CR Manifolds and the Tangential Cauchy- Riemann Complex
STUDIES IN ADVANCED
TWS
Studies in Advanced Mathematics
CR Manifolds and the Tangential Cauchy—Riemann Complex
Studies in Advanced Mathematics
Series Editor Steven G. Krantz Washington University in S Louis
Editorial Board R. Michael Beals
Gerald B. Folland
Rutgers University
University of Washington
Dennis de Turck
William Helton
University of Pennsylvania
University of California at San Diego
Ronald DeVore
Norberto Salinas
University of South Carolina
University of Kansas
L. Craig Evans
Michael E. Taylor
University of California at Berkeley
University of North Carolina
Volumes in the Series Real Analysis and Foundations, Steven G. Krantz CR Manifolds and the Tangential Cauchy—Riemann Complex, Albert Boggess Elementary Introduction to the Theory of Pseudodifferential Operators,
Xavier Saint Raymond Fast
Fourier Transforms, James
S.
Walker
Measure Theory and Fine Properties of Functions. L. Craig Evans and
Ronald Gariepv
ALBERT BOGGESS Texas A & M University
CR Manifolds and the Tangential
Cauchy—Rieman n Complex
CRC PRESS Boca Raton Ann Arbor
Boston
London
Library of Congress Cataloging.in-Pubhcation Data Catalog record is available from the Library of Congress
This book represents information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Every reasonable effort has been made to give reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. All rights reserved. This book, or any parts thereof, may not be reproduced in any form without written consent from the publisher.
by Archetype Publishing Inc., P.O. Bo' 6567,
This book was formatted with Champaign, IL 61821.
Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida, 33431.
©
1991
by CRC PreSS, Inc.
International Standard Book Number 0-8493-71 52-X
Printed in the United States ofAmerica
1 234567890
Contents
Contents
Introduction PART I. PRELiMINARiES
1
1
Analysis on Euclidean Space
2
1.1
Functions Vectors and vector fields Forms The exterior derivative Contractions
2
17
2.2 2.3 2.4 2.5
Analysis on Manifolds Manifolds Submanifolds Vectors on manifolds Forms on manifolds Integration on manift)lds
3
Complexijied Vectors and Forms
39
3.1
Complexification of a real vector space
39
1.2 1.3 1.4 1.5
2 2.1
17 19
23
26 29
Contents
3.2 3.3
Complex structures Higher degree complexified forms
4
The
Frobenius Theorem
5/
4.1
The real Frobenius theorem The analytic Frobenius theorem Almost complex structures
51
61
5.1
Distribution Theory The spaces 7)' and e'
5.2 5.3 5.4
Operations with distributions Whitney's extension theorem Fundamental solutions for partial differential equations
65
6
Currents Definitions Operations with cunents
79
PART II: CR MANIFOLDS
95
7
CR Manifolds
97
7.1
Imbedded CR manifolds A normal form for a generic CR submanifold Quadric submanifolds Abstract CR manifolds
4.2 4.3
5
6.1
6.2
7.2 7.3 7.4
8 8.1
8.2 8.3
The Tangential Cauchy—Riemann Complex Extrinsic approach Intrinsic approach to C9M
41
45
56 58
61
71
74
79 84
97 103 111
120
122 122
130
The equivalence of the extrinsic and intrinsic tangential Cauchy—Riemann complexes
134
9
CR
9.1
CR functions CR maps
140
The Levi Form Definitions
156
9.2
10 10.1 10.2
Functions and Maps
The Levi form for an imbedded CR manifold
140 149
156 159
ix
Contents
The Levi form of a real hypersurface
The Imbeddability of CR Manifolds The real analytic imbedding theorem Nirenberg 's nonimbeddable example
12
Further Results
12.1 12.2 12.3 12.4 12.5
Bloom—Graham normal form Rigid and semirigid submanifolds
More on the Levi form Kuranishi 's imbedding theorem Nongeneric and non-CR manifolds
163
169 169 172
179 179 183 185 187 187
PART 1!!: THE HOLOMORPHIC EXTENSION OF CR FUNCTiONS
189
An Approximation Theorem
191
14
The
14.1
14.2 14.3
Lewy's CR extension theorem for hypersurfaces The CR extension theorem for higher codimension Examples
15
The
15.1 15.2 15.3
15.4 15.5
Reduction to analytic discs Analytic discs for hypersurfaces Analytic discs for quadric submanifolds Bishop's equation The proof of the analytic disc theorem for the general case
16
The
16.1
A Fourier inversion formula The hypoanalytic wave front set The hypoanalytic wave front set and the Levi form
230 237 244
Further Results The Fourier integral approach in the nonrigid case The holomorphic extension of CR distributions CR extension near points of higher type
251
16.2 16.3
17 17.1 17.2 17.3
Statement of the CR Extension Theorem
Analytic Disc Technique
Fourier Transform Technique
198 198
200 202
206 207 208 210 214 221
229
251
254 257
x
17.4
Contents
Analytic hypoellipticity
PART IV: SOLVABILITY OF THE TANGENTIAL CAUCHY-R1EMANN COMPLEX
263
18
Kernel Calculus
265
18.1
18.2
Definitions A homotopy formula
265 272
19
Fundamental Solutions for the Exterior Derivative
and Cauchy—Riemann Operators
277 278
19.2 19.3
Fundamental solutions for d on Fundamental solutions for 0 on Bochner's global CR extension theorem
20
The
20.1
20.4
A general class of kernels A formal identity The solution to the Cauchy—Riemann equations on a convex domain Boundary value results for Henkin's kernels
21
Fundamental Solutions for the Tangential
19.1
20.2 20.3
Kernels of Henkin
Cauchy—Riemann Complex on a Convex Hypersuiface 21.1
A
23.1
23.2
299 303
312 312 317
Local Solution to the Tangential
Cauchy—Riemann Equations
23
294 297
A second fundamental solution to the tangential Cauchy—Riemann complex
22
294
The first fundamental solution for the tangential Cauchy—Riemann complex
21.2
281 291
Local Nonsolvability of the Tangential Cauchy—Riemann Complex Hans Lewy's nonsolvability example Henkin's criterion for local solvability at the top degree
327
334 334 337
xi
Contents
24
Further Results
342
24.1
More on the Bochner—Martinelli kernel Kernels for strictly pseudoconvex boundaries Further estimates on the solution to Wealdy convex boundaries Solvability of the tangential Cauchy—Riemann complex in other geometries
342 345 348 348
Bibliography
354
Notation
359
Index
361
24.2 24.3 24.4 24.5
349
Introduction
theory of complex manifolds dates back many decades so that its origins are considered classical even by the standards of mathematicians. Consequently, there are many fine references on this subject. By contrast, the origins of the theory of CR manifolds are much more recent even though this class of manifolds contains very natural objects of mathematical study (for example, real hypersurfaces in complex Euclidean space). The first formal definition of The
the tangential Cauchy—Riemann complex did not appear until the mid 1 960s with the work of Kohn and Rossi [KR]. Since then, CR manifolds and the tangential Cauchy—Riemann complex have been extensively studied both for their intrinsic
interest and because of their application to other fields of study such as partial differential equations and mathematical physics. The purpose of this book is to define CR manifolds and the associated tangential Cauchy—Riemann complex and to discuss some of their basic properties. In addition, we shall sample some of the important recent developments in the field (up to the early 1980s). In the last two decades, research on the subject of CR manifolds has branched into many areas. Two of these areas that are of interest to us are (I) the holomorphic extension of CR functions (solutions to the homogeneous tangential Cauchy—Riemann equations) and (2) the local solvability or nonsoivability of the tangential Cauchy—Riemann complex. The first area started in the 1950s when Hans Lewy [Li] showed that under certain convexity assumptions on a CR functions locally extend to holomorphic functions. real hypersurface in Over the years, many refinements have been made to this CR extension theorem so that it now includes manifolds of higher codimension with weaker convexity
assumptions. The second area started in the i960s with the work of Kohn. He used a Hilbert space (L2) approach to construct solutions to the tangential Cauchy—Riemann complex on the boundary of a strictly pseudoconvex domain (except at top degree). Later, Henkin developed integral kernels to represent solutions to the tangential Cauchy—Riemann equations. A closely related topic is the nonsolvability of certain systems of partial differential equations. In the 1950s, Hans Lewy constructed an example of a partial differential equation with smooth coefficients that has no locally defined smooth solution. In particular, he showed that cannot replace "real analytic" in
xiu
xiv
the statement of the Cauchy—Kowalevsky theorem. Lewy 's example is closely related to the tangential Cauchy—Riemann equations on the Heisenberg group in C2. His example illustrates that the tangential Cauchy—Riemann complex on a strictly pseudoconvex boundary is not always solvable at the top degree. Later, Henkin developed a criterion for solvability of the tangential Cauchy—Riemann complex at the top degree. The first half of this book contains general information on the subject of CR manifolds (Part II) and the prerequisites from real and complex analysis (Part I). In Parts III and IV, we develop the subjects of CR extension and the solvability of the tangential Cauchy—Riemann complex. This book is not a treatise. We do not discuss the £2-approach to the global solvability of the tangential Cauchy—Riemann equations. This material is contained in Folland and Kohn's book [FK] and our work certainly could not offer any improvements. Instead, the integral kernel approach of Henkin is presented. The local theory dealing with points of higher type (points where the Levi form vanishes) is not presented in detail. The theory of points of higher type is too immature or too complicated for inclusion in a book at this time. Instead, the end of each pan contains a chapter entitled Further Results where some of the recent literature on function theory near points of higher type and other topics are surveyed with few proofs given. Each writer has his own peculiar style and tastes and this author is no different. The reader will notice that I favor the concrete over the abstract. This may offend some of the purists in the audience but I offer no apologies. I firmly believe that a student learns much more by getting his or her hands dirty with some analysis rather than by merely manipulating abstract symbols. In this book, abstract concepts are introduced after some motivation with the concrete situation. It is hoped that the audience for this book will include researchers in several complex variables and partial differential equations along with graduate students who are beyond their first year or two of graduate study. The reader should be familiar with advanced calculus, real and complex analysis, and a little functional analysis (at least enough so that he or she does not faint at the sight of a Banach or Fréchet space). Although this book cannot reinvent the wheel, many of the prerequisites for reading Parts II through IV are given in Part I. We start with a discussion of vectors and forms, both in the Euclidean and manifold setting. A proof of Stokes' theorem is given since it is such a basic tool used throughout the book. Proofs of the smooth and real analytic versions of the Frobenius theorem
are given since these theorems are used in the imbedding of CR manifolds. At the end of Part I, a rapid course in the theory of distributions and currents is given. This material will be essential in Part IV. There are other elementary topics that are not included in Part I. These include the existence and uniqueness theorem for ordinary differential equations and the Cauchy—Kowalevsky theorem for partial differential equations. Even though these topics are no more advanced than other topics covered in Part I (for example, Stokes' theorem), they are not as frequently used in this book and therefore we only give references. Surprisingly
xv
little theory from several complex variables is used. For most of the book, the reader needs only to be familiar with the definition and basic properties of holomorphic functions of several complex variables. However, the book will certainly be more meaningful to someone who has a further background in several complex variables. The Newlander—Nirenberg theorem is the most advanced topic from several complex variables that is used in the book. It is only used in the discussion of Levi flat CR manifolds. A proof of the imbedded version of the Newlander—Nirenberg theorem is provided for the reader who wishes to restrict his or her attention to imbedded Levi flat CR manifolds. Part II covers general information about CR manifolds and the associated tangential Cauchy—Riemann complex. We start with the definitions of imbedded
and abstract CR manifolds. In addition, we present a normal form that gives a convenient description of an imbedded CR manifold in local coordinates. Next, we introduce the tangential Cauchy—Riemann complex. For an imbedded CR manifold, an extrinsic approach is given that makes use of the ambient complex For an abstract CR manifold, an intrinsic approach is given that structure on makes no use of any ambient complex structure (since none exists). In the case of an imbedded CR manifold, these two approaches are technically different, but we show they are isomorphic. Our approach to the tangential Cauchy— Riemann complex makes use of a Hermitian metric. We also mention a more invariant definition of the tangential Cauchy—Riemann complex that does make use of a metric, but this approach is not emphasized because calculations usually require a choice of a metric. CR functions and CR maps are then introduced. We prove Tomassini's theorem [Tom], which states that a real analytic CR function on an imbedded real analytic CR manifold is locally the restriction of an ambiently defined holomorphic function. This theorem does not hold for the class of smooth CR functions. However, we do show that a smooth CR function is always the restriction of an ambiently defined function that satisfies the Cauchy—Riemann equations to infinite order on the given CR submanifold. Next, we introduce the Levi form, which is the complex analysis version of the second fundamental form from differential geometry. An extensive analysis of the Levi form for the case of a hypersurface is given. In particular, we show the relationship between the Levi form and the second fundamental form. We then show that any real analytic CR manifold can be locally imbedded as a CR submanifold of The smooth version of this theorem is false in view of Nirenberg's counter example [Nir], which is also given. In the chapter entitled Further Results we discuss some related results such as the Bloom— Graham normal form for CR submanifolds of higher type, rigid and semi-rigid CR structures, and Kuranishi's imbedding theorem. Most of these results are presented without proof. Part III discusses the local holomorphic extension of CR functions from an imbedded CR manifold. We start with an approximation theorem of Baouendi and Treves [BT 1], which states that CR functions can be locally approximated by entire functions. Their theorem is more general but we restrict our focus to CR
xvx
functions to simplify the proof. Next, we state the CR extension theorem, which is a generalization of Hans Lewy's hypersurface theorem alluded to above. In addition, the convexity assumptions of this theorem are discussed and examples are given. We present two techniques for the proof of this theorem. Both of these techniques are used in today's research problems and thus these techniques are as important as the CR extension theorem. The first technique involves the use of analytic discs and it was originally developed by Lewy [Li] and Bishop [Bil. This technique together with the approximation theorem yields an easy proof of Hans Lewy's CR extension theorem for hypersurfaces. An explicit proof is also given in the case of a quadric submanifold of higher codimension. The proof for the general case requires an analysis of the solution of a nonlinear integral equation involving the Hilbert Transform (Bishop's equation). The second, more recent, technique involves a modified Fourier transform approach due to Baouendi and Treves. The idea here is to obtain the holomorphic extension of a given CR function from the Fourier inversion formula — suitably modified for CR manifolds. This technique is applicable to CR distributions and points of higher type. However, to avoid technical complications, we give the details of this technique only for the case of smooth CR functions on a type two CR submanifold. Some of the extensions of this technique to CR distributions are discussed at the end of Part III in the chapter entitled Further Results. Part IV deals with the solvability and nonsolvability of the tangential Cauchy— Riemann complex on a strictly pseudoconvex hypersurface in The approach taken here involves Henkin's integral kernels although we use the notation and kernel calculus set down by Harvey and Polking [HPI. We give two fundamental solutions for the Cauchy—Riemann complex on — the Bochner—Martinelli
kernel [Boc] and the Cauchy kernel on a slice. These kernels together with the kernels of Henkin yield a solution to the Cauchy—Riemann equations on a strictly convex domain in Furthermore, these kernels provide an easy proof of Bochner's theorem, which states that a CR function on the boundary of a bounded domain with smooth boundary globally extends to the inside as a holomorphic function. Next, a global integral kernel solution for the tangential Cauchy—Riemann complex is given for a strictly convex hypersurface. These kernels are then modified to yield Henkin's [He31 local solution to the tangential Cauchy—Riemann equations. We then present Henkin's criteria for local solvability of the tangential Cauchy—Riemann complex at the top degree. Results on the solvability of the tangential Cauchy—Riemann complex on hypersurfaces under other geometric hypotheses are given in the chapter entitled Further Results. My point of view in mathematics has been influenced by a number of people whom I have the pleasure to thank. First, I owe a lot to my thesis advisor, John Polking. He along with Reese Harvey has shaped my mathematical development since my early graduate school years at Rice University. Even though I have
never met him, 0. M. Henkin has provided a lot of inspiration for much of my work. Other mathematicians who have influenced my mathematical point
xvii
of view include Salah Baouendi, Al Taylor, Dan Burns, and Alex Nagel. The reviewers did an excellent job of finding errors and making helpful suggestions. I also wish to thank Steve Krantz (who initially encouraged me to write this book) and the rest of the editorial staff at CRC Press for having the confidence in me to complete this project. I wish to thank Texas A&M for support during the preparation of most of this project. I did the final editing while visiting Colorado College and I want to thank their mathematics department for their hospitality during my visit.
In addition, I wish to thank Robin Campbell, who typed portions of this manuscript and answered many of my questions concerning I also wish to thank my son for putting up with me during the preparation of this manuscript. Finally, I wish to thank Steve Daniel and the rest of the Aggieland Paddle club for convincing me that from time to time, I need a break from the book writing to go whitewater kayaking.
Al Boggess, October 1990 Colorado Springs, CO
Part I Preliminaries
In this first part, we provide most of the prerequisites for reading the rest of the book. We start with a review of certain aspects of function theory, vectors, vector fields, and differential fonns on Euclidean space. These concepts are then defined in the context of manifolds. Proofs are given for Stokes' theorem and its corollaries — Green's formula and the divergence theorem. A proof of the Frobenius theorem is then given. The real analytic version of this theprem is also discussed since it will be used for the imbedding theorem for real analytic CR manifolds in Part II. We discuss distribution theory as applied to partial differential equations. Fundamental solutions for the Laplacian on and the Cauchy—Riemann equations in one complex variable are given. They are used in Part IV, where we discuss fundamental solutions to the Cauchy— Riemann equations on and their analogue on a real hypersurface of — the tangential Cauchy—Riemann equations. These systems of partial differential equations act on differential forms. Therefore we shall need a distribution theory for differential forms, i.e., the theory of currents. This and related topics are reviewed at the end of Part I. Excellent references are available for all of the topics in Part I. These include Spivak's volumes on differential geometry [Sp], Krantz's book [Kr] or Hörmander's [Ho] for several complex variables, Yosida's book [Y] for functional analysis and distribution theory (see also Schwartz [Sch}), Federer's book [Fe] for geometric measure theory, and John's [Jo] or Folland's book [Fo] for partial differential equations.
1 Analysis on Euclidean Space
Here, we discuss some function theory and define the notions of vectors and forms on Euclidean space.
1.1
Functions
There are several classes of functions we shall use. For an open set 11 in let CIc(1l) = the space of k-times continuously differentiable real- or
complex-valued functions on
=
space of infinitely differentiable real- or complexvalued functions on ci, the
= the space of elements of compact support.
with
We shall make use of a special class of mollifier functions {x€; V(ci). This class is defined as follows. Let
for x E
with IxI
> O} C
1
and let
= tx Each
2
is smooth. Here,
.
IIcI(RN) denotes the usual C'-norm of a function.
