on supp 'l/Jj, so instead of (3.10) we have
(3.56)
p(x, e)
E
C()..) sro
=?
E
defined in the same fashion as (3.1) and (3.19), with C()..) replaced by the space C[w 1, which is defined by (1.61). This is particularly natural, since (3.15) sharpens to (3.57)
III -
J,/IIL~ ~
11/1101"']
·w(E.).
as
<
00,
~
L'
~
1
I'L'
4'.
1 so that
lei ----;
00,
then Proposition 2.7 applies to pb(x, e)(e>-(-l) replaced by w((e>-6) in (3.59), Le., (3.62)
(0 -,pW = B(IW~(e)w( (e> -6),
T(e) = _ ... "
1
,,,-,
Co'
We illustrate Proposition 3.4 with a family of examples parallel to (3.28)-(3.32). Given r, s > 0, take (for h E (0,1/2]) (3.63)
w(h)
=
( log2
h1 )-s '
~(e)
=
(log2(e>r,
and W
-,pro =
S
(log2(e> r- +
1
/2+c,
T(e) = (10g2(e>
r-
12 E / - ,
B(2 k ) = k 1/2+E.
Then we have desirable estimates (via Proposition 2.7) on pb(x, D) whenever s-r > 1/2, whereas in the previous analysis we needed s - r > 1. In this case, C[w] = CW, by the remark following (1.63).
IDt D~p#(x, e)1 ~ Ca/3(e>m-lal+111I (log2(e> f'Y'/3I.
C1Sr(I<)' with ~(e) = (e>(10g2W)-'Y. By Proposition 2.3 and (2.37), whenever p#(x, 0 has this property, with m = 0, we have p#(x, D) : V(Rn) ----; LP(Rn), provided'Y > 1. It is useful to consider smoothing of symbols in C[wJSi,o and in C[wJSi,(I<)'
In particular, p#(x,e)
k )2
(3.65)
for example
°
...... ,. I
k
and also
(3.55)
L B(21
(3.61)
(3.54) 00;
=
while
(3.64)
E.j = cTjaU)
B(IW~(e)w((e>-l) ----;
(3.60)
We briefly mention a variant of the symbol smoothing described by (3.4) and (3.9). We take
instead of (3.9), where aU) /
T(e)
If you are able to choose a slowly varying B(IW
Also, if we write Tj = F(x, D), then, for any modulus of continuity w, with AU) = w(2- j ), we have (3.50)
(e>-,pw = B(lw~(e)w((e>-l),
4. Operator estimates on Sobolev-like spaces Let A(e) ~ 1 be a slowly varying monotonic function of lei, with A(O) = 1. Assume
IDr A(e)1 ~ CaA(e)(e>-la l.
(4.1)
This implies that, for all s E R,
IDtA(~)SI ~ CasA(ene>-la l.
(4.2) We set (4.3)
Ha,p(Rn ) = {f
E
U(R n ) : A(D)I
E
U(R n )},
A(e)
= (e>a(~).
38
L
Since Qk'(x)lI'dD)f has spectrum in I~I ~ C2 k , we can dominate this by
Parallel to Lemma 2.1, we have the following result. LEMMA
4.1. Let
Ik
E
2: A(2k )IIQk'(x)lkIILP ~ 2: ,8(k)A(2 k )lllkll
S' (JRn) be such that, for some B j > 0,
k
supp Jk C {~ : B 0 2k- 1 ~ I~I ~ B I 2k+1},
(4.4)
k::::: 1.
00
If fk
00
112: fkllHa,p ~ CII{2: k=O k=O = 'l/Jk(D)f,
k
2 A(2 )2Ifk I }
~ C2:{J(k)IIA(D)lkllL p
(4.11)
k
~ CA 21IA(D)fIILP,
1/2
provided
IILP'
L (J(k) ~ A 2 < 00.
(4.12)
the converse inequality also holds.
k2: 0
We can use this to parallel the analysis of §2, to produce conditions under which p(x,D): Ha,p(JRn) ----> Ha,p(JRn). As in §2, we can reduce the problem to examining elementary symbols, taking the form
j-4
k=O
2: Qk(X)lI'k(~),
q(x,~) =
Ilq3(X, D)fIIHo,p
k=O
(4.13)
where lI'k is supported on (~) '" 2k and bounded in (4.7)
IQk(X)1 ~ AI,
~ (3(k,j),
~
j-4 j cll {2: A(2 )212: Qkjlkl} j=4 k=O
~
k-4 k cll {2: A(2 )212: Qkjlkl} k=4 2
00
1/2
j=O
(4.8)
00
~ cll{2:A(2k)2I1QklliexolfkI2}
Qk'(x) =
tp tp
2: ITp,Ab(j)1 2 ~ A~ 2: Ib(j)1
(4.15)
j2:0
2:
Qkj(X),
(J(k) =
2:
(3(k,j).
j=k-3
Ilq3(X, D)fIIHa,p
~ CA311{2: A(2 j )2IfjI2 j
L k
j '>"(j)A(2 ) ",",
2: '>"(j)II;(2 j
IIA(D)Q;:'(X)'Pk(D)fIILP.
.
f/2tp ~
CA311f11Ha,p.
= .>..(j)1I;(2 k ), for k
lI;(j) /, A(j)
we can deduce the £2-operator bound (4.15) from (4.12), i.e., from (4.17)
IIq2(x, D)fIIHa,p ~
2
It follows from (4.13) that
(4.16)
We have (4.10)
tp'
j2:0
As in Proposition 2.3, if we also have (3(k,j)
k+3
j=k-3
k=O
1/2
Tp,Ab(j) = A(2 j ) 2:{3(k,j)A(2 k )-lb(k) k=O
We next tackle q2(X, D)f. As in §2, set
(4.9)
j=4
2
defines a bounded linear operator on £2 :
1/2
k=4
~ CA 1 1IA(D)fIILP.
k+3
j-4
tp
j
(4.14)
= ql (x,~) + q2(X, 0 + q3(X, 0,
as in (2.8), and do separate estimates of qj(x, D), 1 ~ j ~ 3. To estimate ql (x, D)f, we apply Lemma 4.1 with Ik = 'Pk(D)f. Since (~) '" 2j on the spectrum of Qkj, we have
Ilql(X, D)fllHa,p
00
1/2
This time, make the hypothesis that
where again {'l/Jj} is the partition of unity (1.2). We make the decomposition
q(x,~)
2
00
~ cll{2: A (2 j )2(2:{3(k,j)llkl)}
S?, and Qk(X) satisfies
II'l/Jj (D)Qk Ilu'"
I: Qkjlk, to get
To estimate q3(X, D)f, we apply Lemma 4.1 to 9j =
00
(4.6)
Lp
k
Say Jo has compact support. Then, for p E (1,00), we have (4.5)
39
4, OPERATOR ESTIMATES ON SOBOLEV-LIKE SPACES
OPERATORS WITH MILDLY REGULAR SYMBOLS
This proves the following.
j
)
< 00.
,
~ j, and
,""
L OPERATORS WITH MILDLY"REdULARS-YMBOLS
40
_A.
Assume ).,(j), 1\;(0 are as in Proposition 2.3. Let A(~) = (~)aW ;::: 1 be a slowly increasing function of I~I and satisfy (4.1). If (4.16)-(4.17) hold, then PROPOSITION 4.2.
p(x,O E eUI,) Sr,(I<)
(4.18)
for 1 < P <
====}
p(x, D) : Ha,p(JR n )
---;
_? - _. . .
4.
LEMMA 4.5.
Let /k
-
(~)aW
k
k 2 11f: fkllHu,p ~ ell {I: A(2 )2IfkI f/ 1ILP' k=O k=O
(4.27)
).,(j)I\;#(2 j ) <
).,(j)I\;#(2 j )
00,
).,(j)A(2j )
A(j) "0 I\;#(j) '" ,
'\"
'\, .
PROOF.
Set
(4.21)
L 1](j),B(2
COROLLARY 4.3.
I\;W
)
<
A(D)/k. The left side of (4.27) is 00
00
£=0
k=£
2
1/2
tp 2
00
00
£=0
k=£
~ II{LA(2£)21~e(D)Lfkl} 00.
Thus, with Wk
The implication (4.18) is valid if
= A(~),
Uk =
~ II{LI~£(D) LUkl}
Then (4.18) holds provided 1\;#(2 k ) ;::: 1\;(2 k ) and j
L A(2£) ~ A 4 A(2 k ). £'5,.k
and
k~£
).,(j) = A(2 j )-I1](j), 1](j) '\, 0 as j ---; 00, 1\;#(2 k ) = A(2 k ),B(2 k ), ,B(2 k ) / 0 0 as k ---; 00.
(4.20)
00,
L A(2 k )-1 ~ A 4 A(2£)-1,
(4.28)
for some 1\;# ;::: 1\;. To restate the sufficient condition for (4.18), suppose
(4.22)
o.
;::: 1 is a slowly varying monotonic function of I~I, satisfying
provided that also, for some A 4 < L
k;:::
2
00.
More generally, (4.18) holds if we replace (4.16)-(4.17) by the hypothesis that (4.19)
_ ... -41
S'(JR n ) be such that, for some B > 0,
E
supp fk C {~ : I~I ~ B2},
(4.26) Then, if A(~) = (4.1), we have
Ha,p(JR n ),
oPlmAT·oR. ESTU,iATES'ON"SOBOLEV-LIKE SPACES
1](j) '\, 0,
L1](j) <
2
00
II{LA(2f)21~£(D)
k l LA(2 )- wk l} k=f
£=0
00.
II£P·
A(2 k )fk, we need to show that
00
(4.29) ).,(j) = A(2 j )-I1](j),
=
1/2
1/2
tp ~ el!{L I 00
Wk
I2 }
k=O
1/2
tp·
In other words, we need 00,
PROOF. It is a simple exercise to pick ,B / and then we just set I\;#(~) = I\;(~),B(~).
00,
slowly, such that
I: 1](j),B(2 k ) <
r(D): P(JRn, £2)
(4.30)
(4.23)
e(>')Sr,(I<)
====}
p(x,D): H 8 ,P(JRn )
---;
H 8 ,P(JR n ),
1
00,
=
(~)8,
).,(j) = T s j1](j),
L
IDerkf(~)1 ~ ea(~)-Ial,
L 1](j) <
and this follows from the hypothesis (4.28).
00.
(4.33)
The treatment of Q2(X, D)f in (4.10) is cruder than the analogous treatment in [Hour] (and also in [Mal], [T2]) for operators on Sobolev spaces H8,p, for s > O. Such a treatment, needed for an analysis of operators with symbols in S'1\' is based on the following sort of estimate, which, as we will see, does not extend far beyond its original setting.
IDerk£(~)1 ~ ea(~)-Ial,
£
Note that, if we set a(t)
1](j) '\, 0,
L
k
with (4.25)
k < £.
Now (4.30) holds provided (4.32)
I\;(~)
A(2£)
= A(2 k ) ~k(~), k;::: £,
o
provided (4.24)
rkf(~)
(4.31)
4.4. Given s > 0,
p(x,~) E
U(JR n ,£2),
where
Taking A(~) = (~)8, we have: COROLLARY
-->
00 a(t) dt
1 T
=
A(2t ), then (4.28) is equivalent to
A
~ a(~)'
iT a(t) dt ~ A a(t). 1
4
Such an estimate holds for a(t) = 2st , S > 0, i.e., A(I~I) = (~)s, which defines the standard Sobolev spaces, but not for a(t) = r, for any r > 0, i.e., not for A(IW = (log 1~lr (for 1~llarge). The following is a little sharper than Corollary 4.4.
.."
1.
Vf'l:';RKIURS-WITFi-IVUr;O-LY-REGULi\.U-SYMHUr;S
PROPOSITION 4.6. Assume A(j) = 2- sj 7](j) with s > 0 and K:(2 k ) = 2sk T(2 k ). Then, for 1 < p < 00, (4.34)
p(x,~) E C(>,) Sp,(r<)
==}
"
2: A(j) < 00.
Set
p(x, D) : HS,P(JR n ) --+ HS,P(JR n ),
A3
j
I: T(2 k) ::; 7](j)' I: 7](j) ::; T(2 k)'
k=O
p(x,e) E ESP,l
43
(e)a(~) ?: 1 is as in Proposition 4.2 and s E
=
==}
:=
p(x,D): Hsa,p(JR n ) --+ Hsa,p(JR n ).
The following stands in a similar relation to Corollary 4.4 as Proposition 2.7 does to Proposition 2.6, except that, for simplicity, we restrict attention to integral orders of smoothness.
A3
00
O;ERA;:OR ESTIM'ATES'ON SOBOLEV·LIKE SPACES
PROPOSITION 4.7. If A(e) JR, p E (1,00), then
(4.42)
provided
(4.35)
4.
PROPOSITION 4.8. Given s E Z+, 1 < p < 00,
j=k
(4.43)
PROOF. As usual, it suffices to consider elementary symbols, of the form treated in the proof of Proposition 4.2. The estimate (4.8) of q] (x, D)f works s in this case. To estimate q2(X, D)f, instead of using (4.10), we use the A(~) = case of Lemma 4.5 to write
w
p(x,O E CiwJSp,(r<)
==}
p(x, D) : H 5 ,P(JRn ) --+ HS,P(JR n ),
provided (4.44)
K:(~)
=
w(2- j ) = 2- sj 7](j),
(~)S,
and (4.36)
IIQ2(X, D)fIIH·'.P ::;
cll{I: 22kSlfkl2} 1/2tp ::; CllfllHs,P.
(4.45)
I: 7](j)2 < 00.
7](j) \. 0,
k=4
Finally, to estimate
(4.37)
Q3(X,
D)f, we need to verify the estimate (4.15), when
js T 13 ,Ab(j) = 2
j
I:
A(j)K:(2 k )T ks b(k) =
k=O
j
I: 7](j)T(2 k )b(k).
PROOF. We give details for s = 1; similar arguments work for other s E Z+. In this case, we need to estimate the V-norm of Djp(x, D)u, given u E H1,P(JR n ). The analogous estimate on p(x, D)Dju follows from Proposition 2.7. Pushing the D j past p(x, D), we are left with an LP-operator norm estimate on Qj(x, D), where
k=O
(4.46)
The desired bound follows if (4.35) holds,
Qj(x,~) = op(x,~) ~~. (~)-1, ]
Note that (4.35) holds when 7](j) we can replace (4.24)-(4.25) by (4.38)
K:(~) = (~)s(log(l + (~))r-\
Also, (4.38) holds when 7](j) the result of [Bou] that (4.39)
= j-r,T(2 k ) = k r-l, provided r > 1. Thus,
p(x,~) E C:+sSPl
r> 1, s> 0.
= 2- jr , T(2 k ) = 2kr , provided r > O. Then we recover ==}
p(x,D): HS,P(JR n ) --+ HS,P(JR n ),
assuming 1 < P < 00. The symbol classes defined as follows: (4.40)
A(j) = T S] . rr,
p(x,O E C:Si~" ~
r,s > 0,
C:Bl,\' and more generally C;Sr,'", are
IDfP(x,OI ::; ca(~)m-Ial, and IIDfp(',Ollc; ::; Ca(~)m-Ial+or.
Hsa,p(JR n ) = {f E S'(JR n ) : A(D)S f E LP(JRnn,
Moving away from symbols of type (1,1), we have the following simple but useful result. This result actually belongs as part of Proposition 2.1.E of [T2], but it escaped our attention at the time. PROPOSITION 4.9. Assume k E Z+, Ii E [0,1), p E (1,00). Then, for (4.47)
As mentioned in §3, operators with symbols in ESf\ have special properties. Here is a simple extension of (3.40). For any s E JR, we ~an set
(4.41)
and this in turn also follows from Proposition 2.7.
A(e) = (e)a(~).
There is an obvious extension of Lemma 4.1 and then an estimation parallel to that in (4.8) establishes the following.
p(x,~) E
Ck -
1
,1 Sr,'o,
we have (4.48)
p(x, D) : Hs+m,p(JR n ) --+ HS,P(JR n ),
-(1 - Ii)k < s ::; k.
PROOF. The only part not contained in Proposition 2.1.E of [T2] is the end point result, s = k. We treat explicitly the case k = 1. It suffices to take m = -1; we need to show that
(4.49)
p(x,O E Lip1S;-L u E LP
==}
Dj(p(x,D)u) E LP.
That p(x, D)Dju E LP is clear from standard results. On the other hand, if Dj falls on the coefficients of p(x, D), we need only note that
(4.50)
a(x,O E Loosl;g, a>
°
==}
a(x,D): LP
--+
which follows from elementary integral kernel estimates.
LP,
1::; p::;
00,
-.n--~ -~~~-
- =~~~croPERATORScwIT~HMIDSLYREGuLAfCsYMBoLs~-"'~c=-----
.co_.=~._.~~"~."_".~._
The following limiting cases are also noteworthy. Set
(4.51)
LEMMA
H S ,I(lR.ll ) = (1- Ll)-s/2~1(1R.1l), Hs,oo(JR n ) = (1- Ll)-s/2bmo(JRn ),
where ~1(I~n) is the localized Hardy space and bmo(JRn) is the localized John
Nirenberg space, introduced in §2.
PROPOSITION 4.10. Given t5 E [0,1), we have also for P = 1,00, (4.52)
p(x,~) E
LipS!!:8
==}
p(x, D) : Hm+l,p
-+
45
5.
(5.4)
5.1. Let fk E S'(lR. n ) be as in Lemmas 2.1 and 4.1. Then
II'EikIIC().J ::; Os~p
lIikl u<>,
'x(k)-l
k=O
and if ik = 1Pk(D)f, then the inequality is reversible. To estimate ql (x, D)f, since (~) ,..., 2j on the spectrum of Qkj, we have, by Lemma 5.1,
H1,p.
k-4
Ilql(x,D)fll c ().) ::; 0 sup 'x(k)-l
PROOF. It suffices to take m = -1 and show that (4.53)
p(x,~) E LipS~l
==}
Djp(x, D) : ~l
-+
k~O
(5.5)
~1, bmo
-+
::; 0 sup
bmo.
k~O
DjP(x, D)u = Pj(x, D)u + qj(x, D)u,
(5.6)
Pj(x, D)
=
p(x, D)Dj E OPLipS~,8 : ~l
-+
~1, bmo
-+
qj(x,~) = DXjP(x,~)
E
~ s~p
Ilq2(x, D)fllc().)
bmo,
Now Q'k ik has spectrum in I~I
L oo S';'6
1
-
8 ) =? qj(x,D): L I
-+
~1, bmo
-+
L oo ,
by an elementary estimate.
L Q'k(x)iktoo'
(5.7)
1jJ£(D)
::; 02 k , so,
for some finite N,
L Q'k(X)fk = 1jJ£(D) L k~O
1
00
q(x,~) =
< 0 sup ,x (e) £
0(>')
Here we want to obtain conditions under whichp(x, D) : 0(>') (JRll) -+ O(>')(JRll). Our analysis will be somewhat parallel to those in §2 and §4. As there, it suffices to consider elementary symbols, taking the form
L QdX)ipk(~), k=O
(5.8)
k~£-N
If jj(k) ::; 1 and 'x(k) is slowly varying, we can write
(5.9)
Ilq2(X, D)fllc(>.) ::;
provided that, with D 2 < 00,
(5.2)
(5.10)
IQk(X)1 ~ AI,
111P) (D)Qkllv>o ::; (3(k,j),
:L IIQ'k ik I Loc
k~£~N
1 '"' ~ < 0 sup 'x(e) L.J (3(k)'x(k) ·llfllc().)· f
where ipk(~) is supported on (~) ,..., 2k and bounded in Sp,o, and Qk(X) satisfies
OD 2 1Ifllc().),
:Ljj(k)'x(k)::; D 2,X(e),
V e ~ O.
k~f
where {1Pj} is the partition of unity (1.2). Again we make the decomposition (5.3)
Q'k(x)fk.
k~f-N
Thus the right side of (5.6) is
5. Operator estimates on spaces
(5.1)
'x(e)-l II1jJf(D)
k
by Propositions 2.10 and 2.13, and
(4.56)
'x(k)-lllfkIILoc
To tackle q2(X, D)f, we take Q'k and jj(k) as given in (2.10) and (4.9), and start with
where (4.55)
IIQkllLoc
q(x,~)
oc L
::; Ollfllc().)·
In fact, given u E ~l or bmo, we have
(4.54)
IILQkjikl1 j=O
= ql (x, 0 + q2(X,~) + q3(X, ~),
as in (2.8), and do separate estimates of qj(x, D), 1 ::; j ::; 3. As in (2.8), we set Qkj = 1Pj(D)Qk' In place of Lemma 2.1 or 4.1. in this case we have, essentially from the definition, the following:
Note that, if
(5.11)
(3(k,j) = 'x(j)1I:(2 k ),
then the condition (5.10) becomes (5.12)
:L 1I:(2 k )'x(k)2 ::; D 2'x(e). k?,f
w,_...
_.,._~
46
_",__'.. _
1. OPERATORS WITH MJLDLY REGULAR SYMBOLS
j-4
Finally, as gJ
=
I::
k=O
Qkjfk has spectrum in
Ilq3(x,D)fllc(.\) '5. C
s~~ J_
(5.13)
(~)
.-.J
====--------------------
2j , we have, by Lemma 5.1,
p(x, D) : C~(JR.rl) ----. C~(JR.n).
Atj) IILQkjfkllu" k=O j-4
Let us consider our favorite examples, starting with
J k=O
Ilfll c (.\)·
Consequently,
provided that, with D 3 <
j
v j.
(5.26)
L K:(2 k)A(k) '5. D3. k=O
(5.27) J-t(j) '5. A(j),
then the condition (5.15), controlling the estimate of Q3(X, D), becomes
j A( ') k L K:(2 )A(k) '5. D3 (~)' k=O J-t J
(5.18)
LK:(2 k )A(k)J-t(k) = L T kS ,
k~j
k~j
k~j
p(x,~) E C:S?,l(JR. n )
(5.21)
~
o '5. J <
L k
1,
,.;;(2 k )J-t(k) <
C;(JR.n ),
A(k) = k- s ,
1 r > 1 _ J'
0
< r < s,
r> 1.
s '5. r.
L K(2 k )A(k) ~ L k-(s-r") ~ jl-s+,." if
===}
p(x,D): C(>') (JR.rl) ---.
K~j
and
(5.30)
log j
if
s - rJ
= 1,
C
if
s - rJ
> 1,
CP,) (lR. n ),
L K:(2 k)J-t(k)A(k) '5. D 2 A(j),
while A(j)/J-t(j)
k> _J
=
y-s. Hence the first part of (5.21) holds provided that, either
rJ'5.s
and
l-s+rJ'5.r-s(i.e"r~l/(l-J)),
or else 00,
and
A(j) / . J-t(j)
0'5. s - rJ < 1,
k=l
0,
L K:(2 k )A(k) '5. D 3 A(~), k<_JJ-t(J)
---.
j
conditions that hold whenever (5.22)
p(x,D): C:(JR. n )
Let us examine (5.21) directly. We have
5.2. Assume IDfp(x,~)1 '5. CCI:(~)-ICl:I. If J-t(j) '5. A(j), then
provided that, for all j
===}
K:(2 k ) = (kr",
J-t(j) = rr,
(5.29)
in place of (5.12). We therefore have the following result.
p(x,~) E C(IL)S?,(K) (JR.rl)
k~j
a result first established by E. Stein [82]. Next, consider
k~P
(5.20)
A(j) = Tjs.
Note that (5.22) holds if and only if
L K:(2 k )J-t(k)A(k) '5. D 2 A(£),
PROPOSITION
K:(2 k ) = 2kr ,
LK:(2 k )A(k) = L2 k(,.-s),
(5.28)
V j.
Note that, if IJ(k,j) is given by (5.17), then, for the condition (5.10), controlling the estimate of Q2(X, D), we have (5.19)
p(x,D): C;(JR. rl ) ---. C;(JR. rl ),
so (5.21) holds. We have
If, more generally, ,B(k,j) is given by (5.17)
===}
Note that, when (5.25) holds, even though (5.22) fails, we still have
00
IJ(k,j) = J-t(j)K:(2 k ),
J-t(j) = 2- jr ,
(5.25)
If IJ(k,j) is given by (5.11), this condition becomes (5.16)
p(x,~) E C:S?,,,(JR. rl )
if J E [0,1), which is a well known result. Let us look at the case
00,
LIJ(k,j)A(k) '5. D 3A(j), k=O
(5.15)
in which case C(IL)Sr':(K) = C;'Sr:", defined as in (4.40). Then (5.22) holds as long as J E [0,1). Thus, for r > 0, (5.24)
Ilq3(X, D)fllc(.\) '5. CD 31Iflb.\),
K:(2 k ) = 2knl ,
J-t(j) = A(j) = Tj,.,
(5.23)
,
'5. C sup A( ') LIJ(k,J)A(k) J~O
== 1 here, in which case the conclusion of (5.20) is
We can take A(j)
REMARK.
47
that
j-4
1
(5.14)
5, OPERATOR ESTIMATES ON SPACES c(.\)
s
= 1 + ra,
and
r> s,
and
r
or else
s>
1+
ra,
~
s.
48
1.
OPERATORS WITH MILDLY REGULAR SYMBOLS
5.
Meanwhile,
(5.31)
L K(2 k )A(2k )JL(k) ~ L k ro - r- s ~ k-(s+(l-o)r-l),
o ~ 8 < 1,
(5.32)
and
1
r8 ~ s < r (and s > 0),
1 + r8 < s ~ r.
or
Then the two conditions in (5.39) take the form (5.41)
k5.j
===}
p(x,D): C(>') (IR n ) ~
k~j
PROPOSITION 5.6. If 0 < s < r, then
In summary, when JL, K, and A are given by (5.28), and (5.32)-(5.33) hold, then p(x,~) E C(/l)Sp,(><) (IR n )
L ,(k)2(r-s)k ~ Dn(j)2(r-s)j, L ,(k)T Sk ~ D 2,(j)T sj.
Note that the first inequality in (5.41) always holds when ,(k) /", while the second inequality in (5.41) always holds when ,(k) \, . In particular, if there exist Sl, S2 E (O,r) such that 2S1k A(k) \, while 2S2k A(k) /" (for large k) then both conditions in (5.39) hold. For example, we have the following.
r;::: 1- 8'
and either
0 < s < r.
A(k) = Tsk,(k),
(5.40)
k~j
if s > 1 - (1 - 8)r. Thus the second part of (5.21) holds provided s > 0 and s - 1 + r(l- 8) ;::: s, i.e., provided r ;::: 1/(1- 8). Hence, given (5.28), the condition (5.21) of Proposition 5.2 holds, provided
(5.34)
49
Suppose for example that k~j
(5.33)
OPERATOR ESTIMATES ON SPACES cP,)
n
C(>') (IR ).
(5.42)
p(x,~) E C;SP,l
===}
p(x, D) : C(,),)(lR n ) ~ C(,),) (IR n )
for Recall from the discussion of (2.43) that this class of operators is bounded on LP(lR n ), for 1 < P < 00, provided r> 1 + r8, i.e., provided (5.32) holds, with strict inequality. If we take K(j) = 1 and JL(j) = A(j) in (5.20), we have the following consequence of Proposition 5.2. Note however that in case JL(j) = y-r, r < 1, A(k) = k-S, 0 < s ~ r, the conclusion (5.34) from (5.32)-(5.33) (with 8 = 0) is stronger then Corollary 5.3, specialized to A(k) = k- s . COROLLARY 5.3. If A(k) \, 0 is slowly varying and
(5.35)
p(x,~) E C(>')SP,o
==}
:E A(k) < 00,
then
k"IT sk ,
any,
E R
Propositions 5.5-5.6 do not apply when A(j) decreases to zero very slowly. It is useful to note the following result, which follows directly from the analysis in (5.5): PROPOSITION 5.7. Whenever A(j) \, 0 is slowly varying, then (5.44)
(5.45)
L A(k)2 ~ D 2A(j),
LA(k) ~ D 3 , k5.j
=
p(x,~) E BSp,l
==}
p(x, D) : C(,),) (IR n ) ~ C(,),) (IR n ).
Hence the conclusion of (5.44) holds if p(x,~) E Sp '" with 8 E [0,1). For another important example, note that if Tf denotes the p~raproduct (3.51),
p(x,D): C(>') (IR n ) ~ C(>') (IR n ).
PROOF. The two parts of (5.21) are (5.36)
A(k)
(5.43)
f
E LX
===}
Tf : C(,),) (IR n ) ~ C(,),) (IR n ).
We next study mapping properties of Rf, defined by (3.48), the remainder term in the expansion (3.47) of a product into paraproducts.
k~j
in this case. The hypotheses of the corollary imply these results, since, under these A(k)2 ~ A(j) :Ek>' A(k). hypotheses, Lk>' _J _J
PROPOSITION 5.8. Assume f E C(,),). If JL(k) \, is slowly varying, JL(k) ;::: A(k), and
COROLLARY 5.4. IfA(k) \, 0 is slowly varying and:EA(k) < is a Banach algebra, i. e.,
(5.46)
u,v E C(>')(lR n )
(5.37) If we set JL(k)
p(x,~) E C:S?,l
===}
00
then
uv E C(>') (IR n ).
and 2- rk ~ A(k), r > 0, then
p(x,D): C(>') (IR
n
)
~ C(>') (IR
L kScJ
A(k)2 rk ~ D 3 A(j)2 rj ,
and
L A(k)JL(k) ~ CA(£), k~£
n
(5.47)
Rf : C(/l) (IR n )
-t
C(,),) (IR n ).
PROOF. By (3.51) we have, with w(2- k ) = A(k),
),
provided (5.39)
then C(>') (IR n )
= 2- rk , K(2 k ) = 2rk , and apply Proposition 5.2, we have:
PROPOSITION 5.5. If:E A(k) <
(5.38)
==}
00,
Rf E cws~t),
(5.48)
L A(k) ~ D 2A(j).
with
k~j
(5.49)
(~)-,p(~)
=
B(IWw((~)-l),
T(~)
=
(~),p(~),
50
1.
where B(I~I) ~ 1 is a slowly varying function we are free to choose. We choose it so that, with a(2- k ) = J-L(k),
B(2
j ) =
1 J-L(j)'
(~)-1/J(~) = w((~)-l)
hence
a((~)-l)
L
rpj(D)
=
j
L
~k(D),
Ij-kl9
L
11~£(D)RfgIILC<>::;
Ilrpj(D)fIILoc II~j(D)gl\L~
j?£-N
(5.51)
yielding the conclusion that (5.46)
=}
(5.47).
We mention some cases in which (5.46) holds. Assuming as always that .\(j) and J-L(j) are decreasing, we have (5.46) whenever
LJ-L(k) <
It also easily follows that, if T
==}
P : cr+m,(A)
-+
Cr,(A).
> 0, /-L(j) = 2- rj .\(j).
Cr,(A) -- C(/1-) ,
Also, we note that, if k E Z+, Ck,(A) C C k ,
(5.60)
L .\(j) <
if
00.
PROPOSITION 5.10. Suppose p(x,~) E C(A) S~,(!>. Assume that 7(~) = (~)1/J(O, and that
.\(j)7(2 j
) '\.
0,
L .\(j)7(2
j
)
< 00,
IDf7(OI::; Ca7(~)(~)-lal.
Then
k?£
-----+
C(/1-) ,
J-L(j) = 7(2 j )-1.
More generally,
This includes the classical case, .\( k) = 2- rk , r holds for
J-L(k) = k- 1 ,
p(x, D) : C:
(5.62)
L.\(k)::; C.\(£).
> 0, J-L(k)
.\(k) = k-',
=
1. In addition, (5.46)
s> 1.
Using the expansion (3.47), i.e., fg = Tfg
P E OPS';o
(5.58)
(5.61 )
00.
Also, we have (5.46) whenever
(5.55)
Cr,(A) = {(I - D.)-r/2 u : u E C(A)}.
Results so far in this section have examined operators mapping a space C(A) to itself. From these results one can deduce further results, about operators between different spaces of the type C(A), or more generally Cr,(A). We will not formu late general results here, but will record one case that will arise in the proof of Proposition 1.1 of Chapter III.
::; C L .\(j)J-L(j), j?£-N
(5.54)
CllfilLoo IIgllc(A) + Cllfllc
+ Cmin{ Ilfll c<,,) IlgIIccA), Ilfllc(A) IIgllc(~·)}·
It is useful to consider such spaces as Ck,(A), for k E Z+, consisting of U E C(A) such that Da u E C(A) for 10:1 ::; k. More generally, we can define, for T E JR,
(5.59)
and, using the fact that the jth term in this sum for Rfg has Fourier transform supported in a ball I~I ::; C2 j , we have
(5.53)
51
It follows from Proposition 5.2 that this is equivalent to the characterization above when T = k E Z+, and also that
L rpj(D)f ~j(D)g,
(5.52)
Ilfgllc(>') ::;
(5.57)
7(2 k).\(k)2 ::; C.\(£),
and since 7(2 j ) = J-L(j)j.\(j), this is equivalent to the hypothesis (5.46). A more direct approach to the proof of Proposition 5.8 is to write
=
c(>')
.
k?£
Rfg
OPBRATOR ESTIMATES ON SPACES
COROLLARY 5.9. n.'hen the hypotheses of Proposition 5.8 hold,
(5.56)
Now the analysis of R f is controlled by the analysis of q2(X, D) in (5.6)-(5.12). We see that (5.47) holds as long as
(5.50)
5.
OPERATORS WITH MILDLY REGULAR SYMBOLS
+ Tgf + Rfg = Tfg + Tgf + Rgf,
we deduce the following refinement of Corollary 5.4:
(5.63)
p(x, D) : Cr,h)
-+
C(I/),
v(j) = max(J-L(j)t(j), .\(j)).
PROOF. If A(~) = (~r-1/J(~), then A(D) : C; -+ C(/1-) , and p(x, D) = q(x, D) A(D), with q(x,~) E C(A)S?,(r)' Now, Proposition 5.2 (with the roles of J-L and .\ reversed) implies q(x, D) : C(/1-) follows similarly.
-+
C(/1-) , so we have (5.62). The property (5.63)
The following is a useful compactness result.
52
1.
'~";'O'~ERATOR ESTIMATES ON SPACES
OPERATORS WITH MILDLYR.EGI)LAR SYMBOLS
PROPOSITION 5.11. Let )..(j) = w(2- j ), p,(j) = 0"(2- j ). Assume that wand 0" are tame of all orders (as defined before Proposition 1.8). Furthermore, assume that
)..(j) "" p,(j)
(5.64)
Ilq3(X, D)fllc(~) ::;
o.
(5.65)
A(~) =
W((~)-l)
']['n.
'---+
-1
j-4
c~~~[)..tj)
( ; IIQkjIILoo)"(k)]
Ilfllc(~)
since q(x,~) E crS~,l
=::::}
IIQkjllLOO ::; C2(k-j)r.
Picking r sufficiently large that 2ks )"(k) / for some s E (0, r), we have j-4
then
L2 kr )"(k)::; C2 jr )..(j), k=O
(5.73) (5.66)
This result is consistent with (5.68). Finally,
::; Cr sup [)..(1.) L2(k-j)r)"(k)]llfllc(~), )20 J k=O (5.72)
,
53
j-4
C(/1) (M) is compact.
Next, it suffices to show that, if
B(~) = A(~)
11"\_1\'
::; C.
(5.71)
Then, for a compact manifold M, the inclusion C(>.) (M) PROOF. First, it suffices to consider M =
since q(x,~) E S~,l ::::} IIQ);" 11£'' ' parallel to (5.13), we have
C(A)
B(D) : c(>.)(']['n)
----+
C(/1) (']['n) , and hence
and
Ilq3(X, D)fllc(~) ::; Cllfllc(~),
(5.74) (5.67)
A(D): C(/1) (']['n) ----+ C(/1) ('][''') is compact.
again without the restriction that
Of these, (5.66) follows from Proposition 5.10 and (5.67) follows from the fact that A(D) is a norm limit of finite rank operators, under the hypothesis (5.64). We have seen that OPSP 1 maps C(>.) into itself sometimes, but not when )..(j) decreases to zero at only an ~lgebraic rate. We record how OPSP,l acts on C(>.) in a greater variety of cases here. PROPOSITION 5.12. Let )..(j) "" 0 be slowly varying and assume Then (5.68)
P E OPS~,l
===}
P: C(>.)
--+
C(/1),
p,(j)
=
I: )..(j) < 00.
L)..(£). £2j
PROOF. It suffices to estimate qv(x, D)f, for v = 1,2,3, as in the proof of Proposition 5.2. An examination of (5.5) shows that (5.69)
\Iq1 (x, D)fllc(~) ::; Cllfllc(~),
even without the restriction that I: )..(j) < 00. (This has already been noted in Proposition 5.7.) Parallel to (5.6)-(5.8), we have
Ilq2(X,D)fllc(/L) ::; Csup (5.70)
£
(1£) P,
L
IIQkhIILC'<>
k2£-N
s:csup [_(l£) )..(£)]llfllc(~), £ p, k2£-N
L
I: )..(j) < 00.
It is natural to sharpen Proposition 5.12, making use of symbol classes defined
by
(5.75)
C(>.) St1
= C(>.) St,(T) , with
T(2 k ) = )"(k) -1.
We have the following. PROPOSITION 5.13. Let v(j), )..(j) "" 0, v(j) ::; )..(j), and assume I: )..(j) < Then
(5.76)
P
E
OPC(v)SP,l
=::::}
P: C(>.)
--+
C(/1) ,
00.
p,(j) = L)..(£), £2j
provided (5.77)
f-. )"(k)
< Cp'(j) .
~ v(k) -
v(j)
k=O
PROOF. We need only revisit the estimation of q3(x,D)f. In place of (5.71), we have 1
(5.78)
j-4
(.)
Ilq3(X, D)fllc(/L) ::; C sup [-(.) L v(~) )"(k)] Ilfllc(~), ) p, J k=O v
which is ::; Cllfllcc~) as long as (5.77) holds. Note that (5.77) holds when (5.79)
v(k) = )"(k),
p,(j) . )..(j) ~ CJ.
------:.--=::-
:::.--:--::=-=--
_._-'-------------~~~-'------------
..
---~~----_~:":':C.~~:~.-"~.:_'~.~.~.~~ ~_~"':"'~~:-':~_~.-.~~_'::"'",:_~~~
.
~~.-"c'::~';_-~,.~:__:__::~~.~-
vU) = AU) =
(5.80)
s
r
f.L(j) =
,
r(s-I),
s> 1.
Here we develop some operator calculus for the operators arising as A#(x, D) in the symbol decomposition A(x,~) = A#(x, 0+ Ab(x,~) of a symbol with limited regularity in x. We begin with a variant of a well known analysis of products a(x, D)b(x, D) = p(x, D) when a(x,~) E I and b(x, E BSf\, the symbol class defined in (3.36)-(3.37). We are particularly interested in estim~ting the remainder Tv(X,~), arising in
°
a(x, D)b(x, D) = pv(x, D)
(6.1)
Here we allow w
+ Tv(X, D),
pv(x,~) =
(6.2)
l: lal~v
_z- ,
a.
Tv(X,~)
=
(2~)n J[a(x,~+ry) -
l:
=}
Bj(x,~,y) = 0 for f)j-I ~ pl~l.
IIAvj(x,~, ·)llu :::; C1lrjAvj(x,~, ·)IIHK,
(6.10)
where rjf(ry) = f(f)jry) , so rjAvj is supported in Iryl :::; f). Let us use the integral formula for the remainder term in the power series expansion to write
AVj(x,~,f)jry)
'Pj( f)jry) (27r)n
'"' LJ
v
lal=v+1
+1(
a!
[1 (1 _ s t+l &fa(x, ~ + sf)j ry) dS) f)jlalrya.
io
Since Iryl :::; f) on the support of rjAvj , if also f)j-I < pl~1, then If)jryl < pf)21~1. Now, given p E (0,1), choose f) > 1 such that
Proposition 6.1 below is a variant of a number of results, including results in [Bon], [Meyl], [H3], and [AT]. To begin the analysis, we have the formula (6.3)
c {Iryl < pl~l}
We next estimate the LI-norm of AVj(x,~, .). Now, by a standard proof of Sobolev's imbedding theorem, given K > n/2, we have
(6.11)
&fa(x,~)· &~b(x, ~).
1, in which case C(,x) =~. Furthermore, given that b(x,O E
supp h(ry,O
where '-Iad
=
55
BpSrl' (6.9)
6. Products
Sr
~·~~u
6. PRODUCTS
(We continue to assume f.LU) is given as in (5.76).) In particular, Proposition 5.13 applies to
:~&fa(x,~)]eiX·T/h(ry,Odry.
pf)3 < 1. This implies (~)
rv
(~+
sf)jry) , for all s
E
[0,1]. We deduce that the hypothesis
l&fa(x,~)1 :::; Ca(~)Jl2W-lal
(6.12)
lal~v
for lal ~ v
+1
implies
Write
Tv(X,~) =
(6.4)
l: TVj(X,~)
Avj(x,~,ry)Bj(x,~,ry) dry = J AVj(x,~,y)Bj(x,~, -y)
~
(6.6)
1
[
dy,
l:
rya a
]
a!&~a(x,~) 'Pj(ry),
and if also (6.9) applies, we have ITv(X,~)I:::; Cv(~)Jl2(~)-r(~)llb(-'~)lle(A)'
(6.15)
provided w(h) has the following property:
l:
(6.16)
=
h(ry, ~)'Pj(ry)eix'T/.
w(f)-j)f)i(v+1):::;
C1~lv+l-r(~).
79j-1
lal~v
Bj(x,~, ry)
< pl~l.
IT vj (x, ~) I :::; C v w( f)- j) f)j(v+I) (~)Jl2(~) -v-III b(-, ~) II e(A) ,
(6.14)
where the terms in these integrands are defined as follows. Pick f) > 1 and take a Littlewood-Paley partition of unity {'P] : j ~ O}, such that 'Po(ry) is supported in Iryl :::; 1, while for j ~ 1, 'Pj(ry) is supported in f)j-I :::; Iryl :::; f)i+ I . Then we set
AVj(x,~,ry) = (27r)n a(x,~+ry) -
for {jJ-I
Now, when (6.8) and (6.13) hold, we have
with
TVj(X,O = J
IIAvj(x'~")llu :::; Cvf)i(V+1)(~)Jl2(~)-V-1,
(6.13)
j:::O
(6.5)
... ,-.,~' .. ~_:"._ :•. ''::":_-..,'~:~~~.--''"~~.-., •.~ ••• ---.~
1. OPERATORS WITH MILDLY REGULAR SYMBOLS
54
To estimate derivatives of Tv(X,O, we can write
Note that Bj(x,~,
(6.7) Thus, with AU) = w(2- j (6.8)
Y) = 'Pj(Dy)b(x
+ y, ~).
),
IIB)(x,~, ')lll,~ :::; Cw(f)-j)llb(',~)lle(A)'
(6.17) D~DlTVj(X,~)
l:
l: (~) (~) JD~l Dl' AVj(x,~,y)· D~2 Dr Bj(X,~, -y) dy.
(31 +(32 =(3 ')'1 +')'2 =')'
~~~~-
~~~-~-~~~~-~~~
~~~~'=-=:-~~~-7"~~==,,~~~~~--:..~-=ec'~~--'-------:~-~'~~-"~
~-=-~~~~-~~=~~-~~,.,-~-:~-~~~~~~
,-,~~-~~"~~~~~~~--,.~-~~~---
~~~-=o~
:--~~~~~-o==~~
-''''"o~~--~-='''~~~-===c''''"-'==c~~--'''~::-~~~~
1. OPERATORS WITH MILDLY RBGULAR SYMBOLS
6. PRODUCTS
Now D~1 Dr AVj(x,~, y) is produced just like AVj(x,~, y), with the symbol a(x,~) replaced by D~ID'ta(x,~), and D~2DrBj(x,~, -y) is produced just like the sym bol Bj(x,~, -y), with b(x,O replaced by D~2 Drb(x, ~). Thus, if we strengthen the hypothesis (6.12) to
Typically, m2(~) = m in (6.26) and (6.28). However, some examples arise where m2(~) > m, but m2(~) - r(~) < m. A typical case in which (6.26) arises is via symbol smoothing, of any of the sorts studied in §3: when L AU) < 00,
56
la~aea(x,~)I::; Cn,8(~)1-'2W-lnl+I,81
(6.18)
for 10d2:
v+ 1,
we have
IID~1 D't AVj(x,~, ·)11£1 ::; Cv'I9j(v+1)(~)1-'2(e)-h',I+I,8d-v-1,
(6.19)
w(t)t- S
w('I9- j )19i(V+1+1,821)::;
CI~IV+1+1,821-rW,
19 j - 1
( 1)
L
(~)1-'2(O+If3I-hll-rWIIDrb(',~)llc(A)'
b(x,~) E BSr'\. ,
p(x, D) E
opsttm .
Assume furthermore that (6.25)
la~aea(x,~)1 ::; Cn,8(~)1-'2(eHnl+I,8I, for
lod 2: v + 1,
with P2(~) ::; p, and that, with A(j) = w(2- j ), (6.26)
II Dfb(-' ~) Ilc(A)
L
::; Cn (~) m2(eH nl.
w('I9- j )'I9 j (V+1)::;
C1~lv+1-r(O.