Functions
3
The following properties can easily be shown: (i)
lxi
(ii) JXEE' x(x)dx = 1. These two properties allow the construction of cutoff functions as described in the following lemma. LEMMA 1
Given a compact subset K of an open set Cl C to V(fl) with 1 on a neighborhood of K.
there is function
belonging
PROOF We first choose a compact set K1 C Cl so that K is contained in the interior of K1. Let
ifxEKi
I I = o
'K1 (x)
x
Choose c > 0 small enough so that 2€ is less than the smaller of the distance between K and Cl — K1 and the distance between K1 and RN — ci Let *
Here, * is the usual convolution operator in RN, so
J x€(s—y)1K1(y)dy. yERN
is smooth, clearly is smooth. Property (1) for xe shows that vanishes outside an c-neighborhood of K1. So 1j has compact support in Cl by our choice of e. Property (ii) for xf and our choice of e imply that q5 1 on an c-neighborhood of K. Therefore, is our desired cutoff function. I Since
As an immediate corollary, we can construct partitions of unity, as described in the following lemma. LEMMA 2 Suppose
U3 is an open subset of
set in Ri" with K C Ui U ...
U
for j 1,. . , m. Suppose K is a compact Urn. Then there is a collection of functions .
such that (i)
1
on a neighborhood of K.
Often, the collection cover {U3}.
is called a partition of unity subordinate to the
Analysis on Euclidean Space
4
PROOF
First, we find open subsets Uj',... ,
with
C U3 so that K C
E D(U,)
Next; we choose cutoff functions a neighborhood of LI3. Then let U
U
1 on
with ii-',
.0(x)
where
with 0
E
easily show that the set
}
1
on a neighborhood of K. The reader can
satisfies the conclusions of the lemma.
I
The key idea in the proof of Lemma 1 is that the characteristic function for a compact set K (IK) can be approximated by the sequence of smooth functions x satisfies properties (i) and (ii) listed just before Lemma 1. Convolving with x€ can be used to approximate other types of functions as well. For example, if I is a continuous function on RN, then properties (i) and (ii) can be used to show that the smooth sequence * 1) converges to f uniformly on compact sets as e —' 0. { Another important class of functions is
=
space of real analytic functions on real or complex valued).
(either
the
A function f is real analytic
on
f
an open set
if in a neighborhood of each
point in can be represented as a convergent power series. It is a standard fact that A(11) is a subset of e(IZ), i.e., real analytic functions are smooth. is given by the following. Let A real analytic version of the
e(x) = The
power series for e(x)
ii._N/2e_1x12
about the
origin
for
x
E
converges
for all
x,
so e belongs
to
For€>0,
€.
e set of functions to (i) and (ii) for
The
{
} satisfies the
following properties
(i) Given t5 > 0,
f
which are analogous
0.
IvI6
(ii)
JRN eEQJ)dy
=
1, for each
0.
follow from a standard polar coordinate calculation after the change of variable t = y/€. we now prove the classical Weierstrass theorem, which states Using the that is a dense subset of 6(e) in the following topology for E(11): a These properties
Functions
5
is said to converge to f in sequence f,,, E compact subset K of and for each multiindex negative integer)
D°f
as n = (ni,...
uniformly on K as n —p
,
00 if for each (o, a non-
00.
Here, and
= THEOREM
1
.. .
lal
=a+
+ aN.
WEIERSTRASS
Suppose f belongs to Then there is a sequence of polynomials P1, P2,. that converges to f in the topology of e(11).
Let K be an arbitrary compact subset of 11 and let E be a cutoff function that is identically 1 on a neighborhood of K. If I belongs to then belongs to V(1l). For > 0, define PROOF
F€(x) =
= =
*
J
YERN
J yEW"
—
y)f(x
—
y)dy
(by a translation).
Note that
=
J yEW"
—
y)dy.
In view of property (ii) for
=
=
Therefore, for x E K (so
-
=
f
J
yEW" 1),
we have
- y) -
YERN
For 6 > 0, we split the integral on the right into the sum of an integral over
6} and an integral over 6). The first of these integrals can be With made small by choosing 6 small using the uniform continuity of this choice of 6, the second of these integrals converges to zero as —p 0 in uniformly on converges to view of property (1) for eE. Therefore,
K as
0.
Analysis on Euclidean Space
6
The power series for each FE about because F( can be written
FE(x)=CN
origin converges
the
for all
x E Ri"
J
yEE'
the power series for e(.) can be integrated term by term. By truncating the power series of F( about the origin, we obtain a sequence of polynomials n = 1,2,... such that for each multiindex and because
sup
—
D°f(x)I
0
as
n —p
xE K
Now let K1, K2,... be an increasing sequence of compact sets with Let be a polynomial with sup
Clearly,
!.
—
f in E(ci) as n —p
and
= ci.
the proof of the theorem is complete. U
The final class of functions to consider is the class of holomorphic functions.
If ci is an open set in C'2, then let
O(ci) = the space of holomorphic functions on ci. A function is holomorphic on ci if it satisfies the Cauchy—Riemann equations on
ci,
where
.0 0z3 Here,
2 \0x3
Oy,
we have labeled the coordinates for C'2 as (z1,... ,
with z3
x2 +
(i = We assume the reader knows some basic complex analysis. If I is holomor. then the reader should know phic in a neighborhood of a point p that f can be expressed as a convergent series in powers of (z1 —Pi), . .. , — pa). This and a connectedness argument imply the identity theorem for holomorphic functions: if f is holomorphic on a connected open set ci and if f vanishes on an open subset of ci, then f vanishes everywhere on ci. This is often expressed by saying that an open set is a uniqueness set for holomorphic .
.
Vectors and vector fieWs
7
functions. Other types of sets are also uniqueness sets for holomorphic func-
tions. For example, if f is holomorphic on and if I vanishes on the copy of given by {(x1 + iO,. .. + iO);(x1,. .. then f vanishes E everywhere. This is because all the v-derivatives of f vanish on this copy of The Cauchy—Riemann equations can then be used to inductively show that all x, y derivatives of I vanish on this copy of In particular, a power series expansion of f about the origin must vanish identically and hence f 0. —' Ctm is called a holomorphic map if each A function f = (f',.. . , fm): I,: C is holomorphic, I <j <m. This definition can be recast in terms of the real derivative of f (denoted Df) as a map from to R2m. Define the complex structure map
J:
—.
by
By the Cauchy—Riemann equations, a C' map f: Il if and only if for each point p E
—, Ctm
is holomorphic in
Df(p)oJ=JoDf(p) as linear maps from to R2m. The J on the left is the complex structure map for whereas the J on the right is the complex structure map for R2m.
1.2
Vectors and vector fields
Now, we turn to the topic of vectors and vector fields on RN. These concepts will be generalized to the manifold setting in Chapter 2. A vector at a point p E RN is an operator of the form
v,ER. The set of vectors at p is denoted (RN) and is called the (real) tangent space of RN at p. An element L in can be viewed as a linear map L: ((RN) —. R by defining
L{f}
=
Analysis on Euclidean Space
8
A vector field on an open set Il C is a smooth map that assigns to each point p E ci a vector in TP(RN). Here, the concept of smooth means that each vector field L can be written
L= where each v3 is an element of S(ci). We let
be the value of the vector field
L at p, i.e.,
= We also write
(Lf)(p) =
=
(p E ci),
Thus, Lf is also an element of e(ci). RN' is a smooth (C') map near a point p E RN. F Now suppose F: R1" TF(p)(RN') called the push forward of F at p. induces a map F.: It is defined by
for f E e(rz).
F.(L){g} = L{goF} for L E
and g E e(RN'). If L is a vector field on Il, then the above
equation reads
= for g E e(RN') and p E ci. If F : ci '—' Il' is a diffeomorphism, then F.(L) is a vector field defined on ci'. Let us compute F.(O/0x3), I <j ai(P)(/_) x3
for pEU
j=I
where each a,: U —p R is a smooth function. Note that if f is a smooth function
on X and L is a vector field on X, then L{f} is a smooth function on X. We call L C T(M) an rn-dimensional subbundle of T(X) if L assigns to each point p E X an rn-dimensional vector space IL,, C These vector spaces are required to fit together smoothly in the sense that near each fixed point po E X, there are smooth vector fields L1,... , Lm so that (Lm)p} forms a basis for As an example, let X be an open subset of RN and suppose
Vectors on manifolds
25
pl,...,pN_t: X
Rare smooth functions with dp1 A...AdPN_t Oon X. with We let IL,, be the set of vectors = 0 for 1 i N — £. We
leave it to the reader to show that L = UPEXL,, is a subbundle of T(X). By Lemma 1, each is the tangent space at p for the submanifold given by the unique level set of p = (p1,... pN—t) that passes through p. ,
The tangent bundle is an example of a vector bundle over a smooth manifold. We shall not need the definition of a vector bundle in its most abstract form. Instead, we refer the reader to [W2]. Often, different coordinate patches are used near the same point po E X and so it is useful to know how the description of a given vector in (X) changes RN be another as the coordinate chart changes. Let Y = (yi, ... ,yN): U' coordinate chart. For p E U fl U', we describe the vector in terms E
of 0/Oxi,..., O/OXN and O/OYi,.
.
O/OYN
=
Since
(0/Ox,)
we have
b, = = a relation between the a's and the b's. Let L = O/Oxk (so ak = and all other a3 = 0); we have
= and so
fO\
=
1
(O\
N(O\ u—)
{y,}
(O\
We now describe the push forward of a vector under a smooth map between manifolds. This is given in Chapter 1 in the Euclidean setting and the definitions easily generalize to the manifold setting. Suppose F: X —+ Y is a smooth map
between the smooth manifolds X and Y. This induces the push forward map —. TF(p)(Y), which is defined by
=
o
F}, for
E
Analysis on Manifolds
26
Here, g is any smooth function defined in a neighborhood of F(p). If L is a vector field on X then the above equation reads
=
0
F}.
for each p E X. Note that (FOG). = F, oG. for two smooth maps F and C. If F has a smooth inverse, then From the definition of 8/8x3 given at the beginning of this section, 8/ôx,
is the push forward of 0/Ot) (from RN) via the coordinate chart
i.e.,
In Chapter 1, we gave a formula for the push forward of a vector in terms of the derivative of the map. The analogous formula holds for maps between manifolds. Let F: X Y be a smooth map. Suppose p is a point in X. Let RN be a coordinate chart for X with p E U and let U x = (x1,. . .
beacoordinatechartforYwithF(p)E V. Here Y=(yI,...,yN'): N' = dimR Y. We let F) = o F: U R, 1 <j 0. Using the above change of variables formula for inte..
RN
=J x'YY'ø x'ø RN
desired. We conclude that the integral of a form supported in a coordinate patch is unambiguously defined. Suppose is a compactly supported N-form on X (but not necessarily supported in one coordinate patch). Suppose {U1,... , } is a collection of coRN be the corresponding ordinate patches that cover supp and let U, oriented coordinate charts. Let ... , } be a partition of unity for supp with c5j E V(U,). Then we define as
I
X
=
Again we check that this definition is independent of the choices made. Suppose {Y,: V3 Rh', j = 1,..., n} is another set of oriented coordinate charts whose corresponding coordinate patches cover supp Let . . , } be a
Analysis on
32
partition of unity for supp 0 with
Since
E
t'k
1,
= k=I)=IRN
has support in a single coordinate patch, we have
Since
RN
Summing the right side over j and using
I, we obtain
k=IRN
so our definition of the integral of 0 is well defined. If M is an £-dimensional submanifold of X, then we define
and
10 where
j: M
—'
X is the inclusion map. The point is that
is
an intrinsically
defined form on the £-dimensional manifold M and so the right side can be evaluated as described above. If 0 contains a factor of dp where p: X —' IR vanishes on M, then JM must also vanish since j*dp = dj*p = 0 on M. Also if r $ £, then JM 0 = 0 for 0 77(X). If M is a smooth £-dimensional submanifold with boundary contained in an N-dimensional manifold X, then fM 0 is defined in a similar way to the case when M is a manifold. Here, the only difference is that M has two types of coordinate charts. The first type is a coordinate chart about a manifold point and the second is a coordinate chart about a boundary point. If x: U —' is a coordinate chart about a manifold point, then fl M} is an open subset of = {(tI,...,tN);tt+I = = tN = 0}. If 0 is an €-form with support in U fl M, then by definition,
10 =Jx'(o). If x:
U
—'
RN is a coordinate chart containing a boundary point, then
is an open subset of the half space of IR' given by {(t1, . t,+i by definition,
f0= J tER'
M
ttO
(x'O)(t).
. . ,
tN); t1
0 and
Integration on
33
As with the case of a manifold without boundary, the above definitions are independent of the oriented coordinate chart. If 4> is a compactly supported i-form on M (but not necessarily supported in one coordinate patch), then its integral is defined in the same way as the case where M has no boundary by using a partition of unity. Now suppose X and Y are smooth, oriented N-dimensional manifolds and suppose F: X —' Y is a smooth map. We say that F preserves orientation if preserves orientation on R" for each oriented coordinate chart x Yo F o for X and Y for Y. The change of variables formula for easily generalizes to manifolds.
x'
LEMMA 1
Suppose X and Y are smooth, oriented N-dimensional Suppose F: X Y is a smooth, orientation-preserving map. 114> E D"(Y), then
Jq5=JF*çb. The proof of this lemma follows from the definitions together with the change of variables formula on 1W". Details are left to the reader. We are now ready to state and prove Stokes' theorem, which for an oriented
submanifold M with boundary, equates the integral of dçb over M with the integral of 4> over ÔM. The simplest example of Stokes' theorem is the fundamental theorem of calculus, where M is the interval {a x b} in lit We shall show that the definitions of forms, orientation, and integration reduce the proof of Stokes' theorem for more general manifolds to the fundamental theorem of calculus. THEOREM 1
STOKES
Suppose M is a smooth, oriented e-dinwnsional with boundary that is contained in a smooth N-dimensional manifold X. Suppose 4> then
f
dM4>
Here, ÔM has the induced boundary orientation.
PROOF We first cover supp 4> with a finite number of oriented coordinate C X '—' RN i = 1,... ,m that are either of manifold type (i.e., charts x': xl{Ui fl M} = {(t1,... ,tN);tt+I = = tp,r = 0)) or boundary type (i.e., = {(tj,. ,tN);tt = ... = V(U1)} = 0}). Let . .
be a partition of unity for supp 4>. It suffices to prove JdM{4>i4>}
=1
Analysis on Man(folds
34
for each i = 1,... , m. Stokes' theorem then follows by summing over i and using 1 on supp So from now on, we assume that supp 0 is contained
in a coordinate patch (one of the We first consider the case where x: U fl M —. R1 is a coordinate chart of manifold type. Using the definitions of the integral and the exterior derivative on M, we have
f is a
d refers to the exterior derivative on IRe. Since compactly supported (C— 1)-form on 1R1, we may write
A ... A dt3 A ... A dt1
(to)
belongs to V(Rt). The notation omitted. We have where each
indicates that dt3
has
been
So
...f The last equality follows from the fundamental theorem of calculus in the t3-
has compact support. variable, using the fact that Therefore, we have shown that if has support in a coordinate patch, U, of = 0. On the other hand, we have U fl OM = manifold type, then also vanishes and we have established Stokes' theorem: 0. Therefore, ó in this case. = The remaining case to consider is where o has support in a coordinate patch U of boundary type. In this case, we have
f
J
M
,({UnM}
=
f ttO
Integration on
35
As above, we write
where
E
V(Re). We claim that for each j = 1,...
f
.
f
.Adt1} =
(t,O}
.
(1)
is just the special case of Stokes' theorem where M is the half space {tt O} because, as mentioned earlier, the induced boundary orientation of
This
{tt = 0} differs from its inherited orientation from Rt by the factor (—I )t. If j 0, and a constant C > 0 such that sup
IalM
an
integer
forall fEe(cl).
xE K
In particular, supp T C K. Conversely, if T is an element of D' (Il) and supp T is compact in Il, then T defines an element of e'(cl).
It is a standard fact from topology that a continuous linear complexvalued map defined on a Fréchet space must be bounded in absolute value by a constant multiple of a finite sum of seminorms. Thus, the above inequality PROOF
follows for some integer M, constant C, and compact set K. In particular, supp T must be contained in K, for if not, then there must exist an element
f
K) with (T,
0. This contradicts the above inequality. For the converse, suppose T belongs to D'(cl) and supp T is compact. Let
e V(1l) be a smooth cutoff function (see Section 1.1) with neighborhood of supp T. Define
= 1 on a
f
=
The right side is well defined because çbf belongs to in 6(e), then in to D'(fl). Also, if this definition of T makes it a well-defined distribution in vanishes on a neighborhood of supp T, clearly
f
foreach
and T belongs Therefore, Since 1 —
fEV(fl).
Hence, the above definition of T as a linear map acting on 6(11) agrees with the original definition of T as a linear map acting on D(11) C 6(11). I
spaces V'(11) and 6'(ll) are endowed with a topology called the weak n = 1,2, . topology. This is defined by declaring that a sequence converges to T in V'(11) (or 6'(11)) if The
.
for each in D(1l) (or 6(11)). This convergence is not required to be uniform As an example, if in is a sequence of continuous functions on 11 that converges to the function T uniformly on compact subsets of 11, then the sequence
also converges to T as distributions in D'(11). However, the topology on
v'(cl) allows for much weaker types of convergence. For example, define the sequence e€(x) = e(x/f)f_", for > 0, where e(y) = This sequence was used in Section 1.1 for the proof of the Weierstrass theorem. Using the ideas from its proof, the reader can show e( m 2)' (Rh') as —p 0. Note that —p o for x 0 (but not uniformly) and eE(0) oo.
Opemlions with dislrlbWlons
5.2
65
Operations with distributions
All of the definitions of operations with distributions are motivated by considering the case where the distribution is a smooth function.
Multiplication of a distribution by a smooth function If T and
are
smooth functions on
then T• i11) defines a V' (11)-distribution
by
=
f
E D(1l).
for
So if T is an arbitrary V' (11)-distribution and Note that (Tm, = (T, by if belongs to 6(11), then we define the distribution
=
for
E D(1l).
With this definition, satisfies all the required properties for a distribution. Moreover, this definition agrees with the action (as a distribution) of Til' when both T and are smooth functions.
Differentiation If T is a smooth function on 11 and
=
f
E D(11), then
=
f
T is an arbitrary D'(Il)-distribution, then we define the distribution
by
= If
q5
in D(fl) then the sequence
is continuous and so Thus, As an example, we have
=
in D(1l). also converges to is a well-defined V' (11)-distribution. for q5 E 6(RN).
If a sequence of distributions } in V' (Il) converges to the distribution This follows from the then DaTa also converges to T as n —p oo, definition of the weak topology on D'(Il). This is in contrast to other types of topologies on function spaces where this does not hold. For example, the is not even sin nx converges uniformly to 0 but = sequence pointwise convergent.
Theor,
66
Convolution
=
Here, we specialize to the case
IRN.
If T belongs to e(RN) and t& belongs is the smooth function defined by
to D(RN), then the convolution T * (T *
Ø)RN
=
f
VERN
f(f - Y)ø(X)dx) dy.
T(y)
With the notation
T(y)i,i'(z — y)dy.
we have
As a distribution on
(T *
=
=
this becomes
=
J T(y)(il'*Ø)(y) VERN
= and belong to D(RN). Therefore, well defined for T in Ds(laN). If T is an arbitrary DI(RN distribution and if ii' E D(RN), then we define Now
(T,
belongs to D(RN) because both
* 65)RN
is
=
for
D(RN).
inV(RN). Therefore,
Øin
is
a well-defined element of Dl(RN). From the above discussion, this definition agrees with the action of T * as a distribution when T is a smooth function. * then belongs to and ó belongs to If belongs to
So the same definition can be used to define the convolution of an ('(R' )-distribution T with an element v of ((Rh'). The result is a V1(RN distribution.
Let us examine the definition of T * t more carefully. If T is an element of and 0 are elements of V(Rv), then
P!(Rv) and
(T *
= (T.
*
= (T(y).
f v(r -
z€Rs
/
Operations with distributions
67
As already mentioned, the above notation indicates that the distribution T pairs with the function — in the y-variable. The term — y)Ø(x)dx can be approximated by Riemann sums in x. Moreover, this approximation can be accomplished in the )-topology (in the y-variable). Therefore, using the linearity and continuity of T, we can interchange the order of the pairing of T with the integral in x to obtain (T *
- Y))YERN
= J (T(y),
Ø)RN
XERN
= ((T(y), Therefore, the distribution T *
,
is given by pairing with the following function:
(T(y), i,b(x
x
—
—
This formula generalizes the usual convolution formula when T is a smooth
function on RN If is an element of D(RN), then a difference quotient for converges in V(IRN) to the corresponding derivative of So if T E DI(IRN), then
(T(y),
* i1b}(x)
—
This shows that T *
is a smooth function on RN. By the definition of the derivative of a distribution, we also have = (DAT,
*
—
can be written as either T * is summarized in the following lemma. So
Y))YERN.
or
*
*
This discussion
LEMMA I
Suppose T is an element of D!(RN) and is an element of distribution T * is given by pairing with the smooth function T*
= (T(y), D°T *
=T* continuous linear map from V(RN) to Moreover,
*
—
Then the
y))YERN.
The operator
i—s
T
*
is a
We can also define the convolution of a 2)'-distribution with an e'-distribution by mimicking the above definition of the convolution of an element of V' (RN)
Distribution Theory
68
with an element of V(RN). First we define 't for T E V'(R1") by
for
= T(—x) when T is a smooth
This definition agrees with the formula function. For T E and S E
the distribution T * S is defined
by
for
is a smooth function (Lemma 1) with compact support, the right side is well defined. The same definition can be used if T is an )-distribution and S is a Since
*
LEMMA 2
For T E 1Y (Rh'),
*
T = T.
PROOF We have for
= —
(by Lemma 1)
Y))YERN
= (T(y),
(by Definition of
= Hence, öo *
T = T and the lemma follows.
I
If T is a smooth function on RN, then Lemma 2 can be established more directly by using Lemma 1 ((5o
* T)(x) = (öo(y), T(x —
T(x).
Lemma 2 shows that convolution with öo represents the identity operator. This will be important in Section 5.4 when we discuss fundamental solutions for partial differential operators with constant coefficients. THEOREM I
For an open set
C Ri', D(1Z) is a dense subset of V'(cZ) with the weak
topology.
Let T be an element of of compact sets in Il. Let
by an increasing sequence be a cutoff function that is 1 on
Exhaust
PROOF
E
Operations with distributions
69
converges to T in D'(Il) as n oc. So Clearly it suffices to approximate any element T of 6'(Il) by a sequence of smooth functions on Il' where is a neighborhood of supp T. Let > O} C be the mollifler sequence defined in Section 1.1. It follows from the ideas in * 4) Section 1.1 that as 0, provided 4) is an element of 4) in a neighborhood of
in
is a smooth function whose support is contained From Lemma 1, T * provided 0 is suitably small. From the definitions, we have for
4)
Therefore, T *
T in e'(cl'),
as
desired.
e(cl').
4) in
I
Tensor products
Suppose T is an element of Dl(IRN) and S is an element of V/(Rc). We wish
to define the tensor product T ® S E
in such a way that the
x
following holds:
((T® for 4)
V(IRN),
=
(T,Ø)RN
.
E V(IR"). Let g be an element of V(RN x
for XERN
(1)
The function (2)
is smooth with compact support. This follows from the fact that S is linear and that difference quotients of g(x, y) in the s-variable converge in D(RN x to the corresponding s-derivative of g(x, y). Therefore the pairing
(T(x), (S(y), g(x,
)XERN
(3)
x then the corresponding sequence of functions defined in (2) with g replaced by gn is a convergent sequence in D(R"). So, the corresponding sequence of complex numbers in (3) also converges. We define (T ® 5, g)RN by (3). By the above discussion, T ® S is a well-defined element in D#(RN x Rk). If g(x, y) = 4)(x)ib(y), then (3) reduces to (1), as desired. Instead of (3), we could have used
is well defined for T E 2Y(RN). Ifg,2 —+ g in
(S(y), (T(x),g(x, Y))XEaN )yERk
(4)
as the definition ofT® S. This formula also yields (1) in the case g(x,y) = So the expressions in (4) and (3) agree for functions of the form Since the set of functions that are finite sums of terms g(.r. y) =
Distribution Theory
70
D(RN x Rk) for E of the form by the Weierstrass theorem (see Section 1.1), clearly (4) and (3) agree for all g E D(RN x As an example, consider the tensor product of in DI(RN) with the function I as an element of 1Y(R") For g E D(RN x Rk), we have ® 1, g(x, Y))(x.y)ERN
= (6o(x),
(1(y), g(x, Y))YERk )XERN
y)dy
/ =
f
g(O,y)dy.
YERk
Composing a distribution with a diffeomorphism —* is a diffeoSuppose and are connected open sets in RN and F: morphism. If T is a smooth, complex-valued function on Il', then T o F is a smooth function on ft As a distribution on Il, we have
(To F,
=
f
=
f T(x)Ø(F' (x)) I det DF' (x) dx
for
0E
by the change of variables x = F(y). So
(T0F,0)n = (T,(OoF')IDet DF'I)c'. F is a diffeomorphism, Det DF' is either always positive or always DF'I is a smooth function with compact So (0° support in el'. We therefore define, for T E D(1l'), the distribution T o F E Since
negative on
D'(fl) by the above formula. As an application of these last two operations, we show how to extend a distribution defined on a submanifold of RN to a distribution defined on all of Suppose N = e+ k where Lis the real dimension of the submanifold in RN. By a partition of unity, we can localize the problem. By a coordinate change and the above change of variables formula, we may assume that the submanifold is the copy of Rt given by {(x,O) E Rt x Rk;x E A distribution on can then be extended to all of R1 x by tensoring it with the function I on Rk.
Whitney's extension theorem
5.3
71
Whitney's extension theorem
In its simplest form (due to Bore!), Whitney's extension theorem states that given
any infinite sequence of real numbers {ao, a1,...), there is a smooth function for n = 0,1 Here, f(n) is the nth derivative f: R —' R with = of f. We wish to prove this theorem along with its natural generalization to higher dimensions. First, we need a preliminary result from distribution theory which is interesting in its own right. THEOREM I
Let fl be an and y Let (x, y) be coordinates for RN with x = {(x,0) 1l;x Rk}. Suppose T is an element open set in R" and let with support in Then there is an integer M 0 and a collection of with M} such that of distributions {T0 e'(clo); a = (a', .. . ,
T(x,y)= aI<M
is all of RN and leave to the reader the minor We assume that modifications required in the case where fl is not all of RN. Since supp T C {y = 0}, by Lemma 1 in Section 5.1 there is a compact set K in RN, a constant C > 0, and an integer M 0 such that PROOF
l(T,f)RNI C sup
for
I E e(RN).
(I)
IczIM (x,O)EK
is a partial differential operator in both x and y of order al S M. For a = (ai,..., aNk) with al <M, define the distribution T0 eI(Rk) by Here,
(Ta(s),
e(Rk).
for
=
We claim {TQ } satisfies the requirements of the theorem. its Taylor expansion in the y-variable is Given f
f(x,y)= V31M
The Taylor remainder e satisfies 0)
=
0
1131
S
M.
So in view of (1) (T,f)RN =
.ij. I/31M
.
(x,y)ERN
(2)
Diiiribution Theoiy
72
On the other hand, from the Taylor expansion of f, we have
1
1
\IaIM
RN
= xER"
IoIM Clearly —
f0 =
if Therefore ®
= By the definition of
the right side equals
(z,y)ERN
By (2), this equals (T, f)RN. This completes the proof of the theorem.
I
Now we prove Whitney's theorem. THEOREM 2
Suppose RN has the coordinates (x, y) with x E and y E RN* Let be an open set in RIC. Suppose is an element of e(110) for each index = (cr1,... , with 0, j N — k. Then there is a function x RN_Ic) such that for each IE 1
(x,0)=a0(x) for The function
f is unique modulo the space of smooth functions on x {0}.
X
which vanish to infinite order on
PROOF The uniqueness assertion is obvious. So we shall concentrate on the existence part of the theorem. As in the proof of Theorem 1, we shall assume is all of Rk. if N = I and k = 0, then the theorem reduces to showing that for a given sequence of real numbers {ao, ai,. . .}, there is a smooth function f: R —' R
Whitney's extension theorem
73
with
forn=O,1. This
can be done explicitly by first choosing a cutoff function 0
E
D(R) with
iflyll
1
10
iflyl
2.
Then we let (3)
= There
exist
(depending on
I) so
that for any integer a 0
C is a uniform constant depending only on a. Therefore, the sum in (3) converges in the topology of e(R). Since 0 in a neighborhood of the for n 0, 1,2,..., as desired. origin, = The reader can modify this proof so that it will work in the general case. where
1
However, a slicker proof is available with the help of a little functional analysis. We consider the space
fle(Rk) where fl0 denotes the infinite cartesian product indexed by a (as, . .. , where each a1 is a nonnegative integer. Elements of this space are infinite tuples
(a0) where each a0 is an element of e(lRk). This space is given the product topology, that is, a sequence (az) for n = 1,2,... is said to converge to (a0) in as n —' oc if for each fixed a0 in e(Rk). This makes 110 e(lRk) into a Fréchet space (since each e(Rk) is a Fréchet space and since the index set {a = (aj,. . ,aN_k);a1 is a nonnegative integer} is countable). x RN_k) Define the map by fi .
(irf)0(x) =
1 01°1f
(x,0)
for each
a.
We want to show the map ir is surjective. By looking at the subspace of polynomials on RN, it is clear from the definition of the topology of fl0 e(Rk) that the image of ir is dense. So it suffices to show the image of ir is closed. By the closed range theorem for Fréchet spaces, it suffices to show the range of the dual to ir ir':
{He(Rk)}
74
Dtstribwion Theory
is closed. Here, ir' is defined by
(lr'(T),f)RN = (T,irf)
forTE {fl e(Rk)}'
and
f E e(IRN). We have
el(lRk) are infinite tuples (Ta) With where, by definition, the elements of eF(Rk) such that only a finite number of the T0 are nonzero. An element Ta E (Ta) in E061(Rlc) acts on an element E by
((Ta),
=
fa)Rk.
The sum on the right is well defined since only a finite number of the Ta
are
nonzero.
We now compute ir'. We have
(lr'{(Ta)},f)RN = ((Ta),irf)
=
laif
/
(Ta(x)
®
Therefore
(_l)1k1
&
( ) ôyOaY
which has its support in the set 11o = {(x,o) E E R"}. This equation together with Theorem 1 implies that the range of ir' is the set of all elements RN_k) whose support is contained in in eF(RIc This set of distributions is closed with respect to the weak topology on el(Rk x RN_k). As mentioned earlier, the closed range theorem implies that the range of ir is closed. Since ir also has dense range, ir must be surjective and so the proof of the theorem is complete.
I
54 Fundamental solutions for partial differential equations As mentioned at the beginning of this chapter, one of the reasons for the introduction of distribution theory is that it provides a convenient language to discuss
Fundamental solutions for partial differential equations
75
solutions to partial differential equations. We start by defining a fundamental solution for a constant coefficient partial differential operator
P(D)= I
a0EC.
T E IY(RN) is a fundamental solution for P(D) if
P(D){T} = The reason for the name "fundamental solution," is that a solution to the equation P(D){u} = for E D(RN) can be found by convolution with T,
as the following theorem shows. THEOREM I
Suppose P(D) is a partial differential operator with constant coefficients. Supis a E D(IRN) then u =
pose T is afundamental solution for P(D). If solution to the differential equation P(D){u} = PROOF
From Lemmas 1 and 2 in Section 5.2, we have
= (P(D){T})
P(D){T * Therefore,
P(D){T *
= 0, as desired.
*
=
*
=
I
We remark that if P(D) has variable coefficients, then this theorem is not true. This is because the step P(D){T * = (P(D){T}) * is not valid; for in order to apply the derivatives to T, an integration by parts is required and expressions involving the derivatives of the coefficients of P(D) will appear. In Part IV, we shall need fundamental solutions for the Cauchy—Riemann operator on C and the Laplacian on RN.
THEOREM 2
The distribution T(z) = /irz is a fundamental solution for the Cauchy-. Riemann operator = l/2(ô/ôx + iO/Oy) on C. 1
Note that T(z) = l/irz is a locally integrable function on C and so T defines an element of D'(C). To prove this theorem, we must show PROOF
all) —
az
iirzj
=
Distrthutkn Theory
76
which is equivalent to —
for
q5(O)
(1)
zEC
This is a generalized version of the Cauchy integral formula which is established
R < oo, let
by the following safety disc argument. For 0
0.
With this choice of a, b, t, we have q, (w, 11) = lwi 12 + 1w2l2
lawi
Define the following linear change of variables:
z2=z2+tz1 w2=aw,+bw2.
£1=z1
This is a nonsingular linear map since b Im
0. We have
= Im = q,(w,iD) = wi + =
(if (z,w) EM)
1w212
+
—
= alth,12 + 2 Re(-yth1th2) + i31th212
where a and
are positive real numbers and 'y is a complex number. Similarly
Imz2= Imz2+tlmz, =
=law,+bw212 =
(if (z, w) E M)
q2(w, tip) + tq1 (w,
lti'212.
by(1)
CR Manifolds
118
ti)), then Therefore, if M is the image of M under the linear map (z, w) i—' with = where in the new coordinates, M = {Im =
= alti'112 + 2 =
+ /31 W2I2
After dropping the A, we obtain
Now we complete the square in
qi(w, ti)) = Ia"2w1 +
+ (/3 — F112a1)Iw2I2
(recall that a > 0). We make one further linear change of coordinates
i,
2
a
—
Z2—Z2
)Z2,
= a"2Wi + 'ya"2vJ2,
W2 =
W2.
This change of coordinates is nonsingular since a > 0. Again, let M be the image of M under the map (z, w) (1, t1). The defining equation for M in the new coordinates is given by — Im z2 =
which is the normal form given in (i) of part (b) in the theorem.
Case 2. (w, = wi 12 — 1w212 (qi has eigenvalues of opposite sign). In this case, we make the following nonsingular linear change of coordinates: Z1=Z1
In
Z2=Z2
the new coordinates, (the image of) M is given by (after dropping the A)
I bn z1 = z2 = AIw1I2 + 2 Re(AwitD2) + B1w212
forsomechoiceofA,BER,andAEC. Let
A=r+si and r,sER. After the change of variables = is given by (drop the A) {Im z1 =
z1,
=
— 2rz1,
=
w1,
Im z2 =
Re(with2)
=
A1w112
—28 Im(w1i12) + B1w212.
In matrix form, we have -
-
-
q2(w,w) = (wt,w2)
(A is
—i8\(Wi
B)
= w2, M where
Quadrk submanifoids
119
If the determinant AR —
82
is positive, then the matrix of
is positive or
negative definite and so this falls under Case 1 above with the roles of and reversed. This leads to the normal form given in (i) of part (b). So we assume AB —
s2
0.
We first show that we can force the coefficient of Iwi 2 in change of variables of the form
z1=21 w1=th1
to vanish by a
z2=22 w2=tZ)2+itiI'1
for an appropriate t E R to be chosen later. Such a change of variables preserves qi. We obtain q2(w, tD)
(A + 2st + Bt2)1t11 12 — 2(s + Bt)
+
BItu'212.
is a real root t to the quadratic equation A + 28t + Bt2 = 0 because its discriminant 4(s2 — AB) is nonnegative. With this choice of t, the coefficient of itiii 2 vanishes and so we may assume (after dropping the A) There
qj(w,th) = Re(wiiD2) q2(w, tii) =
+ /31w212
and 0 are real numbers. 0. After a rescale in If = 0 then must be nonzero, for otherwise the z2-variable, M is in the normal form given in (ii) of part (b) with the roles of Wt and w2 and the roles of z1 and z2 reversed. if 0, then we can force the coefficient of 1w212 to vanish by a change of variables of the form where
ZIZ2
Z2Z2
w1—w1+itw2
W2—W2
t is a real number. Again, any change of variables of this form preserves We have
where
q2(w,w) =
+
CR Man Ifoldc
120
0, we may let t =
Since
which forces the coefficient of ki)212 to
vanish. After a rescale in the z2-variable, M is now in the normal form given in (iii) of part (b).
= lwi 2 (qi has a vanishing eigenvalue). We let
Case 3.
q2(w,w) =
A1w112 + 2 Re(Aw1i12) + B1w212
where A and .8 are real and A is complex. We make the change of variables
z2=z2—Az1 W1 =W1
W2
In the new variables, M is defined by I Im
= Itl,i
Im
2
2
Re(Athith2) + B1th212
Let q2(w, ti') = 2 Re(Awiü)2) + B1w212. The matrix that represents
is
(0 0, then the determinant of this matrix is —1A12 which is negative. Hence, the eigenvalues of the matrix of have opposite sign and this falls under Case 2 If A
above with the roles of and reversed. If A = 0 then, after a rescale, M has the normal form given in (i) of part (b). The proof of the theorem is now complete.
7.4
I
Abstract CR manifolds
So far, we have been dealing with CR submanifolds of In this section, we define the concept of an abstract CR manifold which requires no mention of an or complex manifold. ambient Let M be an abstract C°° manifold. As defined in Part I, Tc (M) denotes the complexified tangent bundle whose fiber at each point p E M is ® C. then from Lemma 3 in Section 7.1, If M s a CR submanifold of
(i) H"°(M) fl H"°(M) = {0} (ii) HLO(M) and H°"(M) are involutive. These two properties make no mention of a complex structure on other than to define the space H"°(M). Therefore, we define an abstract CR manifold to (M) which satisfies the above be a manifold together with a subbundle of two properties.
Abstract CR man(foldc
121
Let M be a C°° manifold and suppose L is a subbundle of Tc (M). The pair (M, L) is called (an abstract) CR manifold or CR structure if DEFINITION I
(a)
L is involutive, that is, [L1, L2] belongs to L whenever L1, L2 E IL
It is clear from the above discussion that if M is a CR submanifold of — then the pair (M,L) with L = H"°(M) is a CR structure. By analogy with the imbedded case, we call dimc {Tc (M)/L L} the CR codimension of (M, L). There is a complex structure map J defined on the real subbundle which generates L so that the eigenspaces of the extension of J to L L are L (for the eigenvalue +i) and L (for the eigenvalue —i). This follows from Lemma 3 in Section 3.2. In Section 4.3, we said that a pair (M, L) is an almost complex structure if L L = {O}. L is a subbundle of Tc (M) with Thus, an involutive almost complex structure is an example of a CR structure. As mentioned in Part I, the Newlander—Nirenberg theorem [NN] states that a manifold with an involutive almost complex structure is a complex manifold. (by definition), Now since a complex manifold can be locally imbedded into this prompts the analogous question for CR manifolds: if (M, L) is an abstract M CR structure, then does there exist a locally defined diffeomorphism
= so that '1(M) is a CR submanifold of Ctm with This last requirement for 1 implies that the CR structure for M (namely L) is pushed forward to the CR structure for 4{M} (namely H"°('I{M}). The answer to this question is a qualified yes. If M is real analytic, then there is a real analytic imbedding, as we will show in Section 11.1. If M is only smooth, then the answer, in general, is no, as we will show in Section 11.2, where we present Nirenberg's counterexample. There are further conditions on a smooth CR structure that will guarantee a local imbedding and we will briefly discuss these in Chapter 12.