19j-l
Then we have (6.28)
(6.1)~(6.2)
with rv(x, D)
The result (6.32) follows easily from the identity
J
A(7J,~) = ih(7J-(,~+7J)p2(,~)d( for A(x, D) = P1 (x, D)P2(X, D). One easy consequence of Propositions 6.1--6.2 is the following result of [Meyl], which will be useful in the next section. Here, B~S~'~i is as in (3.39). PROPOSITION
6.3. If Pj (x,~) E B~S~t and P ::; 1/3, then
[P1(x,D),p2(X,D)] E OPBSr,'I+ m2 - 1.
Regarding the hypothesis (6.25), the following describes a class of operators for which this is relevant. It consists of operators of the form
(6.35)
m r OPSI-'2+ 1,1 2- .
Tfu(x)
=
L !k(x)1/Jk+1(D)u(x),
!k(x) = Wk-5(D)f(x),
k
which give rise to paraproducts. Note that (6.36)
E
then
p1(x,D)p2(x,D) E OPBpS~7I+m2.
(6.34)
We assume that either w(h) is a modulus of continuity or w = 1. Finally, assume that w(h) has the property (6.27)
+ P2 + P1P2 < 1,
Tn
(6.33)
=
21
and
6.1. Assume
a(x, D)b(x, D)
for 0 < h ::;
pdX,~)p2(X,~) E B P S 1,1l+ m 2,
Then (6.24)
-8
(6.27) holds with (~) -rCe) = (log(~)) - 8 .
(6.32)
These estimates lead to the following result.
a(x,O E Si1' ,
= w(I~I-1).
6.2. Ifpj(x,~) E BpjS';t and p = P1
PROPOSITION
"11 +"12="1
(6.23)
+ 1)
The class OPBS;,l is not quite an algebra, but the following result is a useful substitute:
so, as long as (6.9) applies, (6.19)-(6.20) yield
PROPOSITION
for some s E (0, v
w(h) = log h
(6.31)
ID~Dlrv(x,~)1 ::; C
"'"
In the version of Proposition 6.1 given in [AT], w(h) = h r , and one needs v+ 1> r. As another example, given s E (0,00),
=}
L
IIDfp#(',Ollc(A) ::; Cn(~)m-Inl.
(6.27) holds with (~)-rCe)
=}
j IID~2 Dr Bj(X,~,·)Ilv>o ::; Cw('I9- )'I9 1,82Ij IIDrb(.,~) Ilc(>·)·
(6.21)
=}
Another (related) case is given by b(x, D) = T f , f E CC),); see (3.50). Regarding the condition (6.27), note the following sufficient condition:
for 'I9 j - 1 < pl~l.
Now, if (6.16) holds, then
(6.22)
p(x,O E CC),)Sr,'o
(6.30)
Furthermore, extending (6.8), we have (6.20)
(6.29)
57
f E L oc
=}
Tf = F(x, D) E OPBS?,l,
f E C~
====:>
Tf = F(x, D) E OPBSf,l'
(~)'P(e) = log(~).
';;::"":==':::=~=="'=':C_:.=::.'="''''-=_",,'''=~--'''~;;;;;~;'''"'''.=''''_''~''''-''''''''''''''-"
58
1.
OPERATORS
PROPOSITION 6.4. If f E (6.37)
WITH
MILDLY REGULAR SYMBOLS
e(>'), with
-; -~.,:,,-
.. 0::".
.
_:.."-_:""'-"''''''''''''''' -""'="''''''''''''>"'''''.''''''"''''''':r:,,,,.:,,,",r,?,~:: ,oe":="'· ""':':":':'~:::"'~";~C~MM:UTATO~:~~~{;:;A:;i;:'=:~:==='cc~:"'~~._:~:- _::~:~~,:~";;";'
We have
)..(j) "'" 0, then
lal:::: 1 ==} ID~DfF(x,~)1 ~ e"J1)"(log(~)) (~)-I"I+IJ1I.
(7.4)
In other words, (6.25) holds with M2(~) = -'l/J(~), where (~) -,pce)
(6.38)
PROOF. If ~£
(7.5)
lal :::: 1 ==} Df F(x,~) = ~]ft-j(x) - ft-6(X)]'l/Jt~+6(~)'
on ~e.
8
e" L
Bb(x t) E , <"
e C).) S-l I ,C T)'
R
= A bB#
+ R,
- B# Ab + A # B b - B bA #
+ A bB b -
B bA b.
To analyze [A # , B#], we can use Proposition 6.3 which, together with Proposition 4.7, implies
Hence IDf F(x, 01 ~
B#(x,~) E B~Sr.l'
[A,B] = [A#,B#]
with
j=4
(6.40)
Ab(X,~) E eC).)S?,CT),
we have
8
(6.39)
A#(x,O E B~St.l'
with T(O = (E,). Here, the classes B~Sl.'r are defined by (3.39). One can take p as small as desired; make sure p ~ 1/3, so Proposition 6.2 applies. We expand the commutator [A, B] :
= )..(log (~)).
= {~ : 2£ ~ I~I ~ 2l+ 1 },
A=A#+A b, B = B#+B b,
Ilft-j - fe-611LOC (~)-I"I on ~e,
(7.6)
[A#,B#] : HS,P(JR b) --+ HS,P(JR n ),
'if s E JR, p E (1,00).
j=4
Thus, to check (7.2), it remains to investigate the LP-boundedness of the six terms in R. By Proposition 2.6 we have
for Ia I :::: 1. However, 8
(6.41)
llfe-j - fe-611Loo ~
L
II'l/Je-lI(D)fIILoo ~ ellfllc(>.) . )..(e).
A b : U(JR n )
(7.7)
--+
U(JR n ),
1I=4
This yields (6.37) for j3 = O. Estimates for general j3 are similar. When e U,) = e;, such a result arose in [AT]. Proposition 6.4 is useful in the analysis of Tjb(x, D). As for an analysis of a(x, D)Tj, we have noted that the following version of (6.26) holds: (6.42)
f
E e().) ===? IIDfF(·,~)llc(>.) ~ e(~)-I"I.
These two analyses will be put together for a commutator estimate, in §7.
7. Commutator estimates In many cases, the commutator of operators with symbols in e().) Sr.~ is less singular than a general operator with symbol in e C).) Sn+'P2. We begin our explo ration of this phenomenon with an investigation of whe~ (7.1)
A(x ,<"t) E
e C).) Sl1,o,
B(x ,<"t) E
e C).) SO1.0
implies (7.2)
[A, B] : U(JR n ) --+ U(JR n ),
for 1 < p < 00; for short, we set A(x, D) = A, etc. To tackle this, we use the paradifferential symbol decomposition defined in §3, of the form (3.9) with b = 1 and c large, giving rise to (3.44). Thus, assume that (7.3)
2J )..U) "'" 0,
L 2 )..(j) < 00. j
for 1 < P < 00, since the crucial part of hypothesis (2.37) is the same as hypothesis (7.3), when 1\;(0 = T(O = (~). As noted in Proposition 3.3, the inclusion B#(x,O E B6S? 1 implies that B# is bounded on LP, indeed on HS'P for all s E JR, so we have the desired bounds on the first two terms in R. Also, B b is clearly bounded on LP, so the last two terms in R are under control. It remains to look at the terms A# B b and B bA#, Now, by Proposition 3.2,
(7.8) A#(x,~) E B 1St,1
==}
A# : Hs+l,P(JR n )
-t
HS,P(JR n ),
'if s E JR, p E (1,00).
Hence the desired LP bounds follow if we establish that, for 1 < p < 00,
(7.9)
Bb(x,O E eC).)S~(T)
==}
B b : Hs-l,P(JR n ) - t HS,P(JR n ),
0 ~ s ~ 1.
Setting p(x,~) = Bb(x,O(~), we obtain this in turn from the implication (7.10)
p(x,~) E
e
C).)
S?,CT) ===? p(x, D) : HS,P(JR n ) - t Hs.P(JR n ),
0 ~ s ~ 1,
given 1 < p < 00, and given T(~) = (~), )..(j) satisfying (7.3). Now, (7.10) follows from the s = 1 case of Corollary 4.4, together with Proposition 2.3 (for the case s = 0 in (7.10)), and interpolation. We have the following result. j PROPOSITION 7.1. If A(x,~) E eC).)st,o, B(x,O E eC).)S?,o, with )..(j)2 "'" 0 and
(7.11)
L 2 )..(j) < 00, j
then (7.12)
[A,B] : LP(JRn)
--->
U(JR n ),
1
< p < 00.
1. OPERATORS WITH MILDLY REGULAR SYMBOLS
60
"_....,.-
-----·-gr
···'8.'~It~~6L~~C'OOm~IENJ!fi's
The following is the analogous result obtained by replacing C(>,) by C[wl. Com pare [Ma2] for a similar result, in the framework of
with the same 'IjJ as in (7.17), i.e., 'IjJ given by (7.18). We have established the following. (For a variant, see Proposition 10.2.)
PROPOSITION 7.2. Assume A(x,~) E C[wJSt,o, B(x,~) E C[wJSP,o, with j 2 w(2- j ) ~ 0 and
PROPOSITION 7.3. Let w be a modulus of continuity such that w(t)C" ~ for some s E (0,1). Set >"(j) = w(2- j ), and take f E C(A). Let p(x, D) E OPBI/2Sr~\. Then
cw.
I:[2 j w(Tj)]2 < 00.
(7.13)
j
[Tj,p(x, D)] E OPBS~I-,p
(7.22)
with
Then [A, B] : U(JR
(7.14)
n
) -+
U(IR
n
),
1 < P < 00.
PROOF. We have the symbol decompositions in (7.4), with the difference that, in this case,
Ab(x,~) E C[wJSf,(T)'
(7.15)
Bb(x,~) E C[wJS~tT)'
again with T(~) = (~), by (3.47)-(3.48). The result (7.6) still applies in this case, as does (7.8). For the rest of the estimates of the terms in (7.5), we need only replace the application of Proposition 2.6 in (7.7) by an application of Proposition 2.7, and to replace the application of Corollary 4.4 in (7.10) by an application of Proposition 4.8. We now analyze the commutator [Tj,p(x, D)], given p(x,~) E BS1,\ and f E CP.), where T j is the paraproduct (3.46). We see by (6.36) that f E ciA) =} Tj = F(x, D) E OPBS'i I' with (O'l'W = log(~), and if >"(j) < 00 we can take V' = O. Applying Proposition 6.1, with 1/ = 0, we have
z=
Tjp(x,D)
(7.16)
= a(x,D),
a(x,~) = F(x,~)p(x,~)
where -'IjJ
= 1-L2
+ ro(x,~),
ro(x,~) E S~I-,p,
is the analogue of (6.25) for F(x,~), i.e., by Proposition 6.4, (~)-,p(e)
(7.18)
=
>"(log(~)) ~ w((~)-I).
On the other hand, as noted in (3.50) and again in (6.42), we have
IIDeF(-'~)llc(),) ~ C",llf"c(),)(~)-I"'I,
= w((~)-I).
A special case of Proposition 7.3 is that, for 0
f E C r ===} [Tj,p(x, D)] E
(7.24)
< r < 1, p(x, D)
E OPB I / 2S'!:I,
OPBS~lr.
The following result treats the r / 1 limit of (7.24). The implication (7.25) was established in Proposition 4.2 of [AT]. An r ~ 0 limit of (7.24) will be discussed in §1O. PROPOSITION 7.4. Given p(x,~) E OPBI/2Si~I'
f E Lipl ===} [Tj,p(x, D)] E OPBS~ll,
(7.25)
and (7.26)
f E C;
==}
[Tj,p(x, D)] E OPBS;r\-l+'I', ,
(~)'I'(e) = log(~).
PROOF. We have (7.16) with
a(x,~)
(7.27)
with
(7.17)
(~)-,p(e)
(7.23)
when
f
(7.28)
=
F(x, ~)p(x,~) + ro(x, ~),
ro(x,~) E SiY I
E C;, as a special case of (7.17)-(7.18). This time we have (7.19) with
b(x,~) = p(x,~) - i
I: oeP(x,~) . oxF(x, 0 + rt (x, 0, 1"'1=1
and (6.27) holds with 1/ = 1, r = 1, yielding rl(x,~) E SrI-I, provided f E C;. , The results (7.25)-(7.26) now follow from the observation that
(7.29)
f E Lipl
=}
oxF(x,~) E SP,I'
f E C;
=}
oxF(x,~) E Sr.I·
i.e., (6.36) holds for F(x,~) with m2(~) = O. If we assume w(t)C" ~ 0 for some 8 E (0,1), then (6.27) holds with 1/ = 0, and Proposition 6.1 also gives (7.19)
p(x, D)Tj
= b(x, D),
with b(x,~) =
(7.20) where
(7.21)
r(~)
p(x, ~)F(x,~) + ro(x, ~),
ro(x,O E Siyr,
is as in (6.27). Thus, by (6.30),
w(t)t-··'" for some SE(O,l)===}ro(x,';)ES~'r-1/!,
8. Operators with Sobolev coefficients Here we record some facts about a class of operators studied in [Mal] and [Ma5]. Given q E (1,00), r > 0, 0 E [0,1), we define Hr,qSf'/j to consist of , functions p(x,~) satisfying (8.1)
IDeP(x,~)1 ~ C",(~)"'-I"'I, IIDeP(',~)llw,q ~ C",(.;)"'+8 r -I£>I.
~.
- ' _.-
1.
OPERATt:rn:s Wll'H
Mtt:DLYREGULXlrSymroLs
-.
We will require
63
We mention that Theorem 2.2 of [Mal] also establishes Proposition 8.1 in the cases q = 1,00 and p = 1,00. In these cases we take
n (1 - 6")r > -.
(8.2)
9. OPERA'l'"CrR"s WITH Dous'iE"SYI->fBOLS
q
(8.13)
Hs,l(~n)
= (1- A)-s/2~) (~n),
Hs,oo(~n) = (1 - ~)-s/2bmo(~n).
The following result is contained in Theorem 2.2 of [Mal]. PROPOSITION 8.1. Let p(x,~) E HT,qS'r" , with 6" E [0,1), and assume (8.2) holds. Then p(x,D) ; Hs+m,p
(8.3)
--->
Here ~) (~n) is the localized Hardy space and bmo(~n) is the localized John Nirenberg space, introduced in §2. In particular, taking p = q = 00, one has, for 0 ~ 6" < 1,
HS'P, (8.14)
p(x,~) E
HT,oo Sro
===}
p(x, D) : Hs+m,oo -+ HS'oo,
-(1 - 6")r
< s < r.
provided
(8.4)
1 1)+ p+ q-
n(1
(1 - 6")r
<s
q- p1)+ .
~ r - n (1
One can compare the limiting case s = 1, r "" 1 with the following result, estab lished in Proposition 4.10: p(x,~) E LipSr"
(8.15)
p(x, D) : H m+1 ,oo -+ H)'oo,
We refer to [Mal] for a proof, but mention that the proof involves treating the case of an elementary symbol. We also mention [BR] for the result in the case p = q =: 2; their proof is also given in §2.1 of [T2]. For use in Chapter III, we record a proof of the following result on symbol smoothing.
given 6" E [0,1).
PROPOSITION 8.2. Assume that p(x,~) E HT,qS1'O! with r > n/q. Then the symbol smoothing operation given by (3.3)-(3.4), with 'Cj as in (3.9), yields
Here we consider symbols of the form a(x,y,~), which yield operators via the prescription
p(x,~)
(8.5)
= p#(x,~) + pb(x, ~),
===}
9. Operators with double symbols
(9.1)
Au(x)
= (27f)-n
JJ
a(x, y, ~)ei(x-y),f.u(y) dy d~.
with
(8.6)
p#(x,O E S~','",
pb(x,~) E HT,qS':',,,-a,,,
n
a=r--. q
We say a(x,y,~) E XS1'o , (and A E 0PXS1'o) , when
(9.2) PROOF. Parallel to the proof of Proposition 3.2 (or the proof of Propositions 1.3.D-1.3.E in [T2]) we use the following elementary inequalities: (8.7) (8.8)
(8.9)
IID~Jdllw,q ~ Ci3 C1i3l ll/llw,q
Jdllw,q III - Jdllu>o
III -
~ CII/llw,q, ~ Cacall/llw,q,
n a = r - -. q
The estimate (8.7) implies
(8.10)
IID~DeP#(-'~)lIw,q ~ CQi3(om-[QI+<5Ii3I,
which gives p#(x,~) E S1'", since r
(8.11)
> n/q. The estimate (8.8) implies
IIDfpb(-'~)IIHr,q ~ c(~)m-IQI
= C(o(m-a,,)+a"-IQI,
and the estimate (8.9) implies
(8.12)
IDfpb(X,~)1 ~ c(~)m-IQI-a",
and together (8.11)(8.12) yield pb(x,~) E HT,qsr;r ao .
IIDfa(.,., ~)llx ~ CQ(~)m-IQI,
where X is this time a Banach space of functions on (2.52), we say
(9.3)
a(x,y,~) E XSl:(I<)
~n X ~n.
IDfa(x,y,~)1 ~ CQ(~)m-lal,
{::::::;>
Similarly, as in
and
IIDfa(., ',~)llx ~ CQf\;(lw(~)m-IQI.
As in (3.18)-(3.19), we could replace m E ~ by a function m(O with appropriate behavior. As is well known, if a(x, y,~) is smooth and in one of Hormander's classes S;'o (with 0 ~ 6" < p ~ 1) then A can be rewritten as A = p(x, D), with p(x,~) E This reduction does not work for symbols with limited smoothness, and these operators require a separate (though somewhat parallel) treatment. We begin with some LP estimates, which are variants of results in [Ma4].
S;,,,.
PROPOSITION 9.1. If E w(2- k )
(9.4)
a(x, y,~) E clwJsf,o
===}
< 00,
then
A : £p(~n)
-+
£p(~n),
1
< P < 00.
~~~.:: - _.._ -
';""
-=-:::::::
--'~~:~~-~.I;~~~~rri-r=~:I:iii-i''''·--~~~~~-:-'';:''~_'
--'-'!!i".-
~-:-:~~t:':~'''·''--f-··
PROOF. As in §2, it suffices to consider an "elementary" double symbol:
-'----'-"::' -~-.,
9.
DOUBLE SYMBOLS
To estimate Q2, use (9.7) to write (Xl
00
q(x,y,~) = Lqk(X,Y)
(9.5)
IQ2 U(X)j :s:
(9.14)
k=O with {rpd bounded in Sr and supported on shells (0 '" 2k (say on 2k 2k +2 , for k 2": 1), and with Ilqkllc[wj :s: c. Make the decomposition
(9.6)
qk(X, y)
= qkl(X, y) + qk2(X,y),
qk1(x,y)
2
:s:
I~I
:s:
= Wk-4(D x,y)qk(X, y),
with Wk as in §2 (but acting on functions of both x and y, so Wk E C8"(lR 2n )). Thus
>'(k) = w(T k ).
/lqdlLoo :s: C >'(k) ,
(9.7)
Let Q be the operator associated to q(x, y, ~), and denote the operator decomposi tion resulting from (9.6) by Q = Ql + Q2. To estimate Ql, write
(9.8)
QI U (X) =
fl k=O
We have
[/k(() = C
qkl(X,y)rJ;k(X-y)u(y)dy= f9k(X). k=O l7)
~)u(( - ~ -17) d~ dl7.
gk(X)
=
I
qkl(X, y)rJ;k(X - Y)Uk(Y) dy,
as long as
E >'(k) < 00.
Note that if a(x, y,~) E C[w]S~,(I<) is decomposed into elementary double sym bols of the form (9.5), we have the decomposition (9.6) with
IIqk211Loo :s: C>'(k) ~(2k).
(9.16)
Thus the proof of Proposition 9.1 extends to yield: PROPOSITION 9.2. If Ew(2-k)~(2k) <
a(x, y,~) E C[wl S~,(,,)
jgk(x)1
Uk
=
k+3 L 'l/J£(D)u, £=k-3
:s: IlqklllLoo ·!rJ;kl * IUk[(X) :s: C MUk(X),
where MUk is the Hardy-Littlewood maximal function of Uk. Due to the support property of 9k, we have
2 2 IIQI Ulb :s: cil (L 19kI f/ 1ILP'
(9.11)
k
We can estimate the right side of (9.11) using (9.10), together with the following vector maximal estimate, established in [F81]; see also [RRT] or [83] for a proof:
II (L IM fkl 2 f/
(9.12)
k
for 1 < P <
(9.13)
00.
2
tp :s: Cpll (L likI 2f/2tp' k
In concert with the previous estimates, this yields
IIQIU/ILP :s: cil (L IUk[2f/2tp :s: ClluliLP. k
==}
A :
then
00,
P(lR
n
) ----t
U(lR n ),
1
< P < 00.
Note that the conditions on w(2- k ) in Propositions 9.1-9.2 are more stringent than those in Propositions 2.5 and 2.7. This is necessary in view of the following example, given in [Ma4]. Consider
q(X, y,~) = L >'(k) ei2k (x-y) 'l/J(2-k~),
(9.18)
k2:0
and also that 9k (~) is supported on a shell (~) '" 2k . We have
(9.10)
lrJ;kl * lul(x).
/lQ2 uIILP :s: CllujlLP,
(9.15)
Now the restriction on the support of qk1 implies (9.9)
L >'(k) k=O
This implies the desired estimate
(9.17)
I qk1(~,
65
where 'l/J E Ctf(lR) is supported on 1/2 < I~I < 2 and 'l/J(O = 1 for 3/4 < I~I < 3/2. One sees that q(x,y,~) E C[w1Sr,o, with w(2- k ) = >'(k). Now, if u is supported on 3/4 < I~I < 3/2, then Qu = E >'(k)u. Hence E >'(k) < 00 is necessary for even very weak boundedness of Q. We next establish some special properties of operators produced by double symbols that vanish at x = y. Let C 1 ,[w} consist of C 1 functions whose first-order derivatives belong to C[wl. PROPOSITION 9.3. Assume Ew(2- k ) < 00. Then (9.19) a(x, y,~) E Cl,(w1si,o, a(x, x,~) = 0 ==} A : LP(lR n ) ----t
P(lRfl ),
1
< P < 00.
PROOF. Write
(9.20)
a(x,y,~)
=
Lbj(x,y,~)(Xj - Yj),
bj(x,y,~) E
c[w1si,o'
j
Then writing (Xj - Yj )ei(x-y),~
(9.21)
= -iO~j ei(x-y),~
Au(x) = (27r)-fli L
and integrating by parts yields
II :~~ (x,y,Oei(x-y),~u(y) dyd~.
J
Since 8~j bJ (x, y,~) E C[w]S~,o, the conclusion follows from Proposition 9.1.
.....~ -
rrPERATt.m:S'-wrdrMILD'r,v'REffirT;AR SYMBOLS-
m>~~
PROPOSITION 9.4. Assume (9.22) a(x,y,~) E cl,(wJsf,o,
L w(2- k ) < 00.
a(x,x,O = 0
PROOF. We have
8 j Au(x) = (21r)-ni (9.23)
==}
ff 11
Then
A: LP(lR n )
->
H 1 ,P(lRn ),
1
< P < 00.
a(x,y, ~)~j ei(x-YHu(y) dyd~
+ (21r)-n
8 x]a(x, y, ~)ei(X-Y)'~U(Y)
dyd~.
1
s:: s < r < 1. Then
a(x, y, ~), V' ya(x, y,~) E C Tsi,o s n n S ==} A : Hl+ ,P(lR ) -> H ,P(lR ),
(9.29)
1
00.
PROOF. We may as well treat a(x, y,~) = b(x, y, O~j, with b(x, y, ~), V' yb(x, y, Using ~jei(x-y),~ = i8Yjei(X-Y)'~ and integrating by parts, we have
o E CTSf,o'
Au(x) = - (21r)-n
a(x, x,~) = 0 = } A : LP(lR n ) -> H S,P(lR n ),
67
PROOF. If a(x,y,~) satisfies the hypothesis of (9.28), then we know by Corol lary 4.4 that p(x,~) = a(x, x,~) E CT Sf,o satisfies p(x, D) : HS,P -> HS'P for the range of sand p considered. Meanwhile b(x,y,O = a(x,y,~) - p(x,O satisfies the hypotheses of (9.24), so this yields an operator satisfying (9.25). This proves Proposition 9.6.
s:: s < r < 1. Then
PROPOSITION 9.5. Assume 0 (9.24)
9. OPERATORS WITH DOUBLE SYMBOLS
PROPOSITION 9.7. Assume that 0
Proposition 9,3 applies to the first term on the right and Proposition 9.1 applies to the last term.
a(x, y,~) E C TSP,o,
,~
-
00.
- (21r)-n
11 ib(x,y,~)ei(x-YH8ju(y)dyd~ 11 i(8Yjb)(x,y,~)ei(X-YHu(y)dyd(
Moreove'f, under the hypotheses in (9.24), (9.25)
A: H a,p(lRn )
---;
H a+s,p(lR n ),
PROOF. With A = (1- ~x (9.26)
1 < p < 00, -r <
(J
s:: O.
If a(x, y,~) E Cl+T Sro, then Proposition 9.7 applies to A and A * , so we have, . for 1 < P < 00,
~y)1/2, set
-
bAx, y,~)
=
(9.30)
A -(z-s)a(x, y, ~),
fl = {z
E C :0
s:: Re z s::
(9.31 )
I}.
Then set az(x,y,~)
= bz(x,y,O -
===?
A: HHS,P(lR n ) -> H S,P(lRn ),
A*u(x) = (21r)-n
11
a(y,x, _~)*ei(x-YHu(y)dy~.
Using Proposition 9.5 we can obtain further results for symbols that do not
vanish at x = y.
PROPOSITION 9.6. Assume 0 < r < 1 and -r < s < r. Then
a(x, y, 0 E C TS?o
==?
A: H S,P(lRn ) -> H S,P(lRn ),
a(x,y,~) E
C:+aS[,o
s:: s < r < 1 and 0 s::
==}
(J
s::
-1- r
< s < r.
1. Then
A: H a+s ,p(lRn ) -> H S,P(lRn ),
1
< p < 00.
PROOF. By duality, it suffices to show that, when a(x, y,~) satisfies the hy pothesis in (9.31), then
bz(x,x,O·
Let A z denote the associated family of operators. Now, given u E LP(lR n ), Propo sition 9.1 implies Azu E LP(lR n ), for all z E fl, with an exponential bound as IImzl -> 00. Also, Proposition 9.4 implies Azu E H 1 ,P(lR n ) whenever Rez = 1, again with an exponential bound. By complex interpolation, Asu E HS,P(lR n ). Since a(x, x,~) = 0 implies As = A, we have the desired result in (9.24). The result in (9.25) follows by duality and interpolation. The duality argument works since
(9.28)
a(x,y,O E cl+Tsi,o
PROPOSITION 9.8. Assume 0
for z in
(9.27)
The result now follows from Proposition 9.6.
1
< p < 00.
(9.32)
A: H- s,P(lR n ) ---; H- a- s,p(lRn ),
As in the proof of Proposition 9.5, set A = (1- ~x
1 -
< p < 00,
~y)1/2. For each 6
> 0, set
az,.s(x, y, 0 = A -(z-a)a(x, y,~) (oz-a e-61~12 on fl = {z E C : 0 s:: Re z s:: I}, and denote by Az,.s the associated family of operators. If Rez = (, we have az,.s(x,y,~) in C:+(S~go, for each 6> 0, and in C: H sf,o, uniformly in 6 E (0,1]. Take u E H-S,P(lR n ). Then, for each 6> 0, Az,.su belongs to H-S,P(lR n ), for all z E fl, with an exponential bound as Izi -> 00 (as a very special case of Proposition 9.6). Furthermore, we have (by (9.28) and (9.30))
A·ZY,u 00. From here a use of the Hadamard three lines lemma and complex interpolation theory yields Ad,.su E H-s-a,p(lR n ), uniformly in 0 E (0,1], and taking 0 -> 0 gives (9.32).
MILDLY REGULAR,
1.
We next derive some estimates on operators associated with double symbols of negative order. PROPOSITION 9.9. Assume LW(2- k )"'(2 k ) < (9.33)
a(x,y,E)
E
c[wJSl,tl<) ,
00.
'Vxa(x,y,E)
E
OPERATORS WITH DOUBLE SYMBOLS
9.
If C[w]S~,(I<)'
69
where /-l varies over 0 ::; /-l ::; k - 4 for Q = 1, over k - 3 ::; /-l ::; k + 3 for Q = 2, and over /-l ~ k + 4 for Q = 3, and v varies over similar ranges for (3 = 1,2,3, respectively. We then need to estimate (9.40)
fJ
QaI3U(x) =
qkal3(X, Y)(Pk(x - y)u(y) dy
= Lgkal3(x),
k=O
then A: LP(lRTI.)
(9.34)
~
H1,P(lRTI.),
1
00.
PROOF. Look again at the formula (9.23) for ajAu. The two amplitudes on the right belong to C[wJS~,(I<) under the hypotheses (9.33), so the boundedness of ajA on LP(lRTI.) follows from Proposition 9.2.
k
in these 9 cases, by examining supports of Fourier transforms of various functions. For example, for (a, (3) = (1,1), we see that we can replace u by Uk (as in (9.9)), and that 9kll(E) is supported on (E) '" 2k . Hence IIgkllllu'" ::; C1lqkllllu",lIukllu"'. Thus, (9.41)
II'ljJf(D)QllUllux> ::; C(L IlqkllIILc>o>'(k)) Ilullc(),)· kROf
Note that Proposition 9.9 applies when
a(x, y, E) E
(9.35)
cl+ r Sl~' ,
We have the same bounds on the supports of 9k12 and u by Uk. Hence we have estimates
r > 0,
9k13, but we cannot replace
where, as is natural, we say
a(x,y,E)
E
(9.42)
C r Sf,'8 ~ IDta(x,y,E)I::; Ca(E)m-1a l , and IIDta(.,', E)llc r ::; Ca (E)m- 1a l+r8.
By duality and interpolation we have, for r > 0, r5 E [0,1), (9.36)
a(x, y, E)
E
cl+ r S~~
provided -1 ::; s ::; 0, 1 < p < to obtain:
00.
==}
on
We can apply interpolation in a different way,
a(x, y, E)
E
c r Sl,~-s8 =? A : LP(lRTI.) _
HS,P(lRTI.),
1
(9.43)
We next study the action of operators of the form (9.1) on the spaces C P.). Again we can consider symbols of the form (9.5). We begin by splitting the associ ated operator Q into 9 pieces: Q = L~,I3=l Qa{3, by splitting qk(X, y) into 9 pieces:
qkaO(X, y)
=
L 'ljJp(Dx)'ljJv(Dy)qk(X, y), J.l,LJ
11'ljJJ'(D)Q21 UIILC>O::;
C( L
Ilqk21IILoc>.(k))llullc(),).
k~f-3
11'ljJJ'(D)Q2I3ullux> ::;C( L
~y)l/2. Then
on 11 = {z E C : 0 ::; Re z ::; I}, and denote by A z the associated family of operators. Given u E V(lRTI.), we have Azu E LP(lR"), for all z E 11, with an exponential bound, by Proposition 9.2. Also we have Azu E Hl,p(lRTI.), for Rez = 1 (with an exponential bound) by Proposition 9.9. Interpolation gives Au = Asu E HS,P(lRTI.).
(9.39)
9k21 is supported
00.
az(x,y,E) = A-(z-s)a(x,y,E) (E)-(z-s)
qk = L~,I3=l qkal3' Take
2,3.
We have the same bounds on the supports of 9k22 and 9k23, but we cannot replace u by Uk. Hence we have the estimates
set (9.38)
=
For (a, (3) = (2,1), we can again replace u by Uk; this time Lk~f-3gk21; hence
(9.44) PROOF. As in the proof of Proposition 9.5, let A = (1 - ~x
{3
lEI::; 2k+3, so 'ljJf(D)Q21U =
A : HS,P(lRTI.) _ Hl+s,P(lRTI.),
PROPOSITION 9.10. Assume 0 ::; s < r < 1 and 0::; r5 < 1. Then (9.37)
II'ljJf(D)QlI3uIILC>O ::; C(L Ilqkll3l1LC>O ) IlullLoo, kROf
IIqk2131ILc>o)lluIILC>O,
(3=2,3.
k~f-3
To analyze Q313u, we make a finer decomposition, Q313 ting qk313 into pieces: (9.45)
qk313(x, y)
=
L 'ljJk+v (D x )qk313 (x, y) v~O
=
=
Lv>o QV 313, by split
L qV k313(x, y). v~O
Then (9.46)
Q3I3u(x) = L
J
qV k313(x, y)'-Pk(X - y)u(y) dy = L
k,lI~O
k,lI~O
In all cases, 9v k313 has support on (E) '" 2v+k , and (9.47)
'ljJf(D)Q3I3u =
L g" k313'
"+kROf
gV k313(x).
uta
i#ww,,·~
it
70
i i i S.
~.
..~.
~£j=
OPERATORS WITH MILDLY REGULAR
1.
If f3 = 1, we can replace u by ih, so
i.~oiiiii!iZZt=.iil~?£Paiimc
IIg.. . k311IvX> -s; CllqVk31\\Loo\luk\ILoc,
and
L Ilq.. . k3111Loc Ilukllu'" ::; C( L: II¢R(Dx)qkllu>oA(k))lluIIC(A).
11~£(D)Q31uIlLoc ::; C
.... +k:od
(9.48)
£+3 Lj=£-3 ~j' For f3
=
=
L:
11¢£(Dx)~k(Dy)qkIlLoc) IlullLoc,
k~£+3
11~£(D)Q33UIIL'X> ::; C(
L kS£+3
11~£(Dx)(I -
Wk-4(Dy»)qkIILoc) IlulluX>.
A(k) <
00.
that A(k)/Jl(k) /" and that
IDf1/J£(DxHI - WdDy»a(x,y,OI s; CaA(k)Jl(£)(~)-lal, Then A: CC>') (lR. n ) _ CC>') (I. n ).
(9.59)
V k,£.
/Df(I - Wk(Dx,y»a(x,y,~)1 S CaA(k)Jl(k)(~)-lal,
(9.60)
i.e., provided
a(x, y, 0 E c[als~.o, a(2- k ) = A(k)Jl(k).
Considering the special case Jl(k) = 2- r'k , A(k) = 2- sk , we have, for r > 0,
(9.62) a(x,y,O E c;rs?,o ===? A: C:(lR n ) - C:(lR. n ), 0 S s Sr. (9.61 )
PROPOSITION 9.11. Let w be a modulus of continuity, and A(k)
L
71
Note that (9.59) holds provided
Using the estimates (9.41)-(9.50), we obtain:
Assume
00,
satisfies
and (9.50)
_
OPERATORS WITH DOUBLE SYMBOLS
Regarding the conditions (9.51)-(9.54), note that (9.53) is actually a conse quence of (9.54). Furthermore, when" = 1 and LJl(k) < 00, (9.51) and (9.52) are also consequences of (9.54), if it is strengthened to hold for all k and £. Thus we have the following special case of Proposition 9.11. a(x,y,~)
2 or 3, we cannot replace u by Uk. We have
111/J£(D)Q32 U IILoc ::; C(
(9.49)
n
9.
COROLLARY 9.12. Assume that LJl(k) <
k~H3
Here, ~£
__.
SYMBOLS~,
=
w(2- k ).
Assume a(x, y,~) satisfies the following conditions:
(9.51)
IDfa(x, y,OI S; Ca(~)-Ia\,
(9.52)
IDt(I - wk(Dy))a(x,y,OI S CaA(k) ~~(j) (o-\a l ,
We will sharpen these results below. We next examine the behavior of operators of the form (9.1) on various Zyg mund spaces, when a(x, x,~) = 0, paralleling Propositions 9.3-9.5.
PROPOSITION 9.13. Assume r > O. Then, for 0 S s::; r,
IDe~£(Dx)a(x, y,~)1 S; CaJl(£),,(~)(()+>I,
(9.53) (9.54) (9.55)
IDe~£(Dx)(I - Wk(D.y))a(x,Y,()I::; CaA(k)Jl(£),,(~)(~)-lal,
L:
,,(2 k )Jl(k)A(k)::; CA(£),
k~R-3
L
(9.63)
k S £+4,
,,(2 k)A(k)Jl(£) S CA(£).
k:'O£+3
A: C(>,) (IR n ) ~ CC).) (IR n ).
(9.56)
PROOF. Note that (9.51)-(9.53) (supplemented by (9.55)) apply to (9.41) (9.43), respectively, that (9.54) applies to (9.44), (9.49), and (9.50), and that (9.55) applies to (9.48). Note that, when a(x, y, 0 is independent of y, this result reduces to Proposition 5.2. As noted there, the condition (9.55) holds provided
L ,,(2
(9.57)
In particular, if (9.58)
K, =
k
)Jl(k) <
00,
A(k) and Jl(k) /' .
1, then (9.55) holds provided
2: Jl(k) <
E
c;+2r'Si,o, a(x, x,~)
=
0 =;. A : C; (R. n ) _ C: (R.").
PROOF. We have (9.20)-(9.21) with o(;bj(x, y,~) E C2r'S~.o, so the conclusion in (9.63) follows from (9.62). PROPOSITION 9.14. Ifr> 0, then, for 0 S s::; r, (9.64)
Then
a(x, y,~)
a(x, y,~) E C;+2r SP,o, a(x, x,~) = 0 =* A : C:(JR") _ C; I S(R. n ).
PROOF. As in the proof of Proposition 9.4, to analyze ojA we use (9.23). We see that Proposition 9.13 applies to the first term on the right and (9.62) applies to the last term. PROPOSITION 9.15. If 0 < a < 1, then, for all s < a, (9.65)
a(x,y,O E C~S?,o, a(x,x,O
=
0 =* A: C~(lR.n) _ C:(l. n ).
PROOF. Assume s E (0, a), and set a = s+2r, r> O. Define A z as in the proof of Proposition 9.5, via (9.26)-(9.27). Take u E C~(lRn). Then (9.62) implies Azu E n C2(lR ) for all zED, and Proposition 9.14 implies Azu E C;(lR n ) when Re z = 1 (with exponential bounds). By complex interpolation, Au = Asu E C:(lR n ), as desired.
A(k)
00,
and Jl(k) /' .
We can now improve (9.62), at least for r E (0,1):
72
L
OPERATORS WITH MILDLY REGULAR SYMBOLS
9.
a(x, y,~) E CrS?,o
===}
A: C:(lRn )
-+
C:(Rn ),
0::::: s < r.
PROOF. As in the proof of Proposition 9,6, consider p(x,~) = a(x, x,~) E cr Sr.o. We have p(x, D) : C: -+ C: for s E [0, r), by (5.24), plus a related (well known) argument for s = 0, or an appeal to (9.62). Furthermore, Proposition 9.15 applies to b(x, y,~) = a(x, y,~) - p(x, ~). This proves (9.66). Having this improvement of (9.62), we can sharpen Propositions 9.13-9.14 slightly, replacing C;+2r by c;+r in (9.63) and (9.64), at least for r E (0,1). We omit details on sharpening these results for larger values of r. It is worth noting that we can use Proposition 9.5 to extend Proposition 9.16, as follows. PROPOSITION 9.17. IfO < r < 1, then a(x,y,~) E CrS?,o
(9.67)
===}
73
and Lk;>-£-3 ,(2 k )K(2 k )tL(k)>'(k) :::: C>'z(£), i.e.,
PROPOSITION 9.16. IfO < r < 1, then
(9.66)
OPERATORS WITH DOUBLE SYMBOLS
A: C:(JRn )
-+
C:(lR n ),
-r
< s < r.
L:
(9.73)
k
K(2 )tL(k)>'2(k):::: C>'z(£).
k?-£-3
Furthermore, Lk?-£_3I1qk2,i3IILoo :::: C>'z(£) for j3
=
2,3, provided
IDf'l/J£(Dx)(l - wk(Dy))q(x,y,OI::::: Co:>'(k)tL(£h(~)K(~)(~)-lo:l
(9.74)
and (9.73) holds. Examining (9.48)-(9.50), we see that Lk'(k) :::: C>'z(£) provided
IDf'l/J£(Dx)q(x, y,~)1 :::: CtL(£h(~)K(~)(~)-lo:l
(9.75)
and Lk:SH3,(2 k )K(2 k )>'(k)tL(£) ::::: C>'z(£), Le.,
L
(9.76)
K(2 k )>'z(k)tL(£)::::: C>'z(£).
k:S£+3
PROOF. Make the decomposition a(x,y,~) = p(x,~) + b(x,y,~) used above. We actually have p(x, D) : C: -+ C: for -r < s < r (d. Proposition 2.1.E of [T2]). Meanwhile, Proposition 9.5 applies to the operator B associated with b(x, y, ~); in particular (9.25) holds for B, and this in turn implies B : C: -+ C; for -r < s < r.
Finally, we have Lk:SH311~£(Dx)~k(Dy)qkIILoo ::::: C>'z(£) and a similar estimate with ~k(Dy) replaced by 1- Wk-4(D y ), provided
In fact, it is useful to record the following improvement of Proposition 9.15, whose proof follows from the argument just presented.
and (9.76) holds. Note that (9.72), (9.74), (9.75), and (9.77) are implied by
PROPOSITION 9.18. IfO < a < 1 then, given r E (-a,O], s E (O,a),
(9.68)
a(x,y,~) E C~S?o, a(x,x,~) =
°
===}
A: C:(JR n )
-+
C:+s(lR n ).
We return to the level of generality of Proposition 9.11, and consider conditions on a(x, y,~) that imply A : C(>,) -+ C(>·2). For this, the estimates (9.41)-(9.44) and (9.48)-(9.50) are still effective. This time we want to estimate II~h(D)Qo:,i3ullu>a by >'z(£). Let us set (9.69)
,(~)
>'z (logz (~) ) =
>'(logz(~»)
.
Ilqkllllu>a >'(k)::::: C>,z(k)..;:= IDfq(x,y,~)I::::: Co:,(~)(~)-lo:l,
(9.70) =
If K =
K,
this also implies (9.71). We have the following result:
PROPOSITION 9.19. Given>. and >'z (decreasing and slowly varying) we have A .• C(.X) -+ C(>·2) provided a(x , y ,<"C) E C[O'JS-,p with 1,(K)
(~)-'iJ(~)
(9.79)
=
,(~)
=
>'z(logz(~»)
>'(logz (~») ,
a(T k ) = >'(k)tL(k),
(9.80)
(9.81)
L K(2 k )tL(k)>'z(k) :::: C>'z(£), L K(2 k )>'z(k)tL(£) ::::: C>'z(£). k?-£
2,3,
Ilqkl,i3IILoo ::::: C>'z(k) (9.71) {=
>'(k) IDf(I - wk(Dy))q(x, y, ~)I ::::: Co: K(2 k ) ,(OK(~)(O-lo:l.
Next, Lk;>-£_31Iqk21I!U>O >'(k) :::: C>'z(£) provided
(9.72)
IDf(I - wk(Dx,y))q(x, y,~)1 :::: Co:tL(k)>'(kh(~)K(~)(~)-lo:l.
(9.78)
and
We see that
while, for j3
IDf'l/J£(Dx)(l - wk(Dy))q(x, y, ~)I ::::: Co:>'(k)tL(£h(~)K(~)(~)-lo:l
(9.77)
ID{'1P£(D r )q(x, y,~)1 :::: Co:tL(£h(~)K(~)(~)-lo:l
k:S£
In particular (9.80)-(9.81) hold if (9.82)
a(T k ) = >'(k)>'z(k), and
L K(2 k)>'z(k) <
00.
As in §3, it is useful to apply symbol smoothing to double symbols with limited smoothness. Thus, if a(x, y, 0 E X Si,(K) , we can write, parallel to (3.3)-(3.4), (9.83)
a(x, y,~) = a# (x, y,~) + ab(x, y, 0,
-_...
,
74
_---------------~-~"'"."
1. OPERATORS WITH MILDLY REGULAR SYMBOLS
.._".
,,_.--'--"'''-'''''"''''~'-'-'"'~'-'--'''-'~iO:~'THECRW~COMMUTATOR
for some 6
with (9.84)
a#(x, y,£,) =
L 'l1
0
(2- j6 Dx,y)a(x, y,£,) 1fj(£,)'
E (0, 1).
Then a(x, y, £,) E C[w] S?,o
(9.91)
-ESTiMATE·-'--"--_·"'''·-·'-~''~-
===}
A : CC>.)
---+ CC>.).
j
PROOF. Replace A by J-L and A2 by A in the result of Proposition 9.20 about A b. We have A b : CC/1) ---+ CC>.). Hence Ab : CC>.) ---+ CC>.), if A(k) :::; J-L(k). The result for A# is clear.
Pick 6 E (0,1). As in (3.10), as long as X c Loo, we have (9.85)
a#(x,y,£,)
E
S'i,o'
We also have estimates on ab(x, y,~) parallel to those on pb(x,~) derived in §3. For example (d. (3.27)) if 0 < r < 1, a(x,y,~) E crSro
(9.86) More generally,
8.<;
in (3.58)-(3.59),
a(x "<,, y C)
(9.87)
ab(x,y,~) E c r S'!:i r6 .
===}
E
C[w]S'"1,0
===}
ab(x "<,, y C)
E
C[w]S",-lj; I,Cr)'
with (O-'fiC';) = B(I~J)w((~)-6),
(9.88)
1
T(~) -
n I l .. "
I ' .. ,
-0)'
and a variety of choices of B(I~I) 2: 1. Let the operator decomposition associated with (9.83) be denoted A = A# + A b. Note that, if a# satisfies (9.85), then (9.89)
A# = b(x, D)
E
OPS'i,6'
PROPOSITION 9.22. Let w, A, and A2 be as in Proposition 9.20. Also assume A2(k)jA(k) 2: C2- CI -"Y)k for some "y E (6,1). Then
(9.92)
a(x, y,~) E C[w]S?,o, a(x, x,~) = 0
w(2- k ) = A(k)A2(k) and ET(2 k )A2(k) < 00. Since (9.88) gives T(2 k ) = A(k)jA2(k), this is equivalent to E A(k) < 00. Let us summarize: 9.20. Assume that A2(k) :::; A(k) and that (9.90) is 2: 1. We CC>'2) provided ab(x, y,~) E C[wJS~t) with w(2- k ) = A(k)A2(k),
have Ab : CC>.) ---+ (~)-'fi given by (9.79), T(~)
=
(~)'fi, and EA(k)
Using this we can establish that A : CC>.) (usually) weaker than (9.61). PROPOSITION 00.
< 00. ---+ CC>.)
under a condition that is
9.21. Assume that w(2- k ) = J-L(k)A(k) with J-L(k) 2: A(k) and
Also assume that
~01
1
>1
A(6k) J-L(k)J-L(r5k) -
in addition to a#(x,y,~) E SP,6' In particular p#(x,~) = a#(x,x,~) satisfies
IDE D.~p#(x, ~)I :::; Ca6(~)-lal+6ll3l.
Interpolation with (9.93) gives p#(x,~) E Sl,~ for all "Y E (6,1). Hence p#(x, D) : CC>.) ---+ CC>'2). Finally, a#(x,y,~) - p#(x,~) = b#(x,y,£,) belongs to Sl,~I-"Y), so also B# : CC>.) ---+ CC>'2).
In this section we establish the following estimate of [CRW]: THEOREM
(10.1)
10.1. Given P
E
OPS?,o, 1 < p <
Ilf(Pu) - P(ju)IILP :::;
00,
we have
CpllfllbmollullLP'
More generally, this holds for P E OPS?,6' 0:::; 6 < 1, and also for P E OPBSP,I'
Since [CRW], there have been a number of proofs of this result. For example, we mention one in [Jo]. The proof we give is taken from [AT]. It exploits the representation of a product in terms of the paraproduct,
fu
(10.2)
=
Tfu + T,,,!
+ R(f, u),
used here already in several places, such as (3.47) and (8.6). We also make use of the following estimates:
(10.3) ,
---+ CC>'2).
10. The CRW commutator estimate
The necessity to have B(I~J) 2: 1 puts a mild restriction on the relation between A and A2. If ab(x, y,~) E C[w]S~:t), we see that Ab : CC>.) ---+ CC>'2) provided
PROPOSITION
A: CC>.)
IDEa#(x,x,~)I:::;Ca(~)-'fi-I(>I,
(9.93)
B(2k) _ A2(k) _ 1 - A2(6k) A(k)A(6k)
(990) .
===}
PROOF. Use the symbol smoothing (9.84) and associated decomposition A = A#+A b. We have ab(x, y,~) E C[w]S~t), and Proposition 9.20 implies Ab : CC>.) ---+ CC>'2). Also a(x, x,~) = 0 ==> a#(x, x, £,) = -ab(x, x, ~), so we have
(9.94)
Consequently this piece is very well behaved. We can use Proposition 9.19 to estimate A b. Consider the case
EJ-L(k) <
Note that (9.61) looks similar to the hypothesis above, but there we require J-L(k) :::; A(k). We can also use Proposition 9.20 to estimate A when a(x, x,~) = O.
IITf91lLP :::;
CpllfllLPllgllbmo,
IIR(f,g)IILP:::;
CpllflhJlllollgllLP.
These estimates are due to [CM]; proofs can also be found in [T2].
-ro-------------r:-uPEl{m'ORH~LY
REGVLAICSYMBULg----------------------
The space bmo(JRn) was introduced in §2. We recall that it has norm given by
Ilfllbmo = IIfllBMo + Ilwo(D)fIILoo, n
where 'lfo E CO'(JR ) satisfies wo(O) space, with semi-norm given by
=1=
s~p
IlfllBMO =
0, and BMO is the classical John-Nirenberg
V!Q)
JIf - fol
(10.5)
IITpufllLP
==}
TfP = Ql E OPSf,l,
= F(x,D), then Fj(x,~) = 8F(x,OI8xj
"i
E
> 0,
"i
E
> O.
is specified by
Fj(x, D) = Talf,
f E C~(JRn) ==? \l xF(x, {) E B 1/ 16 Si,I'
(10.16)
(10.17)
and P E OPBp SP,1 (with P + 1/8
[Tf'P] E OPBSP,I'
I: 8fP(x,{)· 8';F(x,{),
Q2(X,{) = P(x,~)F(x,~) - i
< 1),
The proof of Proposition 7.3 can be applied to this case, but we will give the proof from [AT], which is slightly different. To begin the proof, we note that, for r > 0,
mod SP,I'
10:1=1 and the sum over lal = 1 here also belongs to sp l' Together, (10.13) and (10.17) establish that [Tf, P] E OPSP,I' and invoking Pr~position 6.2 we can replace SP,1 by BSP, 1 , proving Proposition 10.2. As a consequence of Theorem 10.1, we now establish the following "super commutator" estimate, which will be convenient for applications in Chapter III. Let f be an i-form, and set Wfu = f 1\ u, and
n
f E c;r(JR ) ==? T f E OPB1/ 16 Sl,I'
(10.18)
and hence f E C~(JRn) ==? Tf E OPB1/ 16 Sf,l>
"i
E
> O.
Now, for any F(x,D) E OPSf\, P(x,D) E OPBStl' the product Ql(x,D) F(x, D)P(x, D) belongs to OPS';,:/-" and we have a s~mbol expansion (10.10)
O. Applying
Consequently, again by Proposition 6.1, (10.14) holds with
then
(10.9)
=
so, by (10.8),
In turn, this is a consequence of the following:
(10.8)
1/
P E OPBSP,I' f E C~(JRn) ==? PTf = Q2 E OPSi,I'
(10.15)
II[Tf,P]uIILP:S Cpllfllc~lluIILP'
(10.7)
~ 1,
Also by Proposition 6.1,
Note that, ifTf
E C~(JRn)
lal
Ql(X,O = F(x,~)P(x,~) mod SP,I'
(10.13)
< P < 00, we have
+ IIR(j, Pu)IILP + IIPTufllLP + IIPR(j, u)liLp
10.2. If f
for
and
(10.14)
Thus it remains to estimate [Tf, P]. The desired estimate is a consequence of the following stronger estimate:
PROPOSITION
P E OPBSP, 1 , f E C~(JRn)
+ TPuf + R(j,Pu), + PR(j, u).
:S CpllfllbmollullLP.
(10.6)
ID~DfF(x,{)1 :S Co:iJ({)-lo:l+liJl,
(10.11)
(10.12)
P(ju) = PTfu + PTuf
As long as P: V(JR n ) -+ LP(JRn) for 1
with estimates on Rv(x,{) given by Proposition 6.1. Applying Proposition 6.4 to Tf = F(x,{), we have
dx,
the sup taken over all n-dimensional cubes Q in JRn, where V(Q) is the volume of Q and fQ is the mean value of f over Q. Actually [CRW] established (10.1) with BMO in place of bmo, but the result as stated here is equally effective for local analysis. Our first step in proving (10.1) is to write fPu = TfPu
----------------77--
····-----------iii~-TjIE-nRwc-O-M~ltJTAT6-R-ESff~,lATE---------
so we can use this in hypothesis (6.25) of Proposition 6.1., with this Proposition (with /l-2 = m2 = r = 0), we deduce that
Q
(10.4)
---------- . -----------
Ql(X,O =
i-!al L -, 8fF(x,{). 8';P(x,{) + Rv(x,{), I()!~v
Q.
[[A- 1d, Wf]]
=
[A -ld, Wf]
if £ is even,
{A -ld, Wf}
if £ is odd,
where [A, B] = AB - BA and {A, B} = AB + BA. Here, d is the exterior derivative and A = (I - ~) 1/2. There is the following estimate. PROPOSITION
(10.19)
10.3. For 1 < p <
00,
IIl[A -ld, Wf J]J3IILP
we have ::::;
CpllfllbmollJ3IILP.
... ~...
·l.TIPERNrURsWI'l'H MILDLY REGUr;AK SVMBC>I;lj-··
Write Wf = Then
PROOF.
Ilfllbmo' (10.20)
L:: M f ,We,
where ei are smooth £-forms and L::
[[A-1d,W:f]]P= I:[A-1d,M.rJWe,p+
11. OPERATORS WITH VMOCOEFFICIENTS
Ilfillbmo '"
[[A -ld, We,]]
E
p(X,~) = I:Pj(X,~). j?O
Now the estimate (10.1) applies to the first sum on the right. Since the principal symbol of A-1d is wedge by il~\-l~, we have (10.21)
so we have
(11.9)
I: Mf,[[A-1d,WeJJP.
The operator Po(x, D) has a simple analysis. One can write
Po(X,O = I:Pe(x)eie'Eip(~/2),
(11.10)
e
OPSl,J,
with
so the estimate on the second term on the right side of (10.20) is elementary.
IIPellbmo :::; C'tv(£)-N.
(11.11) 11. Operators with
VillO
coefficients
Thus
Consider a symbol
(11.12) p(x,~) E
(11.1)
I: Pe(x)"l/Je(D)u = I:Pe(xh~e * u,
where
IIDEP(-'~)llbrno
(11.2)
(11.13)
:::; Ca(~)m-Ial.
p(x,r~)
(11.3)
= rmp(x,~),
r:::: 1, I~I:::: 1.
p(x,~) E
(11.4)
bmoSci.
(11.15)
(L 00 n vmo) Sci,
L 00 Sci,
J
ip(~) = 1 for I~I
:::;
1/2, and Po(x,~) is supported on I~I
(11.16)
[B,pj(.7:, D)]u
IlfJllbrno :::; CN(j)-N.
Now write
=
IJ (x )Wj (I~I ) 1~lm (1- ip(~)) = fJ (x)ajm (~),
LP(lR n ) n bmo(lR n ),
'if P E [1,00).
= b(x,D)
E
OPSP.J, 0:::; J < 1.
:::;
1.
= [B, Mfjaj(D)]u = fj(x)[B,aj(D)]u + [B, Mfj]aj(D)u.
Since (11.17)
Cj = [B,aj(D)] E OPS~~l-J),
and there are polynomial bounds (in j) on the relevant seminorms of the symbols, we have
II!J[B, aj(D)]uIILP :::; CN.K(j)-NlluI/LP,
1 < P < 00,
given supp u C K, compact. Also, by the fundamental commutator estimate (10.1), we have (11.19)
(11.7)
Pj (x.~)
B
(11.18)
p(x,~) = Po(x,~) + I:!J(x)Wj (I~I) 1~\m(1- ip(~)), ),
---t
Take m = 0 above, and use the notation aj(~) instead of ajo(~), Then
defined in the obvious way, where vmo denotes the closure in bmo of the space of smooth functions with bounded derivatives of all orders. We aim to study operators in such symbol classes, obtaining some results of [CFL] and some extensions, some of which have been discussed in [T4]. These results will be useful for some elliptic regularity results in Chapter III. We start with the following decomposition of a symbol p(x,~) E bmoSci. Let {Wj : j :::: I} be an orthonormal basis of L 2 (sn-1) consisting of eigenfunctions of the Laplace operator D.s on sn-1. We can write
where ip E CO"(lR Furthermore,
+ f).
We use this decomposition to establish some commutator estimates. First, suppose
Related symbol classes include
n
Po(x,D): LP(lR n )
(11.14)
We say
(11.5)
;j;e(x) = CtP(2.7:
Hence
Assume furthermore that
(11.8)
Po(X, D)u =
bmoSro,
i.e., such that
(11.6)
79
II[B,Mfj]vl\LP:::;
CllfjllhrnollvllLP,
and hence (11.20)
II[B,pj(x,D)]uIILP :::; CN(j)-NlluliLp,
for j 2: 1. Summing over j, we have:
80
1.
OPERATORS WITH MILDLY REGULAR SYMBOLS 11.
PROPOSITION 11.1. I/p(x,~) E bmoS~1 and BE OPSP,8' 8
[B,p(x,D)]:
(11.21)
L~omp - +
1
£P,
< 1, then
81
Hence
< 00.
If p( x,~) E vmoS~1 has compact x -support, this commutator is compact.
OPERATORS WITl! VMo COEFFICIENTS
p(x, D)q(x, D) = Po(x, D)q(x, D)
(11.29)
+
[Mg,p(x, D)] = [Mg,po(x, D)] + ~)Mg,pj(x, D)J.
and the first two terms on the right are compact. The double sum is equal to (11.30)
L Mf;gkaj(D)ak(D) j,k
j~1
Clearly, for g E bmo,
[Mg,po(x, D)J : L~omp
(11.23) Next, for j
(11.24)
~
-+
£P,
1
~ P
<
L
M!j [aj(D), ij(x, D)],
j
[Mg,pj(x, D)J = M!j [Mg, aj(D)].
\J[Mg,pj(x,D)JuIiLP ~ CllfjllL'",lIgllbmolluIILP,
[Mg,p(x, D)J : L~omp
-+
1 < P < 00.
£P,
If also 9 E vmo and p(x,~) is supported on x E K compact, then this commutator is compact. The following is the key result of this section. PROPOSITION 11.3. Assume that
q(x,~) E (L oo n vmo)S~I'
p(x,~) E LooS~l'
with compact x-support. Then p(x, D)q(x, D) = a(x, D)
+ K,
where K is compact on LP(JRn), for 1 < P < PROOF. We decompose
p(x,~)
a(x,~)
= p(x, ~)q(x, 0,
00.
where q(x,~) = q(x, 0 - qo(x, t). Recall that aj(~) = (1 -
IIM!j [aj(D), ij(x, D)]uliLv
for some M <
(11.33)
as in (11.6), and make a similar decomposition
00.
~ Cpllfj 11L'>o (j)MlluIILP,
This follows from the expansion
[aj(D),q(x, D)J = L[aj(D), M9k]ak(D), k
the estimate
(11.34)
II[aj(D), MgkJIIC(LP)
from (10.1), the norm
s: Cpllgkllbmolllajlll,
IIlajlll
depending on some finite number of derivatives of Having (11.32), another application of (11. 7) completes the proof of Proposition 11.3.
aj(~), and the estimate IIgkllbmo ~ CN(k)-N, as in (11.7).
The following well known result, whose proof the author learned from P. Auscher, is useful in the study of operators with coefficients in £00 n vmo. PROPOSITION 11.4. The space Loo n vmo is a closed subalgebra of £00 (JRn). PROOF. First, assume Ii E £00 nvmo, so there exist fi'" smooth with bounded derivatives of all orders, such that ii" ----+ Ii in bmo-norm as j/ ----> 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"=.~~="='=-'<~"
It is a general fact that, if 21 is a C* algebra and s.B a closed *-subalgebra of 21, containing the identity element, and if f E s.B, then f is invertible in s.B if and only if f is invertible in 21. To see this, consider h = f and expand (h + 1 - z) -1 in a power series about z = O. Consequently we have
r
(11.37)
a E LX
n vmo,
a-I E
L
OO
==}
a-I E
L oc
n vma.
"n~_~_~.-~;~~~;;S ON A CLASS OF BESQV SPACES
This is enough to establish (12.2) for all P any effort to general q E [1,00].) Next we look at
q2(X, D)f
(12.7)
This holds for matrix-valued a(x). Hence, as another consequence, if p(x,O E (LOO n vmo)S~1 is elliptic, i.e., Ip(X,O-ll ::::: C l for I~I ? C2, then p(x,~)-l(l
=
(12.1)
f
E
B;,q(JRn )
-.,
In this case the Fourier transform of Qk'
7/J£(D)q2(X,D)f
(12.8)
=
(12.3)
ql (x, D)f = L
Q'k
Q'k = L Qk)' j-::;k-4 k Since Q'k
(12.4)
so if IIQkIlL~
7/Jl(D)ql (x, D)f =
: : : AI,
LI12£87/Jl(D)q2(X,D)fIILP ::::: CL
L 7/Jl(D) (Q'k
we have
=
IIQrIILP,
L
~(k)11218
£20
=
LL~(k)11218
k';::O£=O ::::: CL~(k)112k8
< 1. the last inequality holding provided s (12.11)
IIQk'IIL~
> 0, and we have
: : : C ==? q2(x,D): B;,l(JRn ) -., B;,l(llitn ).
Next we look at (12.12)
q3(X, D)f =
L
L Qkj
k';::Oj';::k+4
In this case Qkj
7/J£(D)q3(x,D)f = L L 7/J£(D)(Qkj
k-::;£+5j=£-,5
hence
l+5
IIQkllLX : : : Al < 00
==}
ql(x,D) : B;,l(lRn )
-.,
B;,l(lR n ).
(12.14)
117/J£(D)q3(X, D)fIILP ::::: C
L L
k9+5j=£-5
j
'" 2 . We have
£+5
(12.13)
117/J£(D)ql(X, D)fllu ::::: CAl L 1I
~(k)
k2£-4 k+4
£
Hence (12.6)
~(k)ll
C L
so
£+5
(12.5)
117/J£(D)q2(x, D)fIILP :::::
(12.10)
It is straightforward to reduce to the case m = O. As in §§2, 4, etc., it then suffices to consider the action of q(x, D) when q(x,~) is an elementary symbol, with decomposition as in (2.8). We first look at PROOF.
: : : C2 k , and we
L 7/Jl(D) (Qr
B;,l (lR n ),
for all P E OPBSf\. Hence (12.2) holds whenever P E OPSl',/j'O ::::: 0 Furthermore, if s > 0, then (12.2) holds for all P E OPC'--Sl',l with r > s.
Qkj.
j=k-3
k21-4
JR, 1 ::::: P::::: 00, we have
P: B;tm(JR n )
(12.2)
L
Qr =
k
(12.9)
with 7/Jj as in (1.2). We aim to look at a subclass of this family of Besov spaces, with q = 1. We begin with a result whose extension to general q is well known (d. [Run]), but for simplicity we stick to the case q = 1, with a view to some applications in Chapter III. PROPOSITION 12.1. Given m, s E
k+3
Hence
00 L 11 2j8 7/Jj(D)flltp < 00, j=O
{=}
OPBS~I.l' (This part extends without
L Qk'
12. Estimates on a class of Besov spaces Assume s E JR, 1::::: p, q ::::: 00. The Besov spaces B;,q(llitn ) are defined by
E
83
IIQkj
84
1. OPERATORS WITH MILDLY REGULAR SYMBOLS
12. ESTIMATES ON A CLASs OF BESOV SPACES
85
The Besov space B~,l (JR n ), defined by
SO
L) 2i8 7fi(D)q3(X, D)fIILP
00
f E B~,l(JRn) ~ L II7fj(D)fIIL~ < 00. j=O
(12.23)
i i+lO
(12.15)
i8 ::; C L L IIQkillu>o 112 ipk(D)flb i2:0 k=O 00
::; CL( L 112(i-k)8QkiI/U>O) 112 k8 ipd D )flb· k=O i2:k-lO
(12.24)
Thus we have
(12.16)
is of particular relevance for one of the points of view we take in this chapter, namely to study spaces of functions very close to the space C(JRn) of continuous functions, but on which pseudodifferential operators (with rough symbols) act. It is clear that
L 112(i-k)8QkiIIL~::; A 2 ==? Q3(X, D) : B;,l ~ B;,l' i2:k-lO
Since Q(x,~) E crs~ ,1 sition 12.1.
=}
IIQkillu>o ::; C2(k-i)r, this completes the proof of Propo
C(A)(JRn ) C B~,l(JRn) c C(JR n ),
PROPOSITION 12.3. Given s > 0, P E [1,00], the space B;,l (JRn) n Loo(JRn) is an algebra under pointwise multiplication. We have
IlfgIIB;.l ::; CllfIIL~llgIIB;,l
(12.17)
L II7fi(D)Q2(X, D)fIILP ::; CL L ;3(k)llipk(D)fIILP i2:0 k2:i-4 i ::; C L(k + 4);3(k)llipk(D)flb, k2:0
B;,l(JR n ) c Loo(JRn ),
C IIQrllu>o ::; k + 1
==?
Q2(X, D) : B~,l (IRon) ~ B~,l (JR n ).
When s = 0, (12.16) continues to hold. If Q(x,~) E C(.>")S~,(K)' so that
(12.19)
IIQkllu>o ::; AI,
IIQkjIIL~::; C)..(j)K(2 k ),
C
k )"(k)K(2 ) ::; k + l'
L
)..(j)K(2 k )::; A 2 < 00.
We have the following result.
+ Tgf + R(f,g),
p(x,~) E C(A) S?,(K)
and
R(f,g) =
L
7fj(D)f· 7fk(D)g.
It is readily verified that
(12.30)
==?
IITfgIIB;,l ::; Cllfllu>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)
<
00.
(12.33)
s
> 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 <
(14.10)
io
00.
)
T
Ilu llc,)
IlulIL= ::; Cw(2- k ) Ilu!lclwj + Ck Ilullc2'
(14.6)
assuming w(h) satisfies the Dini condition:
As is well known, elements of OPSP 0 are not typically bounded on L"'C. It has proven useful to have estimates of the f~llowing form: 14.1)
Ilull£''oO ::; L 11~£(D)ull£',,) + 11(1 -Wk(D))uII L "", £=0
Hence, by the definition (1.61),
(14.8)
E 13.
Generally, whenever all operators in OPSP,1 are bounded on B, the paradifferential operator calculus gives (13.17); see the introduction to Chapter II to see how this works. This works for the spaces given in (13.15). A more elaborate argument, given in §9 of Chapter II, is needed when B = C(>')(M).
(
(14.5)
We see from (14.7) that, if >'(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 <
00.
The BKM-type estimates were used to show that (14.28) does guarantee that the solution does not break down. The proof given in §5.3 of [T2] made use of an auxiliary space C~(M) c Cl(M) having the following two properties: (14.29)
P E OPS~1
=?
P: C~(M) -7 C1(M),
and
D>k-2
:::; C{
L
Aj(t,x,v,D:IJ =
lal<:":m-j
(14.28)
PROOF. We have
11(1 -
L
with
> O.
for this class of spaces C()..). As we have stated, the original point of the sort of estimates discussed above was to obtain LOCJ estimates. However, slight variants of these arguments are actually effective in obtaining estimates in a slightly stronger norm, namely the B~.l-norm, as we now demonstrate.
(14.18)
a;n u =
)=0
If we apply (14.15) and take k ~ (1Iullc(A)/llullc~)l/(l+s), we obtain
(14.17)
log Ilullcr) Crllullcz ( 1 + 1OQ' II 1/, II " 0 , r > 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
~
~ 1, P E OPsp,o(JRn), then
II[Mf , P]ulle; ~
Cllfller (1IuIIL~ + IIPuIIL~)'
The necessity to restrict r to (0,1] when the dimension n is greater than 1 is due to well known results on symbol calculus. The special features that allow the estimates (15.2) and (15.4) to hold for r > 1 will be apparent when we rederive (15.4), at the end of this section. The proof of Proposition 15.2 will use techniques of paradifferential operator calculus. We begin with the following representation of the product: (15.6)
fu = Tfu
+ Tuf + R(J, u),
as in (3.47), where T f is Bony's paraproduct. We have (15.7)
f E L'X)
===}
Tf = F(x, D) E OPBSP,I,
where BSf,\ denotes Y. Meyer's class of paradifferential operators, defined in (3.36) (3.37). When m = 0, these operators are bounded on all the standard Sobolev and Zygmund spaces. Thus we have
+ TPuf + R(J, Pu), = PTfu + PTuf + PR(J, u).
f Pu = TfPu
(15.8)
P(Ju)
Directly using (15.7), we have
(15.9)
Ru E OPSP,I'
(15.10)
IIR(J, Pu)lle~
+ IIPR(J, u)lle~
~ Cllflle~ Ilulle~,
\j
r
> O.
Hence, to prove (15.5), it suffices to have the following. PROPOSITION 15.3. If 0
(15.12)
~ 1, P E OPsp,o(JRn), then
II[Tf,P]ulle~ ~ Cllfllerllulle~.
(15.11) In fact, for 0
~
1, f EC
r
===}
[Tf, P] E OPSl.r.
This result is proved in [Bon] and also contained in results of [Mey2] and of
We can replace the mates hold for all r > O. Our first goal is to establish the following multi-dimensional version:
(15.5)
===}
so
k2:0
PROPOSITION 15.1. Ifu E R(P+), r>
93
IITPufllc;
+ IIPTuflle~ ~ Cllflle; (liullu>o + IIPuIIL~),
[AT]. Furthermore, (15.12) is a special case of Proposition 7.3 for 0 < r < 1 and Proposition 7.4 for r = 1. Next, we rederive the estimate (15.4). Note that, to prove (15.4), for all r > 0 (with cr replaced by C:), since we have the estimates (15.9)-(15.10), it suffices to prove: PROPOSITION 15.4. Ifr
(15.13)
> 0,
f E L'x;(SI), then
II[Tf' P+]ulle; ~ CllfllL~ Ilulle~.
In fact, [Tf,P+] E OPS- oo .
(15.14)
In this case, we analyze TfP+ = A(x, D) and P+Tf since P+(O is independent of x, we have (15.15)
A(x,~)
Next, using Proposition 6.1, with (15.16)
B(x,~)
1/
= B(x, D) as follows. First,
= F(x, ~)P+(~).
= 0, we have
= P+(~)F(x,~) + i'o(x, ~),
with the following analysis of the remainder. The symbol P+(~) has the following special property: (15.17)
1001
~
I:::} afP+(~) E S~oo.
Thus (6.25) holds for any /-l2, e.g., /-l2 = -r. Since have (15.18)
i'o(x,~) E S~;',
\j
r
f
E
Loo :::} F(x,~) E SP,l' we
> O.
This proves (15.14), which in turn implies (15.13). We next establish the following extension, to estimates in C(Al-norms.
~MC_-~==""=~~=~T.-=-c:JpERKtu=Rs=Wli'IrWCbLy=It'EGUCXR SYMBOLS~~~~--~~~~~--------
PROPOSITION 15.5. Let w be a modulus of continuity, >'(k) = w(2- k ). Assume
that
M(k) "'-., M(k) =2: >'(k), and
2: )"(k)I~(k)
(15.19)
s; C>.(£).
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")
<;=}
Cllfllc(>.) (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")
<;=}
r- n
J
If(x)[ dx S; Cr-n/p,
for all balls B r of radius r E (0,1). Morrey's imbedding theorem is that Vu E MP(JR n ), p> n
==}
u
E
C 1 -"/p.
We will consider the following Morrey-type spaces. Let w be a modulus of continu ity. We say (16.3)
f
E
Mp,W(JR")
<;=}
r- n
(16.9)
f
E
M~,w(JRn)
¢=}
J
If(x)1 dx S; Cr-n/pw(r),
(r- n
If(xW dX) 1/q S; Cr-n/pw(r),
for all balls of radius r E (0,1). Clearly Mg,W(JR TI ) c Mf,W(JR n ) = MP,W(JR n ). One advantage of spaces of the form (16.9) is that large classes of pseudodifferential operators act on these spaces, provided 1 < q S; P < 00, as we will see shortly. In [P] and [Sp] the space M:'w is denoted Lq,'P, with (r) = r n(1- q/ P )w(r)q. We also define homogeneous versions of the spaces considered above. Namely, we say f E Mg,W(JR") if and only if the condition (16.9) holds for all r E (0,00). As a check on the sharpness of (16.4), consider the following family of examples:
us(x) = (log Ixl)"
for x E JR", [xl S; 1/2, cut off smoothly to be supported in ( 16.11)
IVus(x)1 =
lsi' Ilog Ixll
8-1
1
'~'
(16.12)
1
Vu s E Mn'W(JR"),
with (16.13)
w (h) -- -n1-1 h
-
l
0
h
{Ixl
Ixl S; 2'
and hence
R,_
for all balls B r of radius r E (0,1). We will demonstrate the following.
J Br
(16.10)
Br
(16.2)
the condition that defines membership in C{w}. In addition to the spaces NJP,w (JRn), it is useful to consider the following ex tension of the class of spaces M:(JRn). Given 1 S; q S; P < 00, we say
r n-l[logr 18-1 dr
r '
S; I}. We have
L OPERATORS WITH MILDLY REGULAR SYMBOLS
96
for 0 < h :S 1/2. If n = 1, one has w(h) Us E CW in this case, if s < 0. However,
= foh IlogrI
n> 1 ~ w(h) ~ Iloghl S -
(16.14)
S
-
16. ESTIMATES ON MORREY-TYPE SPACES
Ir - 1 dr ~ I log hl\ i.e.,
(16.15)
Us
Us E CJ1.,
(16.25)
p(x,~) E
E C""(lR n ),
r J
f; w(t)t-
1
dt <
o
00,
L
gj
+ h,
i. e.,
= XB 2r (zJ!,
gj
= XArJ,
A rj
= {x: Ix - zl
E [2
IITfollLq(lR n ) :S ClifoIILq(lR n ) :S Craw(r),
n n a=-- q
w(t) dt. (16.27)
t
E w(2- k )
IIThIILOO(B'/2(Z)) :S C1lhIIMt,W(lRn).
It remains to estimate
E Tgj
(16.28)
<
00,
on Br(z). To do this, write
XBr(z)Tg j
we
=
Tj(XArjf),
where Tj has integral kernel 00.
(16.29)
kj(x,y)
=
XBr(z)k(x, Y)xA rj (y).
Now, using (16.22) for Ix - yl :S 1, we have =
k- s ,
M n,w,l(lRn ) C C""(lR n ),
J
s> 1.
On the other hand, if w(t) satisfies (16.20), we have
jkj(x, y)1 dx
E
Proposition 2.4 is more effective than Proposition 2.5 (and also Theorem 16.1 is more effective than Corollary 16.2). Next we examine the action of various classes of pseudodifferential operators on the spaces M{W(lR n ). We begin with the following general result, extending a result of [T4].
J
=
Ik(x, y)IXA rj (y) dx
Br(z)
(16.30)
:S C(2 j r)-n. vol Br(z)
a(T k ) = k- s + 1 .
If we were to apply Proposition 2.5 to LP-estimates instead of Proposition 2.4, we would need k- 2 (s-1) < 00, i.e., s > 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 <
w(T k )
(16.21)
+
f = fa
fo
(16.26)
Note that (16.19) applies when (16.20)
M~,w(lRn) ----+ M~,w(lRn).
where
Let us say U E M n,w,l if U, V'u E Mn,w. Note that, if we apply the LP-operator estimate Proposition 2.4 together with Theorem 16.1, we get PROPOSITION 16.3. As long as have
(0, nip).
2J r$1
1
U
E
PROOF. Let f E Mg,W(lR n ). Pick z E IR n , r E (0, IJ, and write
COROLLARY 16.2. Provided fo w(t)r 1 dt, we have
(16.19)
T: Lq(lRn ) ----+ U(lR n ) ===} T:
(16.24)
r;. Cw.
This optimizes the containment C{w} C CJ1. in (1.65), for this family of moduli of continuity. We record the result on modulus of continuity which follows from Theorem 16.1 combined with the containment (1.65). Such a result, with a different proof, is contained in [Sp].
(16.18)
for some {3
J.L(h) = I log his , h:S 1/2.
V'u E M n,w(1R n ) ~
yl)-M
then
Note that, by Proposition 1. 7, if s < 0, then
(16.17)
+ Ix _
and w is a modulus of continuity such that
00,
r f3 w(t) ",",
(16.23)
C{w} ,
as long as s < 1 (a fact that can also be verified directly from the definition (1.64)), clearly (16.16)
Ik(x,y)l:s Clx - YI-n(l
for some M > 0. If 1 < q :S p <
for h :S 1/2, if s < 1. We see here that, while, by Theorem 16.1, Us E
PROPOSITION 16.4. Assume the Schwartz kernel k(x, y) of T satisfies (16.22)
1
97
:S CTjn, and
J
Ikj(x,y)1 dy =
(16.31 )
J
XBr(z)lk(x,y)1 dy
A rj
:S C(2 j r)-n. vol A Tj
:S C.
p'
~~'''''~~_~"<'Il;;'''''''l''1~",_,<",
98
1. OPERATORS WITH MILDLY REGULAR SYMBOLS
Hence, if
f
E
Me,WORn),
If 1 < q :::; p < (16.39)
II T j(xA r jf)IILq :::; CTjn/qllXArjfllLq :::; CTjn/q(2jr)aw(2jr)
(16.32)
(16.40)
00,
L
(16.33) where
log2(2/r)
L
T j n/ p w(2 j r)
j=l
(16.34)
~
f
1l r
1
= r n/
dT
r-n/pw(rr)
p
T
1 1
t- 1 - n/ p w(t) dt.
If t- f3 w(t) ",", then
a(r) :::; r n/ p - 13 w(r)
1c 1
1
-
n/ p + 13 dt :::; Cw(r),
and the proposition is proved. If
f
E M~,w(JRn),
one can replace (16.24) by (Xl
f
= fo
+ Lgj j=1
and repeat the estimates above, obtaining: PROPOSITION
16.5. Assume the Schwartz kernel k(x, y) of T satisfies
(16.35)
If 1 < q :::; p < (16.36)
Ik(x,Y)1 :::; Clx _ yl-n.
and w is a modulus of continuity satisfying (16.23), then
T: Lq(JR n ) --t Lq(lR n ) ==} T: M~,w(lRn) --t M~·w(lRn).
00,
By the results of §2, we then have the following: PROPOSITION
(16.37)
16.6. Let p(x,~) satisfy either
p(x,O E
C(>') Sf,(t<)
L A(j)f;:(2
(JR n ),
j
)
<
00,
j
or (16.38)
p(T.~) E C[oTJS\\,,) (IR n ),
2:]a(2- k )f;:(2 k )]2 < k
00
99
and w is a modulus of continuity satisfying (16.23), then p(x, D) : M~,w(Rn)
~ M~,w(lRn),
p(x, D) : M~,w(JRn)
~ M~.w(IRn).
For convolution operators, a similar result is contained in [P], which also studies more general spaces, of Campanato type.
IITgj!iLq(Br(z)) :::; a(r)r a,
a(r) =
,,_
and
:::; Crjn/Praw(2jr),
so, if p <
__ ,,~._._~.__,•.._.
-.-,-,_""._,~",,_,,~,~
16. ESTIMATES ON MORREY-TYPE SPACES
00.
CHAPTER 2
Paradifferential Operators and Nonlinear Estimates Introduction This chapter presents a variety of estimates on compositions of functions F 0 u, typically where U = (Ul,"" Uk)' Estimates on such compositions have long been important for nonlinear PDE. Various analytical tools have been brought to bear on such estimates, such as the Gagliardo-Nirenberg inequalities. Seminal work of J.-M. Bony [Bon] and Y. Meyer [Meyl] introduced the use of the paradifferential operator calculus. This has been quite effective, especially when F is smooth. S. Alinhac [AI] extended paradifferential techniques to cases where F has limited (though a fair amount of) regularity. Some more recent papers, including [CW], [KPV], and [Sta], have established useful results about F 0 U under weaker regularity hypotheses on F. We will reca pitulate some of their results, and produce some more results along these lines, in §§1-5. In §1 we establish estimates of [CW] on IlfgIIHs,P, i.e., on IIAs(Jg)lb, where A = ~. In §2 we establish estimates on [KPV] on AS(Ju) - fAsu. In both cases, we make use of Bony's paraproduct, in addition to techniques used in [CW] and [KPV], involving maximal functions. Using the decomposition (0.17) given below, we reduce the latter problem to an estimate on [AS, Tf]. Part of the analysis of this commutator involves estimates made in §3. Our choice of order of arguments was made in order to motivate some of the rather technical arguments of §3; some of them are taken from [CW], and others are extensions, which will be useful in §5. In §4 we estimate F 0 u in HS'P, when F ' is bounded and 0 < s < 1, obtaining part of a result in [CW]. While many of the results of §§1-4 have been known before, our main goal in presenting our approach to their proofs is to lay the groundwork for the result of §5, in which we estimate Fou under more general hypotheses on F'. The usefulness of such an estimate for problems in nonlinear PDE was perceived in [CW], and a similar result is stated there. However, the proof given there has a gap. This gap was filled for an important class of functions F in [Sta] , and the result there is useful for many problems in nonlinear PDE. However, in the course of work in [K], T. Kato formulated a more general estimate. The work presented in §5 was done by the current author in response to a question posed by Prof. Kato. In §6 we digress from applications of paradifferential operator calculus. We discuss some results on the continuity of u f---+ f(u) in H1,p, due to [MM]. We include this partly to put in perspective some of the results of §5 and §7, and also to derive some implications for Sobolev spaces on irregular domains. This latter
102
._--------------_.
~:::::::::::.::..=.-~o.=:---;:::~.,;;;;::=~:::..=:=':'::':'~~~.
__
-="''::::::::::::::~~'':~~,~~-::-
category of results, in the case p = 2, played a role in some results of the author on scattering of waves by very rough obstacles, given in Chapter 9 of [T3]. In §7 we return to paradifferential operator calculus and obtain estimates on F(u) - F(v). These estimates generalize and sharpen some given in [Sog]. Also we correct a small error occurring there. In §8 we discuss a pseudodifferential operator estimate which to some degree combines methods of §1 with those of [Bour], which have played an important role in paradifferential operator calculu..'l. In §9 we analyze F( u) in terms of paradifferential operators, in yet a different setting. We assume F is smooth but u belongs to a space CC)..) , introduced in Chapter 1. We show that F( u) produces a paradifferential operator (by a process we recall below) that acts on CC)..) , given u E CC)..) , for slowly varying sequences A(j) "" o satisfying a condition just slightly stronger than the Dini condition L A(j) < 00. This is less straightforward for such cases as j-S, S > 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
£
k
3. SOME HANDY ESTIMATES INVOLVING MAXIMAL FUNCTIONS
2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
108
PROOF.
Set bR= 2R8 aR. Then the left side of (2.19) is equal to
2::: 2::: 2::: 22k82R-kTR82j-kTj8bl)j k R
(2.20)
j,R
C
(2.21)
S
< 1, so (2.20) is equal to
k l'l/JkV(X) - 'l/JkV(Y) I ::; C· 2
(2.23)
L: Ib k l2, and this
gives (2.19).
Cllullu'" II (2::: 22k8IM'l/JdI2) 1/2tp'
::;
k
Ix -
yl ::; T
I (2:::
I
M 9k
2 1
)
1/2tp ::;
cil (2:::
k
""
I
Ix - yl·I"J;.j(x - y)1 dy
1
Next,
CiluilLoc IlfIIH8,P,
3. Some handy estimates involving maximal functions Here, take 'l/J E S(lR n ) and set 'l/Jj(~) = 'l/J(2-j~), 'l/JjU = 'l/Jj(D)u = "J;j * u. We establish some results of lew], and some extensions. We begin with a result that is obvious.
I Ix-YI2:2-
(3.7)
::; CN
l'l/JkV(X) - 'l/Jkv(y)I·I"J;j(x - y)1 dy
k
I
Ix-yl2: 2 -
(l'l/JkV(X) I + I'l/Jkv(y) I) 2j (1
+ 2j lx -
yl) -N-1 dy.
k
Using (1 + 2jlx - YI)-l ::; 2- j lx - yl-1 ::; 2k- j on the region of integration, we dominate this by
C2 k-. j M('l/JkV)(X),
(3.8)
Hence:
3.1. We have
I
(3.1)
LEMMA
(3.9)
l'l/JkV(X) - 'l/JkV(y) I =
II
v(z) ["J;k(X - z) - "J;dy - z)] dzl
: ; 11
1
\V(z)\·ID"J;k(sx+(1-s)y-z)ldsdZ' IX-YI.
3.3. We have
I
Iv(x) - v(y)I' l"J;j(x - y)1 dy ::; CMv(x).
Next, write
Hence:
yl.
::; C2 k- j M(tpkV)(X).
and Proposition 2.2 is proved.
(3.2)
k
::; C2 kM(tpkV)(X)
k
LEMMA
s)y) ds ·Ix - yl·
l'l/JkV(X) - 'l/Jkv(y)I'I"J;j(x - y)1 dy
Ix-yI9-
(3.6)
19k 2) 1/2tp
k
CllullLOC II (2::: 22k81'l/JdI2f/2tp
+ (1 -
k ==} l'l/JkV(X) - 'l/JkV(Y) I ::; C2 kM(tpkV)(X) ·Ix -
I
of [FSl]; compare the use of such estimates in §9 of Chapter I. Thus we dominate the right side of (2.23) by (2.25)
Mv(sx
We next use (3.4) in an estimate of J l'l/JkV(X) - 'l/Jkv(y)I'I"J;j(x - y)1 dy. First, we have
Now there is the vector maximal estimate (2.24)
1
Ix - yl::; T k ==} l'l/JkV(X) - 'l/JkV(Y) I ::; C2 kMv(x) ·Ix - yl·
(3.4)
Returning to (2.18), we hence have, for s E (0,1), p E (1,00),
II[A8, Tj]u II £I'
1
Furthermore,
(3.5)
= 2(1-8)[jH-2max(j,R)] = T(1-8)!j-RI.
It is then elementary to dominate (2.21) by
3.2. We have
Note that, if tpk(~) = tp(2- kO, tpk = Ion supp 'l/Jk' then we can substitute tpk V for v, obtaining
2::: AjRbRbj
provided s < 1, where
A jR
(3.3)
k>max(j,R)
The inner sum is equal to C . 2- 2(1-8) max(j,R), provided
(2.22)
LEMMA
109
l'l/JkU(X) - 'l/Jku(y)I'I"J;j(x - y)1 dy::; C2 k- j M(tpkU)(X).
This is useful when k < j. Otherwise, use (3.1), with v = 'l/JkU. In §5 we will need analogous estimates when a function H(y) > 0 is thrown into the integrand. We begin with the following obvious extension of Lemma 3.1. LEMMA
(3.10)
I
3.4. We have
Iv(x) - v(y)\·I"J;.j(x - y)1 H(y) dy::; Clv(x)IMH(x)
+ CM(vH)(x).
110
2. PARADIl'FERENTIAL OPERATORS AND .NONLINEAR ESTIMATES
4. A COMPOSITION ESTIMATE
Next, extending (3.6), we have
f
I'lPkv(x)
-7/Jkv(Y)I'I~j(x
!x-yl::::2- k (3.11)
f
k :::; C2 M(
Using Lemma 3.3 and the comment following it, we dominate the right side of (4.4) by
- y)/ H(y) dy
:::; CKL2k-j M
f
I
2js '" (L 2 l7/Jj F 0 u!2 f/2 tp'
IIF 0 ull Hs,p
]
- y)1 H(y) dy
and (neglecting the term j
(l7/Jk V(X) I + /7/Jkv(y)I)2j
(1 + 2j lx -
ylr (2 k- j )H(y) dy N
jx-yl2:2- k
(4.7)
:::; C2k-jl7/JkV(X))' M H(x)
k~
Now
!x-yl2:2- k
(3.12)
+ CKLM
k~
(4.6)
-7/Jkv(y)I'I~j(x
l7/Jk U(X) -7/Jk u(y)!·I¢](x - y)1 dy
k
(4.5)
Ix - yl . I~j(x - y)1 H(y) dy
and, extending (3.7)-(3.8), we have
l7/Jk V(X)
f
CKL
:::; C2 k- j M(
f
=
0) we have
L
22js l7/J]F 0 u(x)1 2 :::; CK 2 L
j
j
22] .. (L 2k-] M
k<j
+ CK2 L
+ C2 k- j M(I7/JkvIH) (x).
22 ]S
j
Hence there is the following extension of Lemma 3.3: LEMMA
f
LEMMA
l7/Jk U(X) -7/Jku(y)1 . I¢j(x - y)1 H(y) dy
If s
L 22]SIL 2k-jak/2 :::; CL 22ksjakl2. ] k<j
> 0, then L22]SILakI2 :::; CL22kslak/2. j k2:j
(4.9)
4. A composition estimate
Our goal here is to establish this result of leW]:
4.1. Assume F(O) = 0 and
IF'I :::; K.
Then, for
S E
(1,00),
(4.1)
=
when j
~
E
PROOF. The inequality (4.8) was established already in (2.19). To prove (4.9), we again set bk = 2ks ak, so the left side of (4.9) is equal to
L j
In the proof, we will want to estimate
7/Jj(D)F 0 u(x) =
(0,1), p
(4.10)
IIF 0 ullHs,p :::; CKlluIIH"'Po
(4.2)
k2:j
4.2. If s < 1, then
(4.8)
k j :::; C2 - M(
PROPOSITION
(2: M
We use the following:
3.5. We have
(3.13)
111
L
L
k2:]f~]
22jsTksTfsbkbf = L Bkfbkbf' kJ
with
f F(u(y))~JCx f
Bkf
Tslk-fIC.