8 The_Tangential Cauchy—Riemann Complex
For a CR submanifold of C'2, there are two ways to define the tangential Cauchy—
Riemann complex and both approaches appear in the literature. The first way The secis an extrinsic approach that uses the 0-complex of the ambient and ond way is an intrinsic approach that makes no use of the ambient therefore generalizes to abstract CR manifolds. In this chapter, we present both approaches. In the imbedded case, these approaches lead to different tangential Cauchy—Riemann complexes but in Section 8.3, we show they are isomorphic.
8.1
Extrinsic approach
Here, we assume the reader is familiar with the bundle of (p, q)-forms on C'2, over an The space of smooth sections of denoted Basic facts about the bundle of (p, q)open set U in C'2 is denoted forms and the associated Cauchy—Riemann complex 0:
—'
are
given
in Section 3.3. As also mentioned in Chapter 3, for each point p0 E C'2, the (C'2) is defined by declaring that the set Hermitian inner product on {dz' A = p. IJI = q, I, J increasing} is an orthonormal basis. Let M be a smooth, generic, CR submanifold of C'2 with real dimension to be the restriction of the bundle
2n—d. We define
is the union of where p0 ranges over to M, that is, M. This space is different from the space E AP'QT(C'2)}, where is an j: M —+ C'2 is the inclusion map. A smooth section of element of the form
f= 111p JI—q
whose
122
coefficient functions, fjj, have been restricted to M.
Extrinsic approach
123
For 0 p, q n, define the ideal in jp,q =
which is generated
by p and Op where p:
R is
any smooth function that vanishes on M Elements of jp,q
+
If {p',.
. . ,
are
sums of fonns of the type
A Op,
E
E
pd} is a local defining system for M, then {Pi, . .. , Pd} locally gen-
erates the ideal of all real-valued functions that vanish on M as shown in Lemma 3 of Section 2.2. Therefore, jp,q is the ideal in that is locally generated by
P1,...,Pd,OPI,••.,OPd. The restriction of jp,q to M, denoted is the ideal in locally generated by Op', .. , '9Pd. Since M is CR, the dimension of the fiber is independent of the point pij E M. Thus, is a subbundle of
Let
= I
the
orthogonal complement of jp,q I M in
L.
Elements in for E M are orthogonal to the ideal in (po)._ Let k be the number of linearly indepengenerated by Opi (P0),. .. , dent elements from (Po), . , (9pd(po)} (i.e., k is the CR codimension of M). Since M is CR, k is independent of the point p0 E M. Therefore, the dimension of is independent of the point p0 E M. Hence, the space is a subbundle of Note that =0 .
either p> n or q > n — k. If M is generic, then k = d (the real codimension of M) by Lemma 4 in Section 7.1. The space is not intrinsic to M, i.e., it is not a subspace of the exterior algebra generated by the complexified cotangent bundle of M. This is due to the fact that if p: R vanishes on M, then Op = (l/2)(dp+iJ*dp) is not orthogonal to the cotangent bundle of M due to the presence of J* dp.
For s 0, let =
..• e
where some of the summands on the right may vanish. The space is not the same as the space A8T*(M). The latter space is intrinsic to M whereas the former is not. As an example, let_M = {(z, w) E C2; Im z = 0}; then We have Op = p(z,w) = (2i)'(z — is and so the space of (p, q)-forms on M that are orthogonal to the ideal generated by
The Tangesnial Cauchy—Riemann Complex
124
In particular, A2.IT*(M) is generated by the form dz A dw A dii) and = A2.IT*(M), whereas A3T(M) is the 0. Therefore, space generated by dx A dw A dii) where x = Re z. Note that {jw;w E = A3T*(M). More will be said about f{AP.QT*(M)} in Section 8.3 AL2T*(M) =
where we discuss the relationship between the extrinsic and intrinsic tangential Cauchy—Riemann complexes.
For an open set U C M, the space of smooth sections of (M) over U will be denoted (U) will denote the space of compactly sup(U), and ported elements in If the open set U is not essential for the discussion, then it will be omitted from the notation.
For s 0, we let C8
—
where some of the summands on the right may vanish.
Again, note that
be the orthogonal projection map. For a form f E AP,qT* we often write ftM for 1M (f) and call this the tangential part of f. If / is a smooth (p, q)-form on then ftM is an element of Conversely, any form f E (U) can be extended to an where U is an open set in with Un M = U. This is element f E accomplished by writing Let tM:
fjjdz1 A
f=
with
fjj E E(U)
'I =p
JI=q and
extending each coefficient function fjj to an open subset U of
We now define the (extrinsic) tangential Cauchy—Riemann complex. DEFiNiTION I For an open set U C M, the tangential Cauchy—Riemann is defined as follows. For_f E complex OM: let with U fl M = U and let f E with ftM = / U be an open set in
on U fl M = U. Then OMf = The form OAif is calculated by extending / ambiently to an open set in Ci', then applying 0 and taking the tangential part of the result. Since there are many possible ambient extensions of a given element of we must show
that the definition of '9M is independent of the ambient extension. LEMMA
I
OM is well defined, that is, if fi and hare elements in
(f2)tM on MflU, then
= (0f2)tM on MnU.
with (f1)tM =
Extrinsic approach
125
Note that (f' — /2) is an element of Jp,q• l'herefore, it suffices to If show that 0 maps smooth sections of jp,q to and E U—+lRvanishesonMflU,then PROOF
=
+ fi A
+
+
A Op.
The iigbt side is clearly an element of
As already mentioned, the spaces (M) are not intrinsic to M. Thereand the resulting tangential Cauchy—Riemann operator are fore, the spaces
not intrinsic to M. For this reason, we refer to the above-defined tangential Cauchy—Riemann complex as being extrinsically defined. We shall specialize to the It is useful to have a procedure for computing case of a real hyperswface. LEMMA 2
p(z) =
Suppose M = {z
O}
is a real hypersuiface in
where
R is smooth with IdpI = 1 on M. Let N = 4(Op/Oz)
=
Then
= Nj(Op A
for
A
J is any ambiently defined smooth (p, q)-form with ftM = / on M. Recall that denotes the contraction operator of a vector with a form (see Section 1.5). The hypothesis that = I on M can easily be arranged by replacing p by p/IdpI. PROOF
Since Idp) =
1,
the vector field N is dual to the form Op. From
Section 1.5, if
and A t,t')
where (.) is the Hermitian inner product on
Therefore
=0 on M if and only if From the product rule for
(see Lemma 1 in Section 1.5), we have A
Since
=
q5 e
=
—
A
IdpI2 = 1 on M, this becomes (1)
The Tangential Cauchy-Riemann Complex
126
is an element of AP'QT(M). The ideal generated by p and lip).
0 and so Na(Op A form Op A (N1Ø) is an element of jp,q Now,
(the
Therefore, equation (1) provides an orthogonal decomposition of an element q5 of into its tangential part btM and the component of In particular, 4tM = A as claimed. The formula for and the definition of 8M• I OMf follows from the expression for The term NaØ is called the normal component of 0 and it is denoted by OflM.
Equation (1) then reads on M. 0= OtM + Op A This equation provides an orthogonal decomposition of into its tangential and normal components. If is an ambiently defined (p, q)-form on then both and are ambiently defined because p and hence N are ambiently defined. The equation = OtM + 0j A OflM also holds ambiently provided IdpI = I ambiently. This can be arranged, for example, if p is the signed distance function to M. ° At this point, the reader may wonder whether or not 00 = where j: M is the inclusion map. However, the right side of this equation does
not make sense because the domain of the tangential Cauchy—Riemann operator
which is not contained in A*T*(M), which is the range of j. The
is
right side of this equation does make sense if the tangential Cauchy—Riemann operator is defined intrinsically (see the next section). In Section 9.2, we shall
discuss CR maps (such as j) and the validity of commuting their pull backs with the tangential Cauchy—Riemann operator.
-
The following lemma follows easily from the analogous properties of 0 (see
Section 3.3). LEMMA 3
Suppose M is a smooth CR
of C's.
(a)
(b) OMOOM=O. From part (b), if OMf = g, then = 0. An important question is to ask whether or not the converse holds: if 0M9 = 0, then does there exist a form f with 8Mf = g? This solvability question for the tangential Cauchy—Riemann operator will be discussed in Part IV. It is useful to interpret the equation 0Mf = g in terms of currents (see Chapter 6 for basic facts about currents). If M is a smooth, oriented submanifold of of real dimension 2n — d (1 norq > n—d. Therefore, iff
andg
en
(f,g)M = where and
is the piece of g of bidegree (n — p,n — q — d). If I E then f A 9 has bidegree (ii, n — d) and so
(f,g)M =11 Ag
=
A
LEMMA 6
Suppose M is an oriented, CR, generic 2n — d. Let f E and g E
=
of
with real dimension Then
Extrinsic approach
PROOF
129
From the discussion preceding the ststement of Lemma 6, we have
t
—
\VMJ , 9/M — \1LVJ J
where I E LP'Q(cn) of f and g. We have \(JMJ,9/M =
and
A g,'cn
,
are any ambient extensions
E
-'
,r,.xiO,d A
= = =
(by Lemma 4)
A Of,
(since
Af
0)
A
where the last equation follows from the definition of 0 applied to a current. = A Therefore By Lemma 4, we have [M]0.d A
= = as desired.
(—
IA
I
This integration by parts formula allows us to extend the definition of OM to currents on M. The space of currents on M of bidimension (p, q) is the dual of the space and it is denoted by }'. By adapting the proof of Lemma 1 in Section 6.1 to this context, the reader can easily show that
which is the space of currents of bidegree }' is isomorphic to n—q—d (n — p, n — q — d) on M. An element of is a form of bidegree (n — p, n — q — d) on M with distribution coefficients. {
Suppose M is an oriented, CR, generic with real dimension 2n — d. Let T E Dr', then aMT E
DEFINITION 2
of is the current
defined by
(OMT,g)M = From Lemma 3, we have
for g E
(T) = 0 for T E
In addition, Lemma 5
holds for currents, although we should say a word about the definition of [M]O*dA It suffices by a partition of unity argument 7 when 7' is an element of A Tlocally. Let p', . . . , Pd be a local defining system for M. to define is Hausdorif measure on M. A Then, A ... A 0Pd where = typical element of is T = T1 where T1 is a distribution on M and where with There is an element E 4e = on M. To define A T, it suffices to define the distribution 4UM . 7'1. By using a (smooth) x Cn_d; = 0}. local coordinate system, we may assume M = {(z, w) E In these coordinates, we have 4UM = where y = Imz. A distribution T1
The Tangential Cauchy—Riemann Complex
130
M acts in the variables x = Rez and w. Therefore, we define the distribution (ILM Ti)(x,y,w) = 6o(y) ® T1(x,w) where ® denotes the tensor product of distributions (see Section 5.2). The proof of Lemma 5 for currents now proceeds by approximating a current by a sequence of smooth forms and then using Lemma 5 for smooth forms. on
If M is not generic, then the CR codimension of M is less than d. In this case, the reader can show
= where
=
=
=
k is the CR codimension of M. Lemmas 4 through 6 hold with [M]0"
replaced by
8.2
Intrinsic approach to
Our treatment of the intrinsically defined tangential Cauchy—Riemann complex
is similar to that in [PWJ. We assume that (M, L) is an abstract CR structure (see Definition in Section 7.4). Therefore, L is an involutive subbundle of Tc (M) and L fl L = {O}. It will be necessary to choose a complementary subbundle to L L. In order to do this, we assume that M comes equipped with a Hermitian metric for Tc (M) so that L is orthogonal to L. If M is a submanifold of then the natural metric to use is the restriction to Tc(M) of the usual Hermitian metric on Tc (Ce). If M is an abstract manifold, then a metric can be constructed locally by declaring a local basis of vector fields to be orthonormal. This metric can be extended globally by a partition of unity. For each point p0 M, we let be the orthogonal complement of in C. Clearly, the spaces M} fit together smoothly (since p0 the do) and so the space 1
X(M)= U poEM
forms a subbundle of Tc (A'!)
—
then L = H"°(M) and L = H°"(M). In this case, X(M) is the totally real part of the tangent bundle.
If Mis a CR submanifold of Define the subbundles
T"°(M) = L
X(M).
We emphasize that T"°(M) is not analogous to H"°(M) for an imbedded CR manifold (unless X(M) = {O}).
Intrinsic approach to
The
131
dual of each of these spaces is denoted T°'3 (M) and T'°(M), respec-
lively. Forms in T*°" (M) annihilate vectors in T"°(M) and forms in T*'°(M) annihilate vectors in T°" (M) (by the definition of dual). Define the bundle
=
(M)).
This is called the space of (p, q)-forms on M. Unlike the extrinsic approach, these spaces are intrinsic to M (in the abstract case, they cannot be anything but intrinsic since there is no ambient space). Let m = dimc L and d =
dime X(M). Recall that d is the CR codimension of M. If p > m + d or q > m, then AP'QT*(M)
0.
The pointwise metric on Tc (M) induces a pointwise dual metric on T*C (M) in the usual way. Let be an orthonormal basis for and let
be an orthonormal basis for T°" (M). The metric for T* (M)
. . . ,
extends to a metric on A
Ill = p, IJI = q, I, J are increasing multiindices}
is an orthonormal basis. We also declare that M"?T*(M) is orthogonal to if either p r or q s. We have the following orthogonal decomposition ArT*C
(M) =
...
with the understanding that some of these suminands vanish if r > m. Let ArT*C
(M) —i AP.QT*(M) for p + q = r
be the natural projection map. The space of smooth r-forms on an open set U C M is denoted by and The space of smooth sections of AP'QT(M) over U is denoted by is the space of compactly supported elements of The "U" may be omitted from the notation if it is unimportant for the discussion at hand. The intrinsic definition of the tangential Cauchy—Riemann operator can now be given in terms of the exterior derivative dM: er'. DEFINITION 1 The tangential Cauchy—Riemann operator o is defined by 0M =
This definition of 8M is analogous to the definition of 8 on C" (or any other complex manifold). —p We will show that OM: is a complex, i.e., OM oOM = 0. This will follow from the equation dM o dM = 0 and type considerations. First, we need a preliminary result.
The Tangential Cauchy-Rlemann Complex
132
LEMMA I
If M is a CR
then
,,
icp,qi
çp+2,q—I
çp+i,q
Conceivably, the exterior derivative of a (p, q)-form, 0, might be a sum of forms of various bidegrees. The point is that the only possible nonvanishing components of dM0 have bidegrees (p+ 2,q — 1), (p+ 1,q) and (p,q + 1).
The case p =
PROOF
1,
q=
(dMO,iJi AL2) = 0 where
(,
0
will be handled first. We must show
This is equivalent to showing
7r°'2(dM(b) = 0 if 0
Et =
for all
) denotes the pairing between forms and vectors. From Lemma 3 in
Section 1.4 (dMq5,Tj AL2) —
Since
E
and L1,L2
(0, [Li, L.2]).
T°"(M), we have
0.
By the
definition of a CR structure, L is involutive, and so [Lj, L2] L. Therefore (0, (L1, L2]) 0, from which (dM0, L1 A L2) = 0 follows, as desired. This proves the lemma for the case p = 1,q 0. Note that the lemma automatically holds for p = 0 and q = I. For p, q 1, is generated by the following terms: OIA ... A
A
...A
is a smooth (1,0)-form and each is a smooth (0, 1)-form. The general case now follows by using the product rule for dM and the lemma for the case of a (1,0)-form or (0,1)-form. I where each
The
key ingredient of the proof is that T°" (M) = L is involutive. Since
T"°(M) is not necessarily involutive, we cannot conclude that 0
for 0 E
so
=
If M is a complex manifold, then T"°(M) is involutive and = 0 for 0 E This is a key difference between the
class of complex manifolds and the class of CR manifolds that are not complex manifolds. LEMMA 2
If M isa CR manifold, then ÔM o8M =0.
intrinsic approach (0
133
OM
is a smooth (p, q)-form; then ÔMØ =
PROOF Suppose Lemma 1 implies
=
dM0 —
(dM0).
+
Therefore
= =
(dM0) +
the term on the
dM
right vanishes. Therefore, OMÔMq5
=0, as desired.
I
The following product rule for OM follows from the product rule for the
exterior derivative. LEMMA 3
If f
and 9 E
then
OM(f Ag) = From Stokes' theorem and the product rule for 8M we obtain an integration
by parts formula for 0M• LEMMA 4
If I E
and g is any smooth form on M with compact support, then
=
Let m be the dimension of L (and L); let d be the dimension of X(M). Therefore, = m. dimc TLO(M) = m + d and dimc Tc (M) has complex dimension 2m + d which is the same as the real dimension of M. So a form of top degree on M has bidegree (m+d, m). 1ff then and so t9Mf pairs with forms of bidegree (m+d—p,m—q— 1). ÔMf From the product rule, we obtain Let g PROOF
Ag The
=JÔM(f Ag) +
A
8Mg.
(1)
bidegree of fAg is (m+d,m— 1). Since top degree on Mis (m+d,m),
we have
OM(f Ag)
=dM(fAg).
The Tangential Cauchy-Rlemann Complex
134
Since g has compact support on M, we have
fdM(f Ag) =0 by Stokes' theorem. This together with (1), establishes the lemma.
I
Just as with the extrinsic case, the integration by parts formula allows us to extend the definition of 0M to currents. By definition, the space of currents of bidimension (p, q) on an open set U C M is the dual of the space By adapting the proof of Lemma 1 in Section 6.1, the reader can easily show that this is isomorphic to the space of currents of bidegree (m + d — p, m — q) which is denoted by Elements in this space are (m+d—p,m—q)forms on U with coefficients in D'(U). If the open set U is not essential to the discussion, then it will be omitted from the notation.
DEFINITION 2
if T E
then the current OMT E
is defined by
(bMT,g)M = (_1)(T,9Mg)M. together with Lemma 2 shows that - An easy argument using this definition 'pq
OM(OMT)
8.3
=0 for a current T VM
The equivalence of the extrinsic and intrinsic tangential Cauchy—Riemann complexes
For a CR submanifold M of there is a choice of viewpoints for the tangential Cauchy—Riemann complex — the extrinsic and the intrinsic. These two complexes are different, but in this section, we show they are isomorphic. Before we establish this isomorphism, let us precisely define an isomorphism between two complexes. DEFINITION 1
(a) A complex is a collection of vector spaces A = {Aq; q E
o Z,q 0} with maps such that = Ofor q 0. (b) Suppose A = {Aq, dq; q 0} and A = {Aq, Dq; q 0} are two com-
plexes. These complexes are isomorphic if there exists a collection of isomorphisms of vector spaces Pq: i.e., Aq that intertwine dq and 0 Dq
= dq 0 Pq.