=
Hence (4.10) is dominated by Cl:: Ib k l , which gives (4.9). 2
- y) dy
[F(u(y)) - F(u(x))]¢j(x - y) dy,
Using Lemma 4.2 to dominate the right side of (4.7), we get
(4.11)
1. Using
IIF 0 U/IHs,p
:::; IIWo(F 0 u)IIHs,p + CK/I (L 22kSIM
(4.3)
IF(u(y)) - F(u(x)) I :::; Klu(x) - u(y)l,
we have, for j
~
(4.4)
17/J](F 0 U)(X) I :::; CK
and, as in the estimation of (2.23), we can use (2.24) to dominate the last term by
1,
f
(4.12)
/u(x) -
u(y)I'I~j(x -
CKII (L 22kSI
y)1 dy.
proving Proposition 4.1.
112
2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
6. CONTINUITY OF
5. More general composition estimate
---+ JRf
is a C 1 map, satisfying F(O)
IF'(TV + (1 - T)w)1 :::; JL(T) [G(v)
(5.1)
(5.8) =
0 and
IIF 0
uIIHB,P :::;
C1IM HIILOl I (L: 22k81
(5.9)
1
P
q1
q2
- = - + -,
(5.3)
Thus Proposition 5.1 is proved.
q1 E (1,00], q2 E (1,00).
IF(u(y)) - F(u(x)) I :::; C[H(x) + H(y)] lu(x) - u(y)j,
H(x) = Go u(x),
and hence we replace (4.4) by
l'lh(F 0
+C
lu(x) -
lu(x) -
u(y)I'I~j(x -
PROOF. Use
(6.1)
u(y)I'I~j(x -
(6.2)
y)1 dy
y)1 H(y) dy.
+ C(MH(x))2 2: 22j8 (2: M
k?j
j
C2: 22j8 (2: 2k- j M(I1/Jku IH )(x)) j
+ V)) -
f(u(x)) I :::; Mlu(x
+ y)
- u(x)l,
j
Jsing Lemma 4.2 to dominate the right side of (5.6), we have
+ CII(.MB) (L: 22k8IM
+CII (L: 2
2ks
k
jM(I1/Jk u I H )
In [Zie] can be found the following result, for f Lipschitz, u E H1,p(n) :
(6.3)
n
\7fou(x) = j'(u(x))\7u(x),
a.e.
In particular, j'(u(x)) is well defined a.e. on {x En: \7u(x) '# O}. Of course, (6.3) implies (6.2). A particular case of this is that, if u E H1,p(n), then
\7lul(x) = \7u(x) on {x En: u(x) :::: O} -\7u(x) on {x En: u(x) :::; O},
and
(6.5)
k?,j
k
MII\7uIILP'
2
k<j
5.7)
II\7f(u)IILP :::;
A simple localization allows us to obtain the following result for arbitrary open n , even with very rough boundary: PROPOSITION 6.2. Ifu E H 1,p(n), then f(u) E H 1,p(n), and (6.2) holds.
(6.4)
+ C2: 22j8 (2: M(I1/JkuIH) (x)) 2.
uIIH"'P :::; II wo(F 0 u)IIHB,P
If(u(x
n c JR
2:22j811/JjFou(x)12:::; C(MH(x))22:22j8(2:2k-jM
IIF 0
f(u) on H 1 ,p
Throughout this section, we assume 1 < p < 00. We assume f is Lipschitz, with Lipschitz constant M, and f(O) = O. All functions are assumed to be real valued, except f; we will find it convenient in (6.8)-(6.11) to allow f to be complex valued. PROPOSITION 6.1. Ifu E H 1,P(JR n), then f(u) E H 1,p(JRn).
The first term on the right has an estimate of the form (4.5), while Lemmas 3.4-3.5 apply to the last term in (5.5). Thus, as in (4.7) (again neglecting the term j = 0) we have
+
f---'
where M is the Lipschitz constant of f. In that way, one sees that
J J
u)(x)1 :::; CH(x)
(5.5)
C1IM HIILQ' IluIIHB,Q2.
k
CIIG 0 ullL ql IluIIHB,q2,
This was established in [Sta], in the case that (5.1) holds with G(v) = IF'(v)l. The more general statement above is from [K]. In the proof, we again want to estimate (4.2). This time, we replace (4.3) by
(5.4)
:::;
cll(2:22k811/JkuI2H2r/2tp:::; C1IHIILQlIIUIIH8,Q2.
6. Continuity of u
1
113
k
provided 1
feu) ON H1,P
Similarly, we estimate the last term in (5.7) by
+ G(w)],
given G > 0, JL E £1([0,1]). Then, for s E (0,1), p E (1,00), (5.2)
f->
Using (2.24), we dominate the second term on the right by
In this section, we establish the following result.
PROPOSITION 5.1. Assume F : JRk
U
1/2tp'
\7u(x) = 0 a.e., on {x En: u(x) = O}.
We mention that (6.3) is straightforward if also f is C 1 • The results (6.4)-(6.5) can be obtained from this by a simple limiting argument; d. [GT], pp. 151-152. However, (6.3) seems to be more delicate for general Lipschitz f. We next establish a fairly simple, but weak, result on continuity of the map u f---' f(u). PROPOSITION 6.3. If Uj ---+ U in H 1,P(n)-norm, then f(uj) ---+ f(u) weakly in H1,p(n).
·-
_
;;
~=.,,
.
..
_.-::::..:::~ -:~,:='::: :.:~- -~--:_:=:=--:::--~~'---=:~:=:::
Pi
PROOF. By Proposition 6.2, {f(Uj)} is a bounded subset of Hl,P(f!). Now, given p E (1,00), the Banach space Hl,p(f!) is reflexive. Hence, there is a sub sequence f(ujJ which converges weakly, say to 9 E H1,P(f!). However, it is clear from
In other words, 'f}j(s)
=S -
(j (s) = s if
1 Jl ' f 1 S> J
(6.13)
PROOF. It suffices to show that (j(u) ---4 0 in H1,p-norm. Clearly (j(u) ---40 in LP-norm. Meanwhile, by (6.3),
'V(j 0 U = (;(u)'Vu,
(6.14)
'V(j(U) = 'Vu on Sj = {x
o
each £ E {1, ... ,n}, Now Sj "" Z
ad(u) in LP-norm.
We will first show that, if
(6.16)
11'(8)1
(6.8)
Ilad(Uj) IILP
(6.9)
lIad(u) IILP,
----+
for each £. Since LP(f!) is uniformly convex for each p E (1,00), this together with the weak convergence established in Proposition 6.3, will imply (6.7); d. [Ko]. (This type of argument is rather familiar in the case p = 2, where L 2 (f!) is a Hilbert space.) In fact, using (6.3), we have jad(Uj) I = \agujl for a.e. x E f!, whenever (6.8) holds, and hence Ilad(uj)IILP = IlagujllLP, so (6.9) is clear, and hence we have (6.7), whenever (6.8) holds. We now establish (6.7) for all Lipschitz f. To do this, it suffices to assume the Lipschitz constant of f is :::; 1, so If' (s) I :::; 1 a.e. It is then routine to write
I'(s)
= ~[91(S) + 92(8)],
with 9k measurable, and, upon setting h(s)
(6.11)
f(s)
19k(S)1
=1
a.e.,
= J; 9k(CT) dCT, we have
= ~!I(s) + ~h(8),
and the argument above applies to fk(Uj) for each k. This finishes the proof. Next, define functions 'f}j on lR by
'f}j(s) = (6.12)
{x
E
E
f! : ju(x)1 :::;
t}
on f! \ Sj'
f! : u(x) = O}, and, by the monotone convergence theorem,
11'V(j(u)111p
=
J
J
Sj
Z
['Vu(x)IP dx""
l'Vu(x)IP dx
= 0,
the last identity by (6.5). This proves the proposition.
then, as j ---4 00,
(6.10)
=
a.e.,
= 1
J
PROPOSITION 6.5. Ifu E H1,P(f!), then 'T/j(u) ---4 U in H1,p-norm.
PROPOSITION 6.4. Ifuj ---4 U in H1,P(f!)-norm, then f(uj) ---4 f(u) in H1,P(f!)so norm. (6.15) PROOF. It is clear that f(uj) ---4 f(u) in LP-norm. It remains to show that, for ----+
IsI :::; ]
J
The following gives a significant improvement in the conclusion of Proposition 6.3. The proof differs only slightly from one in [MM]; that paper also treats the case p = 1.
ad(Uj)
115
feu) ON H1,p
_.! if s < _.!.
that f(uj) ---4 f(u) in LP(f!), so 9 = f(u). The consequent uniqueness of the weak limit proves the result.
(6.7)
u>-+
(j(s), where
If(Uj) - f(u)1 :::; Mluj - ul
(6.6)
--------- - :_-- - .
6. CONTINUITY OF
2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
114
We apply some of the propositions above to deduce some results about the structure of Ht'P(f!), defined as the closure of Cgo(f!) in H1,P(f!). These results are of interest when of! is perhaps very rough; d. Chapter 9, §12, of [T5] for applications in the case p = 2. LEMMA
(6.17)
6.6. Let 'R C f! be open. Then
U E H6,P(f!) n C(f!), U = 0 on an \ of!
if
lsi:::; J
s-
ifs>.!J
s+
ifs<-.!. J
uln
E
Ht'P('R).
PROOF. It suffices to assume U is real valued. Then the hypotheses apply to u+ and u-, so it suffices to assume U :::: 0 on f!. Take Uv E Cgo(f!), Uv ---4 U in H1,p-norm. Then ut ---4 U in H1,p-norm. Now, with 'T/j as in (6.12), set Vv =
(6.18)
min (ut, 'T/v( u))
In'
Since min(u, w) = (u-w)- +w, we see that Vv E H6'P('R) and Vv ---4 U in H1'P-norm, so we have (6.17).
6.7. Let f!j be open in Rn, Uj component of f!l n f!2' Then LEMMA
0
=}
(6.19)
Ul
= U2 = U on
0
=?
E
H6,P(f!j). Let 0 be a connected
U E H6'P(O).
116
2, .PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
7.
PROOF. It suffices to assume Uj are real valued. The hypotheses imply uj E H~,p(nj) and ut = ut = u+ on O. Thus it suffices to assume in addition that Uj 2: 0 on j . Now we can find Vv E c(j"'(nd, W v E c(j"'(n 2), such that Vv --t Ul in H 1,p(n 1) and Wv --t U2 in H 1,p(n 2). Hence v;j- --t Ul and w;j- --t U2 in H1,p-norm. Now 'Pv = min(v;;,w;;)lo
has the properties
n
lPv E
H~'P(O),
'Pv
--t
U in Hl,p-norm,
Actually, to apply previous estimates to G(u,v), we need G(O,O) = 0, i.e., F'(O) = O. This can be arranged by replacing F(u) by F(u) - F'(O)u, a change that does not affect the validity of the estimates (7.3) and (7.5). Next, we give a counterexample to the following:
IIF(u) - F(v)IIH 1 :s C(F, B)llu - vllHl.
This is a special case (with n = 2) of the estimate (3.8') in [Sog], which is consequently not correct. On the other hand, as mentioned above, the estimate (3.8) of [Sog] is correct, so none of the major results of that paper are affected. We take F(u) = u2 • Then F(u) - F(v) = u2 - v 2 = (u + v)(u - v). Note that a correct result, and a special case of Proposition 1.1, is 2 2 (7.7) IIu - v 11Hl Cllu + vllLoo Ilu - vlIH' + Cllu + vllH1llu - vllv"'.
We can estimate F(u) - F(v), using
F(u) - F(v) = G(u, v)(u - v),
with
1 1
G(u, v)
=
F'(ru
+ (1- r)v)
:s
dr,
together with estimates on G(u, v) and product estimates. The simplest case is the following.
IR n ,
PROPOSITION 7.1. Assume F is Coo. Then, whenever p E (1,00), s we have
> 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<;j-4
:s CIIq~~[ILTl'
Ilq(x, D)uIIHs,p :s Cllq* IILql IluIIH8,Q2
+ Gllq~~ IILTl Ilull HS~("-T),T2'
The dependence in (8.20) on ~-derivatives is hidden, but implicit in the hy pothesis that q(x,O is an elementary symbol; the "constant" C in (8.20) depends on bounds on (OloIDf'Pk(~), for some finite range of 0: (as well as qj, Tj, and other parameters) . A general symbol can be written as a sum of elementary symbols (8.21 )
We then dominate the right side of (8.8) by CIIQ2(E2
IIQ211L Tl
(8.19)
Consequently, when q(x,~) is an elementary symbol, 1 < P < 00, 0 < s < a, q* and q~~ are given by (8.14) and (8.18), and qj, Tj satisfy (8.7) and (8.12), we have
Next, in view of the spectral properties of ('l/JjQk2)'PkU, we have
(8.10)
2ja (O-T 1'l/J)(Dx)q(x,~)I.
Note that we can set 2ja 'l/Jj(D x) = ~j(Dx)IDxla. It follows that
l
k
(8.9)
;:::IW
where
(8.5)
(8.8)
:s Gllq*IILql'
On the other hand, we have
k
where
(8.7)
Mq*(x),
q*(x) = sup Iq(x, 01.
{(j,~):2]
k
(8.6)
:s c
Hence, for ql E (1,00],
(L: 22kSIQkI121'PkUI2) l/211LP Ql
l\I1k(Dx)q(x,~)1
~
Thus, q(x, D)u = ql (x, D)U+q2(X, D)u, and we estimate these two terms separately. First, for all s > 0, P E (1,00),
:s
Tl E (1,00], T2 E (1,00).
T2
Tl
(8.14)
Qkl supported on I~I:S 2k+5 Qk2 supported on I~I ~ 2kH .
Ilql (x, D)uIIH8,P
P
where M is the Hardy-Littlewood maximal operator and
with (8.3)
111
- = - + -,
For an elementary symbol q(x, ~), we have
(8.13)
Qk(X) = Qkl(X)
(8.2)
s < a,
provided that (8.12)
2k + l (for k ~ 1) and bounded in S?,o(lRn).
:s
:s Gil Q211ul IluIIHs-(0'-r),T2'
Ilq2(X, D)uIIH8,P
(8.11)
k;:::O
with 'Pk(~) supported in 2k- 1 Let us write
119
p(x,~) =
2: qR(X,~) = L: L: QkR(X)'PkR(O, R
R k;:::O
with
2kTl'PkUf) 1/2tp'
(8.22)
(~)loIIDf'PkR(~)1
:s Co(£)a1o l ,
AND NONLINEAIt ESTIMATES
for some a < 00 and Qkl(X) obtained from a Fourier analysis of p(x,~) on (~) Details on this are recalled in §2 of Chapter 1. Thus, we have:
9. PARADIFFERENTIAL OPERATORS ON THE SPACES cC>") 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
e
+ T'l/Jk+l(D)u, O.s T .s 1.
and
lal:SN
jal:SN
(9.12)
Then
(8.26)
Vk = Wk(D)u
Note that, parallel to (9.4), we have
T2
and
(8.25)
\Ve want to draw an analogous conclusion when we assume u E CCX) (IRn). To do this, we need to estimate Ilmk Ilc(>.), which is essentially equivalent to an estimation of IIF'(Vk)llc(),) , where
(9.9)
For sufficiently large N, take (8.24)
121
IIF'(v)llc(),)
.s CF'(llvllu ,)(1 + \Ivll c (>'),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<)'
1\;(2 k) =
u E C(>")
==}
F(u) = MF(U;x,D)u + R(u),
R(u) = F('l/Jo(D)u) E Coo,
and
(9.17)
p(x,~) E C().,) S~,(I<)
==}
p(x, D) : C().,)
---7
C().,),
provided
2: 1\;(2 )A(k) < 00, k
(9.18)
k
i.e., in the setting of (9.16), provided 00
(9.6)
MF(u; x,~) =
2: mk(x)'l/Jk+l(~).
A(k)
~ A2(k)
(9.19)
k=O
We have
(9.7)
< 00.
Thus, if (9.13) and (9.19) both hold, we have
uEL
oo
A2~k)'
We recall from Proposition 5.2 of Chapter I that
where
(9.5)
.s
Hence, if (9.13) holds, and
Hence
(9.4)
Il vkllc(),),2)
so we have
(9.15)
where
(9.3)
F(Uk)]
(In fact, (9.12) can often be seen to hold via a modulus of continuity argument.) Now
=}
MF(U;X,~) E S?,I'
As we noted in the introduction to this chapter, if T > 0, (9.8) u E C r ==} MF(u;x,O E CrS?,o'
(9.20)
u
E
C().,)
==}
MF(u;x,D): C().,)
---7
C().,),
so
(9.21 )
IIF(u)llc(),) .s CF (ll uIILoo)(1
+ Ilullc(),)t
122
2, PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
For example, we can take
r
s
'x2(j) =
,
r
1
,X(j)
=
r> O.
'x2(j) = 1,
TiT,
'x(.k)~\k) < 00.
L 'x(P)M(P)'x2(P) ::::; C'x(j)M(j), L £~i
(9.33)
T(2 k )
=
1
M(kp'
'x(k)
~ M(k)2
< 00.
===}
M F (u; x,~)
p(x,~) E C('\",,) Sf,(T)
provided LT(2 k )'x(k)M(k)
~
E C(>.. ",,) 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) <
IIF(u)llc(A) : : ; C(ll uIILoo)(l + IlullctA) (
PROPOSITION 9.1. Let 'x(j) \, 0 be slowly varying. Suppose there exists slowly varying 'x2(j) \, such that
1 M(k)2'
00,
k
for some v E Z+. Then
C(,\),
(9.38)
u E C(,\)
===}
M F ( u; x, D) : C(,\)
----+
C(,\),
and we have
(9.39)
00.
This condition fits together with (9.24) best when M(j) = 'x2 (j) 1/2. We thus see that if
L ,X(e)'x2(p)3/2 ::::; C,X(j)'x2(j)1/2, L 'x(k)'x2(k)-1/2 < £~i
00.
k
We see that (9.35) holds for ,x, 'x2 as in (9.22), provided s Inductively, we have the following.
< 00, i.e., provided
(9.29)
L 'x(P)'x2(p)5/3 ::::; C,X(j)'x2(j)2/3, L 'x(k)'x2(k)-1/3 <
(9.36)
: : ; C(ll vk IILoo)(l+ Mtk)ll vk I C(A))2.
C(,\)
= 'x2(k)1/6, we see that (9.32) and (9.34) both hold provided
Then (9.20) holds and we have
This time, Proposition 5.2 of Chapter I gives
k
then (9.20) holds, and we have (9.31)
MF(U;X,~) E C('\"")S~,(T)'
£~j
Thus, when (9.24) holds,
(9.30)
==}
Another invocation of Proposition 5.2 of Chapter I gives (9.28) in this case, provided
(9.35)
IIP'(Vk)llc(AI') : : ; C(llvklluX>)(l + Ilvkllc(AI'))2 (9.26)
(9.28)
u E C(,\)
Taking M(k)
IIF(u)IIC(AI') : : ; C(llullu>c)(l + Ilullc(Al'l)2.
u E
Then we have (9.25), with the exponent 2 replaced by 3, and similarly for (9.26). Thus, when (9.32) holds,
k
Hence we can replace (9.12)-(9.15) by
(9.27)
00.
(9.34)
Then, parallel to (9.21), we have (9.25)
£~j
(9.32)
k
Of course, in this case, we already have OPSP,l acting on C;, so the exponent of 2 in (9.21) is not needed. We want to extend the collection of ,X(j) for which it can be shown that (9.20) holds. As a first step, assume we have M(j) \, such that 'xM satisfies (9.13) and (9.19), i.e., (9.24)
123
L 'x(P)M(P)'x2(P)3/2 ::::; C,X(j)M(j)'x2(j)1/2,
L 'x(k)M(k)'x2(k)-1/2 <
.
Then (9.13) holds for s > 0, and (9.19) holds for s > 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
=
0
(Wk U)),
F
0
u(x) = O
0
where
RF,u(x)
(A.15)
1
=
=
With Wk+T = TWk+l
LAk(X)'lj;k+1(~)' k?>O
=
L Fk
F
(A.16)
In particular, for 13
L
u*F =
(A.17)
(D~vWk+TU) DV(WkF)'
0
u).
We estimate the difference of (A.14) and (A.17), given by
(Wk+TU),
R},u = u* F - u* F =
(A.18)
L
with
L
l:Slvl:Slm
I/(WkF)'llclvlllwk+lUllclJ311" ·II Wk+lullclJ3vl.
gk(X)
II A kIIL= ::; II(WkF)'IIL= ::; CIIF'IIL=.
One consequence of (A.7) is the following, in case
U
LEMMA
Fk(u) - Fk(WkU) = bk(X)(1 - Wk)U,
bk(x) =
1
FHTU + (1- T)WkU) dT.
A.I. Given s > 0, we have
Ilu* F - u* Fllc;+s ::; C(llu'IIL=) 1IF'llc:llullc:.
(A.20)
is Lipschitz. Note that, for
=
1
(A.19)
Of course, (A.8)
0
k
> 0,
IID~AkliL= ::; C
+ RF,u(x).
In fact, the operator defined as paracomposition in [AI] is
T)Wk, we can estimate
L C(D~lWk+TU)'" f31 +"+f3v=f3
(WkU)(X) - u* F(x),
u(x) = F(U; X, D)u + u* F(x)
0
D~(WkF)' 0 (Wk+Tu) = (A.6)
0
which will be specified in Appendix
then, from (AA), we have
(Wk F )'(TW k+1 U + (1 - T)WkU) dT.
+ (1 -
II u'll L=,
k
1
Ak(x)
'lj;£(~).
IJI-kl'SN
Here, N is a number, determined by B. If we set
(WkU)(X),
L
=
k
(A.4)
(A.7)
F(U;X,~) E S?,l(JR n ).
Thus, for s > 0, P E (1,00),
L Fk(WkU). k?>O
Meanwhile, (A.3)
I/D~AkIIL~ ::; C(llullLipl )1!3I IIF'I/u>o .2k1f31 ,
(A.H)
[Fj (\l1 ku) - Fj (Wk-1 u)].
j,k
127
A. PARACOMPOS1TION
1131'12:1, PROOF.
Ilwk+1ullclJ3,,1 ::; C2 k(If3,,1-1)lluIILipl .
(A.9)
We have
Ilgkllu>o::; 11F£IIL=II(1 - wk)uIILoc.
(A.21)
Hence (A.7) implies
Now
IID~AkIIL= ::; C (A.lO) =
L
11(\l1kF)'llc:vlllullt~" 2klf3l-klvl
(A.22)
l'Slvl'S 1f3! C2 k1f31 L rklvll/(WkF)'Ilclvl '1Iullt~l' 1 <:: Ivl $1f31
s
>
°
===}
11(1 - Wk)uIIL= ::; Cllullc: . T ks ,
so (A.20) follows.
'l,~"."
,111,', ,
We next estimate u* F. The following is due to [AI].
2. PARADIFFERENTIAL OPERATORS AND NOl'*LIl'*EAR ESTIMATES
128
LEMMA
A. PARACOMPOSITION
Next, we have, for k
A.2. We have
Ilu*Filer ~ C(llu'llu"') 1lFller'
(A.23)
~
129
K,
II'l/Jj(D) L Wk-N(Fk 0 Wk U) too k
PROOF.
By (A.14),
~C L
(A.32)
'l/Jj(D)u* F =
(A.24)
L
'l/Jj(D)ipk(D)(H
0
Ilwk-N(Fk 0
~
Ik-jl:'ON
C(llulle s)
so (A.25)
j+N II'l/Jj(D)u* FllLoo ~ C L k=j-N
RF,u = L(I -wk+N)(Fk 0 WkU)
+ LWk-N(Fk 0
II'l/Jj(Fk 0
+ N, IIDu- 1 1ILoo ),
k
1/
wku)IILoo
> s-
Ilwk-N(Fk
0
~
Cv(llulle.) Tjv 2k(v-s+l) IlFkllLoo,
1 ~ 0, and if also k ~ K (depending on
0
Wk U) too
k
~C L
(A.30)
II'l/Jj(Fk 0 Wku)IILoo
k:'Oj-N
~
Cv(llulle')
L
T jv 2 k(v-s-r)
k:'Oj-N by (A.28). Taking
1/
1IFII e:+l,
u(x) = u* F(x)
+ F(U;X, D)u + R},u(x) + RF,u(X),
Ilu*Filer
(A.35)
IIDullLoo
and
(A.36)
F,u
2-(r+s)j.
~
C(llu'IILoo) 1lFller'
E
Lipl
===?
F(u;x,D)
E
OPS?,l,
and the remainder terms satisfy the following estimates.
IIR},ulle:+ s ~ C(llu'IILoo) IIF'ller Ilulle:,
(A.37)
and, for s
~
1,
l'
s > 0,
+ s > 0,
(A.38)
IIRF,ulle:+ s ~
C(llulle s,IIDu- 1 1ILoo) 1IFIIe:+l.
In [AI] there are estimates on L 2 -Sobolev norms of the various quantities in (A.34). Here, we produce some LP-Sobolev norm estimates. Such estimates on F(u;x,D)u are already recorded in (A.I3). We next give LP-Sobolev estimates on the paracomposition. PROPOSITION
> s + T, we dominate the last espression by
Cv(llulle') IlFll e;+l
0
the second term has the property
wku)IILoo ~ C(llulle., IIDu- 1 1ILoo) Tk(s-l) IIHIILoo.
II'l/Jj(D) L(I -Wk+N)(H
F
where the paracomposition u * F, given by (A .17), satisfies estimates
As will be seen in Appendix B, the hypothesis that u is a diffeomorphism is needed to establish (A.29). We can estimate 'l/Jj (D)RF,u in two parts. First,
(A.3I)
We summarize what has been done above:
(A.34)
Wk U).
We use the following two estimates of [AI], which will be discussed further in Appendix B:
(A.29)
T(r+s)j,
PROPOSITION A.4. Assume u is a diffeomorphism of class Cs, s > 1, that u' is uniformly continuous, and (U')-l is uniformly bounded. Assume F is Lipschitz. Then
C(llulles, IIDu-11ILoo) 1IFII e:+l.
k
for j ~ k
C(llullc-) 1IFII e:+l
Note that
(A.27)
(A.28)
1IFII e:+l,
which gives (A.26), upon taking note of the dependence on N in these estimates, and crudely estimating the last sum in (A.27) over k < K.
LEMMA A.3. Assume u is a diffeomorphism of class CS, that u' is uniformly continuous, and that (u')-l is uniformly bounded. If s > 1 and l' + s > 0, then
PROOF.
II'l/Jj(D)RF,uIILoo ~
(A.33)
We next estimate the "remainder" RF,u in (A.16), folowing [AI].
IIRF,ulle:+s ~
L T(r+s)k k?:j+N
by (A.29). Combining this with (A.30)-(A.31), we have
IlFkllLoo ~ CNTjr 1lFller·
This gives (A.23).
(A.26)
wku)IILoo
k?:j+4
u),
(A.39)
A.5. If 1 < P <
Ilu* FIIHs,p
~
00, we
have
C(IIDulluov, IIDu- 1 1ILoo) 1IFIIHs,P.
130
2. PARADIFFERENTIAL OPERATORS AND NONLINEAn ESTIMATES
PROOF. In view of the location of the spectrum of the various terms in the sum (A .17) defining u * F, we have
(A.40)
l/u*FIIH"P :s;
cil (2: 22ksIFk
(A.41)
:s;
0
u1 2)
1/2IL·
IIGoullLP :s; C(IIDu- 11Ivx», so the right
cil (L:: 22ks lFk1 2) 1/2tp :s; CI/FIIH8,P,
(A.42)
l/u*F - u~FIIH8+r,p :s;
p.Q
II4>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)l
(A.44)
u)IIIp
which gives (A.49).
CIIF~llu>oM(I - Wk)U(X),
we have
lIu* F -
IIVJj(D)(Fk 0
!k-jISN
l
:s;
L:: !k-j!SN
Using (A.17) and the estimate
(A.43)
k
(A. 54)
s
> 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 <
Ilu*FIIBSP,ll :s;
(A.49)
> 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 <
s+r :s; IIRF,uIIB Plq
(A.48)
k
with C
PROPOSITION
(A.47)
k
Now, since u is a diffeomorphism, we have side of (A.40) is
A. PARACOMPOSITION
Bb(~n) ~ Hs,2(~n),
132
2. PARADIFFERENTIAL OPERATORS AND NONLINEAR ESTIMATES
133
B. ALINHAC'S LEMMA
so the L2 -Sobolev estimates of [AI] are contained in these. Whether the set of LP-Sobolev estimates can be completed (to include RF,u) is a matter we will leave uIlBettled, recommending it as a problem for the reader to consider.
with Akv(y,1J,~)
(B.g)
= (Lit Ao(Y)·
Note that Akv(y,1],~) is homogeneous of degree -1/ in (1],~), and smooth on the support of 'l/J(2-j~)'l/J(2-k1]). Also (on the cone generated by this support)
B. Alinhac's lemma Here we prove an estimate which was used in Appendix A in the estimation of the remainder term in the decomposition (A.16). As before, take Fk = 'l/Jk(D)F.
B.1. There exists N sufficiently large, determined by Ilu'IIL"", such that the following holds. For j 2 k + N, 1/ 2 s - 1 > 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"...".;<''''."... ,·~''''
~ ......~:a.i:
...,...:.......i:,,..:..... ,~'''''''·,,''_~'".~"'c..,"- ..._~;;"
CHAPTER 3
Applications to PDE Introduction In this chapter we apply some of the results of Chapters I and II to some problems in PDE. We provide a sampling of applications rather than any systematic development, as the main focus of this work is on the internal development of the theory of various classes of operators. In §1 we produce some results on regularity and Fredholm properties of elliptic differential operators with mildly smooth coefficients. An example is the Laplace operator on a Riemannian manifold with metric tensor of limited regularity. Among the various hypotheses on the metric tensor we consider, we mention particularly gjk E
Cr ,
C(A),
H 1.P, or L = n vmo.
In §2 we study some natural first-order differential operators arising on a mani fold with a Riemannian metric that is Lipschitz in local coordinates, making contact with work in [Mor] on the Hodge decomposition on such a class of Riemannian manifolds; we also produce some results when the metric tensor is in the Zygmund class C;. One particular operator we study here yields the trace-free part of the deformation tensor associated to a vector field on a Riemannian manifold. Previous studies of this operator have played a role in works on quasiconformal mappings. In this section we also study the Beltrami operator
EJ EJ B = EJz -A(x'Y)EJz' for IIAIIL'X> ::; k < 1. We establish regularity results in case A E L30 n vmo and in case A E C(A). Some of the methods developed in §§1-2 are pursued further in [MT], which considers the method of layer potentials on Lipschitz domains in Riemannian man ifolds with C1 metric tensors, and also in [MMT]. This point will be discussed further in Chapter IV. In §3 we produce some estimates on solutions to the Dirichlet problem, particu larly with boundary data in C(A) (EJD), for various domains in JR.". We include some results on regions with nonsmooth boundary, such as arbitrary convex domains in lR". Some of these results will be extended in §4. Others will be of further use in §5. In §4 we consider the method of layer potentials for boundary problems on domains whose boundaries are C1,w regular. We obtain results when w satisfies a Dini condition. We also obtain more precise results when the boundary is Cl+T regular, for some r E (0,1). The study of boundary layer potentials on such regions
'"""'"'
.
u
-
__ __ ~
4==4." .
.;r.
- -_.
ftC'FLIGATrONS TO~P1)E
..
~.-
.
is classical; d. [Mik] and references therein. Some of the results of §4 reproduce results there, while others carry the study further. Results of this section can be compared and contrasted with results discussed in Chapter N, for Lipschitz domains. In §5 we study trace asymptotics for the semigroup e- sVL , where L = -.:l+ V and Ll is the Laplace operator on a compact mainfold M, with a Riemannian metric of class ck+r, for some k E Z+, T E (0,1). We produce k+ 1 terms in the expansion. In §6 we study the Euler equations for ideal, incompressible fluid flow, on rough planar domains. We obtain global weak solutions on a class of domains that includes all bounded convex domains on the plane. In §7 we produce a result on persistence of solutions to a class of semilinear wave equations, assuming an a priori estimate on the solutions weaker than required by standard techniques. This complements results on persistence of solutions to quasilinear equations established by the author in [T2]. These conditions are of a different nature from results in [BB] and [KM]. In §8 we discuss div-curl estimates. The most basic div-curl lemmas provide estimates on U . v when U and v are vector fields on jR3 for which there are known (i) norm bounds on u and v, (ii) norm bounds on divu and curl v, estimates that are stronger than those obtainable from (i) alone. There are many ramifications of this. The sort we discuss here pertain to knowing that u E
LV,
v E
LP',
P E (1,00),
1
1
-+= p p'
1
~"
-'II:.. d.;'
_~. U
J~!lL",,,,,,,:::;~.,.
.""n,- ,.
_-__ ~___
1. INTERIOR ELLIPTIC REGULARITY
•
137
to solutions to the wave equation .:lu = 0,
Utt -
when Ll is the Laplace operator on a Riemannian manifold with bounded Ricci tensor. 1. Interior elliptic regularity
In this section we investigate solutions to various elliptic PDE with mildly smooth coefficients. We first take a look at a second order elliptic equation in divergence form:
LOjAjk(X)Ok U
(1.1 )
= f·
We assume Ajk E C(A), where w is a modulus of continuity and '>"(j) = w(2- j ). The equation (1.1) might be an Lx L system. We write it as
LOjAj(x,D)u =
(1.2)
Aj(x,~) E
f,
c(A)si,o,
and then, using the symbol decomposition of §3 in Chapter I, write
Aj(x,~) = A1(x,~)
(1.3)
A1(x,0 E
+ A;(x, 0,
A;(x,~) E C(A)S~~~.
si,.s,
We take <5 E (0,1) and require (as a special case of (3.22) of Chapter I) '
(~)-'i'C~)
(1.4)
2: 'Y.s(~),
7(~)
=
(~)7jJ(~),
where
and using bounds of the form (ii) to conclude that u· v belongs to the Hardy space SJl rather than merely to L 1 • Such a result and many variants were presented in [eLMS]. We prove a number of these results here. We use two unifying principles, the "supercommutator estimate" produced in §1O of Chapter I, and an "abstract div-curllemma," which will be stated and proved in §8 of this Chapter. In §9 we consider the problem of constructing harmonic coordinates on a neigh borhood of a point in a Riemannian manifold with rough metric tensor, including (in two dimensions) the problem of constructing isothermal coordinates. We make constructions for metric tensors rougher than Holder. In §10 we consider Riemannian manifolds with bounded Ricci tensor. We begin with fairly weak assumptions on the metric tensor and conclude that there exists a coordinate system in which the metric tensor has two derivatives in bmo. This result builds on a number of previous works, beginning with [DeTK]. We also estimate the metric tensor under some other hypotheses on the Ricci tensor. In §11 we study propagation of singularities for solutions to linear PDE (or pseudodifferential equations) with coefficients of limited regularity. We obtain re sults on propagation of singularities along null bicharacteristics for operators whose coefficients are of class C1,1, so the associated Hamiltonian vector fields have Lip schitz coefficients. Then we extend these results to the case of operators whose coefficients have "log-Lipschitz" gradients. This class includes operators whose co efficients belong to the Zygmund class C;. We have results both for non-divergence form operators and for divergence form operators. For example, results here apply
'Y.s(~) = o-((~)-.s),
. (1.5)
o-(h) =
ito
w(t) dt. -t-
We make the following hypotheses: (1.6)
'>"(j)7(2
j
) ""
0,
L '>"(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<5 in (1.3). The following is our first regularity result.
j
TO PDE
1.
PROPOSITION 1.1. Assume that u E C; satisfies the elliptic system (1.1), with Ajk E CC),). Assume that the conditions (1.6)-(1.7) hold. Let 5..(k)jA(k) /" and assume that
!J
f = LOjli,
(1.9)
Assume that (with
PROPOSITION 1.2. If Ajk E OC),)(lI'TI), P = conditions (1.6)-(1.7) hold, then (1.18)
139
L: OjAjk(x)Ok
is elliptic, and the
P: oU),)(r) __ O-l,C),) (ll'TI)
E CCA).
is Fredholm.
T(~)
/-lU) =
(1.10)
INTERIOR ELLIPTIC REGULARITY
as in (1.4)) T(2 j )-1
PROOF. The operator E# E OPS~~ maps C~l.()') to 01,C),). Note that
/-lU/,1 ::; c5..(j), for some M.
==}
(1.19)
E# P
= I + E#ojA~(x, D),
PE# = I
+ ojAJ(x, D)E#.
Then Vu E OC)..) .
PROOF. The operator p# = E OjAf(x, D) is an elliptic operator in OPS~,o' Let E# E OPS~~ be a parametrix for p#. Then (1.1) is equivalent to u = E# f - E#ojA;(x,D)u,
(1.11)
mod 0
00
mod 0
OkU 00
•
= OkE#Ojfj - OkE#OjA;(x, D)u,
!J
OC)..)
E
==}
Ab(x t) E OC),)Sl-';' J
,<"
1,CT)
Ab(x D) . 0 1 ---+ OCp) *
J"
/-l2(j) = max(/-l(j)/-ll(j),A(j)).
A;(x,D)u E OCP2),
Iterating this procedure a finite number of times, we obtain Vu E OC)..), as desired.
Note that, in case (1.8), /-lU) ~ (1.16)
AU) =
r
s
,
r Cs -
5..U)
1
- i ).
= Fa,
Hence Proposition 1.1 holds for 0<
(}' ::;
5, 5
> 1.
We also record the more classical type of result to which Proposition 1.1 can be specialized, namely
(1.17)
Ajk EO:, u EO;, f E O;1+ s ==? u E O;+s,
if r 2:
5
> O.
In Propositions 1.4 and 1.10 below, we will treat variants of the regularity result of Proposition 1.1, with other hypotheses on u. Next we tackle some existence results, starting with the following Fredholm result. For simplicity of notation we work on the torus TTl to start with, extending the result to other manifolds shortly. The spaces or,C),) are as in (5.57) of Chapter I.
compact.
Suppose the operator P in Proposition 1.2 is strongly elliptic. Then P has index zero in (1.16). In fact, the family of operators
,
where /-l is as in (1.10). We deduce from (1.12)-(1.14) that Vu E OC,l,) , where /-l1(j) = max(/-lU),5..U))· Applying Proposition 5.10 of Chapter I again, we have (1.15)
0JA;(x, D)E# : 0-1,C),) __ O-l,C),)
(1.22)
Thus E# is a 2-sided Fredholm inverse of P.
OkE#Ojfj E cC.\).
==}
compact,
and
Now, by Proposition 5.10 of Chapter I, under the hypotheses (1.6)-(1.7),
(1.14)
compact.
E#OjA;(x, D) : OU),) __ 01,C),)
(1.21)
Note that OkE#Oj E OPSP.o, so, by Proposition 5.7 of Chapter I,
(1.13)
A;(x,D) : 01,C),) __ OC),)
(1.20) Hence we have
,
using the summation convention, so
(1.12)
Since A.~(x,~) E OC),)S;:C~, we have
(1 - t)P
(1.23)
+ tt:!. : 01,C),) (ll'TI)
__ 0-1,C),)(lI'TI),
0::; t ::; 1,
is a continuous family of Fredholm operators, of constant index, and t:!. obviously has index zero. In this case, ker P consists of constants. In fact, under our hypotheses, E AU) < 00, so 01,(),) C 0 1 C H 1 ,2, and hence u E ker P==}O = -(Pu,u)
so u E ker P
=:}
= '2)Ajk(x)OkU,OjU) 2: 01iVu11E2,
Vu = O. We are ready to prove the following existence theorem.
PROPOSITION 1.3. Under the hypotheses of Proposition 1.2, suppose further that P is strongly elliptic, and scalar. Then given f E O-l,C),)(lI'TI), the equation Pu = f has a solution u E 01,C),) (ll'TI) , if and only if J f(x) dx = O. In such a case, u is unique up to an additive constant. PROOF. From what we have done so far, we know that the range of Pin (1.18) is a closed linear subspace of O-l,(),) (TTl), of codimension 1. Now J f(x) dx = 0 is one linear condition satisfied by all elements of the range of P. Hence this is the only such linear relation, and the proposition is proved. The standard Hilbert space theory implies that, whenever the operator P is strongly elliptic, P : H 1,2(lI'TI) ---+ H-l,2(TTI) is Fredholm, of index zero, with kernel consisting of constants. We have:
140
3.
1.
APPLICATIONS TO POE
PROPOSITION 1.4. Assume Ajk E C(),l(']['n), P = l:&jAjk(x)&k is strongly elliptic, and the conditions (1.6)-(1.7) hold. Then, given f E H- 1 ,2(']['n), the equa tion Pu = f has a solution u E H l ,2(']['n) if and only if J f(x) dx = O. Such a solution is unique modulo an additive constant. Furthermore, if f E C- 1 ,(),)(']['n), then u E C l ,(),). PROOF. It remains only to justify the last assertion. In fact, if Ul E CI,(A) is the solution to PUI = f guaranteed by Proposition 1.3, then, since l: )..(j) < 00 implies C 1 ,(A) C Hl,2, we see that Ul - U is a constant. Note that, if Ajk E Cl,(!-')(']['n), and l:f-L(k) < 00, so the hypotheses of Propo sition 1.4 hold with )"(k) = 2- k f-L(k), then f E C(/1-)(']['n) =? U E C 2,(/l) C C 2. We also mention that Proposition 1.4 will be localized below, in Proposition 1.10. Many natural PDEs arise in a form which is a variant of the divergence form (1.1). For example, suppose M is a compact lliemannian manifold. Then the Laplace operator ~, acting on scalar functions, is given in local coordinates by ~u = g-I/2l:&j(gjkgl/2&kU), so the equation ~u = f is equivalent, in local coordinates, to
L&jAjk(X)&kU = f,
(1.24)
where
Ajk(X)
(1.25)
= gjk(x)g(X)I/2.
In (1.24), f is treated as a distribution. If f is a function (say in L 2 ), then the right side should be replaced by g(X)I/2 f(x). This slightly subtle (though not difficult) point is often glossed over in the smooth case, when distributions and "generalized functions" are identified via the density g(X)I/2 dx. In the nonsmooth case, this identification breaks down for singular distributions. The following extension of Proposition 1.1 is established by the same sort of arguments as used above. PROPOSITION 1.5. Assume that M has a Riemannian metric of class C(A) (in some smooth local coordinate system), and associated Laplace-Beltrami operator ~. Assume conditions (1.6)-(1.7) hold, and that (1.10) holds, with ~ =)... Then
(1.26)
U
E C;, ~u
=f
E C-I,(A)
===} U
E CI.(Al.
This leads to a Fredholm result for the Laplace operator, for a class of compact Riemannian manifolds with mildly smooth metrics. PROPOSITION 1.6. Let M be a smooth, compact, connected manifold with a Riemannian metric which belongs to C(A). Assume).. satisfies the hypotheses of Proposition 1.5. Then the Laplace operator ~ on scalar functions has the property that (1.27)
~ : CI,(Al(M) ---+ C-I,(Al(M)
is Fredholm, of index zero, and kernel consisting of constants. With f E C-I,(A)(M), one can solve ~u = f for U E C1.( Al(M) if and only if (1, J) = O.
INTERIOR ELLIPTIC R.EGULARITY
141
PROOF. Using symbol smoothing (via local coordinates and a partition of unity), we can write ~ = p# + pb with p# E OPSl.o(M) elliptic, and show that a parametrix E# E OPSl~ for p# is a Fredholm inverse of ~ in (1.27), just as in the proof of Proposition (2. If we consider a line connecting the given metric on M and a smooth one, we obtain a continuous family of Fredholm operators connecting ~ in (1.27) to the Laplace operator for a smooth metric on M; hence all these operators have index zero. The rest of the assertions above follow as in the proof of Proposition 1.3. Again, for examples of when Proposition 1.6 applies, we refer to (1.16)-(1.17). The formula (1.24)-(1.25) for the Laplace operator on scalar functions (O-forms) comes from -(~u,v) = (du,dv). For f-forms, with f > 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
< 00,
is Fredholm, of index zero, with kernel consisting of constants.
PROOF. We start by constructing a Fredholm inverse for ~ when M = Tn, with metric tensor gjk (x) defined and Zn-periodic on JRn, so (1.28) is given by (1.29)
~u
= &j(A j k(x)8ku),
Ajk(x) = gjk(x)gl/2(x),
keeping in mind the remark made below (1.25). Now, in terms of Fourier series on Tn, define A : 1)' n ) ~ 1)' (Tn) by
cr
Au(x) = L
(1.30)
(k)u(k)e ik .x ,
kEzn
and set A=
(1.31 )
A-I~A-I,
i.e., (1.32)
A =
L j,k
RjAjk(x)Rk,
R j = A-I&j = &jA -I.
QJ
~-~
&&
ZJ.4!f¥W£.:;;;=::
=~~;z:.
142
;;_~"i~
_~~~~!S.~~!~i-i.WL.~~)1f.~~..I:£iSS,W::"f::R;t~jk;0i"'A:~~~~~~w..'~il.¥:i;"~:~
3. APPLICATIONS TO POE
(1.42)
where K p denotes the space of compact operators on LP(Tn), and
EAi
a(x,() =
(1.34)
k (x)(j(k(()-2
E
=
(E Nk(X)~j~k) -1 (~)2 (1 b(x,~) E (L CXJ
C(>')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
(1.43)
Furthermore, if we set
b(x,~)
B(x,()
under these circumstances. Also we have
(L CXJ nvmo)S~l'
,i,k
(1.35)
143
In fact, Proposition 4.2 of Chapter I gives
1 < P < =,
A = a(x, D) mod K p ,
....-;;---,---,.-..,~,-," .•,_.. ---~
1. INTERIOR ELLIPTiC REGULARITY
Using the results of §11 in Chapter I, we see that
(1.33)
~'l!~ii:'!::.':~;~"'~:YiP.4!f!!!'Nt".ii;'J.."'';';'~~='J''''':,=-~''':''''~;'''='''<-''''''·'=~·--:''''
n vmo)S~I'
It is useful to extend these results to
and, by Proposition 111.3,
(1.37)
B
= b(x, D)
==}
= BA = I mod K p ,
AB
1
=.
Hence, for M = 1[''', a two-sided Fredholm inverse of A in (1.28) is given by
(1.38)
A-I BA -1
:
H- 1 ,P(M)
----+
H 1 ,P(M).
A standard argument using a coordinate chart and partition of unity then yields a two-sided Fredholm inverse to A in (1.28), for any compact M. Given that A is Fredholm in (1.28), arguments as used above show that its index is zero. The description of ker A is clear for p E [2,(0). For p E (1,2), we claim the annihilator of the range of A in (1.28) consists of constants; this follows by the last remark, applied to pi, since (H- 1 ,P(M))' = H 1 ,P' (M). Then the fact that A has index zero and A annihilates constants implies ker A consists exactly of constants for p E (1,2).
L=A-V.
(1.44)
We will assume V E LCXJ(M) is real valued. Then the same argument as above gives
PROPOSITION 1.9. Let M be a compact, connected Riemannian manifold, with metric tensor in LCXJ nvmo. Assume L in (1.45) is an isomorphism for P = 2 (hence for all p E (1,00)). Then, for 1 < p < q < 00, (1.46)
PROPOSITION 1.8. Let M be a compact, connected Riemannian manifold, with metric tensor of class C(>'), and associated Laplace operator A. Assume conditions (1.6)-(1.7) hold, with T(() given by (1.4)-(1.5). Then
1
(1.39)
----+
H-l+sV!,P(M),
1 < P < 00, -1 :::; s :::; 1,
(1.47)
and
PROOF. Bring in the sort of symbol smoothing used in the proof of Proposition 1.1, so A = p# + L: XjAj(x, D), with p# Fredholm on all spaces of interest, X j smooth vector fields on M, and
(1.48)
Ab(x () .7
'
E C(>')SI-'!J
1,(T)"
u
E H 1 ,P(M),
Lu = f
E H- 1 ,Q(M)
Suppose that the metric tensor is of class
is Fredholm, of index zero, with kernel consisting of constants.
(1.40)
=f
E
H 1 ,P(M), Lu = f
E
u E H 1 ,P(M), Lu
u
E
C(>'),
A~(J:, D) :
Hl+ s 1/J,p
----. H S1/!,p
is compact.
=}
u
E H
1
,q(M).
and (1.6)-(1.7) hold. Then, given
H-1+ s,p,q(M)
C- 1 ,(>.) (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
L : H 1 ,P(M) ----. H- 1 ,P(M),
(1.45)
Now we establish a local regularity result.
-------------
144
3.
APPLICATIONS TO PDE
"'" t
--
:~"
1.
;\.~
PROPOSITION 1.10. Let M be a compact, connected Riemannian manifold, with metric tensor in L oo n vmo, and let 0 C M be open. Then, for 1 < P < q < 00,
u E Hl~~(O), Lu
(1.49)
f E HI-;,~,q(O)
=
Suppose that the metric tensor is of class j
C().),
===}
u E Hl~~(O).
and that (1.6)-(1.7) hold. Assume 00, -1:::: s:::: 1,
,\(2- ) ~ C2- 8j for some 8 < 1. Then, given 1 < p :::: q <
(1.50)
s 1/!,q(O) u E HI,P(O) lac' Lu = f E H-l+ lac
===}
Ojgjk Oku = gl/2 f
(1.55)
(1.51)
I,P(,,",) u E H lac v, LU =
f
E
Clac
==;>
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
lad :S m, so since s > 0,
(1.82)
ao: E HS'P
==}
==}
Ell>(a,u) E Hm+s,p.
Thus, under the hypotheses of our proposition, granted that m (1.83)
u E Hm+u,p
==}
u E Hm+s,p
+ + r > 0, (J
+ Hm+u+r,p.
We can iterate this until we get the desired conclusion (1.74). In [Mila] there is a result similar to Proposition 1.16 (though with a different Proof) in the special case that p = 2, s E Z+, and m = 2, and with the hypothesis that u E cm+r. In that paper boundary regularity is also established. Also [Mila]
148
3.
ApPLICATIONS Td POE.
2.
points out applications of this to regularity for some nonlinear elliptic PDE. Such nonlinear results also follow directly from the paradifferential operator calculus. See Theorem 3.3.C in [T2] and Theorem 4.8 in Chapter 14 of [T5] for interior regularity, and Chapter 14, §8 of [T5] for boundary estimates. We remark that there are analogues of Proposition 1.16 for elliptic equations in divergence form. For an example, see Proposition 9.4 later in this chapter. 2. Some natural first-order operators We now look at some first order systems. We emphasize slightly different phenomena than in §1. As a guide, we focus on the trace-free deformation operator VTP, defined on a vector field X on a Riemannian manifold M by (2.1)
- + \7X- t) - -(div 1 VTPX = -1 ( \7X X)9,
2
n
where 9 is the metric tensor, \7 its Levi-Civita connection, and X the I-form asso ciated to X via "lowering indices." The tensor field V T P X is a section of 83, the bundle of symmetric, trace-free, second order, covariant tensors on M. In index notation,
(2.2)
(VTPX)jk
=
1 1 1 £ 2 Xj ;k + 2Xk;j - ;;:X ;£9jk 1
£
= -9 2 J £X 'k ,
1
£
1
+ -9k£X .',J - -X 2 n
In local coordinates, X£;k = 8 k X£ + r£jkXj, where the connection coefficients are expressed in terms of the metric tensor and its first derivatives. Thus, in local coordinates, (2.3)
£1 £1 £ £ 1 (VTPX)jk = 29j£8kX + 29kf8jX - ;;:9jk 8£X + Bjkf X ,
where B jk £ is expressed in terms of the metric tensor and its first derivatives. If the metric tensor is Lipschitz in local coordinates, then the coefficients of the first order derivatives in (2.3) are Lipschitz, while the zero order coefficients belong to
LOCJ . The operator V TP is "overdetermined elliptic" Le., its symbol is injective. If dim M = 2, then V TP is (determined) elliptic. The kernel of V TP consists of conformal Killing fields on M, Le., vector fields whose flows consist of conformal maps of M (granted sufficient regularity). It is of interest to study V TP : C;(M) --t CZ(M). Estimates on such an operator can be established in the following more general context. PROPOSITION 2.1. Let M be a smooth compact manifold and (2.4)
V: C;(M,Fo ) --> C~(M,Fd,
a first order differential operator between sections of smooth vector bundles Fo and Fl. Assume the principal symbol is injective. Assume that, in local coordinates, (2.5)
Vu =
L Aj(x)ojU + B(x)u,
149
with Aj E C r ,
(2.6)
> 0,
r
B E
c2.
Then there exists a continuous linear map
E: C~(M,H) --> C;(M,Fo )
(2.7) .
such that (2.8)
EV
= I + K,
K
cl (M, Fo).
compact on
If the principal symbol of V is bijective (so V is elliptic), then V is Fredholm in (2.4)· PROOF. Using local coordinates, a partition of unity, and symbol smoothing, we can write V
(2.9)
=
V# +V b +B,
with (for some 5 < 1) (2.10)
£ ·£9J k· ,
SOME NATURAL FIRST·ORDER OPERATORS
V# E OP8Ld'
Vb E opcr8~,§Td,
BE C2(M,Hom(Fo,Fl
)).
If the symbol of V is injective, then V# has a left parametrix E E OP8~l. Thus E : C~ --t C;. Hence, modulo a smoothing operator, (2.11)
EV = I +EV b +EB,
where (2.12)
EV b : C;
-->
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) <---t C;/2(M), so we have (2.8). If V is (determined) elliptic, so is V#, so we can take E E 0 P 8~l to be a 2 sided parametrix for V#. Then, in addition to (2.11), we have, modulo a smoothing operator, (2.13)
VE
=
I +VbE+BE.
Now Vb E : C~ --t C;d, and we see that the right side of (2.13) is the identity plus an operator compact on C~, so the rest of the proposition is proved. When (2.8) holds, we know that the kernel N(V) of V in (2.4) is a finite dimensional subspace of C; (M, Fo) and its range R(V) is a closed linear subspace of C~(M, Fd. Also N(V) has a closed linear complement £ in C; (M, Fo), and V maps £ bijectively onto R(V). Thus there is a continuous inverse T : R(V) --t £. We have the following.
·~;~r"",,.,,-r..p,;."iI',",~~~~~~o;~=r~~~Il,'.';"'-:.>"-"<"I.;:;m.,'IA'I.i~~~!1'.l'.."'-~""!:'~'"jll;1;'~-~;r~=t!<_"""'~~"",,
APPLICATIONS TO POE
3.
2.
(2.24)
X E CS, s
> 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 <
COROLLARY 2.2. Let V satisfy (2.4)-(2.6) and have injective principal symbol. Then there exists C < 00 with the following property. If f = Vu E C?(M, F 1 ) for some u E C!(M,Fo), there exists U E N(V) c C;(M,Fo) such that (2.14)
~w,·~k.W.
SA
,p• •
150
1 ,p.
CJ
is the following. An
(2.25) u E ==} lu(x + y) - u(x)1 :::; Clyllog(ly!-l), Iyl -< ~2' PROPOSITION 2.3. Let V be a first order differential operator satisfying (2.5) (2.6). If V has injective principal symbol, then This well known result, which we have noted in (1.22) of Chapter I, was one of the prime motivations for Proposition 1.5 in that chapter. In turn, the significance of (2.15) u E CS, Vu E ==} u E C;,
(2.24) is that, by Osgood's Theorem, vector fields with such a modulus of continuity provided < s < 1 and r + s > 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<5, we have terms of gJk and its first derivatives. In this case, d + 8 is (determined) elliptic. (2.19) v E C fJ =} Vbv E c~+r<5-1, provided - (1 - 8)r < (J + r8 - 1 < r. We hence have the following, which to some extent complements results in Chapter 7 of [Mor] on the Hodge decomposition for a compact Riemannian manifold with For a proof, see Proposition 2.1.E of [T2]. The condition on (J in (2.19) is equivalent Lipschitz metric tensor. to 1 < r + (J < (2 - 8)r + 1. Thus, if r + s > 1, s E (0,1), and Vu E C?, we have PROPOSITION 2.5. Let Al be a smooth compact manifold with a Riemannian u E C S =} EVbu E C:+ r <5 =} u E CSl, S1 = min(s + r8, 1). metric tensor g. If 9 belongs to the Zygmund space C;(M), then Iterating this argument a finite number of times, we have (2.15). The proof of (2.26) d+8: C;(M,A) ~C2(M,A) (2.16) is similar, and (2.17) follows by an even easier application of (2.18). is Fredholm. We have Applying the results of Propositions 2.1-2.3 to V YF , we have the following: (2.27) u E CS(M,A), s > 0, (d + 8)u E C~ ==} u E c;. PROPOSITION 2.4. Let M be a smooth compact manifold, with a metric tensor If the metric tensor belongs to Lip l (M), then, for 1 < P < 00, g. If the coefficients of 9 belong to the Zygmund space C;, then the following results hold: (2.28) u E C;, (d + 8)u E £P ==} u E H 1 ,p.
C;
c2
°
c;.
(2.20)
V TF : C;(M,T) ~ C?(M,S5T*)
In such a case, the null space of d + 8 in (2.26) satisfies
has closed range and finite dimensional kernel. There exists C < 00 with the following property. If X E C;(M, T) then there exists X E C;(M, T) such that (2.21 )
VyFX
= 0,
(2.29)
IIX - Xllc; :::; C1IVYFXllc~·
There is the regularity result: (2.22)
X E CS(M, T), s
> 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)
Then
p(1I'2)
is Fredholm, of index zero, and Ker B consists of constants.
(2.39)
---t
Here is another Fredholm result.
This operator is important in the theory of quasiconformal mappings. The following result is fundamental. Assume that A satisfies (2.33) and has compact support in C = JR2. Then there exists a unique solution to Bu = 0 on C of the form
u(x, y) = z + v(x, y),
B' : U(1I'2)
has one-dimensional kernel, independent of q E (1,2], from which we deduce that B' has one-dimensional kernel for all q E (1,00). This implies that B in (2.36) has one-dimensional cokernel, independent of p E (1, 00), and the index computation implies that its kernel is one-dimensional.
We assume
(2.38)
~ IIV'ulli2('lf2) = (Au z , uz ) S; ~ II Vu lli2(1l"2)'
and hence Vu = O. This plus the index calculation implies, for 1 < q S; 2, that
The next first-order operator we consider is the Beltrami operator, given by
(2.34)
153
2. SOME NATURAL FIRST-ORDER OPERATORS
3. APPLICATIONS TO pbE
152
(2.46)
q(x, D)
E
OPB;,IS~I'
elliptic.
Hence p(x,~) = q(x,~)-I(I -
is dense in B;,l (1['2). It follows from Proposition 13.2 of Chapter I that p(x, D)
is a Fredholm inverse of q(x, D) in (2.45). Thus B in (2.44) is Fredholm. The same argument used in Proposition 2.6 shows B has index zero. Since B~18(1I'2) C Hl,p(1I'2), the analysis of Ker B follows from Proposition 2.6. OC
il~\.
j
i.
I
. .!
t.
.
154
3. APPLICATIONS TOPDE
3. ESTIMATES FOR THE DIRICHLET PROBLEM
Propositions 2.6-2.8 lead to the following global regularity results. PROPOSITION 2.9. Under the hypotheses of Proposition 2.6, if 1 < p
(2.47)
u E H I ,P(']['2), Bu E U(']['2) ===* u E H I ,q(']['2).
Under the hypotheses of Proposition 2.7, if 1 < p < (2.48)
U
x
=
+ iw,
with v and w real-valued.
_i1 +A 1- A U y ,
00,
u E H 1 ,P(']['2), Bu E e(.\) (']['2) ===* u E
a computation gives
e l ,(.\) (']['2).
Under the hypotheses of Proposition 2.8, if s > 0, 1 < q < p :::; (2.49)
< q < 00,
For a complex-valued function u, write u = v If Uz 1= 0 and A = u-z/u z , so
155
00,
sp
~
2,
(2.53)
dv
2
+ dw 2 =
'"
IUyl~.~ (11 + AI 2dx 2 + 4(ImA)dx dy + 11 _ A1 2dy 2),
u E H 1 ,Q(']['2), Bu E B;.I(']['2) ===* u E B~1'5(']['2).
PROOF. The same sorts of arguments as used in the proofs of Proposition 1.4 and Proposition 1.9 are effective here.
which is positive-definite if IAI < 1. Conversely, if (x, y) preserving diffeomorphism and
(2..54)
dv
2
f---t
(v, w) is a e l orientation
+ dw 2 = .A(x, y) (E dx 2 + 2F dx dy + G dy2), .A> 0,
Now we have the following local regularity result. PROPOSITION 2.10. Let 0 C 2.6, if 1 < p < q < 00,
(2.50)
']['2
be open. Under the hypotheses of Proposition
u E Hl~~(O), Bu E LL;(O)
====?
u E Hl~~(O).
Under the hypotheses of Proposition 2.7, if 1 < P < (2.51)
= v + iw,
then the Jacobian determinant of this diffeomorphism is
lu z 12 - lu-z1 2 , and
U-z = A =
(2.55)
U
E
z
E - G + 2iF + G + 2VEG -
F2
00,
u E Hl~~(O), Bu E el~](O) ===* u E el~Y)(O).
Under the hypotheses of Proposition 2.8, if 0 < s < 1, 1 < q < p :::; (2.52)
and we set u given by J =
00,
sp
~
2,
u E Hl~~(O), Bu E B;,I(loc) (0) ===* u E B~1(lOC)(0).
PROOF. The cut-off argument used in the proof of Proposition 1.10 is also effective here. Having this local regularity result, we can return to the set-up in which A has compact support in C, (2.32) holds, and one has the solution to Bu = 0 satisfying (2.33)-(2.34). We can deduce that, if in addition A E L'XJ n vmo(1R 2 ), then 'Vu E L[oc(1R 2) for all q < 00. Furthermore, if in addition A E e(.\)(1R 2 ) with .A(j) \. 0 satisfying (2.42), then 'Vu E eL.\] (1R 2 ). In such a case, the map u is a e 1 quasiconformal diffeomorphism of C onto itself. Actually, at this point we have that u is a e 1 quasiconformal homeomorphism. To deduce that u is a diffeomorphism, we need to know its Jacobian determinant is nowhere vanishing. One way to prove this is to show that each point p E C has a neighborhood 0 on which the Beltrami equation (2.39) has a e l solution ii that is a diffeomorphism (under the hypothesis A E e(.\) as in Proposition 2.7). Then u 0 ii-I is a e 1 homeomorphism that is conformal; it is well known that this must be a diffeomorphism, and this implies that u is a diffeomorphism. The existence of such ii can be deduced from Proposition 9.3 (later in this Chapter), given the relation between solutions to the Beltrami equation and the production of isothermal coordinates on a surface with a Riemannian metric, which we now recall.
Note that the right side of (2.55) has absolute value < 1, as a consequence of the positive-definiteness of (2.54). We make some further comments about regularity results contained in Propo sition 2.10. That the solution u is e l if A is Holder continuous is classical. The regularity result (2.52) contains results in Chapter 3 of [Bli] (with a somewhat different proof). There is a regularity result given on p. 235 of [LV], which can be compared with (2.51), though neither result contains the other.
3. Estimates for the Dirichlet problem We begin with a study of the solution to the Dirichlet problem for the Laplace equation on a half space 1R~+I. Thus, we look for estimates on
u(y,x) = e- yFK f(x),
(3.1) for
f
defined on IR n .
PROPOSITION 3.1. Assume .A(j) = w(2- j ) and u, defined for z = (y, x) E 1R~+1 by (3.1), satisfies (3.2)
L: .A(j) < 00.
IU(ZI) - u(z2)1 :::; ea(lzl - Z21),
where
r w(t) dt + h 1 w~) dt. io h
(3.3)
a(h)
=
1
t
h
t
If f E e(.\), then
~~"i'~~?"~~~~?··~:::':~;"'~~"";:::·'5'.E-,~;;~~-:?!,~:::C~2~"':'?:'~~-;;-:'""~::"·~':"·::':'~:'";;-:~:~0~E""~-:-~;e;;;;;:";-;::·"'';;''~'0''Q"CW~.~~,,,-;;w.;~~:--'0'-"2'~';~~,w~-::;<::~::~_'i':~~?5'!"T:~'':'i:'?~:~:::''~-:::~~~=~~~:;:
156
APPLICATIONS TO POE .
3.
PROOF. From Proposition 1.3 of Chapter I, applied to P(~) 1~le-I~I, per the remark (1.41) given there, we have (3.4)
=
~je-I~I and to
i(h) = w(h)
(3.5)
=
Ilog\xl[-S-l ·Ixl·
===}
'\7u E CWs(JR.~+I).
+ hi
w(t) dt t2
1Iv'-~e-YvCKfllL''''::::; Ci(Y).
+ [U(YI + h, X2) =
(3.16)
Y
To deduce (3.2)-(3.3), note that
r fl w~) dt dy Jo Jy t = 2 fh w(t) dt + h fl w~) dt. Jo t Jh t
fh i(Y) dy = fh w(y) dy +
Jo
Jo
Y
Y
It is of interest to dwell on some specific examples. For s > 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
===}
s
(lR~+l). It remains to inves
gj(x) = Rjojf(x),
cw
In view of Example 1, one might think you could replace -s-l by -s in (3.13), when n :::: 2, but in fact you cannot. The factor x/lxl in (3.14) prevents such an improvement from working out. Proposition 3.1 contains the following result: 1
< 00, and f E cw (JR. n), then u = PI f, defined by (3.1), belongs to CG (JR.~+l ), with (J given by (3.3). COROLLARY 3.2. If w is a modulus of continuity satisfying fo w(t)t- l dt
As shown by Example 1 above, Corollary 3.2 is strictly weaker than Proposition 3.1. All the same, it is interesting to consider a direct proof of Corollary 3.2, so we will give an alternative proof of the estimates (3.4)-(3.6), given f E CW(lR n ). We use the integral formula
u(y,x)=cn
suitably altered for h :::: 1/2.
f(x)
cw
=F s - I .
j
°
+ C fh i(Y) dy.
Jo
I>-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- -
(3.14)
and also
U E CWs(lR~+I).
PROOF. It can be shown that, if n :::: 2, (3.12)
f(x)
PROOF. Note that
l
h
(3.11)
EXAMPLE 2. Let
0, and n :::: 1,
(3.13)
Y
(3.9)
157
II'\7xe-yvQ>fIILco ::::; Ci(Y) ,
with
where Zj
ESTIMATES FOR THE DIRICHLET PROBLEM
3.
f E C{ws+d(JR.n ).
This can be verified directly from the definition (1.64) in Chapter I; alternatively, see the computations (A.1O)-(A.15) of Chapter I (in which s is replaced by -s). Now (3.12) implie..'l f E CU,) , with A(j) = r(1+s) (d. (1.65) of Chapter I), and then Proposition 3.1 gives the implication in (3.11).
J/').
I
Ytl,)\I~-Lnl')f(x')dx'.
IR n
We use the relatively elementary fact that
foo w(r) n 1'\7xv(y, Xo)1 ::::; C y J (y 2 + r2 )(n+3)!2 r dr. o If we divide the range of integration into (0, y] and (y, 00), we readily obtain an estimate of the form (3.4). A similar attack works on loyv(y, xo)l. We next consider harmonic functions on a bounded convex domain in n c JR.n. With little effort we could extend these tesults to the case where n satisfies a
'"="ffi"
~ .-
~.~iCATlONs" TOP~----~ -
"··-···~=~""''''''''='''''''··:7"~~~;~POTENTIALS-ON Cr;-;;:'-'SURFACES
159
?:
uniform ex:terior sphere condition. The operator PI: C(an) -+ C(n) produces the harmonic function in n with given boundary value. We want to provide a sufficient condition that PI f E Lip(n). Let w(h) be a modulus of continuity. We will impose the following condition on w. Namely, let
w(lx'\)lx'l
for
COROLLARY 3.4. lfn is a bounded convex domain in IR n , the mapping property
(3.20) holds for moduli of continuity w(h) of the form (3.10), as long as s > 1. A special case of Corollary 3.4 will be useful in a study of ideal fluid flow in convex planar regions, in §5. In fact, in §5 it will suffice to know that, for such a domain n, we have
1
Ix'i <::; 2'
while 'P(x') = 1 for Ix'i 2: 1. We assume that
'0=
f
E
PI
~u
(3.24)
PROOF. Write u = v
f(O)
+ V'j(O)· x
E
---t
Lip(n).
f(O)
u E Lip(I'i).
(assuming p ,
< 00)
n) v E H l2 ,P(lR oc'
~w =
a
on
n,
w = -von an.
> 0, so (3.23) applies to w,
Here we study the method of layer potentials on domains whose boundaries are a bit smoother than C 1 • We begin with an analysis of a class of singular integral operators on n-dimensional C 1,w- surfaces on IR m , of the following sort. Assume 'P ; IR n -+ IRm is C 1 , that D'P E CW, and that D 0). Let k E COO(lR m \ 0) be homogeneous of degree -n, and odd, i.e., k( -z) = -k(z). We look at the singular integral operator with kernel k(
+ \7f(O)· x + Av(x)
- Av(x):; u(x) :; f(O)
+ w where
4. Layer potentials on C 1 ,w surfaces
1R+, [xl::; diam(n). Given f E C 1 ,W(lR n ), it
- Av(x) ::; f(x) <::; f(O)
+ \7j(O)· x
===}
long as p > n, H(;~(lRn) C Cl~~T(lRn) for some r and we have (3.25).
(4.1)
on an, for sufficiently large A. By the maximum principle, we hence have (3.22)
>n
and
Cw(lx'I)lx'l + CX n 2:: C'w(lx\)lxl,
> 0, whenever x
ul m1 = o.
g on 0.,
~v = 9 on IRn
(3.26)
1
for some C, C' follows that
=
9 E LP(n), p
(3.25)
PROOF. Let fEC ,w (an) and set u = PI f. By the maximum principle, it suffices to estimate V'u on an. Take Po E an. Translating and rotating, we can assume Po = 0 and 0. is contained in {X n > a}. Let v be given by (3.18). The hypothesis there, plus Zaremba's principle,
implies
v(x) 2::
Assume u satis
\7 f E CW(lR n ),
(3.27)
(3.21 )
.
Then
PROPOSITION 3.3. Ifn is a bounded convex domain in IRn and w is a modulus oj continuity for which the hypothesis (3.18) is satisfied, then
PI: C 1 ,W(an)
> 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(
=
k(D
+ k 1 (x,y)
=
ko(x,y)
where
+ V'f(O)· x + Av(x)
(4.2)
on n. Since v is Lipschitz, this is enough to estimate V'u(Po), and hence to prove (3.20).
kr(x, y)
=
II
\7k([D
+ TI}i(X, y)](x -
y)) dT ·I}i(x, y)(x - y),
with If w(h) has the form (3.10), then the Lipschitz hypothesis (3.18) holds whenever s> L as follows from the implication (3.13). Thus we have:
+ k 1 (x,y),
1 1
..
"' ..
···',,···•.
:r
,
i.· ~ .. :
(4.3)
w(x, y) =
[D'P(tx
+ (1 - t)y) - D'P(x)] dt.
4. LAYER POTENTIALS ON cl,w SURFACES
3. APPLICATIONS TO POE
160
Thus we have, for the singular integral operator K o given by
Kof(x) = P.V.
(4.4)
4.1. If'P [O,r), pE (1,00), PROPOSITION
sE
J
ko(x, y)f(y) dy,
K 1 : P(!R n )
(4.13)
Moreover, for
the property
(4.14)
K o E OPcwS~I'
(4.5)
Let us also assume that, for all linear maps B : !Rn ---. !RID of the form B
D'P(x) +TW(X,y),'x,y E !R n , T E [0,1], there is a lower bound B*B
(4.6)
~
KI
E :
=
el,
(4.15)
with'l/J : !Rn ---. !RID-n, of class C 1 ,w, then (4.6) holds with c = 1. Now, under our current hypotheses, we have Iw(x, y)1 ~ Cw(lx -YI), and hence
Krf(x) =
(4.10)
1
HU,P(!R n ) --+ H U+8 ,p(!Rn),
Kf(x) = p.v.
Krf(x)
=
(27T)-n
k 1(x, y)f(y) dy
11 "'1(x,y,~)ei(X-YHf(y)dyd1;.
The representation (4.2) implies that (when (4.6) holds) (4.12)
'P E Cl,w (!R")
===;- "'1 (x, y,~) E
C W S~.o·
Note also that 1I:1(X,X,O = O. Hence we can apply Propositions 9.5 and 9.18 of Chapter I to deduce:
KI
:
C~(!Rn) --+ C~+8(!R"').
1
k('P(x), 'P(X) - 'P(Y))f(Y)Q(Y) dy,
{'P(x) : x E !R n }
r= Hf(x)
=
Q
E CW(!R n ); for example
c !Rm .
1
h('P(x),'P(X) - 'P(y))f(y) Q(Y) dy,
1 +1
OjHf(x) = p.v. (4.18)
is compact from L~omp(!Rn) to Lfoc(!R n ), for all p E [1,00]. If we are interested in local analysis, we study the action of K = K o + K 1 on functions supported on some fixed compact set C c !R n, and we consider K u(x) for x E C. Thus, we can alter k(z) for Izllarge without affecting this action. We henceforth assume k(z) is so altered, to be rapidly decreasing as Izi ---. 00. In such a case, K 1 can be expressed as a pseudodifferential operator with double symbol, such as considered in §9 of Chapter I:
C:(!R n ).
with hew, z) smooth on!R m x (!RID \ 0) and homogeneous of degree -(n-1) in z (for Izi small), and even in z. In such a case, H E 0PCw S~6' It is important to note that ojH has better behavior than would ojT for a typical element T E 0PCw S16. In fact, '
\81
then the operator K 1 given by
(4.11)
(4.17)
is'
1
(4.9)
--+
Another important class of operators has the form
Ik 1 (x,y)1 ~ Clx - yl-nw(lx - yl).
1 1 w(t)C dt < 00,
K 1 : C~(!Rn)
(-r, 0],
(4.16)
In particular, if w satisfies the Dini condition
H 8 ,P(!R n ),
--+
where we have also thrown in a factor Q. We assume Q(Y) dy could be the area element of the n-surface
'P(x) = (x,'l/J(x)),
(4.8)
Cl+ 1'(!R n ), 0< r < 1, and (4.6) holds, then, for all
E
We can extend these results, replacing k(z) by k(w, z), smooth on!Rm x (!Rm\o), odd in z, and homogeneous of degree -n in z (for Izl not large, and cut off for Izi large). We can consider operators of the form
for some c> 0, independent of x, y, T. For example, if'P has the form
(4.7)
(7
161
(OJ'P)hz('P(x), 'P(X) - 'P(y))f(y) Q(Y) dy
(OJ'P)hw('P(x), 'P(X) - 'P(y))f(y) Q(Y) dy.
. Here hz(w, z) is homogeneous of degree -n in z (for Izi small) and odd in z, while hw(w, z) is homogeneous of degree 1) in z (for Izi small). Thus the first term on the right side of (4.17) defines an operator in 0PCwS~,o and the last term defines an operator in 0PcwS~6' Hence ojH E 0PCwS~,o' W~ can decompose the first term on the right side of (4.17) in a fashion similar to (4.1), to get
-en -
(4.19)
OjH=Aj+Bj ,
where (4.20)
Aj = Aj(x, D) E OPcwS~1
and B j E 0PCw S~,o has double symbol that vanishes at x = y. Thus, if w satisfies the Dini condition (4.9), then B j is compact from L~omp to Lfoc' and if'P E Cl+T(!R n ), r E (0,1), the estimates in Proposition 4.1 apply to B j . In the rest of this section we take m = n + 1, so r in (4.16) is a hypersurface. We can as well take r to be an n-dimensional CI,w hypersurface in an (n + 1) dimensional manifold M. The results derived above extend to the following setting.
4. LAYER POTENTIALS ON
Let M be a compact (n + I)-dimensional Riemannian manifold, and let n be a connected open subset of M, with boundary f = an, which we assume is a hypersurface of class Gl,w. Let P E OPScil(M) be an operator whose principal symbol P-l (x,~) is odd in ( Let b E D'(M x M) denote the Schwartz kernel of P, so that, for a fuction u on M,
Pu(x)
(4.21)
=
J
b(x, y)u(y) dv(y),
M
where dv denotes the volume element on M arising from its metric tensor. Then (with da denoting surface area on r)
Bf(x)
(4.22)
=
P.V
J
Apply the construction in the previous paragraph to this operator Q. Note that Sf = Sflr' where (4.28)
Sf (x)
=
J
E(x,y)f(y) da(y) ,
x
E
M.
r
This is known as a single-layer potential. Clearly, given any decent f, the function u = Sf is smooth on M \ an and satisfies the PDE Lu = 0 there. We produce some other important estimates, yielding behavior as one approaches an. PROPOSITION 4.2. Assume
f
(4.29)
b(x,y)f(y) da(y)
163
SURFACES
E
an is of class cHr,
G 8 (an)
==?
Sfl ll ±
E
0 < r < 1. If s E (O,r), then
CH·'(IT±).
r
defines a bounded operator on V(f), for 1 < P < write
00,
if (4.9) holds. In fact, we can
B=Bo+Bl ,
(4.23)
(4.30)
Bo E OPCw S~l(f),
and (4.25)
B l : V(r)
-----.>
LP(r) compact,
XjXkSf(x) =
J
Ejk(x,y)f(y)da(y),
1
00,
(4.31)
Sjk(t)f(x) =
if w satisfies the Dini condition (4.9). Furthermore, if f is of class C Hr , r E (0,1), then B l satisfies the estimates in (4.13)-(4.14), for all s,a E [O,r), P E (1,00). One has similar conclusions if a factor (3(x, y) is thrown into the integrand in (4.20), with (3 E CW(M x M). There is a related extension of the analysis of (4.17). Let Q E 0 P Sci 2 (M) be an operator whose principal symbol is even in ~, let E E D'(M x M) denote its Schwartz kernel, and consider (4.26)
Sf(x) =
J
E(x,y)f(y)da(y),
x E f.
r
S: LP(r)
-----.>
H1,P(r),
1
00,
if w satisfies the Dini condition (4.9). Furthermore, if X is a CW vector field on M, tangent to f, then X S : LP(r) -+ LP(r) has properties similar to those listed in (4.23)-(4.25) for B. Operators of this form arise when treating boundary problems for elliptic PDE in n, by the method of layer potentials. We illustrate this with the following example. Let ~ be the Laplace operator on M (with its Riemannian matric), and set L = ~ - V where V E GOC(M) is 2': 0 everywhere and> 0 somewhere on each = n_. Then L is invertible, with inverse Q E OPSci2(M). component of M \
n
J
Ejkhx(t), y)f(y) da(y),
all where 'Yx is the constant-speed geodesic satisfying 'Yx(O) = x, have
'Y~(O) =
V(x). We
Sjk(t) bounded in 0Pc r si.o(an),
(4.32)
and t Sjk(t) bounded in 0PC r S?,o(an) ,
(4.33)
for t E (-a, a). By interpolation, we have for any
(J E
(0,1),
\tl OSjk(t) bounded in 0pc ·si,c/(an). T
(4.34)
Then (4.27)
Ejk(X,y) = XjXkE(x,y),
all where XjX k acts on E(x, y) in the x-variable. To do this, let V be a smooth vector field on M, transverse to an, and define for t E (,.....a,a) (for a small) the operators Sjk(t) by
with (4.24)
PROOF. Let {Xj : 1 ~ j ~ N} be a collection of smooth vector fields on M such that they span the tangent space T'.",M for each x E M. We estimate on M\an the functions
If we apply Proposition 9.8 of Chapter I to the family of operators in (4.34), we obtain, for any a E (O,s), P E (1,00),
(4.35)
f
E
C 8 (an)
c Ha,p(an) =* Itl OSjdt)f
provided a + (J - 1 E (0,1). Thus, take large enough, we deduce that (4.36)
(J =
IISjk(t)fIILOO(an) ~ C
1- a
bounded in Ha+O-1,p(an),
+ c,
for small c > O. Taking P
Itl-(1-a+E:)llfllcs(i11l)'
In other words,
(4.37)
IXjXk Sf(x)\
:s: G IlfllcB(all) dist (x, an)-(l-a+E:).
164
4. LAYER POTENTIALS ON
3. APPLICATIONS 'I'O PDE
SURFACES
165
defined on an as follows. Pick 'I'J E (0,1). For x E an, let ex dist(y, an) ~ 'l'Jdist(y,x)}. Then we set
This implies that, if f E CS(an), then
Xk8f111± E C a' (n±),
(4.38)
cl,w
Va'
(4.46)
u*(x)
=
so
=
{y E M \ an :
sup lu(Y)I. yEC x
sfl l1 ±
(4.39)
E C
Hs ' (n±),
V s'
< s.
This is just slightly weaker than the asserted result (4.29). We finish the proof as follows. Namely, we borrow an argument from the proof of Corollary 3.2. Fix Sl E (s,r). Let Xo E an be given. Pick fI E C S1(an) such that fI(xo) = f(xo) and IIfIllce1 ~ Cllfllce. Then the argument so far shows that SfI satisfies
IXjXkSfI(x)1 ~ Cdist(x,an)-(1-s).
(4.40)
On the other hand, the same sort of analysis as used in the proof of Corollary 3.2 shows that (4.41)
IXjXkSU - fI)(-rxo(t)) I ~ C Itl-(l-s).
PROPOSITION 4.4. Assume an is of class Cl,w, where w satisfies (4.9). Then, for p E (1,00),
11("ilSf)*lb(811) ~ Cp llfIILP(811)'
(4.47)
While Sf(z) ---+ Sf(x) as z ---+ x E f from either n = n+ or n_, there is the following well known jump relation for the normal derivative:
(;1/ Sf) ± = (=f~I + K*)f(x),
(4.48)
where
These two estimates yield the optimal estimate (4.42)
The following is the basic estimate on ("il S f) *. It is a consequence of the results in (4.21)-(4.25). Such arguments will arise in Chapter IV, in the context of rougher boundaries, so we will not present them here.
IXjXkSf(x)1 ~ Cdist (x, an)-(1-s),
(4.49)
K* f(x)
=
P.V.
aE aI/x (x, y)f(y) da(y).
J r
and the proof of Proposition 4.2 is complete.
A derivation can be found in many places and will be omitted here. (See, e.g., Using the argument in Corollary 3.2 once again, we can establish: PROPOSITION 4.3. Let an be as in Proposition 4.2. Assume w is a modulus of continuity satisfying (4.9) and also w(h) ~ h P on (0,1], for some p E (O,r). Then (4.43)
f E CW(an)
==}
Sfl l1 ± E cl,a(n±),
where a(h) is given by
r J
h
(4.44)
a(h) =
o
w(t) dt
t
+h
1
w(t) dt.
~
Cw(lx - yl)
Ix _ yl-n.
(4.51)
K* : P(r)
----+
P(r) is compact,
~ p ~
1
00,
t2
PROOF. Pick s E (p,r). Given Xo E an, pick fI E CS(an) such that h(xo) = Proposition 4.2 implies that SfI 111± E CHS(n±) satisfies the estimate (4.40), and the previous argument applies to give
f(xo).
(4.45)
lavxE(x,y)1
(4.50)
We deduce that
1
h
[Mik]; compare also calculations in Chapter IV.) The integral in (4.49) is a weakly singular integral. To see this, note that since the leading singularity in E(x, y) is equal to em dist(x, y)-(m-2) (for m = n+ 1 ~ 3, with a similar formula involving log for m = 2), one readily verifies that
-y(t)
IXjXkSU - fI)(-rxo(t)) I ~ C-t-,
with -y(h) as in (3.5). The conclusion in (4.43) follows from this, via calculations parallel to (3.7)-(3.9). For more general boundaries an. of class c 1 ,w, with w satisfying (4.9), we will describe an estimate on the non-tangential maximal function of "ilS f. Generally, for a function u defined on M \ an, the non-tangential maximal function u* is
if w satisfies the Dini condition (4.9). The general constructions described in the beginning of this section apply to K*, to yield K* = Kt + Kt, with Kt E OPCWS~I(r), and Kt E 0PCwS?,o having double symbol vanishing at x = y (mod S16). For this operator there is a further special feature, following from the estimate (4.50). Of course, the Schwartz kernel of Kt also satisfies such an estimate. Hence, so must the Schwartz kernel of Kt. This implies that the principal symbol of Kt is identically zero. Hence, if f = an is of class cHr, r E (0,1), then, for all s E [0, r), p E (1,00),
cw
(4.52)
K* : P(f)
----+
HS,P(r),
K*: C~(f)
----+
C:(r).
Moreover, for a E (-r, 0],
(4.53)
K* : Ha,P(r)
----+
Ha+s,P(r) ,
K* : C~ (f)
----+ C~+s (r).
---_._-_.__ . --_._--------3. APPLICATIONS TO.PDE
4. LAYER POTENTIALS ON cl,w SURFACES
In the study of layer potentials, one considers both the single-layer potentials (4.28) and the double-layer potentials
PROOF. Compactness results on K and K* established above imply that all these operators are Fredholm of index zero. If we can show that ~ I + K* is injective on each space listed above, it will follow that ~ I + K* is an isomorphism on all these spaces; hence, by duality the isomorphism property follows for ~ I + K, in cases (4.61)-(4.62). Thus we tackle injectivity of ~ I + K*. We start with the case where an is of class C1+ r for some r E (0,1). Suppose f E HS,p(an) belongs to Ker(~I + K*), for some s E (-r, r), p E (1,00), so f = -2K* f. Repeated use of the mapping properties in (4.53) (plus the Sobolev imbedding theorem) show that, in this case
166
(4.54)
Df(x) =
aE aV (X,y)f(y)da(y),
J r
x E M\f.
y
In this case, the jump relations are
(Df)±(x)
(4.55)
=
(±F + K)f(x),
x E f,
where
Kf(x) = P.V.
(4.56)
aE . av (x, y)f(y) da(y).
J
y
r
As the notation indicates, K and K* are adjoint operators. Parallel to (4.50), we have
lav,E(x,y)!:::; Cw(lx - yJ)
(4.57)
Ix -
Yl-n,
and the properties (4.51)-(4.53) apply also to K, under appropriate hypotheses on
an.
There are also results parallel to Propositions 4.2-4.4. For example, if an is of class C1+r, r E (0,1), and if s E (0, r), then (4.58)
f E C"(an)
===?
Dfl n ±
E CS(O±).
Dfl n ±
E CCT(O±),
If w is as in Proposition 4.2, we have
f E CW(an) ===>
(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 <
00,
via an argument used in [FJR]. Pick Xo E an. Given 0 > 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,
1
and (4.76)
-~;_-:"''''~
__
.
",..,
''':''''~~''''''''''-''·::_~~"JfZtr'':''"''"''~''"",-"'~"~'~"~~"'~-~'-"
. - ._.- ...
.. --
S: c:(an)
----t
c;+S(an),
-r
_.. ,:_~~_":W"~~:::-_~"_'
.~:':'
< 00,
•• "
4. LAYERpOTENTIALS ON-·~l.w SURFACES
169
PROOF. Our first order of business will be to show that S is Fredholm in the cases listed in (4.74)-(4.76). To do this, we make some modifications. First, we can refine the differentiable structure on an to a smooth structure (via a process simpler for a hypersurface of a smooth manifold than for the general case), impose a smooth Riemannian metric on an (different from the metric we have been using) and consider D = d + d*, acting on sections of A*T*(an) = EB7=oAjT*(an), the adjoint d* defined using this new metric. So far, S acts on sections of AUT*(an). Using local trivializations and a partition of unity we can define operators we also denote S on sections of AjT*(an), for 1 S j ::::: n. Now consider T=DS,
acting on sections of A*T*(an). By (4.18)-(4.20) we have T E 0PCwSPo, with a ' decomposition similar to (4.19), Le., (4.78)
T= To +Tl ,
with To E 0 PCw S~l' and with T l E 0PCw Sr.o having double symbol vanishing at x = y. Now the crucial thing to note is that To E OPC wS~l is elliptic.
(4.79)
in the form
(4.73)
n_
(4.77)
an
(4.70)
'..,,"
. . ~. -~--_. u_~
It follows that To is Fredholm on LP(an), 1 < P < 00, if w satisfies (4.9). It also follows that To is Fredholm on HS,p(an) for -r < s < r, 1 < P < 00, and on c:(an) for -r < s < r, if w(h) = h r , r E (0,1). Also, T is Fredholm on these spaces. Since D = d + d* is a first-order, elliptic differential operator with smooth coefficients, the Fredholmness of the operators in (4.74)-(4.76) follows. Furthermore, since S has a positive, real principal symbol, all these operators have index zero. It remains to establish injectivity of Sin (4.74)-(4.76). Again we first tackle the case when an is of class cl+r, r E (0,1), so cw = cr in (4.74)-(4.76), and in particular T l satisfies the estimates in (4.13)-(4.14). Then applying symbol smoothing to To to write To = Tt + T8 with Tt E 0 P Sp,8 elliptic, T8 E 0 Pcr S-;~8, ,
taking a parametrix in OPSP.8 for To#, and using arguments as in §§1-2, we have that, whenever f belongs to one of the domain spaces in (4.75), then (4.80)
Sf
Thus, considering u
=
0 ==? f E HS,p(an),
\if s
< r,
p
< 00.
= Sf, we have Lu = 0 on n+ and on n_,
ul n ±
E Cl+S(f2±)
for all s < r, and ulan = 0, so u = 0 on M if Sf = O. Then avu± = 0, so f = O. This establishes injectivity, hence isomorphism, in (4.75) and also in (4.76), when an is of class c1+r. The injectivity property in (4.74) follows as in the proof of injectivity of ~ I + K* on L 2 (an) given in Proposition 4.5. This completes the proof of Proposition 4.6. We briefly describe how one can refine the differentiable structure of an as indicated in the proof above. Let V be a smooth vector field on M, transversal to an. Define a function
< s < r. V(
= 1,
J.