Extrinsic and intrinsic tangential Cauchy—Riemann complexes
135
The following commutative diagram describes part (b) of the definition. DQ
Ag
Aq+i
lPq Ag
Aq+i
Ag+2
As an example, let M and N be smooth manifolds and suppose F: M er+I(M)} and N is a diffeomorphism. The complexes {dM: e"(M) Er+I(N)} are isomorphic and the isomorphism is given by {dN: F*: THEOREM 1
Suppose M is a CR submanifold of
The extrinsic and intrinsic tangential
Cauchy—Riemann complexes are isomorphic. PROOF
Fix p with 0 p n. For q 0, let Ag =
— via the extrinsic definition
Ag =
— via the intrinsic definition.
(M) is the space of smooth Sections of the extrinsically defined bundle which by definition is the orthogonal complement of jp,q ) I M. on the other hand, Ag is the space of smooth sections of the intrinsically defined bundle (M) which by definition is (M)}. We let
Dg: Ag
Ag+i be the extrinsically defined 0M
dq: Ag —4 Aq+i be the intrinsically defined 0M.
= The operator Dg is the tangential part of 0 (i.e., tM o 0), and is the projection of dM where dM is the exterior derivative on M and onto AP{H*'°(M) (M)}. be the inclusion map. We will show that j is the desired isoLet j : M —' morphism between the complexes {Dg: Ag Aq+i } and {dq: Aq ' Ag+i }. The following two statements must be shown:
(i)
(ii)
The map
takes Ag OfltO Ag isomorphically.
j*oDg=dqoj.
To prove (i) it suffices to show the following.
The Tangential Cauchy-Rlemann Complex
136
LEMMA 1
For each point p0 the intrinsic
E
(M) isomorphically onto
M, j maps the extrinsic
For notational simplicity, we will prove this lemma for the case where generic. From Lemma 1 in Section 7.2, we can find an affine complex linear change of coordinates A i—s so that the given point p0 E M is the origin and PROOF
M
is
:
M = {(x+iy,w)
C' x (fl_d;y = h(x,w)}
where h: Rd x Cfl_d Rd is smooth with h(O) =0 and Dh(O) = 0. As mentioned in the remark after the proof of this lemma, A preserves the holomorphic tangent space of M, the totally real tangent space of M and the metric for the (M) real tangent space of M. Therefore, the definition of the intrinsic is invariant under this change of coordinates. In addition, the pull back of A is also commutes with 0. Therefore, the definition of the extrinsic invariant under this change of coordinates. l'he following arguments can be easily modified for the nongeneric case by using the remarks that follow the proof of Lemma 1 in Section 7.2. A local defining system for M is given by {pi,... ,pd} where p3(z,w) = Im —h,(Re z, w). Since Dh(0) = 0, we have .9p3(O) = By definition, the extrinsic is the orthogonal complement in
of the ideal generated by bp1,... OPd. Therefore, a basis for the extrinsic at
the origin is given by
+ IJI = p, IKI = q, I, J, K increasing}.
{dz' A dv? A A basis for L,3 = = is given by
is given by {dwi,.
.
.
,di)Yn_d}. Since
onal complement of L e L in
(M), a basis for
and a basis for is the orthog(M) is given by
{dxi,... ,dxd}. Therefore, a basis for the intrinsic A
A
+
= p, IKI = q, I, J, K increasing}.
Since Dh(0) = 0, the following relations hold at the origin:
1
r0
fl=r
Since fdzk = dxk, we have min(d,p)
r0
(1)
II=r
The Tangential Cauchy—Riemann Complex
138
On the other hand mm(d,p) min(d,q+1)
r=0
III=r
8=0
I
Each
dxk belongs to the intrinsic
J 1=8
and
trinsic
A
A
belongs to the inbelongs to the intrinsic
Due to the presence of
the only contributing term to the sum on the right occurs when s = 0. Therefore min(d,p)
>
r=0
III=r
By comparing this with (1), we see that
= on the space
'T
) I M.
(2)
By examining the above argument, we see that
(this is trivial in the case of
both of these maps vanish on
1) due to the presence of We conclude that (2) also holds on the space and so the proof of the lemma is complete. This completes the proof of statement (ii) and hence the proof of Theorem 1 is also complete. I
j*
0
In view of this theorem and in order to keep the notation to a minimum, we shall not distinguish between the extrinsic and intrinsic OM-complexes. It will be clear from the context which point of view will be used. As the final item in this chapter, we make a remark about metrics. Here, a metric is used to choose a complement, X(M), to A different choice of metric leads to a different X(M) and hence a different ÔM-complex. However, given any two metrics, the associated complementary bundles are isomorphic and therefore the two resulting aM complexes are isomorphic. A metric can be avoided by using quotient spaces. For example in the imbedded case, we may let
= the abstract setting, we let APQ(M) be the space of forms on M of degree p + q that annihilate any (p + q)-vector on M that has more than q-factors In
contained in L. Then we define "
Both of these definitions of
"—
(M) avoid the use of a metric.
Extrinsic and intrinsic tangential Cauchy—Riemann complexes
139
In the imbedded case, the Cauchy—Riemann operator maps the space of smooth sections of jp,q to jp,q+ 1• The tangential Cauchy—Riemann operator can then be defined as the induced map of the Cauchy—Riemann operator on the quotient spaces. Similarly, in the abstract case, the exterior derivative maps the space of smooth sections of to (M). We leave the verification of this to the reader. In this case, the tangential Cauchy—Riemann operator can be defined as the induced map of the exterior derivative on quotient spaces. Both of these complexes are isomorphic to the tangential Cauchy—Riemann complexes defined earlier in this chapter once a metric has been chosen.
We have chosen not to emphasize this point of view because computations usually require a choice of a metric. The metric point of view will be especially useful in Part IV of this book.
9 CR
Functions and Maps
In this chapter, we present the definitions and basic properties of CR functions and CR maps. CR functions are analogous to holomorphic functions on a complex manifold. However, there are important differences. For example, CR functions are not always smooth. There are relationships between CR functions on an imbedded CR manifold and holomorphic functions on the ambient For example, the restriction of a holomorphic function to a CR submanifold is a CR function. However, CR functions do not always extend to holomorphic functions. In this chapter, we show that real analytic CR functions on a real analytic CR submanifold locally extend to holomorphic functions. A C°° version of this is also given. The chapter ends with a discussion of CR maps between CR manifolds.
9.1
CR functions
Suppose (M, L) is a CR struaure. A function f: M distribution) is called a CR function if 0Mf = 0 on M.
DEFINITION I
C (or
For most of this chapter, we shall be dealing with CR functions that are of class C'. The above definition applies to any CR manifold — either abstract or imbedded in We now present other characterizations of a CR function. LEMMA I (a) Suppose (M, L) is a CR structure. only jf Lf =OonMforallLEL.
A C' function f: M —i C is CR if and
pd(Z) = 0} is a generic, CR subman:fold of
CRifandonlyifOfAOplA...AOpd=OonMwheref: extension of f.
140
C is
—iCisanyC'
CR functions
PROOF
141
For the proof of (a), recall that
projection of T (M) onto T only if (dM1, L) =
0
for all L
= 7r°'dMf. Since 7r0'1 is the we have (dM1) = 0 if and
(M) = L , L. Part (a) now follows from the equation
(dMf,7) =12{f} which is the definition of the exterior derivative of a function.
--
(b) follows from the extrinsic definition of OMf as the piece of Of IM which is orthogonal to the ideal generated by {Op; p: CN R is smooth with p=OonM}. I Part
If M is a CR submanifold of
then any holomorphic function on a neighborhood of M in restricts to a CR function on M by part (b) of the lemma. However, the converse is not true, that is, CR functions do not always extend as holomorphic functions. This is fortunate, for otherwise the study of CR functions would be much less interesting. The following example illustrates this behavior.
-
Example 1 Let M = {(z,w) C2; Im z = 0). Here, L = H°"(M) is spanned (over e(M)) by the vector field A function f: M C is CR if
(x= Rez). A CR function on M is a function that is holomorphic in w with x held fixed. Since there is no condition on the behavior of a CR function in the x-variable, an arbitrary function of x is automatically CR. Therefore any nonanalytic function of x is an example of a CR function that does not extend to a holomorphic function on a neighborhood of M in C2. [I In this example, a real analytic CR function on M = {y = 0} is always the restriction of a holomorphic function defined near M. A real analytic CR function on M can be represented (near the origin) by a power series in x and w (no zi). The holomorphic extension is obtained by replacing x by z in this power series. This idea will be exploited in the existence part of the proof of the next theorem, which is due to Tomassini [Tom].
THEOREM I
Suppose M is a real analytic, generic CR subman(fold of
at least n. Suppose f: M
with real dimension
C is a real analytic CR function on M. Then there is a neighborhood U of M in and a unique holomorphic function
F: U—CwithFlM=f.
CR Functions and Maps
142
The neighborhood U in this theorem depends on the CR function f. Additional geometric conditions on M can be added to ensure that the neighborhood U is independent of f. With these added conditions, the real analyticity of f is unnecessary. This and other CR extension topics are discussed in Part Ill. The uniqueness part of the theorem requires that M be generic but it does not require the real analyticity of M. We present it as a lemma. LEMMA 2
Suppose M is a smooth, generic CR subman(fold of C'2 of real dimension 2n — d, o d n. If f is holomorphic in a connected neighborhood of M in C'3 and
tf f vanishes on M, then f vanishes identically. PROOF By the identity theorem for holomorphic functions, it suffices to show
all the derivatives of I vanish at a fixed point po E M. Near p,, E M, there is a local basis for H1'°(M) consisting of smooth vector fields L1,. . (for example, use Theorem 3 in Section 7.2). The collection of vector fields {L1,... ,Ln_d} forms a local basis for H°"(M). Let X1,... ,Xd be a local basis for the totally real tangent bundle, X(M). By ambiently extending the .
coefficients, we may assume these vector fields are defined in a neighborhood ofprj in C'3. The vector fields N1 = JX1,...,Nd = JXd (restricted to M)are transverse to M. Since M is generic, a basis for Tc (C'2) near p0 is given by
{L1,...
,
X1,... ,
,
N1,...
,
The vector fields L1,. . . , L1,. . , X1,.. . , Xd will be called tangential (since their restrictions to M belong to TC (M)). The vector fields N1,. . , Nd will be called transverse. To prove D" f = 0 near p0 on M for all differential operators we use a double induction argument on both the order of the differential operator and the number, rn, of transverse vector fields in If m = 0, then D" involves only tangential vector fields and so D'2f = 0 on M because / = 0 on M. .
.
Now we assume by induction that for rn 0, D" / =
0
on M for all
differential operators that involve only rn-transverse vector fields. We will show
on M where
are
any indices from the set {1,... , d}. From the Cauchy—
Riemann equations on C'3, we have (X + iJX)(f) = 0 for X E T(C'2). nearpo in C'3. Thus Therefore, Njm+L{f} = JXjm+i{f} = N21..
= iN,,..
The right side involves a differential operator with only rn-transverse vector fields and therefore it vanishes on M as desired.
CR functions
To
143
complete the double induction, we assume the following: for integers,
N 0, m 1
D°f=0 on M for N and where involves only rn-transverse vector fields. We also assume f = 0 on M for operators of any order that involve at most (rn — 1)transverse vector fields. We must show that = 0 on M for =N+1 and where involves rn-transverse vector fields. We have two cases to consider.
Case(i). Da=ToDQ'. Here, T is a tangential vector field and is a differential operator of order N that involves only rn-transverse vector fields. In this case, =0 on M by the induction hypothesis and so f} = 0 on M, as desired. = N3 oDa'.
Case (ii) Here,
N3 = JX3 is transverse and
N
is a differential operator of order
that involves only (rn — 1)-transverse vector fields. In this case
= = D°'{N3f} + by ] denotes the commutator. The first term equals , the Cauchy—Riemann equations. This term vanishes on M by the induction hypothesis because D°'XJ involves only (m — 1)-transverse vector fields. The second term is a differential operator of order N and so it vanishes on M, again by the induction hypothesis.
where [
From double induction, it follows that D°f = 0 near p,, on M for all differential operators and so f vanishes identically. The proof of the lemma is now complete.
U
For the existence part of Theorem 1, we give two proofs. The first is per-
haps simpler but the second can be easily modified to handle a Theorem 1. Both proofs illustrate important ideas.
version of
The first proof treats both ( and ( C'1 as independent coordinates. We will show that the given_real analytic CR function f extends to a holomorphic function on C2V1 (of and By using the tangential Cauchy—Riemann equations, we will show that the holomorphic extension of f is independent of the coordinate and so it restricts to a holomorphic function on C'1 which is the desired extension of f. Now we present the details. It suffices to holomorphically extend the given CR function to a neighborhood of a fixed point p0 E M. The global extension FIRST PROOF OF EXISTENCE
144
CR Functions and Maps
can then be obtained by piecing together the local extensions. The uniqueness part of Theorem 1 ensures that the local extensions agree on overlaps. From Lemma 1 in Section 7.2, we may assume the point P0 is the origin and
M = {(z
x + iy, w)
E Cd
x
y
= h(x, w)}
where h: Rd x Rd is real analytic in a neighborhood of the origin and Dh(O) = 0. From Theorem 3 in Section 7.2, a local basis for H°" (M) is given by d
d
the (€, k)th entry of the matrix (I + i(Oh/Ox))'. Since h is a real analytic function near the origin, h can be expressed in a
where
is
power series in the variables z, E Cd and w, E Cn_d. By replacing independent variable ( E Cd and by the independent variable 17 E the power series for h, we obtain a holomorphic function h: Cd x Cd x
= h(x,w). Define
with
Mc = Also define
Cd x Cd x Cd
by the m x
x Cn_d
x
Cd
=
—
x Cd x Cn_d x
by
= Mc is a complex submanifold of with complex dimension 2n — d. Moreis imbedded as a totally real submanifold of Mc. If f: M C is a real analytic function on M, then the above procedure of replacing by ( and ii) by in the power series expansion of / produces a holomorphic function f: Mc —p C with / o = f. Similarly, the real analytic coefficients of L, can be holomorphically extended to vector fields over,
L1,...
T"°(Mc) with
4=
I j n—d.
+
=
Since f is holomorphic, we have
on
then
(1,!)
= on
M.
(1)
If / is CR,
145
CR functions
is a 2n — d real dimensional generic (totally real) submanifold of
Since
we have (2)
on Mc by Lemma 2. Each vector O/O(j is transverse to Mc because Dh(O) 0. Define F: Cd x Cd x Cn_d x C'2_d C that is independent C to be the extension off: Mc So
=
1 j
near the origin,
0
d.
(3)
We also claim
=
0
near the origin,
1
j n—
d.
(4)
To see this, note
a (oP\_ a (a?
—
In view of (1) and using OF/aC, = 0, we obtain
OF
-
-
=0
Since 0/0(, is transverse to
on
Mc,
on
Mc
(by (2)).
the previous two sets of equations imply that
= 0, as claimed. is obtained Finally, the holomorphic extension of f on C'2 = Cd x From (3) and (4), F is independent of ( and and by setting F = F o Therefore, F so the power series of F = F o is independent of and is holomorphic on a neighborhood of the origin in C'2. Moreover FIM
=f
because
FIM = I
=f.
This second proof is based on ideas in a paper by Baouendi, Jacobowitz, and Treves [BiT]. We again start with M presented near the origin as SECOND PROOF OF EXISTENCE
M = {(x + iy, w) E C'1 x Cn_d;
h(x, w)},
CR Functions and Maps
146
where h is real analytic and h(O) = 0, Dh(O) 0. If M is flat (i.e., h 0) then a real analytic CR function near the origin is a convergent power series in x and w (no ff.). The desired holomorphic extension is obtained by replacing x with z = x + iy in its power series. We want to mimic this procedure as much as possible for the general case. The problem is that in general, a real analytic CR function will depend on ti). However, its dependence on ii) is closely linked be with the dependence of the power series of h on ti). Instead of letting
an independent complex coordinate as in the first proof, we shall change the complex structure for C'1 so that h becomes holomorphic. Now we present the details. In the power series of h, we replace x by z. So h is defined on Cd x C'1_d and h(z, w) is holomorphic in z near the origin. Define H:Cd x Cn_d x by
H(z,w) =
(z
+ ih(z,w),w).
be the component functions for H. We use H = (H1,..., Let H1,. . as a coordinate chart to define a new complex structure for C" Cd x C'1. A function g is holomorphic with respect to the new complex structure if there exists a holomorphic function G in the usual sense with g = C o H. This complex structure agrees with the usual complex structure in the z-variables since H is holomorphic in z E Cd in the usual sense. The T°' 1-vector fields for this new complex structure are those vector fields that annihilate the coordinate functions Hi,. . , . ,
.
LEMMA 3 A local basis for the bundle 70,I (C" ) for the new complex structure is given by d
where PROOF
1 <j
is the (i,k)th entry in the d x d matrix [I + i(Oh/Oz)]'.
This lemma follows by showing that for 1 £
n, A,{Ht} = 0 for
= 0 for 1 ( j d. Also note that these
< n — d, and
vector fields are linearly independent near the origin because Dh(0) = 0. Let H0 = HI{Im z=0} The map H0: Rd x Rd x Cn_d be for M. Let 7r: Cd x
—+
I
M is a paraineterization
the projection map given by ir(x + iy, w) = (x, w). Clearly, lrIM is the inverse of H0. From Theorem 3 in Section 7.2, a local basis for H°" (M) is given by d
—
L3
=
—2i
-
#Lk
.9w3
Ozi
+
.9
.9w3
1 j
n— d
CR functIons
147
where Pek is the (4 k)th entry of the matrix [I + i(Oh/Ox)]'. A C' function
f: M —' Cis CRon Mif and only if L,f = 0,1 <j < n—d. This is equivalent to ir.L,{f o H0} 0 on Rd x because H0 o ir is the identity map on M. The vector field L, can be computed by using (O/0y3) = We have 0, = 0/Ox, and =
Suppose f: M Cn_d
—' C.
C is a real analytic CR function. Let fo
The function fo is a real analytic function of x E
Let F0: Cd x Cn_d
Rd
fo
x
and w
be the extension of fo obtained by replacing x by z x + iy in the power series expansion of fo about the origin. Since F0 and h are holomorphic in z Cd, we have Oh/Ozk = Oh/Oxk,OFO/Ozk = OF0/Oxk. we obtain Comparing the expressions for A, and C
AjFoI{Im z=O} =
= 0 (since f is CR). Since A,F0 is holomorphic in z Cd and vanishes on {Im z 0}, A,F0 must vanish identically. Since both A,F0 = 0 and OFO/(92k 0, the function F0 is holomorphic with respect to the new complex structure. There is a function F: C that is holomorphic in the usual sense defined in a neighborhood of the origin with F0 = F o H. The restriction of F to M is I because 1 o H0 = F o H0 on {Imz = 0}. This completes the second proof of the existence part of Theorem 1. We have shown by example that CR functions of class C°° are not necessarily the restrictions of holomorphic functions. However, smooth CR functions are the restrictions of functions that satisfy the ambient Cauchy—Riemann equations on M. This is presented in the next theorem. THEOREM 2
Suppose M is a C°°, generic CR submanifold of C.' with real dimension 2n — d,
1 ,r) =0 for all L E T°"(M) = EM where (, ) denotes the pairing between forms and vectors. Since F is CR, F.L is an element of LN = T°"(N) and so 0 = (4>, F,L) = (F'4>, L), as desired. Part (b) follows by writing a typical term in (N) as
where
Note that
E
T"°(N)
and E T'°"(N) and then by using part (a) for generally has nontrivial components of type (1,0) and (0,1).