IU t
LIGAI'IONSro-PPE~--~~'~~~~~~~~~~~~- ~~~~----~--~~~~~.~--~~----
4. LAYER POTENTIALS
ON
Cl,w SURFACES
171
Then 1> E C 1,W(0). Approximate 1> by 'II E Coo(O), using a mollifier, and let E = {x EO: W(x) = O}. Then E is a Cc>o 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, <"t) E c k-£+rs-2-£ cl'
(5.7)
Note that we take F = F(y,~) rather than F = F(x, O. This is technically convenient, to deal with the limited regularity of the symbols we work with. If L(x, D) is a differential (or pseudodifferential) operator with standard symbol rep resentation, then
(5.8)
L(x, D)Fu
=
(27r)-n
II
L(x, ~)F(y, Oei(X-YHu(y) dy d~.
This can be treated by methods used in §9 of Chapter 1. Under our hypotheses on L = -~ + V, we have, in a local coordinate system
(5.9)
+ Ll(X,~) + Lo(x,~), homogeneous polynomial of degree j in C and L(x,~) = L 2(x,O
with L j (x, 0 a coefficients that are functions of x belonging respectively to C k+r , to C k- Hr , and to c(k-2+r)t. (In fact, L2(X,~) = Lgjk(x)~j~k and Lo(x,~) = V(x).) We can then obtain F(y,~), satisfying (5.6)-(5.7), such that
(5.10)
L(x, D)Fu
=
u(x)
+
I
Irdx, y)1 ::; Clx - yl-(n-k-r).
We give the details of this for the case k = 1; there is a routine extension to larger k.
'VyL2(Y'~) + (x - y). A2dx,y,~)
and L 1 (x,~) = L 1 (y,~)
+ AlO (x, y, ~).
We have A21(X,y,~)
(5.14)
=
~.
A 21 (X,y)C
where L2(X,~) E CHrS;1
(5.15)
===}
IA 21 (X,y)1 ::; Clx _ ylr.
Also we have AlO(X,y,~)
(5.16)
Ll(X,~) E C2S~1
(5.17)
= AlO(X,y)~,
===}
IAlO(x,y)! ::; Clx _ ylr.
Note that (5.18)
II =
~
[(x - y) . 'V yL2(Y,~)
IlL
+ (x -
at [a~ L 2(y, 0
y) . A21 (x, y, ~)] F2(y, ~)ei(X-Y)-~u(y) dy ~
. F2(Y,~) + A 21 (X, y, ~)F2(Y, ~)] ei(x-y),~u(y) dy~.
1(>1=1
Now, define F2(Y,~) and F3(Y,~) by L2(y,~)F2(Y,~) = 1,
L2(y,~)F3(Y,~) = -Ll(Y,~) - ~
(5.19)
L
at[a~L2(Y'~)' F2(Y'~)]'
1(>1=1 We obtain
LFu(x) (5.20)
=
u(x)
+ (27r)-n II{~
L
anA2dx,y,~)F2(Y'~)]
1(>1=1 + AlO(x, y, ~)F2(Y,
0 + L 1 (x, OE3(Y'~)
+ Lo(x, ~)F(y,~) }ei(X-YHu(y) dy d~.
rk(X, y)u(y) dy,
with
(5.11)
= L 2(y,O + (x - y).
and
where
(5.6)
L2(X,~)
(5.12)
(5.13)
We can give more precise results on the smoothness of the terms E 2H (y, z), via (5.7) below. Regarding the appearance of Jg(y) in (5.3), note that, in local coordinates, dv(y) = J g(y) dy. As we will see, Theorem 5.1 follows from a variant + L, acting on functions on lR x M. of Theorem 5.2, applied to Note that, if we omit the remainder on the right side of (5.3), we have the integral kernel of an operator F, also given by
(5.5)
In this case, we evaluate (5.8) by setting
0 and homogeneous of degree -(n - 2 - f)
IRk(X,y)1 = o(lx - yl-(n-2-k)).
(5.4)
175
This in turn yields (5.10)-(5.11), in the case k = 1, since the quantity in brackets in the integral on the right side of (5.20) is a sum of terms homogeneous in ~ of degrees -1, -2, and -3, and the terms homogeneous in ~ of degree -1 are O(lx To estimate Rk(X,y) in (5.3), given (5.10)-(5.11), note that
yn.
(5.21 )
LxRdx, y) = -fk(X, y).
1ilJ{'"
~- -
diiti--·'.---
. . ; ; : --~.;;.,
176
;;~--
--
mi"."~:itSi:(
:iiE'...,~·-·--...~~---
-.~:;,,~'i;-,~;~-;;;~'_;;;~~<W.".,;j~'~.ci4"-nJi1f'firf£"~~ii~~t~~f~::"i~:c;rr:" i·?·i'~:',··
3. APPLICATIONS TO PDE
Now (5.11) implies that, for given y in 0, a coordinate chart on M, f
(5.22)
k (·,
y)
E
U(O),
If p
< __ n
n-k-r
Given our hypotheses on the coefficients of L, techniques used in §1 (or even simpler, more classical methods) give (5.23)
Rk(·,y)
E H
2,P(0),
< n_~_r
If p
Hence, by the Sobolev imbedding theorem, (5.24)
n Ifq
Rk("Y) E U(O),
Note that, under our hypotheses, n - k - r - 2 > 0. To obtain estimates on Rk(X, y) of the form (5.4), we do the following. Suppose Ixo - yl = 2p. We want to estimate Rk(x, y) on {x : Ix - xol < p/2}. Shift coordinates so Xo = 0, and introduce the dilation operators (5.25)
If u(x)
up(x) = u(px), =
Rk(X, y) for
t1 pu p - p2Vpu p = - / f kp ,
=
g~k(x)
fkp(x) = fk(px, y),
= gjk(px).
:s
Ilfkpllu>c
(5.29)
:s C p~(n-k-r).
Now the estimates (5.23)-(5.24) on Rk("Y) imply for its dilates
IlupIILq(B1 )
(5.30)
:s C(q) p-n/ q ,
If q
_esVir.p(x),
II\7upIILQ1
:s C(ql) pl-n/
q1
,
If ql
< n_ k~r_
l'
Note that q:S n/(n - k - r - 2) => 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)
'..:<
(5.36)
Ixl:S 1.
Ixl :s p, then up satisfies the equation
(5.26)
.~.
> 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 <
00,
tifj = Wj,
fj
1
OtUj
+ 'VUjUj
= -'VPj,
lao j = O.
Thus, if 0 CC ON, we have
(6.8)
IIUj(t)IIHl,=(O) :::: C(O),
'if j::::: N,
= {u: DUu
lal
where we set
Hk,OO(O)
(6.9)
E BMO(O),
div Uj = 0,
S;
k}.
Hence
(6.10)
II'V Uj(t)Uj(t)IIBMO(O)
::::
'if j ::::: N.
C(O),
In the analysis of solutions to (6.3), it is convenient to rewrite the system as OtUj
(6.11)
+ Pj ('V Uj Uj)
Uj(O) = Vj,
= 0,
where Pj is the Helmholtz projection onto the space of divergence-free vector fields on OJ tangent to the boundary. Also, it is useful to have the identity
(6.12)
'Vujuj=div(Uj0Uj),
if divuj=O.
The pressure Pj is eliminated from (6.11), but it can be recovered as the solution (well defined modulo a constant) to the boundary problem
(6.13)
tiPj = -Tr(('VUj)2)
~:
on OJ,
Vj -+ v E C (0).
We want to construct a weak solution to the Euler equation for incompressible flow on 0, with initial data v, as a limit of (a subsequence of) solutions Uj E COO(1R x OJ) to (6.3)
179
Now div Uj = 0, *duj = Wj is an elliptic system. In fact, we can write
M
(6.1)
..•- - , - _.._ ,._._._,_ ..
,"""",,,,,,,,,,,,~~,",,,,,~~,.=~
6. EULER FLOWS ON ROUGH PLANAR DOMAINS
3. APPLICATIONS TO PDE
(5.42)
=
~_~i'_5~_i'i~'M'ffi~~~~"'~'!Wji\iL
= -Kj IUjl2 on oOj,
where Kj(X) is the curvature of oOj at x. We now assume that 0 and all the OJ are convex domains in lR. 2. We can then use Kadlec's formula,
L J!ok opul 2dx = JItiuI
Uj(O) = Vj'
k,£ OJ
2
dx
+ (n -
JlovuI
2
1)
H(x) dS(x),
aD]
OJ
We require Uj to be vector fields on OJ, tangent to oOj. The functions Pj give the fluid pressure. We will obtain certain uniform bounds on the Uj. First, there is the well-known conservation law
(in n dimensions, where H(x) is the mean curvature of oOj, d. [Gri], p. 139), to obtain a uniform bound IIfjllH2 :::: Cllwjllu, and hence, by (6.5),
(6.4)
This in turn implies
IIUj(t)llu = Ilvjll£2
s; C.
Next, we have a vorticity estimate. Recall that, if Uj denotes the I-form on OJ corresponding to Uj via the Riemannian metric, then the vorticity is given by Wj = *duj' As a consequence of the 2D vorticity equation, OWj - + 'Vu.Wj ]
at
= 0,
(6.15)
Ilfj(t)IIH2(Oj) :::: C 2 <
IIUj(t)IIHl(Oj) ::::
c.
IlfJ(t)IILip(TI
j
)
::::
C3 <
which implies
IIWj(t)llv>o
=
IldvjllL= s;
K.
00.
Furthermore, by Proposition 3.5, we have a uniform bound and hence (6.16)
we have (6.5)
(6.14)
(6.17)
IIUj(t)lIu>o(o]) S; C.
00,
!lfJliLip ::::
CllWjIlu>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 <
00, S
< 1,
where E j is a sequence of extension operators: 1
E j : H (Oj)
(6.23)
-t
1
H (0),
{Ujlo:j~N}
for any T
< 00,
S
2
E j : L (Oj)
-t
2
L (0),
< 1, P < 00. Hence
Now, perhaps passing to a subsequence, we obtain (6.26)
EjUj
-t
U E L OO (JR,H 1 (0)) n LOO(JR x 0),
with convergence in the weak* topology. Furthermore, (6.22) implies (6.27)
EjUj
for any given T
(6.28) for any T (6.29)
< 00,
S
-t
U in norm, in C([-T, T], H 8 (0)),
< 1. Hence
Ejuj®Ejuj-tu®u weak*in
L OO ([-T,T],L 2 (0)),
< 00. Boundedness results above then imply EjUj®EjUj-tu®U weak*in L OO (JR,H 1 (0)) nLOO(JR x 0),
div(EjUj®EjUj)-t div(u®u) weak* in L OO (JR,L 2(0)).
To see that (6.31)
U
solves the Euler equation, we rewrite (6.3) as BtUj
+ div(uj ® Uj) = -VPj,
div Uj = 0,
where (6.32)
OO (JR,H 1 (0)),
VPJ = -(1- Pj) div(Uj ® Uj)
+
div(u ® u)
=-
div
Vp,
U
= O.
U(t) E VO(O)
= L2 -closure of {v
E CO'(O, T) : div v
= O}.
In fact, for each j < 00, Uj(t) belongs to the analogous space VO(Oj), by the standard theory for smooth boundaries. Hence, for each t, there exist Vj (t) E CO'(Oj, T) C CO'(O, T) such that div Vj(t)
j Iluj(t) - vi(t)llp ~ 2- .
= 0,
It follows from (6.27) that Vj(t) - t u(t) in norm in L 2 (0) as j - t established. Regarding other properties of the limit, we have from (6.25) (6.38)
U
00,
so (6.37) is
E C(JR x 0)
and, from (6.8), (6.39)
U
E L oo (IR, H 1 ,OO(O)),
\;/ 0 Cc O.
We derive some further local bounds, starting with bounds on the pressures Pj, which solve (6.13). Given 0 CC ON, the right side of the first equation of (6.13) is bounded in LP(O) for j ~ N, for any P < 00, as a consequence of (6.8). The global bound (6.33) plus local elliptic regularity yields I!VPjl!Hl,P(O) ~ Cp(O),
(6.40) for j
and hence (6.30)
BtU
compact in C([-T,T],H 8 ,P(O)),
{Ujlo: j ~ N} compact in C([-T,T] x 0).
(6.25)
P weak* in. L
We next show that, for each t, (6.37)
with uniform operator norm bounds. Also, interpolation of (6.21) and (6.8) yields, for any 0 CC ON,
(6.24)
-t
and then results above yield
I!Bt Uj(t)llp(flj) ~ C.
(6.21)
(6.22)
Ejpj
(6.35)
Thus we have
~
N. Hence, if we use the equation
(6.41)
BtUj
+ VUjUj = -VPj
plus the bound (6.10), we get (6.42)
IIBtUjIIBMO(O) ~ C(O).
In the limit, these yield associated bounds on u. We summarize what has been established:
7. PERSISTENCE OF SOLUTIONS TO SEMILINEAR WAVE EQUATIONS
PROPOSITION
n be a bounded convex domain 1 v E C (0) n LipeD) n Vo(O)
6.1. Let
be given. Then there exists a solution (6.43)
Ut
+ 'V u.U = - 'VP'
U
divu
in JR2. Let
(6.44)
Ut
E L oo (JR, HI (0))
with
(6..52)
r =
. 4 2p _ P E (1,2], prOVIded pE
=0
in R. x 0,
u(O)
= v,
n LCXJ(R. x 0) n Lip (R., VO(O)),
The second additional tool we need is an LT-operator norm estimate on Pj' In [FMM] it is proven that p] acts on LT(Oj), for 3/2 ~ r ~ 3, with uniform bounds C = Cr, as long as aOj satisfy uniform Lipschitz estimates. That situation can be assumed to hold for our family of convex domains, so we have, in place of (6.20),
IIPj
(6.53)
2
E LCXJ(JR., L (0)),
'VP E L=(R., L 2 (0)).
div (Uj(t) ® uJ(t))IILr(!lj) ~ C,
provided (6.51)-(6.52) hold with r E (3/2,2]. Note that
Furthermore, given any 0 CC 0, we have
(6.45)
U
E
L CXJ (R.,H 1 ,X(O)),
Ut
E
LOO(ffi., BMO(O)),
(6.54)
v p < 00.
'Vp E LOO(ffi., H 1 ,P(O)),
We can extend Proposition 6.1, obtaining some results of a flavor such as ob tained for flows in all ofJR2 in [DiPM]. That is, we can obtain weak solutions to the Euler equations (6.43), with initial vorticity in LP(O), for some range of p. Some of these results follow from the analysis done above. Indeed, given
IIWj(O)IILP
(6.46)
~
v
E
VO(O)
n H1,P(0),
provided p > 2. We can go further, obtaining weak solutions for smaller p. To do this, we need a couple of more tools. One is the following result of V. Adolfsson [Ad], extending the consequence of Kadlec's formula described above. Namely, for our sequence of convex domains OJ, and for Ii satisfying (6.7), there is for each p E (1,2] a uniform estimate
(6.48)
Ilfj(t)IIH2,P(ll j ) ~
CpIlwjIILP(ll
j )'
This gives, in place of (6.15),
(6.49) If p
IIUj(tlIIHl,p(!lj) ~ C.
(6.50)
Iluj(t)IILq(lljl ~ C,
q=
if p E (1,2). Hence, in place of (6.19), we have
(6.51)
11\7 u, u](t)llu(rl
J )
:::;
C,
==?
r E
G,2].
As before, we can push through variants of (6.21)-(6.37), and obtain the following. PROPOSITION
6.2. Let 0 be a bounded convex domain in v E VO(O)
(6..55)
ffi.2. Let
n Hl,P(O)
be given. Assume p E (12/7,2). Then there exists a solution U to the Euler equation (6.43), satisfying U
(6.56)
Ut
E LOO(ffi., Hl,P(O)
n VO(O)),
E Loo(J~,
\7p E
U(O)), L=(ffi., LT(O)),
with r given by (6.52).
We end this section with a remark on planar Lipschitz domains. Namely, the Lipschitz estimate (6.16) and wnsequent L=-estimate (6.17) on the fluid velocity can definitely fail. To see this, start with f(x) = log Ixl on the annulus 0 = {x E R 2 : 1/2 :::; Ixl ~ I}. It is harmonic, and u = *df defines a vector field U on o which gives a steady, irrotational solution to the Euler equation. Now one can easily construct a conformal mapping <11 : 0 ---> 0 of 0 onto a planar region 0 with a corner whose internal angle lies in (1T, 21T), such that 9 = f 0 <11- 1 is not Lipschitz, and v = *dg is not bounded on 0, nor does it belong to Hl(O), though it does still generate a steady irrotational flow. FUrthermore, one can arrange that v not be in Hl,P(O) for p in the range considered in Proposition 6.2. This illustrates that different methods will be required to prove existence theorems for Euler flows on such Lipschitz domains.
< 2 here, we cannot expect the Loo-estimate (6.17). Sobolev's theorem yields
2p 2 _ p'
C72 , 2]
pE
C,
we have Ilwj(t)IILP ~ C. As long as p ~ 2, the bound (6.14) still holds. If p = 2, we do not quite get (6.16)-(6.17), but if p > 2 we have IlfjllLip ~ C1lwjIILP, so we do get (6.16)-(6.17) when (6.46) holds and p > 2. The rest of the arguments involving (6.18)-(6.37) then go through (except we get weaker bounds on 0 CC ON), and thus Proposition 6.1 extends to the case of initial data satisfying
(6.47)
(43,2.]
to the Euler equation
satisfying U
183
7. Persistence of solutions to sernilinear wave equations Let M be a Lorentz manifold. For simplicity, assume M = lR. x X where X is a compact Riemannian manifold, of dimension n, and the metric on M is gM = -dt 2 + 9x· We consider hyperbolic systems of the form
(7.1)
Ou = B(x, u, \7u),
7.
TO PDE
where Du = ElFu - D.u, D. being the Laplace operator on X. Also, V'u denotes (Ot U, V' xu). We take u : M - t IRk. We assume B(x, u,p) is smooth in (x, u) and a quadratic form in p. As is well-known, given initial data
u(O) = f E Hs+l(X),
(7.2)
Otu(O) = g E HS(X),
PERSISTENCE OF SOLUTIONS TO SEMILlNEAR WAVE EQUATIONS
Our next step is inspired by [BKM]. As in Proposition B.1.C of [T2], we have
IIV'uIIHS]
/IV'xullLoo ~ CIIullc; [1 + log ~~ ,
(7.12) given s > n/2. Also
there is a local solution
u E C(I, Hs+I(X)) n C I
(7.3)
(I, HS(X)), 1= (-TI, T2 ),
for some T j > 0, provided s > n/2. Also, given a bound
Ilu(t)llc'(x) + /lotu(t)IILOO(X) ~ K,
(7.4)
tEl,
PROPOSITION 7.1. The solution to (7.1)-{7.2) extends to an open neighborhood
of [-T I , T2 ] provided Ilu(t)llc;(x) + Ilotu(t)llc2(x) ~ K,
tEl.
PROOF. Let J e be a Friedrichs mollifier commuting with D., e.g., Je = ee/l. Set A = ';1- D. E OPSI(M). Now, in general, for sufficiently smooth w(t,x),
~ (IIV' xwlli2 +
(7.6)
Ilotwlli2)
(7.13)
= 2( Wt, Wtt - D.w).
(7.14)
Ilot AS Jeulli2)
=2(AsJeUt,(ol-D.)AsJeu) = 2(A SJeUt, N Jc;B(x, u, V'u)) ~ 211A s J eot ull£2
liAS B(x, u, V'u) 11£2
~ Cs(lluIILoo) IIV'uIILoo{ IIV'uIIHs
+ IIV'ullc;' (1IuIlHs+l + I)}.
Thus, if
Ne(t) = IIV' x As J eu(t)lli2 + Ilot AS J eu(t)lli2,
we have, under the hypothesis (7.5),
(7.10) €
Ne(t) "'"
~C+C
1 t
(11V'u(S)/lLoo + C)No(s) ds,
0, we have
No(t)
dT -00 TlogT ,
We now establish an estimate that implies (7.8), upon setting v = V't,xU and a=1. PROPOSITION 7.2. If B(x, u, v) is smooth in (x, u) and a quadmtic form in v, then, for s > 0, 1 < p < 00, and 0 < a, (7.16) x {llv/lHs,p(X) + Ilvllc;"(x) (1IuIIH"+"'P(X) + 1) }. PROOF. Write B(x,u,v) = B(x,u)v· v = Wv· v, where W matrix-valued function. By (0.22) of Chapter II, we have
(7.17)
~ C + C it (l1V'u(s)lluX> + 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
Hence, under the hypothesis (7.5), if s > 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
00,
(8.4)
l [ f Pu - P(fu)]vdx
=
h
1
f[(Pu)v - u(ptv)] dx,
Qtpu
(Note that h takes values in (8.9)
L Q;PJu E U(]Rn),
=
]Rf.)
l(f,pu. Qv)1 ~
Then, if supp f
1"
> p.
c K is compact in ]Rn,
CpKllfllbmo(llullLr + IlhIILr)llvIILpl.
PROOF. We have
(8.10)
(f, Pu· Qv) =
1
f(Pu) . (Qv) dx =
1
vQt(fPu) dx
= (v, [Qt, Mf]Pu) + (fv, Qtpu).
one obtains t II(Pu)v - u(p v)llj)l ~ CpllullLpllvllLP"
=
P E opsf,a,
of [CRW], which was established in §10 of Chapter 1. Using the identity
(8.5)
= 2)Pj u)(Qjv).
Assume that p E (1, (0) and
Then
(8.3)
find it convenient to replace the BMO-seminorm by the slightly stronger bmo-norm, given by
]Rf,
8. Div-curl estimates
(8.1)
187
1
00,
P E opsf.a,
which in turn was shown in [CLMS] to yield a number of estimates, including the div-curl estimate mentioned above. We recall a number of div-curl type results established in [CLMS], and present proofs. We use two general principles to derive these results. One is an "abstract div-curl lemma," generalizing (8.5). This result, given as Proposition 8.1, was formulated by the author and P. Auscher. The other is a "super-commutator esti mate," given in Propostion 10.3 of Chapter I. Rather than dealing specifically with Hardy spaces, we show directly that the relevant quantities can be paired with elements of BMO. Due to the famous duality established in [FS], this is the same thing, but in fact it is the HMO-pairing that is most directly useful. Actually, we
Now the hypothesis (8.8) implies
(8.11)
l(fv, h)1 ~ CK IlfllbmollvllLP,llhllLr,
while the basic commutator estimate (8.3) implies that (8.12)
II[Qt,Mf]wIILP ~
Cpllfllbmollwlb·
Hence we have (8.9). We note that the div-curl lemma and a number of variants are special cases of this result.
188
8. DIV-CURL ESTIMATES
3. APPLICATIONS TO PDE
1. The div-curllemma.
4. Estimate of du /\ dv.
Here, N = n, k = n, £ = 1. We take U = (UI,""U n ), Pju QjV = {)jA -IV, Le., Pu = u, Qv = V'(A-IV). Then (8.13)
PU' Qv =
189
I: uj{)j(A -IV),
Uj, and
Qtpu = A-I div U,
Let U be a j-form and v a k-form on M, j + k ::; n - 2. Let f be an i-form, £ = n - j - k - 2. We set U = A-lit, v = A-lv, and desire to estimate (8.21 )
/ f /\ du /\ dv = (WfdA -lit, 8A-1
* v).
j
Since WfA -lddA-1 = 0, the right side is equal to
so, taking w = A-IV, we have (8.14)
I(f, U . V'w) I ::; CpK IIfllbmo (1IuIILP +
Iidiv uIIH-l.r) IlwIlHl. p l ,
(A -ldWfdA-lit, *v)
(8.22)
=
([[A-ld, Wf]]dA -lit, *v).
Applying Proposition 1.10.3, we deduce that which implies the result stated in (8.1)-(8.2).
Here, k = £ = n, N = n 2 . We take U = (Ul,"" un), V = (VI,"" v n ), and Pu
(8.15)
=
(OjA-IUi)'
Qv = ({)iA-IVj).
5. Estimate of dUl /\ ... /\ dUk. Assume Uj are £rforms, Then we claim that
Then,withit=A-lu, v=A-lv, (8.16)
Pu· Qv
=
l: ",J
{)it'i {)Vj {)Xj {)Xi '
(8.24)
I(I, I: 'ilj
1/
J
CpK Ilfllbmo(llitIIHl. P + Iidiv itllu) IlvIIH1,p l .
(8.26)
Let U = (Ul,U2) : ~2
with
If we set w = A-luI, V = A-1U2, then
1 1 -+···+-=1, PI Pk
2. We take
Pw = ({)lA-I W, ~A -IW),
Qv = ({)2A -lv, -{)IA-IV).
Then (8.19)
Pw· Qv = det V'u,
so the conclusion is (8.20)
Then, with v = (8.28)
Qtpw = -{)1{)2 A-2 W + {)1{)2A-2 w = 0,
l(f,det V'u)1 ::; CpKllfllbmo ·lluIIIH1,P ·ll u 21IHl,p / .
There are a number of generalizations of #3 involving estimates of wedge prod ucts. We take a look at some of these. While the special case (j + 1) + (k + 1) = n of the next result can be easily deduced from Lemma 8.1, we find it more convenient to use Proposition 1.10.3 to establish the general result.
PkE(I,oo).
IIUIIH1,P ::; ClluIIIH1'Pl .. ·lluk-lIIH1,Pk-l' 1 1 1 ) I - + ... + - - = -, P E (1,00 , Pk = P . PI Pk-l P
(8.27) (8.18)
f be an (n - m)-form.
dUl /\ ... /\ dUk_1 = du,
det V'u(x) = ({)IA -IW)({)2A -2 v ) - ({)2A-lw)({)IA-IV). =
m ::; n. Let
To prove this, note that, since dUl /\ ... /\ dUk-l is exact, we can use Hodge theory to write
3. Estimate of det V'u, n = 2.
Thus, here k = £ = 1, N = 2, n
=
provided Pj E (1,00] and
1
----> ~2.
+ 1)
f /\ dUI /\ ... /\ dUkl ::; CpllfllbmollulllH1,Pl . .. llukIIHl,Pk,
(8.25)
;:i ~~J) I::;
2:7=1 (£j
(QtpU)j = -{)jA -2 div U,
so the conclusion is (8.17)
1/ f /\ du /\ dvl ::; CpllfllbmolluIIH1,pllvIIHl,pl.
(8.23)
2. Estimate of 1r((V'u)(V'v)).
Uk,
we have / f /\ dUl ... /\ dUk = /
f /\ du /\ dv,
The estimate (8.23) applies to the integral on the right side of (8.28). This proves the desired estimate (8.24). The case k = n, £j = 0 yields a Jacobian determinant estimate, which played a particularly significant role in [eLMS]. 6. Estimates on solutions to Maxwell's equations. Let :F be a 2-form on Minkowski space ~3,1, representing an electromagnetic field. Part of Maxwell's equations is (8.29)
d:F = O.
.~ -,-~-190
=
5
~~_~~~~~. __",~",-,=~ _,,__ -_oc~
3.
If FE Lroc' we can write F
If f
(8.30)
AP;-LIC~;-ONS TO POE
.__, ._. ,._" .--
~'?"F="';;~-;--~""""""-=="'''''''='=--'"''-''''"_..
_
8. DIV-CURL ESTIMATES
= dA, A E Hl~c' Thus by #4, we have
PROOF.
Given KeRn compact, suppose supp f
C
K, and write
(1, v . Qu) = (1v, Qu).
(8.37)
F 1\ FI :<:; C K IlfllhmoIIFlli2CO)'
191
Now (8.34) implies
for f supported in K cc O. If E and B are the electric and magnetic fields encapsulated in F, i.e., (with t = xo)
B j EOPs-rn j , REOPS-I.
QU=B1A1U+A~B~u+Ru,
(8.38) Then
3
(8.31)
F
=
L EjdXj 1\ dt + B
1 dx 2 1\
dX3
+ B2dx3 1\ dXl + B3dxl 1\ dX2,
j=l
d*9 =
(8.32)
o.
*
f(F,9)
wi :<:; CKpllflllllnoIIF\lLpco)1191ILplCO)'
where bj(x,~) are homogeneous of degree -mj in ~, principal symbols of B j E OPS-rnj. Here, bj(x,~) takes values in £(lR fj ,lR k ). We desire to estimate v· Qu. 8.2. Assume that u
E
LP(lR n ) , v
E
£P' (IR n ) , 1 < p <
that
(8.35)
A 1u E H-rnl,T(R n ),
E
LT ,
B 2 A 2v
E
LP,
so we have (8.41)
l(1v, B 1 A 1 u)1
+ 1(1 B 2 A 2 v, u)1
:<:; CKllfllbmollulbllvllLpl.
A 2v E H- rn 2,p(lR n ),
r
> p, p> p'.
(8.42)
l(1v, Ru)1 :<:; CKllfllbmollullLP tlvllLpl.
Finally, since B 2A 2 E OPSo, the basic commutator estimate (8.3) gives
II [B 2 A 2 , Mf]vII LP'
:<:; CllfllhmollvllLpl,
so
q(X,~) = bdx, ~)al(x,~) + a2(x, ~)tb2(X,~)t,
PROPOSITION
B 1 A 1u
(8.43)
If 9 is written in the form (8.31), with E j replaced by Dj and B j replaced by H j , then (8.33) is equivalent to an estimate on E· D - B· H. Compare the analysis of (29) in [CLMS]. We now establish some results that include a variable coefficient version of The orem VI.1 in [CLMS], and that complement estimates in [BH]. (See also [LMZ].) This work arose from conversations with P. Auscher, following up our work in [AT]. Let A j E OPsmj be matrices of operators, A j : COO (IR n , IRk) -----+ Coo(lR n ,lR fj ). The principal symbol aj(x,~) of A j takes values in £(lR k , IR fj ). Let Q E OPSo be a k x k matrix of operators, with principal symbol q(x,~), taking values in End(lR k ). Assume that (8.34)
(8.40)
Similarly,
This means d( *9) = 0, where is the Lorentz analogue of the Hodge star operator. Hence #4 provides an estimate on F 1\ *9 = (F, 9)w, where w is the natural volume element on 1R 3 ,l. Hence
If
+ (1B 2A 2v,u) + ([B 2A 2,Mf ]v,u) + (1v,Ru).
(1v,Qu) = (1v,B 1A 1u)
Note that (8.36) implies
then (8.30) is equivalent to an estimate on E· B. The rest of Maxwell's equations (in empty space) is d* F = 0, where d* is the "adjoint" of d, defined by the (indefinite) Lorentz metric. Rather than directly augmenting (8.29) with this, we consider a 2-form 9, satisfying
(8.33)
(8.39)
00, and
(8.44)
1(1, v·
Qu)1
:<:;
cKltfllbmolluliLvltvllLpl.
Such a result was established in Theorem VI.1 of [CLMS], under the following additional hypotheses. It was assumed that Al = A 2 , a first order differential operator, with constant coefficients, Q was taken to be a constant symmetric k x k matrix, and it was assumed that u = v and p = p' = 2. In [BH] , it was also assumed that Al = A 2 was a first order differential operator, but it could have variable coefficients, and [BH] took Qu = Q(x)u, with Q(x) a smooth k x k matrix-valued function; then [BH] derived Besov space estimates for v . Qu. Other extensions, again in the constant-coefficient case, are considered in [LMZ]. Note that, if (8.34) holds, then (8.45)
q(x,~) : Ker al(x,~)
------+
R(a2(x,~)t).
In particular, if al = a2, then v· q(x,~)v vanishes whenever v E Ker adx,~). In Theorem VI.1 of [CLMS], this was part of the hypotheses, rather than (8.34); another part was that adx,~) = a2(x,~) : JRk -----+ R f have constant rank on T*Rn \ o. We record a simple proof of the following result, that (8.45) plus constant rank hypotheses imply (8.34). PROPOSITION
8.3. Let A j
E OPsrnj,
Q E OPSo have principal symbols
Then
aj(x,~), q(x,~). Assume that, for each x,~,
(8.36)
have constant rank on T*Rn \ O. Then there exist B j E OPs-mj with principal symbols bj (x,~) s'uch that (8.34) holds.
v . Qu E fJtoc(JR n ).
(3.12) holds. Also assume that
aj(x,~)
p ". 1!!2
•
,~
~
:
'
.~~.~"
~.
~~.
···J·
""V'
·~".~m.·=·
.•,
~
•••
<>~.:
.•
~.':'""'''...
••'!f!ii
.."-~~~"""">;.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.
<
P
<
00, u E LP(~n), v E Lpl (ffi. n ) are real
valued, and that A1u E F(ffi. n ),
A 2v E LP(ffi. n ),
r > p, P > 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<
K:; 1 I:::;
('k aJ (x) ) :::; KzI,
Ilws/lOaJ'k IlcCA) :::; K 3.
where here and below we set Wb(X) = W(x/b). Again we set u~(x) = p-1U~(px), which solves (9.4), this time with
(9.23)
0 < K:; 1 I:::; ('k a~ (x) ) :::; KzI,
Ilwza~'k IlccA)
:::; K 4.
Again we estimate u~ - Xe, using (9.6), so we need to estimate h~. Now
(9.24)
C(>.)
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<
K;} :::; f(x) :::; K 14 ,
Ilfllc"(B po ):::; K lS '
3.
HARMONIC COORDINATES
9.
APPLICATIONS TOpPE
199
PROPOSITION 9.4. Suppose U E Hl,Z(O) solves the elliptic PDE PROOF. Pick P = p(2, K 11) as in the proof of Proposition 9.2 and let VI = uf, as constructed there, so VI E OU>'). Set PI = p/2. Perhaps decreasing PI (by Ojajk(x)chu = 0 on 0. a controllable amount), we assume lldvl(x) - dXIIILCO(B pl ) < lO-z and Ilgjk (9.42) tljkllLOC(B < lO-z. Then *dVl is a I-form of class 00-, and it is closed, so we can Assume define jk (9.43) a E 01~c(O) n HI~~(O), o < s < 1, 1 < p < 00. x (9.38) VZ(X) = *dVl, Then pj
)
l
the integral being independent of the path in BpI from 0 to x. It is easily seen that Vz E OlP. But also Vz is harmonic, i.e., it satisfies (9.35), so regularity gives Vz E 0 1 ,(>.). The rest of the proposition follows readily. REMARK. The construction of isothermal coordinates for a OS-metric tensor
(8 > 0) was done by [Lie]. The theory of quasiconformal mappings provides a construction of a homeomorphism, of class Hl,z n Oct (for some a: > 0) from B p C lR. z to U~ eM, conformal almost everywhere, given a measurable metric tensor satisfying 0 < K OI 1-:; (gjk(X)) -:; KoI, though such a map is not necessarily a 0 1 diffeomorphism. For a global result along these lines, see [J]. One tool used in the theory of quasiconformal mappings is the Beltrami equa tion. We recall that results on the Beltrami equation produced in §2 tie in with Proposition 9.3. We record some results one gets on harmonic coordinates when the hypotheses on the regularity of the metric tensor are varied. As one example, consider the case when the hypothesis (9.1) is strengthened to (9.39)
gjk(O)
= tljk'
I 0 < K O 1-:; (9jk(X)) -:; KoI,
IlgjkIIHI,P(B 1 )
-:;
(9.40)
uPi E HZ,P(B ) loc p'
Then (9.39) can be seen to hold in the new coordinate atlas. Hence on overlaps U~l n U~2 the harmonic functions are of class HZ'P in both coordinate systems, so we have an HZ'P-coordinate atlas. We can hence deduce that the old coordinate atla..'l must have been of class HZ'P and the two coordinate atlases are HZ'P-compatible. This result has also been established in [8m2]. In view of some results to be discussed in §1O, it is of interest to consider the following hypotheses: (9.41)
0< K OI I -:; (gjk(X)) -:; KoI, Ilgjk 11c-' (B, ) -:; K l ,
PROOF. For simplicity of notation we use HS,P in place of HI~~(O), etc. We already know that u E 01+8. To prove u E HZ,P we use an argument similar to that in our proof of Proposition 1.16. Setting L#u = OjTaikOkU, write (9.42) as
Ilgjk II HI,p(Bd -:; K z ,
given 0 < s < 1 and 1 < P < 00 (not assuming p > 11.). Again Proposition 9.1 applies. Additional estimatps on u~ are provided by the following regularity result, a variant of Proposition 1.16.
L#u
(9.45)
=
-OJ [TOkuajk
+ R(a jk ,OkU)].
Here L# E OPBSf 1 is elliptic and, since a jk E 0 8 , we can apply Proposition 6.1 of Chapter I to obt~in E E ops~i such that
EL# = 1+ F,
(9.46)
F E OPS~r
Then we have (9.47)
u
=
-Eoj [ TakuaJ k + R(aJ'k ,0kU) ] - Fu.
Now
Kl,
where we are given p > 11.. Then of course Proposition 9.1 applies. We have some additional estimates on the functions u~. Namely, by Proposition 1.12,
S u E 01+ loc (0) n .HZ'P(O) loc .
(9.44)
u E 01+s ==> Tak U ' R OkU E OPS?,l (9.48)
+ R(ajk, OkU) E Hl,p jk ==? EOj [Taluajk + R(a , OkU)] E HZ'P,
===}
Takuajk
the second implication using the hypothesis that a jk E Hl,p. Knowing that U E 01+s, we have U E H1+ s- o,q for all E > 0, q < Fu E H1+Z8-o,q. Thus (9.47) yields (9.49)
u E HZ'P
+ Hl+Zs-o,q,
V E > 0, q <
00.
Hence
00.
This in turn implies more regularity on Fu, and iterating this a finite number of times gives the desired conclusion. Using this result, we now establish the following: PROPOSITION 9.5. In the setting of Proposition 9.1, assume also that (9.41) holds. Then the harmonic coordinate system produced there has the property that the metric tensor pulled back to B po via 'l/Jz satisfies (9.50)
Ilgjk II HI,p(B po ) -:; K 15 ·
__
_
_
_
---------'----0-00'--"-0---
---- --
~ ~ ~ ~ ~ ~ ~ ~ ~ ~
-_._._-----
201
3. APPLICATIONS TO PDE
9. HARMONIC COORDINATES
PROOF. We have the diffeomorphism ~p = (ui, ... ,u~) of B I / 2 onto a region containing B I / 3 , satisfying u~ E Cl+s n H 2 ,p. The inverse (p is easily seen to be of class Cl+S; we claim it is also of class H2,p. To simplify notation, write these diffeomorphisms as ~ and ( = ~-l. We have
with J1U) as in Proposition 5.8 of Chapter 1. For example, one can take J1(j) = A(j)a, with a as in (9.20). It follows that, under the hypotheses (9.52),
200
By paradifferential calculus (or more elementary arguments) we have that when A E CS n HI,p is an invertible matrix-valued function and A-I is bounded, then x I-> 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
PROOF. This is a straightforward consequence of the identity D(u 0
Du(
A similar use of the chain rule shows that a composition of diffeomorphisms of class Cl+ s n H 2 ,p is also a diffeomorphism of class Cl+s n H 2 ,p. Hence the metric tensor in a 'IjIz-coordinate system is obtained from that in a
(9.58)
£# = £#(x,D) E OPBS~,I' elliptic,
IIDf£#(·,~)llc().)::; C",(~)-I"'I.
We apply further symbol smoothing to £#; pick 15 E (0,1) and write £#=£0+£1,
(9.59) with (9.60)
£0 E OPS?,0' elliptic,
(~) -<pro = w( (~) -0).
£1 E OPBS-;'{, ,
Then £0 is Fredholm and £1 is compact, so in (9.53) £ is Fredholm, as asserted. The by now familiar argument involving t£ + (1 - t)Ll shows that the index of £ is zero, and the characterization of ker £ is elementary. We also see that f E C-I,(A) (']['n) n LP(']['n) belongs to the range of £ in (9.53) if and only if f f dx = O. The same sort of argument used in Propositions 1.4 and 1.9 now gives the following global regularity result. PROPOSITION 9.8. Under the hypotheses of Proposition 9.1, (9.61)
u
E C1,(A) (']['n) ,
£u
E
u(']['n)
=}
u
E
H 2,p(']['n).
PROOF. Taking f = £u E LP(']['n) n C-L(A) (']['n), since f f dx = 0 there exists v E C1.(A)(']['n) n H2,p(']['n) such that £v = f. Then u - v E cl,(A)(']['n) and £(u - v) = 0, so u - v is constant.
PROPOSITION 9.7. Given AU) ".0, slowly varying and satisfying (9.20), and given p E (1,00), assume (9.52) a jk E C(A) (']['n) n HI,p(']['n),
From here we have the following local regularity result, which extends Propo sition 9.4.
and (a jk ) is positive definite. Then
PROPOSITION 9.9. Retain the hypotheses of Proposition 9.1. Let 0 c ']['n be open and suppose q > 1 and U E Hl~~ (0) solves
(9.53)
£ =
I: ojajk(x)ok : CI,(A) (']['n) n H
2 ,p(']['n)
----t
C-I,(A) (']['n)
n LP(']['n) (9.62)
£u
=
f
E Cl~~,(A)(O) n £foJO).
is Predholm, of index zero, annihilating only constants.
Then
PROOF. We begin by writing
(9.54)
£u
'k J ,OkU)] = £ # U + £ b u. = OjTaikOkU + OJ [Taku aJ' k + R(a
Note that (9.55)
Iloj[Taku ajk
+ R(a jk , oku)J1ILP
::; CllullLipllajkllHl.P
and (9.56)
If OJ [TakuaJk + R(aJk , oku)lllc-uA)
::; Cllulb,(I') Ila
jk
IIc().) ,
(9.63)
1 u E Cloe ,(A)(0) n H 2 ,P(0) loe'
PROOF. First, llsing Proposition 1.10, we have U E Cl~~A)(O). It remains to show that u E Hl:;~ (0). We do this by localizing and applying Proposition 9.8. In fact, given
£(
202
3. APPLICATIONS TO PDE
We need to show that the right side belongs to C- l ,(A)(1fn )
(9.65)
a jk E C(A), U E Cl~Y) a jk E Hl,p, u E Cl~;Y) a jk E C(A), U E Cl~Y)
10. RIEMANNIAN MANIFOLDS WITH BOUNDED RICCI TENSOR
n LP(1fn ).
==}
8j (a jk (8 k i.p)u) E C-l,(A),
==}
8 j (a jk (8 k i.p)u) E LP,
==}
(8 j i.p)a jk (8 k u) E c(A)
Indeed,
c LP n C-l,(A),
with a naturally accompanying estimate (involving the modulus of continuity postu lated for gjk as well as the other quantities).
A consequence of this result is that, under the hypotheses of Proposition 10.1, (10.4) Hence gjk E
so the proposition is proven.
203
Ricjk E LOO(B l )
cr
===}
\12 gjk E U(B l / 2 ),
\j q
< 00.
for all r < 2. We establish a sharpening of this result.
PROPOSITION 10.2. In the setting of Proposition 10.1, assume This result leads to an extension of Proposition 9.5, whose formulation we leave to the reader.
(10.5)
Ricjk E bmo.
Then
10. Riemannian manifolds with bounded Ricci tensor There has been a substantial amount of work in the area of deducing bounds on the metric tensor on a Riemannian manifold given bounds on the Ricci tensor (and other geometrically defined bounds). Since [DeTK], one important tool has been the use of harmonic coordinates. This is effective because, in harmonic coordinates, the metric tensor and the Ricci tensor are related by a quasilinear elliptic PDE, of the form (10.1)
1"
• -"2 L..J 8jg J'k (x)8 k gRm + QRm(g, \1g) = RlCRm,
(10.6)
\12 gjk E bmo.
PROOF. By Proposition 10.1 we have gem E H 2 ,Q,
(10.7)
PROPOSITION 10.1. Assume there is a harmonic coordinate system 1j; : B l ---4 U c M and the metric tensor pulled back to B l , via 1j;, satisfies gjk E C(Bd n Hl(B l ). Suppose given Ko,K l E (0,00) such that
(10.2)
0< K
a !::::: (9Jk(X)) ::::: K o!, l
IlgjkIIHl(B,)::::: K 1 .
Then, given p 2:: n/2, q E (1,00), and s 2:: 0,
(10.3)
RicJk E [}'(B 1 ) n Hs·q(BI)
===}
< 00.
Hence we can deduce from (10.1) that
"k
(10.8)
L..J8jgJ (x)8 k g£m j,k
j,k
where Q£m(g, \1g) is a quadratic form in \1g, with coefficients that are smooth functions of the components of 9 = (gjk), as long as this is positive definite. In effect, one is presented with two problems. One problem is to establish the existence of harmonic coordinate systems with sufficient regularity (and size). The second problem is to establish bounds on the metric tensor in such coordinate sys tems, given various bounds on the Ricci tensor. Regarding the first problem, some analytical conditions for existence of harmonic coordinates have been presented in §9. There is also work on geometrical conditions for existence of such harmonic coordinates, of a deeper nature. We mention particularly [An], [An2], and [Anej, in which there are assumed a priori lower bounds on either the injectivity radius or on volumes of balls. Surveys of work done on these problems can be found in [G] and in Chapter 10 of [Pet]. These works have had important implications for various compactness and rigidity results. As mentioned above, work on the second problem began in [DeTK]. We state a result that extends that work. The following is established in Corollary 12B.5 in Chapter 13 of [T5].
\j q
= F£m
E bmo.
The following result is more than adequate to finish off the proof of Proposition 10.2. PROPOSITION 10.3. Assume u E HI solves the elliptic PDE
(10.9)
I:8j a
jk
(x)8 k u = f E bmo,
with
a jk E Lip.
(10.10) Then
(10.11)
\12 u E bmo.
PROOF. To begin, we will establish the conclusion (10.11) under the stronger hypothesis (10.12)
a
jk
EC
r
,
r > 1,
which is already more than adequate for the proof of Proposition 10.2. As in the proof of Proposition 1.1, we can write (10.13)
u
= E# f -
L
E# 8jA~u,
mod CDC,
where, for some J E (0,1), gjk E H s+ 2 ,q(B l / 2 ),
(10.14)
E# E OPS~l,
A~
E
OPC r S~,bro.
11. PROPAGATION OF SINGULARITIES
3. APPLICATIONS TO PDE
204
Plugging u E H 1,2 into the right side of (10.13) yields the improved regularity u E H1+ ro ,2. A simple iteration gets us
uE
(10.15)
< 2,
'if s
Hs,q,
q
< 00.
....., v q
< 00,
Then we obtain
ro ,q Abu E H1+ ' J
( 10 . 16)
as long as 1 + r8 < r. This implies
(10.17)
E#ajA~u E
n
H(2+ro)f\(1+r-c),q C
qO
n
e(2+rO-c)f\(1+ r - c ),
V 2E# : bmo
(10.18)
PROPOSITION
E# E OPS~~,
A~
A~ : Hs+(l-o),q
-(1 - 8) < s
-----+ Hs,q,
:=:::
1, q <
00,
A) : H 2 -
O,DO -----+
H 1 •DO ,
iro w(t) t dt. h
RiCjk E B~,l'
then V gj k E B~,l 2
c e.
We are already in the set-up of (10.9), with u = gim and a jk = gjk E e for all r < 2. Hence (10.13) holds, with E#ajA~u E ns<2e2+s. Under hypothesis (10.26), f E e(A) in (10.9). Meanwhile Proposition 5.7 of Chapter I gives
HS,DC
=
(1 - ~)-s/2bmo.
In particular we have, for u satisfying the hypotheses of Proposition 10.3,
(10.30)
A~u E H1,x,
(10.23)
E#a·Abu E H 2 ,DC
(10.24)
J
J
V 2E# : e(A)
-----+
e(A),
so (10.27) follows, the inclusion e(A) c CrY holding by Proposition 1.2 of Chapter I. On the other hand, under hypothesis (10.28), the conclusion (10.29) follows because
hence '
and this gives (10.11).
(10.31)
V E# : B~,l 2
-----+
B~,l'
as shown in Proposition 12.1 of Chapter I.
We could carry out the proof replacing (10.21) by the weaker result Lip, which can be established by more elementary means.
11. Propagation of singularities
---t
We also note that, once we have the result (10.6) on the metric tensor, we can deduce from Proposition 1.13 that any harmonic function u on a coordinate neighborhood U~ has the property
(10.25)
=
u(h)
PROOF.
(10.22)
REMARK. o/ 2 ,oo
c CrY,
r
where, by definition,
A~ : H 2 -
V 2 gjk E e(A)
(10.28)
(10.29)
while by Proposition 4.10 of Chapter I we have (10.21)
(10.27)
If instead we make the still weaker hypothesis that
E OPLipSi:;t
We still obtain (10.15). As a consequence of Proposition 4.9 of Chapter I, we have
(10.20)
RiCjk E e(A).
Assume w(h) satisfies the Dini condition. Then
bmo,
-----+
10.4. In the set-up of Proposition 10.2, assume
(10.26)
so the conclusion (10.11) is established under the hypothesis (10.12). Under the weaker hypothesis (10.10), we have (10.13) with
(10.19)
Osgood's theorem, the geodesic flow is uniquely defined. (Compare remarks made below (2.25).) However, we cannot say that the exponential map is differentiable, without stronger hypotheses. We can show that the metric tensor is e 2 (and hence the exponential map is e 1 ) provided that in the coordinates 'Pa (and hence in 'l/Jv), the components Ricjk have a modulus of continuity that satisfies a Dini condition. In fact, we can establish some slightly sharper results. Let w(h) be a slowly varying modulus of continuity for which C f3 w(t) "" for some (3 E (0,1). Take A(j) = w(2- j ).
c>O
where a 1\ b = min(a, b). Meanwhile, since V 2E# E OPS?,o,
205
u
E
H 3 ,q,
'if q
<
00.
Hence the coordinate atlas {WI! : O~ ---t U~} actually provides a H 3 ,q-coordinate system on M, for all q < ()(). We make a further r{'mark on the significance of the conclusion (10.6). It implies that Vgjk has a modulus of continuity w(h) = h 10g(1/h). Hence, by
In 1981 [Bon] burst on the scene with an analysis of propagation of singularities for solutions to nonlinear PDE, introducing the theory of paradifferential operators as an essential tool. Other approaches to such results, given in [RR] and [BR], had an intermediate step of investigating solutions to linear PDE with rough coefficients. Further results on wave equations with rough coefficients include [8m]. Here we take another look at propagation of singularities for solutions to linear PDE (or pseudodifferential equations) with rough coefficients. Assume (11.1)
p(x,~) E
e r Sci'
•
<
11. PROPAGATION OF SINGULARITIES
.PPLICATIONS TO POE
has real principal symbol. In case r > 2, the analysis done in §6.1 of [T2] (though aimed primarily at the nonlinear case) essentially applies here, but we intend to take a closer look at the case Cl,l S~I' and even rougher cases. We begin in a fairly general context. Assume (11.1) holds for some r > 1, and write (11.2)
p(x,D) = p#
+ pb,
p# E OPA~Sr6'
pb E OPcr Sr';6- r6 .
Here A(;Sr,'6 is as in (3.33)-(3.35) of Chapter 1. We can also write p# = A+iB,
(11.3)
A E OPAOSr';6,
A = A',
d(x,~)
E
S~i,
u E H{~t7-rO(n),
supp d
c
d
f,
(11.16)
Hp,d
2:: 0 on
j E H;;'cl(r).
(11.17)
r
Here, is an open conic subset of T'n \ 0, and to say j E H;;'cl (f) is to say that rp(x,D)j E Hlac(n) whenever rp(x,~) E S?o is supported in Under these hypotheses, we have
r.
P#u = g,
9 = j - pb u ,
and pb U E Hl~c(n).
(11.8)
r \ 0,
m a 1 u E H mel + - (f) ,
r,
+ C* BC + C"[B, cnu, u),
with A and B as in (11.3) and C E OPSi,o to be specified below. Until further notice, we assume that m = 1, which involves no loss of generality. Thus B is bounded on L 2 , and we have the following basic commutator inequality: (11.11)
Re ({ -iC*[C, A] - MC"C}u, u) :::;
IICP#uI1 2 + I(Wu, u)l,
where (11.12)
M
1
= IIBII + 4'
2:: 1,
r.
2:: CI~IJ-L > 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
<
E!.. _
- 2
0
1
J-L ::;
1
2 - 20(1 -
IIE(x, D)v1112
- KliIvll12 ::;
IICEP#vIlI2
+ I(WEv, v)l·
Here WE is given by (11.12), with C = CE, so in light of the behavior of B recorded in (11.3) we see that WE is bounded in OPSi~-l+f>(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'<5-1)(0),
f
E H~cl(r).
Assume 0 E (0,1), ro> 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' <
u
E
Hence, if p(x, D)u =
1+a -- rJ H loc
f
==}
OPA(j-1Si,";srJ,
a
a Abj u E H 10<;
< r, then ==}
a- I pb u E H loc'
with
u E HI~~a-rJ(o),
(11.52)
E
f E H;;,~/(r),
we have
p#u E H~~/(r).
(11.53)
From here, an argument parallel to that given above yields the following. PROPOSITION 11.4. Letp(x,D) be given by (11.48) withAjk E cr, 1 < r:S 2. Assume p(x,D)u = f with u and f satisfying (11.52). Assume 0 < 8 < 1, r8> 1, and -(1 - 8)1' < a < 1'. Furthermore, assume that P1(X,O = Ajk(x)~j~kl~I-1 Isatisfies the hypotheses of Proposition 11.1. Then
uE
101.54)
'f Ajk
E
e 1,1,
H~cI(O)
==}
uE
H~el(r).
this is valid for -2(1- 8) < a:S 2.
The conclusions of Propositions 11.2-11.4 are primarily interesting when r hugs he characteristic set of p(x, D). In fact, if p(x, D) is elliptic on r then, in the setting f (11.1)-(11.6), we deduce from (11.7)-(11.8) that
11.55)
u
E
H;:cta(r),
211
without any need for hypotheses such as those on r j , r j given in Proposition 11.1. Of course this is stronger then (11.47). We now discuss conditions under which one can produce the symbols d, f, and g satisfying (11.14)-(11.17). If PI(X,Ej were smooth, one could simply solve some equations, such as HpJ f = 1, but for PI (x,~) E C r S;] this would not have a smooth solution f(x, ~). However, we need only satisfy some differential inequalities in (11.16)-(11.17), and we can exploit this flexibility. We work under the standing hypothesis that HPI is nowhere radial on the characteristic set char p; in particular it is nowhere vanishing on char p. In cases where one has a uniqueness result for the orbits of H PI (e.g., when this vector field has Lipschitz coefficients) and one expects to prove that smoothness propagates along orbits of HP,' the constructions of d, f, and 9 can be localized. In fact, in the setting of (11.1)-(11.6), suppose / is a null bicharacteristic strip of p(x,D), (0 E /, and u E H;'cta-1(0) for a conic neighborhood 0 of (0. We want to show that u E Hln+a-1 microlocally on a conic neighborhoood of /. Let I be an open interval in / containing (0 and on which u is microlocally in Hln+a -1. It suffices to show that u is microlocally in Hm+a-1 on a conic neighborhood of the endpoints of I. Hence it suffices to make constructions of d, f, and 9 on small conic neighborhoods of these endpoints. (If H pI has non-unique orbits, this localization can fail, to be sure.) Let us hence take (I E / (an endpoint of 1), let V be a sm~ll conic neighborhood of (1, and consider constructions of d,j, and g on V. Here r will be a small conic neighborhood of /, intersected with V, r will be a small conic neighborhood of 7, intersected with V, and 0 c V will be a small conic neighborhood of I. We assume the flow generated by H PI maps (1 into I in small positive time. (If not, simply change (11.4) to -p(x, D)u = -f.) To take care of our homogeneity requirements on d, f, and g, it is convenient to take a "section" W of V, a smooth (2n - I)-dimensional manifold containing (1 and transverse to the radial vector field R = I: ~j a / a~j. We write
H PI
(11.56)
=
X PI
+ (J(x, OR,
where X PI is tangent to Wand its images under the flow generated by R, and (J(x,~) is real valued (and homogeneous of degree zero in ~). Note that our hypothesis that H PI is nowhere radial implies that X PI is nowhere vanishing. Different choices of sections yield different decompositions in (11.56), and it is convenient to note that we can construct a section W such that (J(x,~) ~
(11.57)
o.
In these circumstances, our task of constructing d(x,O reduces to the following. We require
(11.58)
dE Co(W),
supp d
c
Wnr,
d::::: 0,
and we want dlw to satisfy the differential inequalities
+ /-L(J)d ~ 0
on W
n (r \ 0),
+ /-L(J)d ~ C > 0
on W
n r \ o.
(Xp1
(11.59)
(XpI
212
3.
APPLICATIONS TO POE
11.
Given that (11.57) has been arranged (and since /-L > 0), it suffices to have (11.60)
Xp,d;:::O on wn(r\o),
Wn r =
U
PE,
Xp,d;:::C>O on wnr\o.
Wn r
U
=
-b
-a
r#
=
U
we have (11.66)
> 0 on Wnr,
h ;::: C 2 > 0 on W n r,
cc
r# of W n r in r# and
Xp,h> 0 on r#, h:::; 0 on r# \ U.
PROOF. Take!J' E coo(lR) such that !J'(t) = 0 for t < 0 and !J'1(t) > 0 for t > O. Let d(() = !J'(h(()) for ( E r# and d(() = 0 for ( E W \ r#, and then alter d as needed in W n 0 to ensure that d E CO' (W). We now show how Lemma 11.5 can be implemented when X P1 is a C I vector -
-#
field, which occurs when p( x,~) E C 2 S~I' Pick ho E Coo (E) such that supp ho = E and ho > 0 on E#. Pick a E Coo(lR.) such that a(t) = 0 for t < -a', a'(t) > 0 for t> -a', and define hI on W n r by hI (F t ()
= a(t)ho((),
(E
(11.68)
'PE(D)Xp1 h2
--t
r).
f:,
Thus
X p, h 2 locally uniformly,
(11.69)
I [MaJ ,'PE(D)]ullu>c
:::; CE Ila
j
= dim W =
Ilup Ilullu'<>,
and together with (11.68) this gives (11.70)
X P1 ('PE(D)h 2 )
--t
X p1 h 2 locally uniformly.
Thus the results on propagation of singularities along null bicharacteristics mentioned above work for operators with C I .I coefficients as well as for operators with C2 coefficients. The scope of these last estimates can be extended, as follows. Suppose you have h 2 , as above, but with
Then there exists d E CO'(W) satisfying (11.58) and (11.60).
(11.64)
X p1 h 2 (Ft () = a'(t)ho((),
if {'PE(D)} is a Friedrichs mollifier. If
The following gives a relaxed set of conditions to verify in order to construct the symbol d.
Xp,h;::: C I
+ [MaJ , 'PE(D)](ojh 2 ).
Note that
-a'
(11.63)
X P1 ('PE(D)h 2 ) = 'PE(D)Xp1 h 2
so under our current hypotheses X P1 h 2 E Lip(W n
FtE#.
LEMMA 11.5. Suppose there is a neighborhood U h E Coo(r#) such that
X P1 = aj(x)oj,
(11.65)
(11.67)
Pf:.
We also asume Fef: c W nO. Let us also take E#, with smooth boundary, such that E cc E# CC f:, pick -a' E (-a, -b), and set (11.62)
213
cannot be expected to converge in the CI-topology. Nevertheless we can show that X P1 hE converges locally uniformly to X P1 h 2 , as follows. Taking a smooth coordinate system on W in which
Once we have dl w satisfying (11.58) and (11.60), then the unique extension to V homogeneous of degree /-L gives the desired symbol d( x, ~). Let us assume that X P1 has unique orbits and generates a flow :P consisting of a family of local homeomorphisms. We will assume there are smooth (2n - 2) dimensional surfaces E c f:, transverse to XP1 and containing (1, and numbers -a < -b < 0 < c such that (11.61)
PROPAGATION OF SINGULARITIES
(E E.
We have hI E C I (W n r), supported in r#. Inside r#, hI > 0 and X P1 hI > O. taking K = inf r hI > 0, set h 2 = hI - K/2. Approximating h 2 in the CI-topology by elements of Coo(r#) (and perhaps shrinking r# a bit), we obtain h E Coo(r#) satisfying the conditions of Lemma 11.5. Thus we obtain d(x,~) with the desired properties, when (11.1) holds with r = 2. The constructions of f(x,f.) and g(x,~) satisfying (11.14)-(11.17) are rela tively simpler. Thus we can apply Propositions 11.2-11.4 to obtain propagation of singularities results along null bicharacteristics, for operators with C 2 coefficients. Suppose now that X P1 is a Lipschitz vector field. Then the function hI given by (11.64) belongs to Lip(Wnr). and so does h 2 . Hence smoothings hE = 'PE(D)h 2
(11.71)
h 2 , X p1 h 2 EC S (Wnr),
O<s<1.
We are hence dropping the hypothesis that aj(x) in (11.65) are Lipschitz. Let us instead assume (11.72)
a j E C S ' (W n r),
1 - s < s' < 1.
Looking at (11.66), we see that (11.68) still holds. We need to estimate the action of the commutators [Maj ,'PE(D)] on Ojh 2 E C:-I(W n r).
(11. 73)
More generally, if a E cs', P E OPS? 0' and u E C:-1, we examine [Ma, Plu by using the paraproduct expansions: ' (11.74)
a Pu P(au)
= TaPu + TPua + R(a, Pu), = PTau + PTua + P R(a, u).
Now, by (3.53) of Chapter I, for s' > 0, (11.75)
a E C S'
===}
R a E OPS 1,{.
Hence, as long as s - 1 + s' > 0, we have (11. 76)
R(a,Pu), PR(a,u) E C:-1+ s '.
214
3. APPLICATIONS TO PDE
Furthermore, given
8 -
11. PROPAGATION OF SINGULARITIES
1 < 0, u E
(11.77)
In such a case, we have
C:-
l
===}
T u E OPBSi.l s ,
If (11.84) holds with W(8) given by (11.87), we say X P1 ELL. We have
Tp1l.a, PT1l.a E C:~l+s'.
(11.78)
Next, by Proposition 7.3 of Chapter I, for 0 < a EC
SI ,
P E OPSP,o
===}
(11.89)
< 1,
8'
[Ta , P] E OPBSl,{'
modulo a smoothing operator, so [Ta , P]u E C:-l+ s ',
(11.80)
and hence, putting together the various ingredients in (11.74), we have (11.81)
[Ma , P]u E C:-l+
sl
€
:::;
1 we have
[Maj, 4?E(D)]ojh 2 bounded in C:lo~+s' (W n r).
Hence this family of functions in precompact in C~c(W n r). Since it is clear that (11.83)
j aj (x )4?E (D) (ojh 2) and 4?E (D) (a (x)Oj h 2))
---->
a j (x )OJ h 2
S - 1 - 6 (W . C*,loc nr-),
in
for each J > 0, we deduce that locally uniformly on W n
r,
so again (11.70) holds, under the more general hypotheses (11.71)-(11.72). Of course, the hypothesis (11. 72) on the coefficients of the vector field X P1 is not strong enough to imply that X P1 generates a uniquely defined flow P. Osgood's theorem (mentioned already in §2 and again in §10) guarantees that such a unique flow exists provided the coefficients have a modulus of continuity of the following sort: (11.84)
.
-
a J E CW(W n r),
1r
1
d8
W(8)
= 00.
0
In such a case, if laj(xl) - aj (x2)\ :::; I\MJ(IXI - X21) for all Xj E W n r, we have (11.85)
jFtXl - FlX21 :::; 'l9(I X1
x21, t),
-
where 'l9(a, t) is defined by (11.86)
l
{}(a,t) ~ _
a
W (8 )
- ",t.
In particular this applies to the "log-Lipschitz" modulus of continuity given in (2.25): (11.87)
W ( 8) =
S (log
~),
0<
8
===}
IFtxl - FtX21 :::;
IXI - x2IcXP(-Kt).
We henceforth assume that X P1 is log-Lipschitz. Thus the flow P is Holder continuous for each t, though the Holder exponent decays exponentially. However, as we have noted, the task of establishing propagation of singularities along null bicharacteristics involves constructing the symbols d, f, and 9 only in a small conic neighborhood of a given point (1, so in this construction we can keep t small and hence keep the Holder exponent (call it 8) as close to 1 as we like. It is then straight forward to show that cI>(t, () = P( produces a CS-homeomorphism of (-a, c) x ~ onto a neighborhood of (1 in W (which we denote W n r, as in (11.61)), and fur thermore cI>-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
<
00,
!1+
where d(z) = dist(z,r). Let ( , similarly, using f!_. LEMMA
hi+
denote the inner product in 'H+. Define 'H_
1.2. Suppose F is holomorphic in f!+ and vanishes at infinity. Then
1. Cauchy kernels on Lipschitz curves
Let A : lR -+ lR be a Lipschitz function, with Lipschitz constant L, and consider the Lipschitz graph,
r
(1.1)
= {t
+ iA(t) : t E lR},
Denote by f!+ the region in C above r and by f!_ the region below r. We have the Cauchy integral (1.2)
Krf(z)
=
J
1.1. The limits Kf- f(z)
=
lim K r f(z),
±y".,O
z E r,
exist and define operators
(1.4)
Kf- : L 2 (r)
----+
L 2 (r),
satisfying (1.5)
IIKf-IIC(£2(r)) : : ;
For the first step in the proof of Lemma 1.2, we let
Co(l
----t
f!+
z E f!±.
The main result of this section is the following result of [CMM], following work of [Ca2].
(1.3)
for some absolute constant Cr. There is an analogous estimate for F holomorphic on f!_.
(1.9)
f(()
z _ ( dC
r
THEOREM
1IFIIL2(r) : : ; C 1 (1 + L)IIF/I\1ip
(1.8)
+ L)2,
for some absolute constant Co.
It is technically ronvmient to treat first the case when A E CO"(IR) and f E CO"(r), obtaining the estimate (1.5) purely in terms of the Lipschitz constant of
be a conformal mapping so that
(1.10)
0:
1
::::; dist ( ( z ) , af!) ::::;
131 <1>' (z) 1y.
In fact, the sharp values 0: = 1/2 and 13 = 2 are known. We give a proof of this result in Appendix A, at the end of this chapter. Using this distortion theorem, we see that Lemma 1.2 is equivalent to the estimate (1.11 )
f:
IG(x)1 2 I<1>'(x)1 dx
::::; C
J
IG'(zWW(z)ly dx dy,
]R2
+
for holomorphic functions G decaying at infinity. The following lemma will be useful for establishing this.
220
1. CAUCHY KERNELS ON LIPSCHITZ CURVES
4. LAYER POTENTIALS ON LIPSCIlI'l'Z SURFACES
LEMMA
1.3. Let Hand D be holomorphic on
lR~.
Assume ID(z)/ ::;
1
and H
221
where D = eiV . Using this plus Cauchy's inequality, we have
vanishes at infinity. Then
i:
(1.12)
IH(xW dx = 4
/ IGG'"ly dx dy
J
1R 2
IH'(zWydxdy,
+
5: (/ jG'lzl'ly dx dy) l/Z ( / IGl zlV'1 2 l<1>'ly dx dy) 1/2
JR2
+
and (1.13)
/ IH(z)D'(z)IZydxdy::; JR2
i:
(1.17)
IH(xWdx.
JR2
21:
IH(x)D(x)IZdx
=4/
2 IH'D+HD'1 ydxdy.
where the last inequality follows from (1.13) since IIDIIL"" 5: e11'/2. Note that the factor C in (1.14)-(1.15) can be bounded by C't(l + L). At this point we have an estimate of the form (1.18)
]K2
+
The estimate then follows from the triangle inequality, (1.12), and the inequality
IH'DI ::; IH'I· We turn to the proof of (1.11). Let A denote the left side of (1.11) and let B denote the integral on the right side. Note that, since r is Lipschitz, [arg <1>'1 < 7['/2 - E: where E: = arccot(L) > 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)
<1>' = eV ,
so
<1>" = V'e F = V''.
= C1 (1 + L).
12/
Now
f(z)d(z) dxdyl (z - w)3
11+
<
-
2/ Iz -
If(z)ld(z) d d wl3 X y.
11+
We can arrange that 11m V(z)1 < 7['/2.
Hence we will have (1.20) from (1.21) if we can show that
1<1>"1 S
p7T/2[V'e iV <1>'1 =
e11'/zID''I,
(1.23)
S: LZ(O+)
--+
L 2(0_),
---
-
222
_;-_.,~~._~ •. _~~_.::::.~-;;-.,_~::.. __ :;.~=;:;~~~;..~~;;;:.;~-:-.;;,;;,~;..-;~.:=~~.:;:",_-=~~o:-.;;;;,,_,,_.,_~.~:~.::.:,,:.,._:,,~~:=~.,-~-~-~~___
"";;;;;.",;;;;;'~~;_:~~~~:.:.;;;.;.:.-~
(1.24)
F(z)d(z)1/2 wl3 dxdy
11+
wl- 3 .
J
E
fL, we have
J
k(z,w)dxdy:::; 47T,
The general Calder6n-Zygmund theory hence implies the boundedness properties
k(z,w)dA(w):::; 47T,
0+
(1.35)
11_
since d(z) :::; jz - wi. Hence S is a bounded operator from L 2 (D+) to U(D_), with norm:::; 47T, and Lemma 1.4 is proven. We now prove Theorem 1.1 (under the restrictions mentioned after the state ment of that result). Let f E L 2 (r), and let 9 E 1i+, and assume both are smooth and have compact support. Then
(£rf,g)1i+
=-
J(J J
n+
(1.26)
=-
f(() ) (Z_()2d( g(z)d(z)dxdy
r
f(() T(g) d(,
r
where T is given by (1.19). Hence (1.27)
1(£rf,g)1i+I:::;
Ilfllu(r)IIT(g)IIL2(r) :::; C1lfllu(r)llgll1i+,
by Lemma 1.4. Thus
K~: LP(lR) ------; LP(lR),
II£r fll1i+
:::; Cllfllr,2(r)'
",a(s, t) =
",a(s, t)
I KIt fllu(r) :::; C11£r fll1i+,
so we have the desired estimate on KIt f. The estimate on K r We can rewrite the Cauchy integral (1.2) as
(1.30)
Krf(z)=joo -00
1
f
(1.31)
KAg(s)
=
-X)
S- t
+ i(A(s) _ A(t)) + ia dt.
By Theorem 1.1, we haw' limiting operators as (1.32)
.n.~
: L 2 (lR)
---+
±a '" 0:
L 2 (lR).
a
- ((t))
a
'
a
rr
'
where
fO(~~()('(t)dt,
g(t)
+ ",-a(s, t) = ~ w(((s)
",a(s, t) _ ",-a(s, t) = 2i cfl(((s) - ((t))
follows similarly.
z '1!(z) = z2+1'
(1.38)
z- t- z t
00
1
(1.37)
with ((t) = t + iA(t). If z = s + iA(s) + ia, and if we write g(t) then an essentially equivalent operator is a
1
Then
On the other hand, Lemma 1.2 implies (1.29)
IIK~IIL:(u'):::; C(p,L),
= f(((t))('(t),
00.
Also, these operators satisfy weak-type (1,1) estimates. At this point we pass from the case A E C(j(lR) to general Lipschitz A, with Lipschitz constant:::; L. Take such a general function A, and let A v E C(j(lR) have Lipschitz constant :::; L and approximate A locally uniformly. Let K:J denote the associated operators, which by the analysis above have uniformly bounded operator norms on L 2 (lR). There exists a subsequence Vk and bounded operators on L 2 (lR), which we will denote K~ , such that K~ ----+ K~ in the weak operator topology. For now we do not assert the uniqueness of these limits, though that can be deduced from results established below. Examining convergence (K~f, g) ----+ (K~ f, g) for f, 9 E Co (lR) with disjoint supports, we do see that the Schwartz kernels of K1 and K A agree on lR x lR \ diag with kA(s,t), given by (1.33). Hence K1 and K A are Calder6n-Zygmund operators. We want to estimate maximal functions associated to K'A., as a ranges over [-1,0) U(O, 1], and to compare K~ with a principal-value singular integral operator. Let us set (1.36)
(1.28)
lV's,tkA(s,t)1 < ~2 - Is - t1 '
Ikit(s, t)1 < ~ - Is - tI'
(1.34) Now for any w
1 ((s) - ((t)'
A(t))
which satisfies "standard estimates"
k(z, w)F(z) dx dy,
with k(z,w) = d(Z)1/2d(w)1/2/ z -
1 kA(S, t) = s _ t + i(A(s)
(1.33)
0+
J
=
The operators K1 and K A are hence Calder6n-Zygmund operators, associated to the kernel
J Iz -
SF(w) = d(W)1/2
223
1. CAUCHY KERNELS ON LIPSCHITZ CURVES
where S is given by
(1.25)
. '. ",,_ ~;;;, ...::,;;""-~;,~~;;,_,,;-;;,,,~_"~- ...~..:.;_::'~'.;..._..;,;;=::::~:.;;.:;;; ;;:;;,-~:~~-=i':;;'~~-:~_~-L.;;-;,'~:'~ ,';;;'.C.,._"",;.,;~'~:_.", -.;~-:~::-:;:::::i=::~~-;~,,"'';';;'.~=~~:=;;;';~'~~..;c::::i:;~'f''~:t;.;~~~:,;.-c;~~~
4. LAYER POTENTIALS ON LIPSCHITZ SURFAOES
1
cfl(z)
=
z2
+ l'
Note that, since A has Lipschitz constant L, we have (1.39)
((s) - ((t) E OL = {z E C:
IImzl :::; LIRezl}.
(j
Note also that
(1.40)
z
E
OL
~
1(z) I :::;
I
EL .,
224
4. LAYER POTENTIALS ON LIPSCHITZ SURFACES
1. CAUCHY KERNELS ON LIPSCHITZ CURVES
so
225
we have
11I:"'(s,t)-II:-"'(s,t)l~ ~L(ls~tl2 +1)-1.
(1.41)
sup IK'Ag(s) - K'Ag(s) I ~ C L Mg(s).
(1.52)
0<",~1
It follows that sup IKAg(s) - K A'" g(s)1 ~ C L Mg(s),
(1.42)
0<",~1
where M 9 is the Hardy-Littlewood maximal function associated with g. We next compare 11:'" + 11:-'" to 2k A, and particularly to 2k"', where we set
o
0<",~1
(1.54) 1I:"'(s, t)
+ II:-"'(s, t)
- 2kA (8 t) =
2
,
((s)-((t)
<Jl(((s) - ((t))
a
with <Jl(z) given by (1.38). Hence
11:"'(8, t) + II:-"'(s, t) - 2k"'(s, t) = _~ O(((s) - ((t))
(1.45)
a
a
+ C L Mg(s).
Cf. [Jo], pp. 56-57, or [83], pp. 34-35. Also, given 9 E COO(IR), the quantity
if I((s) - ((t)1 < a.
We have (1.44)
sup jKA:g(s) I ~ C M(K!g)(s)
(1.53)
if I((s) - ((t)1 ~ a,
k"'(s, t) = kA(s, t)
(1.43)
Given that K! is a Calder6n-Zygmund operator with kernel as in (1.33), and that 1;;'" is defined by (1.49), yielding the operator K'A as in (1.51), it is a general result in Calder6n-Zygmund theory that
'
. -",
00 [
'"
(( S
g(s+t)
+ t) -
(() S
+ ."r( S
g(s-t)] ) r() dt - t -." S
clearly exists at each point s where A is differentiable. In view of the maximal . estimates established above, it follows that there is a bounded operator (1.55)
'
.1
hm KAg(s) = ",-+0 hm ",-+0
KA
;
V p E (1, (0),
U(IR) -) U(IR),
such that, for each 9 E U(IR),
with 1
(1.46)
1
O(z) = - - z Z2 + 1
if
Izl -> 1,
z --z2 + 1
if
Izi < 1.
(1.56)
K'Ag - ) KAg
in LP- norm, as a ...... O. One writes (1.57)
KAg(s) = P.V.
Thus, if KA: denotes the integral operator with kernel k"', Le., (1.47)
J
K'Ag(S) =
n
}(t);I., dt,
we have by arguments as in (1.39)-(1.42) that sup IKAg(s)
(1.48)
0<",~1
+ K A'" g(s) - 2KAg(s) I ~ C L Mg(s).
).50)
given by
o
'{ote that Is -
,
k"',
k"'(s,t) = kA(s,t)
~1.49)
t/
if Is - t/ ~ a, if Is - tl < a.
for each point s where A is differentiable. Hence the maximal function estimates established above imply that, for each 9 E U(IR), p E (1, (0),
(1.60)
if Is - tl < a < I((s) - ((t)l,
." s
otherwise.
rhus, if KA: denotes the operator with kernel =
f IB-tl2:'"
sup IKAg(s)1 ~ C M(K!g)(s)
0<",~1
k"'(s, t) - k"'(s, t) = -;;---( ) 1
KA:g(s)
KA:g -) KAg
in LP- norm, and pointwise a.e., as a ...... O. Note that (1.42), (1.48), (1.52), and (1.53) imply that
~ 1((8) - ((t)/, and that
o 1.51)
KA:g(s) - KA:g(s) - ) 0
(1.59)
A kernel closely related to k'" is
k"',
i.e.,
+ C L Mg(s),
and a similar estimate holds on K A'" g(s). Also, we could replace M(K!g)(s) on the right side of (1.60) by M(KAg)(s). We are now ready for conclusions about various operators considered above. PROPOSITION
(1.61 )
g(t)r l . , dt, ,"1_\
,g(t) .. , dt.
From (1.50) we see that, if 9 E C8"(IR), then, as a ...... 0, (1.58)
{I«( s) -«( t) 12:"'}
i: .
1.5. Given 9
E
LP(IR), 1 < P < 00, we have
lim (K A+ KArr)g
",-+0
=
2KAg,
in LP -norm. The convergence also holds pointwise a. e.
4.
LAYER POTENTIALS ON LIPSCHITZ SURFACES
PROOF. In view of the estimates established above, it suffices to show that, if 9 E CO'(IR), then
(1.62)
lim [KAg(s)
CT~O
+ KACTg(s) -
2KA:g(s)] = 0,
for every point s where A is differentiable. This can be deduced from the formulas (1.45)-(1.46) and the following lemma.
1.
CAUCHY KERNELS ON LIPSCHITZ CURVES
Reversing the passage from (1.30) to (1.31), we can phrase the last results in terms of the Cauchy integral (1.2). Given f E L 2 (f), z E f, set
(1. 71)
(K r f)*(z) = sup {IK r f(z
PROPOSITION 1.8. If 1 < p <
(1. 72) LEMMA 1.6. Let 9 E CoOR). Assume /3 is continuous on C (except perhaps
for a simple jump across {z : Izi = I}), and (1.63)
1/3(z)1 :Slzl2
Consider (1.64)
TCTg(s) =
~
i:
i:
+ i'
/3(((s)
~ ((t))g(t) dt.
/3 (('(s)t) dt = b(s).
lim TCTg(s)
7r
1
1> (('(s)t) dt = (/(S)'
as long as (' (s) = 1 + A' (s), IA' (s) I :S L. This yields the following. PROPOSITION 1. 7. Give 9 E LP(IR), 1 < p < ,(1 .68)
. (vO
-
00,
we have
V--CT) 9 = 27ri g ('
I\..A
in V-norm, and also pointwise a.e. Consequently ).69)
lim K'Ag
±
7rt
= KAg ± 71 g ."
in V -norm and pointwise a. e. While we have taken a E (0, 1] in the analysis of K~
1.70)
KI'f(z) + 7rif(z), K r f(z) - 7rif(z),
-----4 -----?
K r f(z)
=
P.
v.
J
f(() d(.
z-(
We next establish a couple of results from [CDM] that will play an important role in the extension to several dimensions in the next section. ----t
lR is Lipschitz, with Lipschitz constant L. Set
eA(s, t) = _1_ exp i(A(s) - A(t)) s-t s-t'
Then P. V.eA is the kernel of an operator EA on £2(lR), satisfying
00
-00
Kr f(z + ia) Kr f(z - ia)
r
= b(s)g(s).
The proof is an exercise. (Clearly we have not picked a maximal set of functions /3 for which the lemma holds.) This lemma applies to Proposition 1.5 with b(s) = 0, since n (z) in (1.46) is odd in z (and can be altered off th to satisfy all required conditions). Note furthermore that, if 1>(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<'z/
L.
2.
THE METItOD OF ROTATIONS AND EXTENSIONS TO HIGHER DIMENSIONS
is a well defined operator satisfying K : L 2 (lR k )
(2.3)
7(S,t) = I:a"g,,(s,t),
"
(2.4)
K f(x) =
J
Ck
Twf(x) dS(x),
Sk-l
where, for w E Sk-I,
(2.5)
Twf(x) =
i:
k(,¢(x) -
'¢(~ + sw))f(x + sw)lsl
k
-
1
ds.
We estimate the operator norm of Tw on L 2 (lR k ). To do this, let Vw = (w)J.. = {x E IRk : x· w = O}, and note that
IITw fllI2 =
(2.6)
JIIT ,ddI2 d~, w
where /e(t) = f(~ + tw) and Tw,f, is the singular integral operator (acting on func tions on IR) with kernel
(2.7)
with
7w,f,(S, t) = k( '¢(~ + sw) - '¢(~ + tw)) Is
-
tlk-I.
Thus our task is to estimate the operator norm of Tw,f, on L 2 (1R). Note that
g,,(s, t) = _1_ exp(i ",. 'P(s) - 'P(t)) s-t L s-t .
(1.83)
Now Lemma 1.9 implies that P.V. g,,(s, t) is the kernel of an operator G" satisfying (L~4)
~ C(l
IIG"II.c(L2)
7w
(2.8)
_ _1_ k(,¢(E
f, ( s,t ) s-t ,
+ sw) -
'¢(~
+ tw))
s-t
= _1_ Fw ('P(~ + sw) - 'P(~ + tw)), s-t s-t
+ 1",1)3.
where Fw E C N (IR n- k ) is given by
The estimate
la,,1 ~ C(l
(1.85)
+ 1",I)- N II1>llcN
yields the result.
2. The method of rotations and extensions to higher dimensions One can pass from the one-dimensional result of §1 to a useful multi-dimensional result by the method of rotations. Our treatment of the next proposition follows [CDM] and [Dav]. N
PROPOSITION 2.1. Let k E C (IRn \ 0) be odd and homogeneous of degree -k. Assume N > n - k + 3. Let r be a k-dimensional Lipschitz graph, of the form
r = {(x, 'P(x)) : x
(2.1)
(2.2)
L 2 (lR k ).
V'"
Hence
where 'P: IRk
--t
PROOF. Write
" (1.82)
229
---+
IRn-k is Lipschitz. Set '¢(x)
Kf(x) = P. v.
J
=
Fw(Xk+I, ... ,xn )
= k(WI" .. , Wk, Xk+l, ... ,xn ).
Now the function t f-+ 'P(~ + tw) is Lipschitz, uniformly in ~ and w. Hence the desired estimate on IITw ,d.c(L2) follows from Proposition 1.10, and the proof of Proposition 2.1 is complete. As in §1, we see that the operator K in (2.2)-(2.3) is a Calder6n-Zygmund operator, and we have an estimate on K : LP(lR k ) -+ LP(lR k ): (2.10)
IIKII.c(LP) ~ C(p, r) Ilklsn-lIICN'
1
< P<
00.
The operator Kin (2.2) is closely related to the principal-value singular integral operator:
E IRk},
K r f(x)
(x,'P(x)). Then
k('¢(x) - ,¢(y))f(y) dy
Ili.k
(2.9)
= P.V.
J
k(x - y)f(y) dcr(y)
r
(2.11)
= e-tO lim
J
{YEr:lx-yl>e}
k(x - y)f(y) dcr(y) ,
4. LAYER POTENTIALS ON LIPSCHITZ SURFACES
3. THE VARIABLE-COEFFICIENT CASE
where dCJ is the area element of f, induced from the Euclidean structure of JRn. This is also related to the following operator, defined for x E JRn \ f:
is odd whenever bj i- O. With N as in Proposition 2.1 and M sufficiently larger than N, the regularity hypothesis implies
(2.12)
IC r f(x)
=
J
k(x - y)f(y) dCJ(Y)·
r
in a fashion parallel to analogues in §1. We now restrict attention to Lipschitz graphs of dimension n - 1 in JRn, i.e., to the case k = n - 1 of Proposition 2.1. As in §1 we have estimates on nontangential maximal functions. If f is a Lipschitz graph, with Lipschitz constant ::; L, and if fJ > 1 is chosen, then for each x E f, consider the cone Cx = C + x, where C = {x E IRn : fJLlx'1 ::; Ixnl ::; I}. For a function u defined on JRn, we define the nontangential maximal function: (2.13)
u*(x) = sup lu(y)l,
(3.2)
Ilbjll L = II!pjllcN ::;
Bf(x)
(3.3)
11(lCrf)*IILP(r) ::; CllfIILP(r),
with a bound on C of the form (2.10). Extending jump relations given in (1.73), one can show that /'C r f has nontan gential boundary values a.e. on f, which are related to K r f by: (2.15)
(ICr f)±(x)
= =f~iP-l
(n(x))f(x)
+ K r f(x).
Here, (ICrf)+(x) is the limit from above f and (ICrf)_(x) is the limit from below f (within the cone Cx), n(x) is the unit (downward-pointing) conormal to f at x (defined a.e. on f), and P-I (E) is the principal symbol (homogeneous of degree -1 in E) of the operator Pu(x) = fIRn k(x - y)u(y) dy.
3. The variable-coefficient case Our goal here is to extend results of §2 to the variable-coefficient case. As in §2, let f be a Lipschitz graph in IRn, of the form Xn = !P(XI' ... , Xn-I). PROPOSITION 3.1. There exists M = M(n) such that the following holds. Let b(x, z) be odd in z and homogeneous of degree -(n -1) in z, and assume D':b(x, z) is continuous and bounded on IR n x 8 n - l , for lad::; A-l. Then b(x,x - y) is the kernel of an operator B, bounded on LP(f), for 1 < P < 00. PROOF. The classical method of spherical harmonic decomposition due to Calderon and Zygmund works in this case. Thus, we can write ~3.1)
b(x,z)
=
l:bj(x)!PJ(z/lzl) Izl-(n-l), j,;>1
Hence,
(3.4)
IIBII.c(LP) ::; C(p, f) L Ilbj II LX> II!pj IICN
""here {!pj : j ~ I} is an orthonormal basis of L (8 consisting of eigenfunctions )f the Laplace operator on th(' sphere 8"-1. Furthermore, we can assume that !pj n
- 1)
::; C(p, f) sup IID~b(x, z)IIL=(lRnxsn-1). !a:I::;M
and the proof is done. Proposition 3.1 applies to the Schwartz kernels of certain pseudodifferential operators. In fact, operators in OPCo 8;/ have Schwartz kernels that differ from those treated in Proposition 3.1 by kernels with weak singularities, and with a different asymptotic behavior far from the diagonal. For our purposes it is sufficient to use the elementary consequence that the conclusions of these propositions hold, provided one acts on functions with support on a given compact subset f 0 of f, and estimates the norm of the resulting function over fo. In the rest of this section we will restrict attention to this case. We now state the consequence of Proposition 3.1 most directly relevant for the analysis in §4. PROPOSITION 3.2. Ifp(x, E) E C08(~1 has a principal symbol that is odd in E, then the Schwa'rtz kernel of p(x, D) is the kernel of an operator bounded on U(f o), for 1 < p < 00. The operator B in Proposition 3.1 is given by (3.5)
Bf(x)
=
P.V.
J
b(x,x - y)f(y) ddy)·
r
This is related to the following operator, defined for x E IRn \ f: (3.6)
Bf(x)
=
J
b(x, x - y)f(y) ddy)·
r
There is an estimate on the nontangential maximal function for Bf. Under the hypotheses of Proposition 3.1, if B is as in (3.6) then, by (3.3), we have (3.7)
2
L bj(x)Kjf(x).
=
j;:>l
yEC x
(2.14)
Cr 2 .
Note that, if kj(x) = !pj(x/lxl)lxl-(n-l) with !Pj odd, then the operator K j on U(f) with kernel kj(x - y) is estimable by (2.10), and, for f E LP(f),
x E f.
From the analysis above and in §1 it follows that, with IC r as in (2.12), one has, for f E U(f), 1
231
(Bf)*(x) ::;
l: Ilbjll
L '"
(lCjf)*(x).
1::>1
Thus, using estimates of the form (3.2) and (2.14), we have:
'7":0"""'''''_-'''''~'C2'','~~;'';.'''~s.",''?'_~'~'''="''~c~-~'-~'--cc'''-~,-- --~"--=",,,~,c"~""""-T-"-;:~iC;;;;;;~~~:~~-;;';;~~",;;;;':_',;_-;-,.;';-/,.;;;~;;;;;,,,-,;;;,;.;;:;,;;';";;':~';;'-";';;~;;;;;;';;~:"';;;;'---",=,;;;,-;;;.;;,,;;,,;;;;;;;,~;;=;;;;";;;;,;"";o;==~,;;;o==",,,,,;,;;,;;,,",;,,_._,~~_._.