I
CR maps
153
PROOF OF THEOREM 2
Let 0 be an element of
= F0 —
From Lemma 2, we have
+
I
From the definition of t9M, we have
_JF*0} belongs In view of Lemma 1 in Section 8.2, 9+J'Q—3+l Since j 1, the sum on the right must vanish. Therefore
= Using the fact that dM commutes with F*, we obtain
=
(1)
From the definition of 0N and Lemma 1 in Section 8.2, we have dNO = '9N0 + By Lemma 2,
+
(2)
isa sum of terms of type (p+ 1 +j,q—j) for Inparticular
= 0. Similarly
=
0.
These two equations together with (1) and (2) yield
=
The proof of Theorem 2 is complete.
I
We discuss two corollaries. First, we give the extrinsic version of Theorem 2 for imbedded submanifolds.
COROLLARY I
Suppose M and N are CR submantfolds of and respectively. Suppose f: M N is a CR map. Let F: —i Ctm be an extension off with OF = 0 on M. Then as maps from the extrinsically defined
to the extrinsically defined
CR Functions and Maps
154
and ON refer to the extrinsically defined tangential Cauchy—Riemann
Here,
operators. Elements of are not intrinsic to N. Rather, they are smooth seclions of For this reason, it is necessary to have C an ambient extension (F) of the CR map (f) for the statement of the corollary.
One approach to the proof is to show that Ft maps jp,q IN to M (see the definitions of these spaces in Section 8.2). Then, the corollary will follow from the definition of the extrinsic version of the tangential Cauchy— PROOF
I
Riemann complex. The other approach to the proof involves reducing the statement given in the corollary to Theorem 2. We will give the details of the latter approach and leave the details of the former approach as an exercise. Let :M and iN; N be the inclusion maps. By Section 8.3, and are isomorphisms between the extrinsic and intrinsic tangential Cauchy—Riemann complexes of M and N. Therefore, the statement of the corollary is equivalent to
OtM oF Since bF =
as operators on the extrinsically defined serves bidegree for elements of
0
on M, F pre-
and so this equation is equivalent
to o
o Ft =
0 tM 0
o
0
oFo
From Theorem I in Section 8.3, and ON = °9M = ÔM ) 'oON where the OM and ON on the left are extrinsic and the 8M and ON on the right are = intrinsic. In addition, from Lemma 2 in Section 8.3. Therefore, the above equation is equivalent to o
o (F
OjM) =
o (F ojM) o
o
OM and ON are now the intrinsic tangential Cauchy—Riemann operators. o f (since F = Finally, we use the fact that F o on M) to see that = this equation is equivalent to Here,
f
of OON) is an isomorphism between the extrinsic and intrinsic Eu', the proof of the corollary now follows from Theorem 2. I Since
We say that the CR structures (M, L) and (N, LN) are CR equivalent if there is a CR diffeomorphism between M and N. Using Theorem 2, we will show (in Corollary 2) that if M and N are CR equivalent, then the tangential Riemann complex on M is solvable if and only if the same is true for N. To be precise, we say that the tangential Cauchy—Riemann complex is solvable at bidegree (p, q) if for any form f E with OMf = 0, there is a form U E
with OMU =
f.
CR maps
155
COROLLARY 2
Suppose (M, LM) and (N, LN) are equivalent CR structures. The tangential Cauchy—Riemann complex is solvable at bidegree (p, q) on M if and only if the
is true for N.
Since an open subset of a CR manifold is also a CR manifold, the above corollary applies to CR equivalent open subsets of CR manifolds.
Suppose F: N is a CR diffeomorphism and suppose f is an element of with 0Nf = 0. By Theorem 2, we have PROOF
=
=0 on
M.
If the OM-complex is solvable at bidegree (p, q) on M, then there is a form uE
with OMU =
Applying
o
F'
to this equation and using Theorem 2 with M replaced
by N and F replaced by F', we obtain =
(3)
From Lemma 2, we have
= where
r
Ff
-
min(q, n — p). Applying
o
F'
to this equation and using
Lemma 2 with F-' instead of F, we obtain = — —
p,q
=1 Substituting this equation into the right side of (3) yields on
N.
Hence, the solvability of OM implies the solvability of ON. The converse is established the same way. I
10 The Levi Form
In previous chapters, concepts such as the tangential Cauchy—Riemann complex are introduced first for imbedded CR manifolds and then later for abstract CR
manifolds. In this chapter, we take the opposite approach. First, we give the definition of the Levi form for the case of an abstract CR structure and then proceed to give more concrete representations of the Levi form in the case of an imbedded CR manifold. The Levi form for the case of a real hypersurface in is discussed in some detail. In particular, the relationship between the Levi form and the first fundamental form of a hypersurface is presented.
10.1
Definitions
One of the defining properties of an abstract CR structure (M, L) is that L is involutive (i.e., [L1, L2] E L whenever L1, L2 E L). The subbundle L L C Tc (M) is not necessarily involutive. In fact, the Levi form for M is defined so that it measures the degree to which L L fails to be involutive. For p E M, let
0 C —+
0 C}/(L,
be the natural projection map. DEFINITION I
L
156
The
Levi form at a point p E M is the map
in L that equals
at p.
157
DefinitIons
The vector field L] lies in Tc (M) since Tc (M) is involutive. So the Levi form measures the piece of (1/2i)[L, that lies "outside" of IL,, e The factor 1/2i is introduced to make the Levi form real valued, i.e., = In order to show that the Levi form is well defined, we must show that its definition is independent of the L-vector field extension of the vector
L and Z are two vector fields in L with
= Zr,, then
=
ir,,[t,
Fix p EM and let {L1,. . ,Lm} be a basis for L that is defined near p. For some unique collection of smooth functions at,... , am and b1,... , bm, PROOF
.
we have
L= near
p.
The assumption that = Z,, means that a3 (p) = bj (p) Expanding the Lie bracket, we obtain
for 1 j
m.
[t,L] = =
a,ak[L3, Lk] mod
(LeE).
j,k=I Therefore
= a
j,k=1 Since
a3(p) = b3(p), the proof of the lemma is complete.
I
1fF: M NisaCRmapbetweentheCRstructures (M,LM)and(N,LN), Therefore, F.(p) induces a then F.(p) maps (LM eLM)P to (LN map on the quotient spaces
{Tp(M)®C}/(LM
{TF(p)(N)®C}/(LN
The Levi Form
158
LEMMA 2
Suppose (M, LM) and (N, LN) are CR structures and let £M and £N be their respective Levi forms. If F: M —' N is a CR d(ffeomorphism, then for p E M
Ft'I\l ° ,M_,N 1' — as maps from {LM
0
L'f
L
to {Tp(p)(N) ®
LN)F(p).
PROOF The proof follows from the definitions and the observation that I = say a CR structure (M, L) is Levi flat if the Levi form of M vanishes Im z = O}. at each point in M. For example, let M = {(z, w) C x Since A (global) basis for L = H"°(M) is given by 0/Owi,... = 0, M is Levi flat. Also note that M is foliated by the We
complex manifolds
for xER. The complexified tangent bundle of each following more general result.
is given by L
We have the
THEOREM 1
Suppose (M, L) is a Levi flat CR structure. Then M is locally foliated by complex manifolds whose complexified tangent bundle is given by L e L.
PROOF The idea of the proof is to show that L L and its underlying real bundle are involutive. Then the foliation is obtained by the real Frobenius theorem. The Newlander—Nirenberg theorem will then be used to show that the — submanifolds in the foliation are complex manifolds.
L and L are involutive by the definition of a CR manifold, L ® L is involutive if and only if [L1, L2} is an element of L e L whenever L1 and L2 and belong to L Since Mis Levi flat, [T1,L1], +Z2,L1 +L21 belong to L e E. After expanding [7i + L2, L1 + L2], we see that Since
A
belongs to LeL. A similar computation involving that the vector field
+iL2, L1 +iL2} implies
B =[7.1,L2]+[i.1,L2} also belongs to L E. Adding A and B, we see that [L1, L2] belongs to L E and so L L is involutive. The underlying real bundle for L L is the space
H(M)={L+T; L€L}.
The Levi form for an imbedded CR manifold
159
is involutive, H(M) is an involutive real subbundle of T(M). The Since real Frobenius theorem (Section 4.1) implies that M is foliated by submanifolds, {M'}. such that for each p E M'.The complexified tangent Li,. Since L is involutive, space for M' at p is given by 0C= (M', L) forms an involutive almost complex structure. By the Newlander— Nirenberg theorem, there is a complex manifold structure for M' so that the is L This completes the proof of the theorem. resulting The reader should note that if M is a Levi flat CR submanifold of C1', then the easier imbedded version of the Newlander—Nirenberg theorem (Theorem 2
in Section 4.3) can be used to show that each leaf of the foliation, M', is a complex submanifold of C1'.
10.2
I
The Levi form for an imbedded CR manifold
Computations with the Levi form are facilitated by identifying the quotient space This is accomplished with a subspace of
by choosing a metric for Tc (M) and then by identifying the quotient space ® L1, with the orthogonal complement of IL,, L, (denoted ® in Section 8.2).
For an imbedded CR manifold M, a natural metric exists — namely the restriction of the Euclidean metric on TC (C') to Tc (M). In this case, L = H"°(M), L = H°"(M) and the quotient space
is identified with the complexified totally real part of the tangent bundle. As = and so mentioned earlier, the Levi form is real valued, i.e., which is the totally real part the image of the Levi form is contained in of the real tangent space of M at p. With this identification, the Levi form of a CR submanifold, M, at a point p E M is the map —*
given by
L(
\l
p) —
'-'p
1,0 p
is the orthogonal projection and where L is any H"° (M)-vector field extension of the vector Sometimes, it is convenient to think of the Levi form of an imbedded CR which manifold as a map into the normal space of M at p. denoted where irk:
The Levi Form
160
is the orthogonal complement of in composing £,, with J and then projecting onto
This is accomplished by Let
be the orthogonal projection map. The extrinsic Levi form of Al at p is the map given by *, o J o 4).
DEFINITION I
Since
are J-invariant, we have
and
= where
L is any H"°(M)-vector field extension of 4,.
Now, we develop a formula for the Levi form in terms of the complex hessian of a set of defining functions for M. THEOREM 1
Suppose M = .. = Pd(() = O} isa smooth CR subman(fold E with 1 d n. Let p be a point in M and suppose {Vp1 (p) of Then the extrinsic Levi form is Vpd(p)} is an orihonormal basis for given by
4(W) = for W =
Wk(o/ock)
Vp1(p)
E
PROOF We start with the definition of
= for W E H"°(M). Since {Vp1 (p),. . . , Vpd(p)} is an orthonormal basis for —+ N,(M) is given by the projection
= where ( , ) denotes the pairing between one-forms and vectors. So the £th component of 4) (Wy) is given by
J[W,
The Levi form for an i,nbedded CR manifold
Using
the dual of J, denoted J':
161
we obtain
—'
=
£th component of
1W,
Recall thatd=O+8; J*oe9=iO and J*oO=_ie9. Therefore =
£th component of
—
9)pt(p), [W,
(1)
Now we use the formula for the exterior derivative of a one-form in terms of its action on vectors (see Lemma 3 in Section 1.4). For a one-form and vector fields L1, L2, we have
AL2) =
—
L2
W E H"°(M). We have
=
—
W is of type (1,0) and Opt is a form of the bidegree (0,1), the second term on the right vanishes. The first term on the right also vanishes in view of Lemma 2 in Section 7.1. Therefore, we have L2) = 0. Similarly, we have L1) = 0. Equation (2) becomes Since
W) =
—
—((a — b)p1, 1W, W)).
Comparing this with (1) yields
=
£th component of
=
—
(since d=O+O)
= for
=
wk(O/O(k). The proof of the theorem is now complete.
I
Theorem I is often used in conjunction with Lemma 1 in Section 7.2. In that
lemma, the point p M is the origin and coordinates ç = (z = x + iy, w) E Cd x are chosen so that the defining functions for M are pt(z,w) = — h1(x, w), 1 £ d, with ht(0) = 0 and Dht(0) = 0. Note that Vpt(0) = and so {Vp1 (0),. . , Vpd(0)} is an orthonormal basis for No(M). We identify No(M) with via the map y = (y',... ,yd) i—' E .
The Levi Form
162
with Cn_d by the map
We also identify
n-d t9Wk
k=I
With these identifications, the restriction of the action of the complex hessian of
is the same as the action of the (w, Pt to the directions in of pi on d• We obtain the following corollary. COROLLARY I
is smooth and h(O) = 0, Dh(O) = 0. Then the extrinsic Levi form a: 0 is given by
n—d
lit
= 4i
j,k=I
for W = (w1,.
ÔWJthZ'k
W3Wk
E
This corollary can also be established by expanding ILk, L31 where L1,..., Lnd is the local basis for H"°(M) given in Theorem 3 in Section 7.2. Let us specialize to the case of a quadric submanifold Al = {y = q(w, ti)}, where q: Cn_d x Cd is a quadratic form (Definition 1 in Section 7.3). From Corollary 1, the Levi form of M at the origin is given by
k
In other words, the Levi form at the origin of a quadric submamfold is the associated quadratic form q (restricted to {(w, zn); w E Theorem 2 in Section 7.3, which describes a normal form for codimension two quadrics in C4, can now be interpreted in terms of the Levi form. In this
case, No(M) is a copy of R2 and Hd'°(M) is a copy of C2. In part (a) of that theorem, M = {y' = q1(w, ii)), Y2 = 0} and the image of the Levi form of M at 0 is contained in a one-dimensional line (the Yl axis) in R2. In all three cases in part (b), the image of the Levi form is a two-dimensional cone in R2. In case (i), M = {y' = 1W112,Y2 = 1w212} and the image of the Levi form is the closed
quadrant {yi O'Y2 0}. In case (ii), M = {y' =
!w112,y2
and the image of the Levi form is the open half space > 0} together with the origin. In case (iii), M = = Re(w1z12),y2 = and the image of the Levi form is all of R2 No(M). In all of these cases, the image of the Levi form is a convex cone in N0(M). In general, the image of the Levi
The Levi form of a real hypersurface
163
However, the image of the Levi form is not always form is a cone in convex, as we shall see in example (v) m Section 14.3.
103 The Levi form of a real hypersurface One of the best studied classes of CR manifolds is the class of real hypersurfaces
So we shall devote a section to the study of the Levi form for a real —+ JR is hypersurface in Let M = {z E = 0}, where p: smooth. If p is a point on M with IVp(p)I = 1, then from Theorem I in Section 10.2, the extrinsic Levi form is given by in
n
In this case, is isomorphic to wk(O/thk) a real line via the map t '—i tVp(p), t E JR. For this reason, Vp(p) is often for W =
dropped and the Levi form is then identified with the restriction of the complex hessian of p to The above formula requires JVp(p)I = which can always be arranged by multiplying p by a suitable scalar. However, it is important to note that if is 0 on M, then the map another defining equation for M with 1
for j,k=1
a nonzero multiple of the Levi form at p. To see this, first note that ,5 = ap —+ R, which is nonzero near M (see Lemma 3 for some smooth function a: in Section 2.2). Therefore is
n
-
O(J8(I,
n
= a(p)
+2Re
W3tiJk
{(E Op(P) )
+p(p)
(
(k
The third term on the right vanishes because p(p) = 0 for p E M. The second w,(a/th2) E = 0 for W = term also vanishes because H"°(M) (by Lemma 2 in Section 7.1). Therefore, the complex hessian of
The Levi Form
164
differs from the Levi applied to the vector W = E In particular, information about the Levi form form of M at p by the factor such as the number of nonzero eigenvalues can be determined by examining the complex hessian of any defining function for M. A special case worth examining occurs when the Levi form is definite.
A real hypersurface M is called strictly pseudoconvex at a point p EM the Levi form at p is either positive or negative definite, i.e., if there exists a defining function p for M so that DEFINITION 1
n
F-so-p (,
j,k=1
for all W =
(p)w,wk -
>0
w3(O/O(3) E
The above inequality is an open condition. Furthermore, it is invariant under a local biholomorphic change of coordinates. This follows by explicitly computing the complex hessian of poF where F is a biholomorphism or by using Lemma 2 in Section 10.1.
A real hypersurface M is called strictly pseudoconvex if M is strictly pseudoconvex at each point p E M. THEOREM 1
Suppose M C is a smooth real hypersurface that is strictly pseudoconvex at a point p E M. Then there is a biholomorphic map F defined on a neighborhood U of p in C'2 so that F{MflU} is a strictly convex hypersurface in F{U} C C'2.
PROOF The idea of the proof is to holomorphically change variables so that the real hessian of the defining function in the new variables is positive definite. First, wechoosecoordinates (z,w) CxC'2' asinLemma 1 inSection7.2 so that p is the origin and
M = {(z = x+iy,w) E Cx
= h(x,w)}
—i R is smooth and h(0) =0, Dh(0) =0. By Theorem 2 where h: R x in Section 7.2 (with k = 2), we may assume there are no second-order pure
terms in the expansion of h about the origin, i.e., 02h(0)
—
OWjOWk — ÔX,OWk
Let
—0 —
p(z,w) = y— h(x,w). We have
+ 0(3)
p(z, w) = y + j,k=1
Ic
The Levi form of a real hypersurface
165
where 0(3) denotes terms that vanish to third order in x and to (i.e., 10(3)1 C(1x13 + 1w13). We may assume the quadratic expression in w and tD is positive definite (the negative definite case is similar). Now, we modify p and make a holomorphic change of coordinates so that the quadratic piece of the defining function in the new coordinates is positive definite in z, 2 as well as w, ü). Let
,5=p+2p2. Note that
is
also a defining function for M. We have
w) = y + 2y2 +
WiZDk + 0(3)
()
=y— Re(z2)+1z12+
(1)
Define the following change of variables, (2, ti,) = F(z, w)
2=z—iz2,
zEC wE
F is a local biholomorphism which preserves the set {(O, w); w E
2=+
}.
If
then
Re(z2)
= 1z12 + 0(3).
Let M =F{M} and let tion forM is given by
(2)
(2, ti,)) = 15(z, w). A defining equa= 0. Using (1) and (2), we obtain n—i
5 is positive definite in 2, zli. Therefore, M = I {(2,t1);,3(2,tl,) = 0} is strictly convex in a neighborhood of the origin.
If the complex hessian of the defining equation of M is only positive semidefmite on for each p EM, then the hypersurface is called pseudoconvex. The analogue of Theorem 1 does not hold for pseudoconvex hypersurfaces. There is an example (see [KN]) of a real hypersurface that is strictly pseudoconvex everywhere except at one point p0 which is not biholomorphic near p,, to any (weakly) convex hypersurface in
166
The Levi Form
We conclude this chapter with the comparison of the Levi form of a real hypersurface in with its second fundamental form. Our presentation is similar to that in [Tail. First, we review the definition of the second fundamental
form. Suppose M is a real hypersurface in RN that locally separates RN in two open sets D and RN — D. Let N be the outward pointing unit normal vector field to D on M. We assume M is locally oriented according to N, which means that a collection of vectors Xi,..., XN_I in is considered positively oriented if X,,... , } has the same orientation as RN. Suppose W = w3 is a vector field on RN and let be an element of TP(RN). Define the vector by E
In other words, Vv9W is the derivative of W in the direction of and W are vector fields, then [V, = — Vw9V. DEFINITION 2 The second fundamental form is the map lIp: IR defined by
=
.
W,,
V
x
for
where (.) is the Euclidean inner product on RN.
In the next lemma, we derive a formula for the following defining function for M
p(x)
If M is
then p is
I —dist(x, M)
if x if x
= 1 dist(x, M)
in terms of the real hessian of
D RN
—
D.
near M in RN and Vp = N on M.
LEMMA 1 Let p M, and suppose V, and
W are
=
(a)
(b)
respectively. Then
and
If
=
and W,, =
wk(a/Oxk), then N
crp(p) OxOx j,k=1
Ic
V,Wk.
The Levi form of a real hypersurface
167
For part a), note that N• W = 0, since W E T(M) and N is the unit normal. From the product rule, we have PROOF
V{N.W} =(VvN)W+N.(VvW)
0
and part (a) follows. For (b), write N = Vp = >2(Op/Oxk)(ô/ôxk). Then
= Taking the inner product of this vector with
yields the formula in part (b).
Note that part (b) shows that 1I,,(.,.) is a symmetric bilinear form. To compare the Levi form with the second fundamental form, the first problem to overcome is that ii,, is defined on the real tangent space whereas is defined on which is a subspace of the complexified tangent space of Al. If then W,, X,, — iJX,, where X,, = is an element of + W,,) E C T,,(M). If X is a H(M)-vector field extension of X,,, then W X — iJX is a H"°(M)-vector field extension of We have
=
+iJX,X — = So for the purposes of comparing and let us identify — iJX,,. Then £, is identified with the map = for X,, E The projection —. is given by
with W,, = (V
.
Therefore
= —J[X,
N,.
The Lie bracket [X,JX] can be expressed as Vx(JX) — have J(Vx Y) = Vx JY by explicit computation. Therefore
= —J{Vx9(JX)
—
=
+
We also
.
(from Lemma 1).
Note that if
is a vector in the J-invariant subspace C T,(M) and therefore 119(JX9,
also belongs to We have established the following theorem.
then JX, is well defined.
The Levi Form
168
THEOREM 2
where
X,, E
is identified with form.
=
—
iJX,
E
(M) in the definition of the Levi
If M is convex, then the second fimdamental form is positive (or negaand tive) semidefinite. If = 0 then by Theorem 2, both and are null vectors must vanish. In this case, both (i.e., in the 0-eigenspace) for the second fundamental form. Therefore
=
Y,) = 0 for all
We have
0= (3)
for all Y,,
Now for any X,,
is
an element of
because
= 0. Likewise, we have = 0. This fact Therefore (3) implies is has the following geometric interpretation for a convex hypersurface: if a null vector for the Levi form then the derivatives of the unit normal vector field in the directions of X, and JX, both vanish.
11 The_Imbeddability of CR Manifolds
In Section 11.1, we show that any abstract real analytic CR structure is locally CR equivalent — via a real analytic CR diffeomorphism — to a generic, real analytic CR submanifold of C's. Nirenberg's C°° counterexample presented in Section 11.2 shows that without additional hypothesis, the corresponding theorem for C°° CR structures is false. Additional imbedding results will be discussed in Chapter 12.
11.1
The real analytic imbedding theorem
Recall that the CR codimension of M is the number d = dimc {Tc (M)/LeL}. then the CR codimension of M is the If M is a generic CR submanifold of same as the real codimension of M. To say that a CR structure (M, L) is real analytic means that M is a real analytic manifold and that L is a real analytic subbundle of Tc (M), i.e., L is locally generated by real analytic vector fields. Now we state the imbedding theorem, which first appeared in [AnHil]. THEOREM I
Suppose (M, L) is an abstract real analytic CR structure with CR codimension = d 1. Given any point p0 M, there is a neighborhood U of p0 in M so that (M fl U, L) is CR equivalent via a real analytic CR map to a generic real analytic CR subman(fold of complex Euclidean space with codimension d.
One of the defining properties of a CR structure (M, L) is that L is an involutive subbundle of TC (M). One might be tempted to think that this theorem should follow from the real Frobenius theorem (which does not require the real analyticity of L). However, the real Frobenius theorem requires the underlying real subbundle of L (i.e.,H(M) = {L + L; L L}) to be involutive. If H(M) is involutive, then L L is involutive. This is equivalent to saying that M is
169
The Imbed4ability of CR
170
Levi flat which is not assumed here. Instead, we shall use the real analyticity of M and L to complexify L and then use the complex analytic version of the Frobenius theorem. Suppose that m = dimc L = dmc E. Near p0 in M, L is generated PROOF by rn-real analytic vector fields {L1,. .. , Lm}. Since d is the CR codimension of (M,L), we have 2m + d = dinIcTC(M) = dilnR(M). Using a local real analytic coordinate system for M, we may assume that M is an open subset of R2m+d containing the origin and that each L3 is a real analytic vector field in 7'C (R2m+d). Denote the coordinates of We Write by (UI,.. , .
2rn+d
a real analytic, complex-valued function of u
Since
{Li,...,Lm} is linearly independent, the matrix (a3k(O)) I 5 m, 0. The variable w is the coordinate for H"°(M)
and we assign the weight 1 to w and The variable x is the coordinate for the totally real tangent space direction of M at the origin. Since this direction can be expressed as the projection of a Lie bracket at the origin of length m generated by Hc (M), we assign the weight m to the variable x. Likewise, we also assign the weight m to the normal variable y and to the complex coordinates z = x + iy and = x — iy. The weight of a monomial is by definition the number + + m(j + k). By definition, the weight of a smooth function is the minimal weight of all of the monomials appearing in its formal Taylor expansion about the origin. So the homogeneous polynomial p has weight m whereas e has weight greater than m. Note that in an unweighted sense, the polynomial p may vanish at the origin to higher order than does e. For example, suppose
h(x,w) = IwI2Re{w2} +
1w12x1.
Then the origin is a point of type 4 and the term e(x, w) = IwI2xi has weight 6.
In in unweighted sense, e vanishes at the origin to third order. The above discussion for hypersurfaces generalizes to submanifolds of of codimension d> I. In this case, we follow Bloom and Graham [BG] and say
Bloom-Graham normol form
181
a point p E M has type (mi,.
that
. ,
me,) (with
m3 mk for j < k) if the
following conditions hold:
(M) for j 0} and = {z E C'2;p(z) h(x,w)}). A negative eigenvalue of the Levi form means that M is locally concave down along one of the w-directions. In this case, CR functions holomorphically extend below M. If the Levi form has eigenvalues of opposite sign, then the origin is a saddle point for M and CR extension to both sides of M is possible. Since holomorphic functions are real analytic, part (c) of Theorem 1 implies the following regularity result for CR functions. THEOREM 2
of class Suppose M is a hypersurface in (3 k oo) and suppose p is a point in M where the Levi form has eigenvalues of opposite sign. Then each CR function on M that is a priori C' in a neighborhood of p must be of class in a neighborhood of p. If in addition M is real analytic, then an a priori C' CR function defined near p must be real analytic near p.
The Statement of the CR Extension Theorem
200
Theorems 1 and 2 are also valid for CR distributions. This will be briefly discussed in Chapter 17.
14.2
The CR extension theorem for higher codimenslon
The Levi form is also the key geometric object that governs CR extension from a CR submanifold of higher codimension. If M is a generic CR submanifold of real dimension 2n — d, I d n — 1, then the normal space of M at a point p E M (denoted is isomorphic to Rd with p as the origin in
We consider the extrinsic Levi form at p. 4: —i The definition of 4, along with a coordinate description of 4 in tenns of an appropriate system of defining functions, is given in Section 10.2. As with the hypersurface case, the image of the Levi form at p provides information about the second-order concavity of M near p. For p M, let
=
{the convex hull of the image of
C
is a cone, i.e., if v is an element F,,,, then Ày also belongs to for allA 0. If M is a real hypersurface, then R and is either {0} (if 0) or a ray (if 4 is positive or negative semidefinite) or all of R (if C,, has eigenvalues of opposite sign). The translation of Lewy's hypersurface theorem
into these terms is the following: if F,, is a ray, then CR extension is possible to one side of M; if F,,, is all of N,,(M) R then CR extension is possible to both sides of M. If d = codimaM is greater than one, then 4 is vector valued and so is more complicated. As we shall see in Theorem I below, F,, determines the shape and size of the open set to which CR functions holomorphically extend. To state the theorem, we need some additional notation. For two cones
and F2 in N,,(M) we say that r'1 is smaller than F2 (and write F1 < F2) if F1 fl S,, is a compact subset of the interior of {r2} fl S,,, where S,, is the unit sphere in N,,(M). For example, if the codimension of M is two, then N,,(M)
is a copy of R2. In this case, if F1 and F2 are convex cones with F1 < F2, then either F1 = F2 = N,,(M) or else F1 c F2 and the angle formed by the boundary rays for F1 is smaller than the corresponding angle for F2. Note that N,, (M) is always a smaller cone than itself. For denote the open ball in N,,(M) centered at the origin = p of 0, let
radiusf. FortwosetsAandBinC",weletA+B={a+b;aEAandbE B}. THEOREM 1
CR EXTENSION FOR HIGHER CODIMENSION (BPJ
k oo) with Let p be a poim in M so that F,, has
Suppose M is a generic, CR subman(fold of
dimp M = 2n
—
d,
1 d n—
1.
of class
The CR extension theorem for higher codiniension
'7,
17,
F1(;URE 14.1
respect to nonemptv interior ( ) j. Then fyi eveiv in .'t I and an open set !? in p in .1!. there is an opeii set (a)
p
E
(b) in
C
Ti
iu Ii that
:ti c
ope,i ColiC I < Fe,. f/ieee and an E > U so that
± (c)
of
toe cue/i CR Juintion
.1
qt c/usc (
function F defined on
a C(?nhiC'(tC'd li('i,'/iboIh()o(l
ot p
SB,) 5 .11 r/u're is a unique /10/uon uit/i 1and continuous on .1
on
case, the set depends only on and and not Part (b) of the theorem conveys the following on the CR function defined on = 2. picture of which we draw in the case il In Figure 14.1, the picture on the right is a side view with \i going into the fl B. is represented by the shaded region. The quantifiers imply that page. The closer F'1 gets to Fr,. the smaller and and F depend on the cone usually get as shown by the examples in the next section. By allowing and to approach we see that the tangent cone of at p is spanned by the real tangent space of .1/ at p. .11) because In Theorem 1, if V1,(\I ). then we may let fl contains an open set H, .V,,( .\I) is smaller than itself. In this case. As with the
The Statement of the CR Extension Theorem
202
Therefore, if = then which is an open neighborhood of p in each CR function near p is locally the restriction of a holomorphic function defined in a neighborhood of p. This is analogous to the two-sided CR extension result (part c) of Lewy's theorem for a hypersurface. THEOREM 2
oc, and Suppose M is a generic CR submanifold of of class CRC, 4 k suppose p is a point in M with = Then for each neighborhood w of p in M, there is an open set in with p E such that each CR function which is of class C' on is the restriction of a unique holomorphic function defined on Il.
Since holomorphic functions are real analytic and hence C°°, Theorem 2 implies the following regularity result for CR functions. THEOREM 3
(4 k oo) and Suppose M is a generic CR submanifold of of class = suppose p is a point in M with Then each CR function that is a priori C' in a neighborhood of p in M must be of class in a neighborhood of p. If in addition M is real analytic, then each CR function thai is a priori C' in a neighborhood of p must be real analytic in a neighborhood of p. Theorems 1, 2, and 3 hold for CR distributions as well as C'-CR functions. This will be discussed briefly at the end of Part III. is nonempty imIn Theorem I, the hypothesis that the interior of in poses some restrictions on the codimension of M. For example, if dimc
=
then the image of the Levi form is contained in a one (real) dimensional = 1, subspace of Therefore if codimRM 2 and if dime then the hypothesis of Theorems 1, 2, or 3 is never satisfied. By using the bilinearity and conjugate symmetry of the Levi form, it is an easy exercise to in show that if the interior of is nonempty, then m(m + 1) 2d where m = n—d = dime H"°(M) and d = codimRM. The reader should not get the impression that CR extension is impossible if m(m + 1) 4 as illustrated by the following example.
Example 5 Let (z1, z2, Z3, Z4, w1, w2) be the coordinates for C6. Let
M={Im zi=1w112, Im z2=1w212, Im Z3=
Im z4=
Here, M has codimension four in C6. The image of the Levi form at the origin
isthe cone
{YER4;yiO,y2O and This set is not convex and it has no interior in R4. However, its convex hull is
the set
{yER4;yiO,y2O and interior in R4, and so the CR extension theorem
which does have
applies to this example.
U
15 The Analytic Disc Technique
In this chapter, we present the proof of the CR extension theorem using the technique of analytic discs. The rough idea is the following. In Chapter 13, we showed (without any assumption on the Levi form) that a CR function on a CR submanifold M can be uniformly approximated on an open set w C M by a sequence of entire functions. To extend a given CR function to an open set in it is natural to try to show that this approximating sequence of entire functions is uniformly convergent on the compact subsets of Il. This can be accomplished by the use of analytic discs. Let D be the unit disc in C. An analytic disc is a continuous map A: D —÷ which is holomorphic on D. The boundary of the analytic disc A is by definition the restriction of A to the unit circle S' = OD. Often in the literature, the analytic disc and its boundary are identified with their images in Suppose that } is a sequence of entire functions that is uniformly convergent to a given CR function f on the open set w C M. Let us say we wish to show that {F3} also converges on an open set The idea behind analytic discs is to C show that each point in is contained in (the image of) an analytic disc whose boundary image is contained in w. From the maximum principle for analytic functions, the sequence of entire functions {F3 } must also converge uniformly
on ft So our CR extension theorem is reduced to a theorem about analytic discs, which we state in Section 15.1. In Section 15.2, this analytic disc theorem is established for hypersurfaces. The proof for hypersurfaces involves an easy slicing argument and thus we obtain an easy proof of Hans Lewy's original CR extension theorem. In Section 15.3. we prove the analytic disc theorem for
quadric submanifolds. The proof here is harder than for hypersurfaces but it is still relatively easy since the analytic discs can be explicitly described. The construction of analytic discs for the general case requires the solution of a nonlinear integral equation (Bishop's equation). This is discussed in Section 15.4.
In Section 15.5, we complete the proof of the analytic disc theorem for the general case.
206
Reduction to analytic discs
15.1
207
Reduction to analytic discs
The key result concerning analytic discs is the following. THEOREM 1
ANALYTIC DISCS
of class Ck, 4 k $ oc with n—i. Letpbeapoint inMsuchthatthe interior
Suppose M is a generic CR subman:fold of
dimaM=2n—d, I
is nonempry. Then for each neighborhood w of p in M and for of in each cone 1' < there is a neighborhood wr C w and a positive nwnber such that each point in WF + {1' fl B(r } is contained in the image of an analytic disc whose boundary image is contained in W. PROOF OF THE CR EXTENSION THEOREM FROM THE ANALYTIC DISC THEOREM
Suppose p e w C M is the given point in the CR extension theorem and let f be a CR function on the open set w. By Theorem 1 in Chapter 13, there is a sequence of entire functions F3, j = 1,2,... which converges to f on some open set W2 with p w2 C w C M. Now we apply the analytic disc theorem with ci.'
Let
fi
U
r 0 be given; there exist > 0 and a neighborhood W2 of p in M with W2 C such that if F is holomorphic
+ {r1 n Bj and continuous up to w1, then F is the uniform limit on fl of a sequence of entire functions n = 1,2 + By the identity theorem for holomorphic functions, it suffices to show the following: suppose F is holomorphic on + {F1 fl and continuous up to w1; if F = 0 on Wj, then F 0 on an open subset of wi + {I'1 fl So we can assume that the approximating sequence from the previous paragraph converges uniformly to zero on W2. From Theorem 1, it follows that there is such that each point in U is contained an open subset U of w2 + {ri fl in the image of an analytic disc whose boundary image is contained in w2. The maximum principle implies that converges to zero at each point in U. Therefore, F 0 on U, as desired. I
The Analytic Disc Technique
208
The proof of the uniqueness part of Theorem 2 in Section 14.2 is easier. Here, the open set contains an open subset of M. Therefore, uniqueness follows from Lemma 2 in Section 15.1.
15.2
Analytic discs for hypersurfaces
In this section, we prove the analytic disc theorem (and hence Lewy's CR extension theorem) for hypersurfaces. The proof is particularly simple in this case since we can obtain the analytic discs by an elementary slicing argument. Using Theorem 2 in Section 7.2, we can arrange coordinates so that the given point p E M is the origin and
M={(z=x+iy,w) where h: IR x —i R is of class C3 with no pure terms in its Taylor expansion through order 2. From a Taylor expansion of h about the origin, we have
h(x, w) =
qjkwjwk + 0(3)
n—d is the matrix for the Levi form = 1 5 j, of M at the origin. Here, 0(3) denotes terms depending on both w and x which vanish to third order at the origin. Since Q = is a Hermitian symmetric matrix, the w coordinates for can be chosen so that Qis diagonalized. This is accomplished by finding a unitary matrix U so that tuQu is diagonal and then letting ii) = U w. The hypersurface A! divides a neighborhood of the origin in into two where
sets
y>h(x,w)} y 0. From the Taylor expansion of h, we have
h(0,w1,0) = qiiIwiI2 + 0(Iwi Is). Let w be an open subset of M which contains the origin. Since
is positive, any small translate of the complex line {(O, w1, 0); w1 E C} in the positive y direction will intersect the open set in a simply connected open subset of this translated complex line whose boundary is contained in By continuity,
209
Analytic discs for hypersurfaces
y
.
.0
'WI'
FIGURE 15.1
the same can be said for small translations of this complex line in the x an directions. More precisely, there are positive numbers 6, > 0 w2 0 such that U: C'2 (S', C'2(S1,Rd) depends in a fashion on x E Rd and W E with lxi 0 and a map
A: {(t,x,w) E R" x
x
ti, xl, wi 0 there are constants t5 > 0,e' > 0 such that for xl, wi < 5 and
iti < €1
A(t,x,w)(( = 0) = (x,h(x,w),w) + + RN x Rd x Cn_d
where
Rd
is of class C3 and
ie(t,x, w)I S iiiti2. The sum of the first two terms on the right side of the expansion of A(t, x, w)(( = 0) is exactly the expression for the center of the analytic disc in the quadric
case (compare with (2) 15.3). This is not surprising since M can be approximated to third order at the origin by a quadric submanifold. However, the error term in the above lemma is not just any third-order error term. It is crucial for what follows that e vanish to second order in t with coefficients that are small in (x, w, t) (for example, we cannot allow a term such as to be part oLe). PROOF
Write G(t,x,w)(() = u(t,x,w)(() +
W(t,w)(( = 0) =
w,
u(t,x,w)(( = 0) =
Therefore
A(t,x,w)(( =
0)
= (x,v(t,x,w)(( = 0),w).