232
4. LAYER POTENTIALS ON LIPSCHITZ SURFACES
. _
3.3. Ifp(x,e) E COSci l has a principal symbol that is odd in then its Schwartz kernel is the kernel of an operator B, satisfying
PROPOSITION
e,
Now, the unit conormal to (3.17)
(3.8) 00
(3.18)
Given (2.15), the superposition arguments used above yield:
3.4. If p(x, e) is as in Proposition 3.3, with principal symbol
P-l (x, 0, then, a.e. on f, we have nontangentiallimits
E(x, D)u
=
J
E(x, y)u(y) dvg(y)
=
IR n
J
E(x, y)u(y).jg(y) dy,
IR n
J
where g(y) dy is the volume element associated with the Riemannian metric gjk, so g(y) = det(gjk(Y))' The single layer potential associated with this is then
(3.11)
Sf(x) =
J
J
r
r
E(x,y)f(y)dag(y) =
Sci 1 with
=
gjk(x)ejek,
(ajSf)±(x) = ~~iP-l,j (x,n(x))f(x)
+ Kj f(x),
where
(3.15)
A(x) = .jg(x) p(x) 9 jk (x)nk(x)nj(x)G(x, n(x)) -3/2 (3.20)
=
1,
the last identity being a standard formula for the area element of a hypersurface in a Riemannian manifold (which works as well for Lipschitz hypersurfaces as for C 1 hypersurfaces). Hence we recover, in the context of a Lipschitz hypersurface in a smooth Riemannian manifold, the standard formula
(~ Sf) ± (x) = (=r~I + K*)f(x),
(3.21)
(3.22)
=
'" J J
aE (x, y)f(y)p(y) da(y) lIJ(X) aXj
LJ P.V.
r
J
av"E(x, y)f(y) dag(y).
= P.V.
r
(3.23)
D f(x) =
oE
J
ally (x, y)f(y) dag(y),
One has (3.24)
oE
J
aXj (x, y)f(y)p(y) da(y).
r
Note that
P~l,J (x, n(x)) = -i ;~~~) G(x, n(x)) -lnj(x).
(Df)±(x) = (±V + K)f(x)
nontangentiallya.e. on f, with
(3.25) K;J(x) = P.V.
_ p(x) 1 2 - .jg(x) G(x, n(x)r /
r
G(x,O
where (gjk) is the matrix inverse of (gjk). By (3.9), we have, for a.e. x E f, (3.14)
+ lIj(x)KJ f(x),
There is a similar treatment of double layer potentials, defined by
for Bj ofthe form treated in Proposition 3.3, corresponding to pj(x,e) E principal symbol
(3.13)
(lIjajSf)±(X) = ~~A(x)f(x)
K* f(x)
ajSf(x) = Bjf(x),
. p(x) ( C)-1 ( C) =-z.jg(x)Gx,<., P-l,jX,<., ej,
with respect to this metric is given by
lIj(x) = gjk(X)lIk(X).
with
E(x,y)f(y)p(y)da(y),
where dag(y) = p(y)da(y) is the area element on r inherited from the Riemannian metric gjk, which differs from that inherited from the Euclidean metric rS jk by a factor p E £OO(r), a formula for which we give below. In such a case,
(3.12)
nj(x),
with
To consider an example for which the results above apply, suppose IR n is given a smooth Riemannian metric and ~ is its Laplace-Beltrami operator, and E E
OPSci2 is a parametrix for £ = ~ - V, where V is smooth and ~ O. Assume for simplicity that the metric tensor is asymptotically Euclidean. Then Proposition 3.2 applies to ajE and Eaj, so the associated double layer potentials are bounded on V(f o). It is standard to write the action of E = E(x, D) as
(3.10)
r
1 2 /
Thus we have
(3.19)
+ Bf(x),
(Bf)±(x) = ~~ip-l (x,n(x))f(x)
(3.9)
233
with respect to the metric gjk is given by
and the unit outward normal to
(and f supported on f o).
PROPOSITION
r
lIj(X) = G(x, n(x)r
II(Bf)*IILP(ro) $ CpllfIILP(ro)'
for 1 < P <
(3.16)
3. THE VARIABLE-COEFFICIENT CASE
K f(x) = P.V.
oE
J
ally (x, y)f(y) dag(y).
r
The operators K and K* are adjoints on the Hilbert space £2 (r, dag). A variant of the case considered above arises on a smooth, compact, connected manifold M, endowed with a smooth Riemannian metric. If £ = li - V, with a smooth V ~ 0 that is > 0 somewhere, then £ is invertible, with inverse E E OPS- 2 (M). If n is a Lipschitz domain in M, then one can use local coordinates
.
234
4. BOUNDARY INTEGRAL OPERATORS
and a partition of unity to construct operators to which Propositions 3.1-3.4 apply, and then obtain results parallel to (3.10)-(3.25). We collect some of the results discussed above: PROPOSITION 3.5. Given f E V(oO), 1 < P < Si+(x)
(3.26)
00,
we have, for a.e. x E 00,
= Sf-(x) = Sf(x),
and
4. Boundary integral operators The identities (3.27)-(3.29) suggest investigating the invertibility of various operators on £2(80). We take this up here. Let M be a compact, connected, smooth manifold, with a smooth Riemannian metric tensor, n a domain in M with nonempty Lipschitz boundary. We will assume n is connected, but we do not assume its complement M \ n is connected. We take £=~-V,
(4.1)
n,
Vf±(x) = (±~I + K)f(x),
(3.27)
Sf(x)
=
J
f(y)E(x,y)du(y),
K f(x)
= P.v.
oE f(y) ov (x, y) du(y),
J
an
an
y
Furthermore,
8 8v Sf±(x) = (+~I + K*)f(x).
(3.29)
In addition, there are estimates on nontangential maximal functions: (3.30) for 1 < P
with smooth V ~ O. To study ~u = 0 on 0, we would take V = 0 on but it is of interest to consider the more general case. It is technically important to assume that V is strictly positive somewhere on each connected component of 0 = M \ We then define E, S, V, S, and K as in §3. As before, v denotes the unit outward normal to 80, which is defined a.e. on 80. We tackle the boundary integral equations that arise in the analysis of the Dirichlet problem for £u = 0 on 0, when the method of layer potentials is used. These equations were first treated in [Vel when £ is the standard (constant coefficient) Laplace operator on a Lipschitz domain in Euclidean space. The treat ment here follows [MT], though as stated in the introduction we restrict attention here to operators with smooth coefficients, for simplicity. We mention that in [MT] the hypothesis that V ~ 0 on 0 is also relaxed. One goal in this section is to prove that !I + K is invertible on £2(80). We start with the following result.
n.
where, for x E 00, (3.28)
235
11(\7Sn*IILP(8n) :::; Cp llflb(8n),
II(Vn*lb(8n):::; Cp llflb(8n),
< 00. We note that (3.30) plus interior regularity implies that S: £2(80)
(3.31)
---t
H 1 (M) is compact.
PROPOSITION 4.1. The map ~I
(4.2)
+ K*
: £2(00)
---t
£2(00)
is injective. In fact, there is the following simple general result. PROPOSITION 3.6. Let V be a Banach space. Assume that T: V ~ H 1 (M) is a bounded linear map such that, for some 8 > 1, (3.32)
vEV
===}
Tv E Hl~c(M \ (0), (\7Tv)* E £2(80). 1
Then T is a compact operator from V to H (M). PROOF. It suffices to show that, if {Vj} C V is bounded and TVj ~ 0 weakly in H 1 (M), then TVj ~ 0 in HI (M)-norm. Fix e > 0 and let C be a collar neigh borhood of 80, of thickness :::; c. Since TVjIM\C is bounded in H8(M\C), Rellich's theorem implies that II\7Tvjll£2(M\c) ~ O. Meanwhile, the hypothesis (3.32) im plies a bound 11(\7Tvj)*II£2(an) ::; C, and hence II\7TvjII12(c) :::; Ce, so
PROOF. Suppose f E £2(80) and (~I+K*)f = O. Set u = Sf, so (~- V)u = 0 on M \ 80. The estimate (3.30) allows the use of Green's formula, to write (4.3)
J{1\7uI + 2
J-----'lX
= -
o
J ~~ u
du(x).
an
By (3.29), the right side of (4.3) vanishes when f E ker (~I + K*). Thus u is constant on each connected component of 0 and u = 0 on supp V. Hence u = 0 on O. Hence, by (3.26), Sf = 0 a.e. on 80, so, again by (3.26) and Green's formula (justified as before) we have
J
{1\7uI
0.
limsup II\7Tvjll£2(M) :::; Ce.
V\un dv(x)
2
+ Vlun dv(x)
=
J ~~ u
du(x) = O.
80.
Hence u is constant on 0, so 8vu+ = 0 a.e. on 80. Since, by (3.29), f is equal to the jump of 8vu across 80, we have f = 0, so Proposition 4.1 is proven.
This proves the proposition. Our next step is to establish the following estimates.
236
4. BOUNDARY INTEGRAL OPERATORS
4. LAYER POTENTIALS ON LIPSCHITZ SURFACES
4.2. For all
PROPOSITION
(4.4)
f
This can be done provided 80 isLipschitz. We then deduce from (4.8) the inequality
£2(80),
E
IIGI i- K*)fliL 2 S Crl(~I + K*)fll£2 + cll Sf da(x)1 + cllwsfll£2(M),
(4.10)
ao where W
(4.5)
2 2 I\7TuI da(x) SCi 1811 ul da(x)
II(~I + K*)fIIL2 S CII(~I -
K*)fIIL2
+ cll Sf da(x) I + cll wS fll£2(M)'
(4.11)
00
I
~u
+
I
2
I l\7ul
(4.14)
ao
ao
2
2
da(x) s c i 1811 ul da(x)
ao
I (4.15)
ao
2
1811 uj2 da(x) s c i I\7TuI da(x)
ao
where \7TU(X) denotes the tangential component of \7u(x), for a.e. x E 80. Now, pick w, smooth on M, such that (4.9)
(w, II) ::=: a > 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 :<:;
CII(~I + K*)flli2(ao.) +
Cll Sf
2
d(J"(x)
1
max{j 18,.,u1 d(J"(x) , j IVTul 2 d(J"(x)} an an 2 2 :<:; C min{j 18",u1 d(J"(x) , j I'il T uI d(J"(x)} an an
1
-~)
(A
+ C1I WS flli2(o)'
In concert with (4.16), this proves (4.4), and the proof of (4.5) is similar. Then (4.6) follows from (4.4)-(4.5), together with the fact that the last two terms in each of these formulas are dominated by CIISfI17Jl(M)' We mention another useful way of stating the Rellich estimate:
239
PROOF. Take the identity (4.7) and multiply by A - 1/2; take its counterpart with 0 replaced by O~ = 0 = AI \ n and multiply by A + 1/2. Summing the two, we have
(v,w){I'il T u\2 - (8,.,U)2} d(J"(x)
an +
an
- (A
+~)
j (v, w){I'ilTuI2 - (8 u)2} d(J"(x) "
00._
(4.21)
(2A - 1) j ('ilTwU) (8,.,u) d(J"(x)
=
an+
- (2A + 1) j ('ilTwU) (o,.,u) d(J"(x)
2
(4.18)
BOUNDARY INTEGRAL OPERATORS
4.
+ R,
alL
where, here and below, R denotes a quantity satisfying an estimate
j
+ C {1~uI2 + l'iluI 2} dvol(x).
(4.22)
IRI
:<:;
Cll'ilulli2(M) + Cll h lli (M)' 2
0.
One easy consequence of Proposition 4.2 is the following closed range property.
Recall that u
PROPOSITION 4.3. The maps ±~I + K* have closed range on £2(80). PROOF. As shown in (3.31), the map S : £2(80) ---; HI (AI) is compact. That (4.6) then implies the closed range property for ± ~ I + K* is a well known part of the Riesz theory of compact operators.
(4.23)
= Sf and h =
~uIM\aO = Vu. Now the left side of (4.21) is equal to
- j (v, w)I'il T uI 2 d(J"(x) - (A an +
~) j (v,w)I(-~I + K*)fI 2 d(J"(x) an
(A+~) j(v,w)I(F +K*)fI 2 du(x). an
COROLLARY 4.4. The map
(4.19)
~I
+K
FUrthermore, upon writing ±~I + K* last two terms in (4.23) sum to
: £2(80) --; £2(80)
is surjective.
(4.24)
_(A 2
~) j
-
PROOF. Proposition 4.1 implies that ~I +K has dense range, and by Banach's closed range theorem, Proposition 4.3 implies that ~ I + K has closed range.
= (±~
(v, w)lfl 2 d(J"(x)
an
- A)
+ (AI + K*),
+ j (v, w)I(H + K*)fI 2 d(J"(x). an
On the other hand, the right side of (4.23) is equal to R plus The last step in showing that ~ I + K in (4.19) is an isomorphism is to show that it is injective. In [Vel this was done by showing that ~ 1+ K* has dense range. Here (as in [MT]) , we will instead use the method of [EFV], and establish the following extension of Proposition 4.2. PROPOSITION 4.5. Given A E JR., IAI ~ 1/2, there exists C
(4.20)
=
C(A, 0) such that
IIfllp(iJ!!) S ClI(),[ + K*)fllL2(an) + CII S fIIHl(M)'
(2A - 1) j ('ilTwSf) (( -~I + K*)f) d(J"(x)
an (4.25)
- (2A + 1) =
-2
1
an
one sees that the
1('ilTwSf)((~I +
K*)f) d(J"(x)
an
('ilTwSf) . (AI + K*)f d(J"(x).
-~---~-"==~O-'7=~_
~
=
=-==~=~-=_-=---
=
------------ .--------------------~
240
4.
LAYER POTENTIALS ON LIPSCHITZ SURFACES
Thus, from (4.21)-(4.25) we have
2
(A -
(4.26)
~)
!
2 (v, w) If/ d17(x)
80.
= 2
!
J
+ (v, w) (l'VrS fl 2 - I(AI + K')fn d17(x) 80.
('VrwSf) . (AI + K')f d17(x)
+ R.
Now (4.20) follows directly from (4.26) if IAI > 1/2. The cases A= ±1/2 follow from Proposition 4.2, upon writing f = (~I +K')f - (F -K')f and using the estimates (4.4)-(4.5). Of course, the case A = 1/2 is already contained in Propositions 4.1 and 4.3, which imply the stronger estimate
11fl/£2(8n) :::; CII(~I
+ K·)fliL
2 (8n).
As noted in (3.31), S : L 2 (on) ---+ H1(M) is compact. Hence (4.20) implies that, for IAI ~ 1/2, AI + K' : L 2(on) ---+ L 2(on) has closed range and finite dimensional kernel. Thus, for each such A, AI + K' is semi-Fredholm on L 2 (on), with a well defined index. Furthermore, the index is continuous in A, hence constant on (-00,-1/2] and on [1/2,00). Now, for IAI > IIKII, AI +K' is invertible, so we have: L
2
PROPOSITION 4.6. If A E JR, IAI ~ 1/2, the operator AI + K' is Fredholm on of index zero; hence so is AI + K. In particular, the operators
(on),
±~I
(4.28)
+ K,
±~I + K' : L 2 (on) ---+ L 2 (on)
are Fredholm of index zero.
COROLLARY 4.7. The operators
~I + K,
F
+ K' : L 2(on)
---+
PROPOSITION 4.8. If V> 0 somewhere on 0., then -~I + K and -~I + K' are invertible on L2(on). If V = 0 on n, then
-~I + K' : L5(on)
---+
L5(on)
is an isomorphism.
PROOF. We use reasoning parallel to the proof of Proposition 4.1. To begin, 2 assume f E L (on) and (-~I + K')f = 0, and set u = Sf. Parallel to (4.3), we have
J{1'VuI + 2
!l
2 Vl u l } dv(x)
M
M
where the last identity uses the fact that f is equal to the jump of OyU across on. If Co = 0, then the right side of (4.32) vanishes, so u is constant on each connected component of 0 (and each such constant is 0). Thus the jump of OyU across on is zero, Le., f = 0, so -~I + K' is injective on L 2 (on), if V > 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
(5.4)
----+
HI(D).
The next two propositions deal with the case when aD is smooth; these results will provide useful tools in the analysis of the Lipschitz case.
(5.5)
PI: HS(aD)
----+
H +l/ 2(D),
1 V s > - 2'
PROOF. This is standard, as is the fact that, when aD is smooth, (5.6)
v E HJ(D), Lv E HO"(D)
===}
v E H 2+0"(D).
243
and u has a non-tangential boundary trace at almost every point in aD, we have
I lul
(5.10)
2
I
C lul 2 dO'(x).
dv(x) :::;
o
ao
PROOF. Let Dj be a sequence of smooth domains, with bounded Lipschitz constants, increasing to D. Given f E L 2 (D), define Vj by
Vj E HJ(D j ), LVj = f on Dj . 2 Then Vj E H (D j ), and Proposition 5.4 applies to Vj' Applying Green's formula and the estimate (5.7), we have (5.11)
II
(5.12)
ufdv(x)1
=
I u~~~
I
dO'(x) I
aOj
OJ
:::; Cllullu(aoj) Ilfll£2(Oj)' Given that u* E L 2 (aD) and that we have non-tangential convergence to the limit on aD, we obtain
II
(5.13)
Note that, as a byproduct of Propositions 5.2-5.3 and their proofs, we have the unique solvability of Lu = f, given f E L 2(D), for u E HJ(D) n H 2(D), when aD is smooth. This fact will playa role in the proof of Proposition 5.5.
_
THE DIRICHLET PROBLEM ON LIPSCHITZ DOMAINS
PROPOSITION 5.5. There is a constant C, depending only on the Lipschitz character of D, such that, whenever (5.9) Lu = 0, u* E L 2(aD),
PROPOSITION 5.3. If aD is smooth, then 8
.
5.
ufdv(x)l:::;
Cll ull£2(ao)llfll£2(O)'
o for all
f
2
E L (D), which implies (5.10).
Let us temporarily denote the solution operator to (5.1), produced by Propo PROPOSITION 5.4. Let aD be smooth. Then there exists a constant C, depend sition 5.1, by PI, so
ing only on the Lipschitz character of D, such that the following holds: whenever v E HJ(D) and Lv = h E L 2(D), we have a"v E L 2(aD) and
Ilav vIIL2(aO) :::; C1l hllu(!!).
(5.7)
PI: L 2(aD)
(5.14)
----+
{u E COO (D) : u* E L 2(aD)}.
We have the following compatibility result. PROPOSITION 5.6. We have
PROOF. That a"v E H I /2(aD) c L 2(aD) under our hypotheses follows from (5.6). Also, the hypotheses imply II\7vlli2(o) + (Vv, v)u(O) = -(h, V)£2(O), and this plus Poincare's inequality yields II\7vllu(o) :::; C1lhllu(o). Next, since v E H2 (D), the Rellich type estimate (4.11) implies that, with C depending only on the Lipschitz character of D,
Jla"vl
(5.8)
ao since
\7 TV =
2
dO'(x) :::;
J
PROOF. First, consider
C {1\7vI + Ihn dv(x), 0
0 on aD. This proves the desired estimate (5.7).
We now prove an estimate that implies uniqueness of solutions to (5.1). We return to the general case of Lipschitz D.
f
===}
PIf
=
PIf.
E V, the set of restrictions to
C=(M). Well known arguments involving smooth Dj
aD of elements of
D, the maximum principle, iI.nd barrier functions, such as given in Chapter 5, §5 of [T5], yield PI: V ----+ C(D). /'
E V, PI f satisfies all the conditions in (5.11). Hence, by Proposition 5.5, f E V ==} PI f = PI f. Now any f E H I/ 2(aD) is a limit in H I/ 2-norm of a sequence flo' E V. We have simultaneously PI flo' ----t PI f in HI (D) and PI flo' ----t PI f in COO(D), so we have (5.15).
Thus, for
2
f E H I/ 2(aD)
(5.15)
f
Thus we drop the tilde from (5.14) and write (5.16)
PI: L 2(aD)
----+
{u E COO(D) : u* E L 2(aD)}.
. _-244
. -
.-"'-"'=;c--.-.----~==r
="""-~-=
.•
(5.17)
PI: C(aD)
---+
C(O).
E
lem (5.18)
uloo = f a.e. on aD
lI u ll£<>o(o)
~ IlfIILoo(oo),
We can interpolate between the L2 and Loo results, to obtain:
(5.20)
(5.23)
00
Now u
wx(E)
=f
E V(aD).
Ilu*IILP(oO) ~ CIIfIILP(ao).
Lu E
PROOF. Consider the operator T : f f-t (PI f)*, which is well defined and 2 sublinear on L (aD). Since T is a bounded mapping of L 2(aD) into itself as well as of Loo(aD) into itself, Marcinkiewicz's interpolation theorem implies that T is bounded on V(aD) for 2 ~ p ~ 00. In view of (5.17), we know that evaluating PIf at a point xED produces the
"harmonic measure" W x :
J
" . _ . ~~
245
kx E L 2 (aD),
= 0 = } PI XE(X) =
«
wx ,
for each
O.
= 0,
0 ~ u ~ 1, on D,
aD, lim u(z) = XE(Y),
z ..... y
the limit taken nontangentially. If a(E) I' 0, then we deduce that u is not identically zero on D. Then the Hopf maximum principle implies that u(x) > 0 for all xED. This shows that, for each xED, wx(E) = 0 =:} dE) = 0, so the proof is complete. We mention some further properties of the solution to (5.1). For example, one
f E HI (aD)
f(y) dwx(Y),
f E C(aD), xED.
80
We now have the following result, which was established for the Laplace operator on Lipschitz domains in flat Euclidean space by [Dah]. PROPOSITION 5.9. For each xED, the measure W x and the surface measure a on aD are mutually absolutely continuous.
=}
(Vu)* E L 2(aD).
This is proven in §7 of [MT], using techniques that also treat the Neumann bound ary problem. In the case of the Laplace operator on a Lipschitz domain in Euclidean space, (5.27) was established in [Vel. Also, in [MMT] it is shown that
L 2(aD)
==?
f E H1(aD)
==?
f
PIf(x) =
_np~.
= PIXE satisfies
(5.26)
(5.27)
This solution satisfies
(5.22)
,"<'-='=
has there exists a unique solution to the Dirich
Lu = 0 in D, u* E V(aD), ulao
(5.21)
k x a,
=
Wx
and, for a.e. y
PROOF. Given f E Loo(aD) c L 2(aD), we can construct a sequence {fJ} of continuous functions on aD such that f j -+ f in L 2 (aD) as j -+ 00, and IIfJIILoo(oo) ~ Ilfll£<"'(oo), Then, if Uj = PI!; in D, we have that Uj -+ u uni formly on compact subsets of D and, by the maximum principle, lI ujIILoo(o) ~ IlfJIILoo(oo) ~ IlfIILoo(oo), Hence, passing to the limit, we have (5.19).
PROPOSITION 5.8. For 2 ~ p ~ let problem
....= O-.==.=_===··.,._.._•.,
PROOF. From (5.16) we have
(5.25)
satisfies (5.19)
... . .
.~ ..." . _ " · ..P"'·"'"'~._.,~.,,_ ==.,,,..._""'.~"""'",==
L oo (aD) the L2-solution of the Dirichlet prob (5.24)
u* E L 2(aD),
Lu = 0 in D,
c ' , •• ' _".-
and (5.22) holds for all f E L2(aD). It remains to show that a xED. Suppose E c aD is a Borel set. Then
The following is a useful extension of (5.17). PROPOSITION 5.7. For any f
_----_._-----_._.
.".-.....==",,...,... . .""",. .. ,. 5. THE DIRICHLET PROBLEM ON LIPSCHITZ DOMAINS A
"",~
4. LAYER POTENTIALS ON LIPSCHITZ SURFACES
Note also that, by reasoning similar to the proof of Proposition 5.6, when f E C(aD), PIf coincides with the element u E CCri') solving Lu = 0 provided by the Perron-Weiner-Brelot process:
,-.._.""--...,,_....
(5.28)
E
u E H 1 / 2(D), 3 2 U E H / (D).
In [MT2], extending work done in the constant coefficient case in [Vel and [DK], the authors show that, for some e = e(D) > 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)
(1)
1-
f(z) 2 f( )/ =z+ a2+- z + .... zZo Zo
We have
ds =
(A.ll)
7),
this metric
21 dzl l-lzl 2 •
is biholomorphic and if dO' = 1'(z) Idzl is a metric on n, the metric f*(dO') induced on 7) is 1'(f(z)) II'(z)lldzl. Thus the Poincare metric induced on n is of the form
If f :
7) ---->
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.
V. Adolfsson, LP-integrability of the second derivatives of Green potentials in convex domains., Pacific J. Math. 159 (1993), 201-225. L. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, New York, 1966. [Ahl] [AhB] L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. '72 (1960), 385--404. S. Alinhac, Paracomposition et operateurs paradifferentiels, Comm. POE 11 (1986), 87 [AI] 121. S. Alinhac and P. Gerard, Operateurs Pseudo-dijJerentiels et Theoreme de Nash-Moser, [AG] Editions du CNRS, Paris, 1991. M. Anderson, Metrics of positive Ricci curvature with large diameter, Manus. Math. 68 [An] (1990), 405-415. M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, In [An2] vent. Math. 102 (1990), 429-445. [AnC] M. Anderson and J. Cheeger, CO-compactness for manifolds with Ricci curvature and injectivity mdius bounded below, J. Oiff. Geom. 35 (1992), 265-281. P. Auscher and M. Taylor, Paradifferential operators and commutator estimates, Comm. [AT] POE 20 (1995), 1743-1775. A. Bachelot and B. Hanouzet, Applications bilineaires compatibles avec un systeme [BH] differentiel a coefficients variables, CRAS Paris 299 (1984),543-546. H. Bahouri and J. Chemin, Equations de transport relatives a des champs de vecteurs [BC] non-lipschitziens et mecanique des fiuides, Arch. Rat. Mech. Anal. 127 (1994), 159-191. [BKM] T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3-d Euler equations, Commun. Math. Phys. 94 (1984), 61-66. M. Beals and M. Bezard, Low regularity local solutions for field equations, Comm. POE [BB] 21 (1996), 79-124. M. Beals and M. Reed, Microlocal regularity theorems for non-smooth pseudodijJerential [BR) operators and applications to nonlinear problems, Trans. AMS 285 (1984), 159-184. J. Bergh and J. Lofstrom, Interpolation spaces, an Introduction, Springer-Verlag, New [BL] York,1976. N. Bliev, Generalized Analytic Functions in Fractional Spaces, Pitman, Essex, 1997. [Bli) J.-M. Bony, Calcul symbolique et propagation des singularities pour les equations aux [Bon] derivees nonlineaires, Ann. Sci. Ecole Norm. Sup. 14 (1981), 209--246. [Bour] G. Bourdaud, LP-estimates for certain non regular pseudo-differential operators, Comm. POE 7 (1982), 1023-1033. [Bour2] G. Bourdaud, Une algebre maximale d'operateurs pseudo-dijJerentiels, Camm. POE 13 (1988), 1059--1083. A. P. Calderon, Intermediate spaces and interpolation the complex method" Studia Math. [Cal 24 (1964), 113-190. A. P. Calderon, Cauchy integrals on Lipschitz curves cmd related operators, PNAS USA [Ca2] '74 (1977), 1324-1327. L. Carleson and T. Gamelin, Complex Dynamics, Springer-Verlag, New York, 1993. [CG] S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982), 7-16. [Cha] [CPSS] G. Chavant, G. Papanicolaou, P. Sachs, and W. Symes (eds.), Inverse Problems in Wave Propagation, IMA Vol. gO, Springer-Verlag, New York, 1997. [Ad]
250
[Chel [CFL] [Chr] [CW] [CDM] [CJS] [CLMS] [CMM] [CM] [CM2] [CRW] [Dah] [DK] [DKV] [Dav] [DeTK] [DiF] [DiPM] [EFV]
[FJR] [FKV] [FMM] [FS1] [FS2] [FJW] [GT] [GM]
BIBLIOGRAPHY
S. Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. AMS 6 (1955), 771-782. F. Chiarenza, M. Frasca, and P. Longo, Interior estimates for non divergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), 149-168. M. Christ, Lectures on Singular Integral Operators, CBMS Reg. Conf. Ser. in Math. # 77, AMS, Providence, RI, 1990. M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. FUnct. Anal. 100 (1991), 87-109. R. Coifman, G. David, and Y. Meyer, La solution des conjectures de Calder6n, Adv. in Math. 48 (1983), 144-148. R. Coifman, P. Jones, and S. Semmes, Two elementary proofs of the L 2 boundedness of Cauchy integrals on Lipschitz curves, Jour. AMS 2 (1989), 553-564. R Coifman, P. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pure Appl. 12 (1993),247-286. R Coifman, A. McIntosh, and Y. Meyer, L'integrale de Cauchy definit un operateur bome sur L 2 pour les courbes lipschitzeans, Ann. of Math. 116 (1982),361-387. R. Coifman and Y. Meyer, Au dela des operateurs pseudodifferentiels, Aste'f'isque #57, Soc. Math. de France, Paris, 1978. R Coifman and Y. Meyer, Commutateurs d'integrales singulieres et operateurs multi lineaires, Ann. Inst. Fourier 28 (1978), 177-202. R. Coifman, R Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables" Ann. of Math. 103 (1976),611--635. B. Dahlberg, Estimates of harmonic measure, Arch. Rat. Mech. Anal. 65 (1977), 275 288. B. Dahlberg and C. Kenig, Hardy spaces and the Neumann problem in LP for Laplace's equation in Lipschitz domains, Ann. of Math. 125 (1987), 437-465. B. Dahlberg, C. Kenig, and G. Verchota, Boundary value problems for the system of elastostatics on Lipschitz domains, Duke Math. J. 51 (1988), 795-818. G. David, Operateurs d'integrale singuliere sur les surfaces regulieres, Ann. Sci. Ecole Norm. Sup. Paris 21 (1988), 225-258. D. DeTurk and J. Kazdan, Some regularity theorems in Riemannian geometry, Ann. Sci. Ecole Norm. Sup. 14 (1981), 249-260. G. DiFazio, LP estimates for divergence form elliptic equations with discontinuous coef ficients, Bollettino V.M.L 10-A (1996), 409--420. R. DiPerna and A. Majda, Concentration in regularizations for 2-D incompressible flow, CPAM 40 (1987), 301-345. L. Escauriaza, E. Fabes, and G. Verchota, On a regula''f'ity theorem for weak solutions to transmission problems with intemal Lipschitz boundaries, Proc. AMS 115 (1992), 1069-1076. E. Fabes, M. Jodeit, and N. Riviere, Potential techniques for boundary value problems in C 1 domains, Acta Math. 141 (1978), 165-186. E. Fabes, C. Kenig, and G. Verchota, The Di'f'ichlet problem for the Stokes system on Lipschitz domains, Duke Math. J. 51 (1988), 769-793. E. Fabes, O. Mendez, and M. Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains, Preprint (1997). C. Fefferman and E. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107 115. C. Fefferman and E. Stein, HP spaces of several variables, Acta Math. 129 (1972), 137 193. M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Reg. Conf. Ser. Math. #79, AMS, Providence, 1991. D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983. J. Gilbert and M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Univ. Press, Cambridge, 1991.
BIBLIOGRAPHY
[GV] [Gol]
[G] [HW]
251
J. Ginibre and G. Velo, The global Cauchy problem for the non linear Klein-Gordon equation, Math. Zeit. 189 (1985), 487-505.
D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), 27--42. R Greene, Proc. Symp. Pure Math. 54 (Vol. 3) (1994). P. Hartman and A. Wintner, On the existence of Riemannian manifolds which cannot carry non-constant analytic or harmonic functions in the small, Amer. J. Math. 15 (1953), 260-276.
[HI]
L. Hormander, The Analysis of Linear Partial Differential Operators, Vols. 3-4, Springer Verlag, New York, 1985.
[H2]
L. Hormander, Nonlinear Hyperbolic Differential Equations, Springer-Verlag, New York, 1998.
[H3]
L. Hormander, Pseudo-differential operators of type 1,1, Comm. PDE 13 (1988), 1085 1111.
[H4]
L. Hormander, On the existence and the regularity of solutions of linear pseudo-differ ential equations, L'Enseign. Math. 11 (1971), 99-163.
D. Jerison and C. Kenig, The inhomogeneous Di'f'ichlet problem in Lipschitz domains, J. FUnct. Anal. 130 (1995), 161-219.
[JK]
[IN]
F. John and L. Nirenberg, On functions of bounded mean oscillation, CPAM 14 (1961), 415-426.
[J]
J. Jost, Conformal mappings and the Plateau-Douglas problem in Riemannian manifolds, Jour. fiir Reine Angew. Math. 359 (1985), 37-54.
J. Jost and H. Karcher, Geomet'f'ische Methoden zur Gewinnung von a-priori Schranken fur harmonische Abbildungen, Manus. Math. 19 (1982), 27-77.
J.-L. Journe, Calder6n-Zygmund Operators, Pseudo-differential Operators, and the Cauchy Integral of Calder6n, LNM #994" Springer-Verlag, New York, 1983. T. Kato, On nonlinear Schrodinger equations, II. HS-solutions and unconditional well posedness, Preprint (1995).
[JoK]
[Jo]
[K] [K2]
T. Kato, Liapunov functions and monotonicity in the Navier-Stokes equation, LNM #1450 (1990), 53-63.
[KP]
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equa tions, CPAM 41 (1988),891-907.
[Kel]
O. Kellogg, Harmonic functions and Green's integral, Trans. AMS 13 (1912), 109-132.
[Ken]
C. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser #83, AMS, Providence, RL, 1994.
[KPV] C. Kenig, G. Ponce, and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Preprint (1991).
[KM]
S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, CPAM 46 (1993), 1221-1268.
[KS]
S. Klainerman and S. Selberg, Remarks on the optimal regularity for equations of wave maps type, Comm. PDE 22 (1997), 901-918.
[Ko]
G. Kothe, Topological Vector Spaces, Springer, New York, 1969.
[KN]
H. Kumano-go and M. Nagase, Pseudo-differential operators with nonregular symbols and applications, FUnkcial Ekvac. 21 (1978), 151-192. [LV] O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane, Springer-Verlag, New York,1993. [LMZ] C. Li, A. McIntosh, and K. Zhang, Compensated compactness, paracommutators, and [Lie] [LS] [MM] [Mal]
Hardy spaces, J. FUnct. Anal. 150 (1997), 289-306. L. Lichtenstein, Zur Theorie der konformen Abbildung, Bull. Acad. Sci. Cracovie, Cl. Sci. Math. Nat. A (1916), 192-217. H. Lindblad and C. Sogge, On existence and scattering with minimal regularity for semi linear wave equations, J. FUnct. Anal. vol 130 (1995), 357-426. M. Marcus and V. Mizel, Every superposition operator mapping one Sobolev space into another is continuous, Jour. FUnct. Anal. 33 (1979), 217-229. J. Marschall, Pseudo-differential operators with coefficients in Sobolev spaces, Trans. AMS 301 (1988), 335-361.
252
[Ma2]
[MaS] [Ma4] [Ma5] [McM] [MeV] [Meyl] [Mey2] [Mey3] [Mik] [Mila] [Mil] [MiD] [MMP]
[MMT] [MT] [MT2]
[MT3] [MT4] [MY) [Mor] [Mur] [Nec]
[P] [P2] [Pet] [Porn]
[RR]
BIBLIOGRAPHY
BIBLIOGRAPHY
J. Marschall, A commutator estimate for pseudo-differential operators, Proc. AMS 103 (1988), 1147-1150. J. Marschall, Weighted parabolic Triebel spaces of product type. Fourier multipliers and pseudo-differential operators, Forum Math. 3 (1991), 479-511. J. Marschall, Weighted LP-estimates for pseudo-differential operators with non-regular symbols, Zeit. Anal. Anwend. 10 (1991), 493-501. J. Marshall, Parametrices for nonregular elliptic pseudodifferential operators, Math. Nachr. 159 (1992), 175-188. C. McMullen, Renormalization and 3-Manifolds which Fiber over the Circle, Princeton Univ. Press, Princeton, N. J., 1996. M. Melnikov and J. Verdera, A geometric proof of the L 2 boundedness of the Cauchy integral on Lipschitz graphs, Intnl. Math. Research Notes 7 (1995), 325-331. Y. Meyer, Regularite des solutions des equations aux derivees partielles nonlineaires, Springer LNM #842 (1980), 293-302. Y. Meyer, Remarques sur un theoreme de J.M.Bony, Rend. del Circolo mat. di Palermo (suppl. II:l) (1981), 1-20. Y. Meyer, Wavelets, Cambridge Univ. Press, Cambridge, 1997. S. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergammon Press, Oxford, 1965. A. Milani, A remark on the Sobolev regularity of classical solutions of strongly elliptic equations, Math. Nachr. 190 (1998), 203-219. J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, AMS, Providence, R.I., 1999. D. Mitrea, The method of layer potentials for non-smooth domains with arbitrary topol ogy, Preprint (1996). D. Mitrea, M. Mitrea, and J. Pipher, Vector potential theory on nonsmooth domains in ]R3 and applications to electromagnetic scattering, J. Fourier Anal. and Appl. 3 (1997), 131-192. D. Mitrea, M. Mitrea, and M. Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, Memoirs AMS, to appear. M. Mitrea and M. Taylor, Boundary layer methods for Lipschitz domains in Riemannian manifolds, J. Flmct. Anal. 163 (1999), 181-251. M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian man ifolds: U', Hardy, and Holder space results, Commun. in Analysis and Geometry, to appear. M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian mani folds: Sobolev-Besov space results and the Poisson.problem, Preprint (1999). M. Mitrea and M. Taylor, Potential theory on Lipschitz domains in Riemannian mani folds: Holder continuous metric tensors, Comm. PDE, to appear. A. Miyachi and K. Yabuta, LP-boundedness of pseudo-differential operators with non regular symbols, Bull. Fac. Sci. Ibaraki Univ., Math. 17 (1985), 1-20. C. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966. T. Murai, A real variable method for the Cauchy transform and applications to analytic capacity, LNM #1307, Springer-Verlag, New York, 1988. J. Necas, Les Methodes Directes en Theorie des Equations Elliptiques, Academia, Prague, 1967. J. Peetre, On convolution operators leaving LPA-spaces invariant, Ann. Mat. Pura Appl. 72 (1966), 295-304. J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Press, Durham, N.C., 1976. P. Petersen, Riemannian Geometry, Springer-Verlag, New York, 1998. C. Pommerenke, Boundary Behavior of Conformal Maps, Springer-Verlag, New York, 1992. J. Rauch and M. Reed, Propagation of singularities for semilinear hyperbolic equations in one space variabk, Ann. Math. 111 (1980), 531-552.
[RT] [Rei] [RRT] [Run] [Run2] [RuS] [ST] [ScT] [Sem]
ISh] [Sic] [SiT] [Sm] [Sm2] [Sog] [Sog2] [Sp] [Sta] [SI] [S2] [S3] [Tl] [T2] [T3] [T4] [T5] [T6] [T7]
[Tril] [Tri2]
[Ta]
[Vel
253
J. Rauch and M. Taylor (eds.), Singularities and Oscillations, IMA Vol. 91, Springer Verlag, New York, 1997. H. Reimann, Ordinary differential equations and quasiconformal mappings, Invent. Math. 33 (1976), 247-270. J. Rubio de Francia, F. Ruiz, and J. Torrea, Calderon-Zygmund theory for operator valued kernels, Adv. Math. 62 (1988), 7--48. T. Runst, Para-differential operators in spaces of Triebel-Lizorkin and Besov type, Zeit. Anal. Anwend. 4 (1985), 557-573. T. Runst, Mapping properties of non-linear operators in spaces of Triebel-Lizorkin and Besov type, Anal. Math. 12 (1986), 313-346. . T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter, New York, 1996. R. Schrader and M. Taylor, Semiclassical asymptotics, gauge fields, and quantum chaos, J. Funct. Anal. 83 (1989), 258-316. B.-W. Schultze and H. Triebel (eds.), Seminar Analysis of the Karl- Weierstrass-In8titute 1985/86, Teubner, Leipzig, 1987. S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Muller, Comm. PDE 19 (1994), 277-319. A. Shnirelman, Evolution of singularities and generalized Liapunov function for an ideal incompressible fluid, Preprint (1995). W. Sickel, On pointwise multipliers in Besov-Triebel-Lizorkin spaces, pp. 45-103 in [SeT]. W. Sickel and H. Triebel, Holder inequalities and sharp embeddings in function spaces of B~q and F;q type, Z. Anal. Anwendungen 14 (1995), 105-140. H. Smith, A parametrix construction for wave equations with 0 1 ,1 coefficients, Ann. Inst, Fourier 48 (1998). H. Smith, Strichartz and nullform estimates for metrics of bounded curvature, Preprint (1999). C. Sogge, On local existence for nonlinear wave equations satisfying variable coefficient null conditions, Comm. PDE 18 (1993), 1795-1821. C. Sogge, Lectures on Nonlinear Wave Equations, International Press, Boston, 1995. S. Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa 19 (1965), 593-608. G. Staffilani, Well-posedness for generalized KdV equations, Preprint (1995). E. Stein, Singular Integrals and Differentiability Properties of FUnctions, Princeton Univ. Press, Princeton, N.J., 1970. E. Stein, Singular Integrals and Pseudo-differential Operators, Graduate Lecture Notes, Princeton Univ., 1972. E. Stein, Harmonic Analysis, Princeton Univ. Press, Princeton, N. J., 1993. M. Taylor, Pseudodifferential Operators, Princeton Univ. Press, Princeton, N. J., 1981. M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhauser, Boston, lW1. M. Taylor, Analysis on Morrey spaces and applications to Navier-Stokes and other evo lution equations, Comm. PDE 17 (1992), 1407-1456. M. Taylor, Microlocal analysis on Morrey spaces, pp. 97-135 in [RTj.. M. Taylor, Partial Differential Equations, Vols. 1-3, Springer-Verlag, New York, 1996. M. Taylor, Estimates for approximate solutions to acoustic inverse scattering problems, pp. 461-499 in [OPSSj.. M. Taylor, Microlocal analysis and nonlinear PDE, Proc. Syrnp. Pure Math. 59 (1996), 211-224. H. Triebel, Spaces of Besov-Hardy-Sobolev Type, Teubner, Leipzig, 1978. H. Triebel, Theory of FUnction Spaces, Birkhauser, Boston, 1983. A. Tumanov, Analytic discs and the regularity of OR mappings in higher codimension, Duke Math. J. 76 (1994),793-807. G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equa tion in Lipschitz domains, J. Funet. Anal. 59 (1984),572-611.
[WZ] [Yab] [Yl] [Y2] [Zie]
M. Weiss and A. Zygmund, A note on smooth functions, Nederl. Akad. Wetensch. Indag. Math., Ser. A 62 (1959), 52-58. K. Yabuta, Generalizations of Calderon-Zygmund operators, Studia Math. 82 (1985), 17-31. M. Yamazaki, A quasi-homogeneous version of paradifferential operators, J. Fac. Sci. Univ. Tokyo 33 (1986), 131-174; 311-345. M. Yamazaki, A quasi-homogeneous version of the microlocal analysis for non-linear partial differential equations, Japan J. Math. 14 (1988), 225-260. W. Ziemer, Weakly Differentiable F'unctions, Springer, New York, 1989.
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<) X S'[,(I<)
I (3.34) I (2.83) I (2.84) I (2.72), (3.37) I (3.36) I (0.14) 1(12.1) I (0.2), (1.4) I (2.45) I I I I I I I
(5.75) (0.1) (0.3), (1.61) (0.4), (1.64) (1.21) (5.57) (4.40) III (2.1) I (4.3) 1(4.51) 1(4.51) I (2.60) I (2.62) I (11.5) I (16.1) I (16.3) I (9.2) I (3.48) I (0.9) I (3.46); II (0.18) I (0.8), (2.44) I (2.52) I (3.18)
Index
Beltrami operator, 152
Besov spaces, 16, 82
boundary integral equations, 235
Calder6n-Zygmund operators, 223
Cauchy integral, 218
commutator, 58
deformation operator, 148
Dini condition, 89, 160
Dirichlet problem, 155, 168, 217, 241
diy-curl lemma, 186
double layer potentials, 233
double symbols, 63
elementary symbol, 18
elliptic, 137
elliptic regularity, 137
Euler equation, 178
exterior derivative, 77, 151
Fourier multiplier, 1
Fredholm, 139, 240
global regularity, 143
Hardy spaces, 24
Hardy-Littlewood maximal function, 64, 104
harmonic coordinates, 194
harmonic measure, 244
HOlder spaces, 5
index, 139, 240
isothermal coordinates, 154
John-Nirenberg space, 27
jump relations, 230
Kadlec's formula, 179
Koebe-Bieberbach distortion theorem, 219,
246
Laplace operator, 140
layer potentials, 159, 217
Lipschitz domains, 217
Lipschitz surfaces, 217
Littlewood-Paley partition of unity, 4
Littlewood-Paley theory, 17, 104
local regularity, 143
metric tensor, 202
modulus of continuity, 3
Morrey space, 94
Morrey's imbedding theorem, 94
Neumann boundary problem, 172
Neumann operator, 171
nontangential maximal function, 230
null bicharacteristics, 215
Osgood's Theorem, 151
paracomposition, 127
paradifferential operator, 101
paradifferential symbol smoothing, 35
parametrix, 138
paraproduct, 35, 104
propagation of singularities, 205
pseudodifferential operators, 2
quasiconformal mappings, 152, 198
Rellich-type identity, 236
Ricci tensor, 202
Riemannian manifold, 140
Riesz transform, 157
sharp GArding inequality, 208
single layer potential, 232
Sobolev spaces, 2
super-commutator, 77, 186
symbol smoothing, 31
symbols, 2
Triebel-Lizorkin spaces, 16
univalent functions, 246
vorticity equation, 178
Zygmund space, 1
Zygmund spaces, 5