From Lemma 1 x.
The Analytic Dtsc Technique
224
It suffices to examine a Taylor expansion of v(t,x,w)(( = v(t,x,w)(() is harmonic in (, the mean value theorem yields
=0)
v(t, x,
0)
in t. Since
x,
Since the boundary of A is contained in M, we have h(u(t, x,
W(t,
v(t,x,w)(( =
0)
=
Therefore
=
(I)
x Note that W(t = 0,w)(() = w (a constant) and so u(t = is the unique (constant) solution to Bishop's equation in this case. Therefore,
we have
v(t = O,x, w)(( = 0) =
J h(x,w)dØ = h(x,w).
(2)
This is the constant term (in t) in the expansion of v(t,x,w)(( = 0). For the linear term, we differentiate (1) with respect to t; evaluate this at
t=0 and use the fact that u=x and W =w at t = 0. We obtain = 0,x,w)(( = 0) =
J2 Re To
save space, we have written
dØ.
for
and
Now (Ou/Ot,)(t = 0,x,w)(() is harmonic in (for (I 1 (since u = Re C). Furthermore, u(t,x,w)(( = 0) = x and so (Ou/ät3)(t,x,w)(( = 0) = 0. Therefore, the first integral on the right for
vanishes by the mean value theorem for harmonic functions. The same argument shows that the second integral on the right vanishes since w)(( = 0) = 0. Therefore, we have (3)
The proof of the analytic disc theorem for the general case
The second-order part of the Taylor expansion of v(t. x, obtained by differentiating the right side of (1). We have
jk
V
225
=
0)
in t is
(t=0,x,w)((=0)
h
= t.itk
+ e3k(t, x, w)
x, w) is a quadratic term in t whose coefficients involve the secondorder pure terms from the expansion of h, i.e., w), w)}, and (x, w)}. These pure terms vanish at x = 0, w = 0 by our choice of local coordinates (from Theorem 2 in Section 7.2). Therefore, e3k(t, x, w) can be absorbed into the error term, x. w), with the estimate stated in the conclusion of the lemma. The term (02h/OWa&G)3)(X,W) can be written where
02h(0,0)
+
(82h(x,w)
U2h(0,0)
The term in parentheses can be made as small as desired by suitably restricting x and w. Therefore
(t = 0,x,w)(ç = 0) = + where where
is the quadratic form that generates the Levi form at the origin and
provided (t, x, w) belongs to a suitably small neighborhood of the origin.
We have
ow
=
Substituting this into £ and integrating ç =
over
0 0 for
27r, we obtain
226
The Analytic Disc Technique
When j = k, we have (for 1(1 = 1)
=
=x2. Therefore, from (4), we have
t,tk
82V
—o
2
—
f w)
—
if j = k.
Since the third-order Taylor remainder in t can be absorbed into the error term, the proof of the lemma follows from (2), (3), and (5). I Let us summarize where we stand. We have shown (Lemma 1) that each
point in the set
Il = {A(x, w, t)(ç = 0); iti, lxi, wi belongs to the image of an analytic disc whose boundary is contained in the given
open set w for the analytic disc theorem. Furthermore, the Taylor expansion of t in Lemma 2. The constant term in this expansion A(x, w, is (x, h(x, w), w) and the set wr = {(x, h(x, w), w); lxi, wi < 6} is an open subset of M that contains the origin. Therefore, the proof of the analytic disc theorem will be complete once we show that for fixed x, w with xl, wi < the map t
f(x. w)(t)
tE
= parameterizes a set that contains an open neighborhood of the origin in the cone
ti < es', r (here, €' and are as in Lemma 2). Now the map t that parameterizes an open neighborhood of the origin of the cone XN. The hope is that since contains the cone I' by the choice of f(x, w)(t) will x, w)i is small relative to ti2 (Lemma 2), the image oft also contain the desired neighborhood of F.
To carry out the details, we replace t, by
for t, 0. We define two
maps
and
F(x, w)(tj,
.
. . ,
tN) = f(x,
. .
.
,
E(ti,...,tpj) +e(t,x,w) where the error term e(t, x. w) =
x, w) is continuous (but not
The proof of the analytic disc theorem for the general case
227
differentiable at t = 0) and satisfies the estimate
ie(t,x,w)i provided lxi, iwl 0 are given. Then there exist Rd is a continuous map with iF(t)—E(t)i '1 it! 17, e > 0 such that F : Rd for t = (t1,. .. , td) with t3 0, 1 j < d, and < €', then the image of
{t=(tl,...,td);t3Oiti<e'}underFcontainsBEflr.
PROOF
In our context, F is differentiable except at t = 0. We could use
the inverse function theorem to examine the images of little balls which are contained in the set rx, Xa• However, the proof of the above lemma does not require any differentiability assumptions on F. So we offer this purely topological proof. We may assume that the vectors X1,. . . , Xd are the standard basis vectors in Rd. Therefore, we have rXI...Xd = {(x1,. . . , xd); x3 0). Given a cone such that if > 0 and there exists 0 < < rXl...Xd there exists x e r n BE then the Euclidean distance from x to O{rXl...Xd fl B4 is greater than ii'ixi. Since X1,. . . , Xd are the standard basis vectors, F is just the identity map. e > 0 are chosen small relative to and Suppose it — F(t)i S iiiti. If €', then the line segment between t E O{rXl,..Xd fl and F(t) does not intersect (see Figure 15.2). Now suppose the point XE rflBE is not in the image of FXI...Xd fl under F. By the above discussion, the restrictions of F and the identity map to O{rX,...Xd fl are homotopic in Rd — {x}. Since fl is homeomorphic to a (d — 1)-dimensional sphere in Rd which encloses x, the homology class of the image of O{rXI.,.Xd fl } under F is nontrivial in the (d — 1)st dimensional homology group of Rd — {x). On the other hand, rXI...Xd fl BE' is contractible to the origin in Rd. Since F
r
228
The Analytic Disc Technique
t2
image of fl
under F t
€
FIGURE 15.2
is continuous and x is not in the image of F, the image of O{FxI...xd fl under F is also contractible in — {x}. This means that the homology class of F {O{['x..xd fl is trivial in the (d— l)st dimensional homology group of — {x}. This contradiction proves the lemma. I stated earlier. Lemmas 2 and 3 complete the proof of the analytic disc theorem, which in turn completes the proof of the CR extension theorem. As
16 The Fourier Transform Technique
In this chapter we present a Fourier transform approach to the proof of the CR
extension theorem. This technique has the advantage in that it can be more easily adapted to the holomorphic extension of CR distributions, which will be discussed at the end of Part ifi. However, the goal of this chapter is to introduce the technique rather than to prove the most general theorem. Therefore, to avoid some of the cumbersome technicalities of more general results, we shall assume where d = codimR M the given CR function is sufficiently smooth (class will suffice). In addition, we shall assume that the submanifold M is rigid,
which means that near a given point p E M there is a local biholomorphic change of coordinates so that p is the origin and
M = {(z = x+iy,w) E Cd
x Cfl_d;
= h(w)}
with h(O) = 0 and Dh(0) = 0. where h: Cn_d is smooth (say class The point is that the graphing function, h. for a rigid submanifold is independent of the variable x E Rd. The modifications required to handle the more general case where h depends on both x and w will be mentioned at the end of Part ifi. The basic idea of this technique is to use the approximation theorem given in Chapter 13 to derive a modified Fourier inversion formula for smooth functions on M. We will then show that this modified inverse Fourier transform of a given CR function is the restriction of a holomorphic function provided the modified Fourier transform of the CR function is exponentially decreasing. Finally, we show that the set of directions in which the modified Fourier transform of a CR function is exponentially decreasing is related to the convex hull of the image of the Levi form and the CR extension theorem will follow. The ideas presented in this chapter are due to Baouendi, Treves, Rothschild, Sjostrand, et al. (see [BCT], [BRT], [BR2], [Sj]). Our presentation is closest to that in [BRT].
229
The Fourier Transform Technique
230
16.1
A Fourier inversion formula
Our desired Fourier inversion formula will be derived after we present three lemmas. The first of these lemmas is analogous to Lemma 1 in Chapter 13 for the approximation theorem. Instead of an integral over totally real n-dimensional
slices of M as in Chapter 13, we integrate over the following d-dimensional slices. For p = (z, w) E Al let
=
M,,
= {(z',w) E JVI;z' E
Our analysis will be a local one about the origin. So as in Chapter 13, we assume the graphing function h: Cn_d ...+ is suitably cutoff so that < 1/2 on LEMMA I
and Suppose U1 and Uj' are neighborhoods of the origin in C" with U1' Cc Let g be an element of suppose U2 is a neighborhood of the origin in 1 on Uj'. Suppose f: {U1 x U2} fl M —* C is a continuous V(U1) with g function. Then for (z, w) E {U1' x U2} fl M,
f(z, w) = urn
f
g(z')f(z',
z' E
+ and [z — z'12 = (z1 — Moreover, this limit is uniform in (z, w) E {Uj x U2} fl i'vf. where
dz' =
A ...
A
...
+
—
Note the function f is not assumed to be a CR function. We will not assume f is CR until the next section. The purpose of the cutoff function g is to ensure that the integrand has compact support. The reason we do not assume f has
compact support is that we will apply this result in the next section to CR functions, which typically do not have compact support.
PROOF The proof is the same as the proof of Lemma 1 in Chapter 13. The only difference is the dimension of the slice of M over which we are integrating. This changes the constants involved but otherwise has no effect on the proof. Also note that c here plays the role of in Chapter 13. I LEMMA 2 Suppose U1,
U2 andg,f areas in Lemma!. Thenfor(z,w) E
f f where
the limit is uniform for (z, w) E {U( x U2} fl A'!.
A Fourier inversion formula
w=
Here, w u = PROOF
231
= (ul,...,nd) E
(w1
Cd.
by 4f in the statement of Lemma 1. We obtain
Let us replace
f(z,w) =
J
z' E
Lemma 2 will follow once we have established the following identity:
= (f)d/2
f
(1)
To show (1), first note that if w is a vector in Rd (rather than C'), then by a translation, we have
J
=
f
(2)
The left side of this equation is an entire function of w E C' which agrees with the right side when w E Rd. Therefore, equation (2) holds for all w E C' by the identity theorem for holomorphic functions. Now we complete the square in the exponent to obtain
=
J (by letting w =
—
z') in (2))
= The last equality follows from a standard polar coordinate calculation. Multiplying this equation by yields (1) and so the proof of the
lemma is complete. For a continuous E 1W', define
I
function f: M
=
J z' E
C
and for z E Cd, w E
dz'.
and
The Fourier Transform Technique
232
Lemma 2 can now be rewritten
f(z,w) =
urn J
We will show in Lemma 3 below that if f is of class +
on M,
for some uniform constant C. Since E Rd. the dominated convergence theorem
+ is integrable in
then implies
f(z, w)
=
f If(z, w,
for (z, w)
x U2} n M.
eERd
This is analogous to the standard Fourier inversion formula for Euclidean space.
The only difference is that If involves an integral over a slice of M (rather than Euclidean space). Also, it is customary in Euclidean space not to include in the integrand of the Fourier transform as we have done in If. the factor would then reappear in the Fourier inversion formula. The factor For technical reasons which will become clear, we wish to modify the above — z']2 in the exponent. Fourier transform to one with a term of the form Rd define For z E Cd, wE Cn_d, and
If(z, w,
f g(z')f(z',
— z',
z' E
= (1 + i(u .
where
for
E Cd.
LEMMA 3 There exist neighborhoods
and Uj' of the origin in Cd with U CC Uj and a neighborhood (12 of the origin in Cm_d such that if g D(U1) with g = 1 on U, then the following holds. (a)
1ff: Mn {Ui x U2} —'
C
is afunction of class Ca', then there is a
constant C > 0 such that + I)_(d+1) for all
I(If)(z,w,e)I
+ 1)_(d+I)
forall
E Cd
E Rd
and for
(z,w) = (x+ih(w),w) E Mn {Uj' x U2}. (b) More generally, if f: Mfl{U1 xU2} —+ C isafunction of class CN,N> is a jth order derivative (0 j N) in on Ctm, 0, and if then there is a constant C > 0 such that +
for alle E Rd andw E U2,z —x+i(h(w)+v) E Uj' (here v
R").
A Fourier inversion formula
233
We will need the above estimate in part (a) on lf(z, 1L', for E Cd (rather than just e E Rd) because in the proof of the next theorem, we will transform the k-integral over IRd in the Fourier inversion formula for If into an integral over a contour in C" which will yield a Fourier inversion formula for If. In the next section, we will need the more general result in (b), which estimates the derivatives of If for (z, w) in a -neighborhood of the origin.
The proof of the estimate for If in part (a) is a special case of part (b) d + 1, and v = 0). Therefore, we first prove the estimate 0, N l}. So it in (b). This estimate clearly holds on the compact set E suffices to show PROOF
(with j =
1D3{(If)(z,w,e)}I
for
1.
For fixed w E Cn_d the set is parameterized by the map z' = x'+ih(w) for x' For z = x + i(h(w) + v) with v E Rd. we have
(If)(z,w,e) =
f
— x' +
x' ERd
= g(x' + ih(w)) and J(x',w) = f(x' + ih(w), w). If jth order derivative in z, w, ü), then where
{(If)(z, w,
=
f
is a
z,
(3)
Gk (f) is an expression involving kth order derivatives of Moreover, Gk(f)(x', z, has compact x'-support and is a polynomial in and where
of degree j — k (as a result of the derivatives of order j — k of the exponential term).
Now the idea is to integrate by parts with the vector field
As we will see, each integration by parts will yield a factor of this procedure N — k times will yield the lemma. To carry out the details, note
L{ie. (x — x' + iv) — IeJ[x — x' + iv]2} = where
+
x',
Iterating
The Fourier Transform Technique
234
The term
is
homogeneous of degree one in
and satisfies the estimate
+ Ix'I + for some uniform constant C > 0. Choose neighborhoods U1 C Cd and U2 c
of the origin so that if w E U2 and if z = x + i(h(w) + v) and z' =
Cn—d
x' + ih(w) belong to Ui, then
We have (4)
U cc U1
Let
g = 1 on
be a neighborhood of the origin in Cd. Let g e D(U1) with
We have L1
(x—x +zv)—feI[x—z +ivj 2
=
+ 7j(x, x',
(5)
Substituting the left side for the right and integrating by parts, we obtain
f
z,
x' ERd
=
—
I
J
LI
Gk(f)(x', z,
1.
I iei
J
Using (4) together with the fact that Gk(f)(x', z, and ofdegreej — k, we have
Gk(f)(x',z,w)(e)
1
—
—
C is some uniform constant. Since L is a differential operator in x' whose coefficients are homogeneous of degree zero in we have where
s
for
1.
is homogeneous of degree j — k in whereas that IGk(f)(x', z, If f is of class CN, the above term is homogeneous of degree j — k — in then Gk(f) is of class and we may iterate the above procedure starting
Recall
1
A Fourier inversion fonnula
235
with (5) N — k times to obtain
J G,,(f)(x', XIERd
=
f
z,
(6)
x' ER"
is homogeneous of degree j — N in
where
for where C is a constant independent of w
Therefore
1
U2 and z = x + i(h(w) + v)
(7)
U.
For the exponential tenn, we have I=
I
eRe{1 (xx'-4-:v)
= < This estimate together with (6), (7), and (3) yield part (b) of the lemma. As already mentioned, the estimate on If given in part (a) is a special case of part (b). The estimate on If in part (a) is proved in a similar manner. In fact, this estimate is easier since the exponent occurring in If is simpler. Therefore, it will be left to the reader. I We
remark that the estimate in part (b) also holds for e in a
neighborhood of Rd. We now state and prove the Fourier inversion formula for the transform If. THEOREM 1
There exist neighborhoods U; cc U1 of the origin in and a neighborhood 1 on u; and ('2 of the origin in such that if 9 E 'D(Ui) with g
f:
M —* C is of class Cd+ 1, then
f(z,w)
E
{U x U2}nM.
=
J
The Fourier Transform Technique
236
PROOF From Lemma 2 and Lemma 3 part (a) and the dominated convergence theorem, we have
f(z,w)
f
=
R
=
urn
R—.oo
R
f f (?f)(z, w,
-R
..
(8)
.
—R
For (z, w) e M near the origin, we write z = x + ih(w) for some x E Rd. The slice is parameterized by z' = x' + ih(w), for x' e Rd. Therefore
=
J
x' ERd
where as before w) = g(x' + ih(w)) and f(x', w) = f(x' + ih(w), w). Note that is an entire function of e E C" (since g has compact x'-support). We can use Cauchy's theorem to change each in (8) to an integral over the contour — e R} in the complex = + plane (1 = R} appearing j d). The integral over the side contours in the change of contour process disappear as R —' 00, because the measure whereas the integrand is O(R_(d÷l)) by the of these side contours is
estimate on If in Lemma 3. By replacing I
by
+
—
x') in the
exponential term in If, we obtain
=
(9)
which is the exponential term in the definition of If. Moreover, we have
d
iC(x—x') '
=
—
x',e)de.
)deiAdei+IA...Aded
('°)
237
The hypoanoiytic wave front set
From (8) and Cauchy's theorem, we obtain
f(z, w) = lim J 7 If(z, w, =
iimf
7 I
R-.ocJI
= urn
=
(with
=+
- x'))
(by (9) and (10))
J
J
This completes the proof of Theorem 1.
I
The hypoanalytic wave front set
16.2
To summarize our progress so far, we have shown in the last section (Theorem 1) that
f(z,w)= f
for
(z,w)EMneartheorigin
(1)
where
(If)(z, w,
1ff:
=
Al —+C is of class + l)_(d+1) for
f g(z')f(z',
— z',
then we have shown (Lemma 3) that I(If)(z, w, E
and (z,w) €M and hence the integral in (1) is
well defined. Note from its definition that (If)(z, w, is analytic in z ECd. If (If)(z, w, is exponentially decreasing in E Rd, then the right side of (1) also defines an
analytic function of z E Cd. Later, we will see that if f is a CR function near the origin on M, then the right side of (1) also defines an analytic function of w E Cn_d near the origin. In this case, (1) shows that f is the restriction of an ambiently defined holomorphic function. All of this is to serve as motivation for examining the set of vectors e E Rd in which (If)(z, w. is exponentially decreasing. Roughly speaking this is the complement of the hypoanalytic wave
238
The Fourier Transform Technique
front set of the function f at the origin. More precisely, we make the following definition. Fix a smooth function g: Cd —p
JR
with compact support which is identically
one on a neighborhood of the origin.
Let F be a set of continuous functions on M. A vector is not in the hypoanalytic wave front set of F at the origin if there exist DEFINITION 1
E Rd
a cone F in JRd containing (b) a neighborhood U of the origin in (c) a constant 0 (a)
such that if f belongs to F, then there is a constant C > 0 such that
0 both depend on Fi. If r1 0 and an open cone F which is slightly larger than F (i.e., F 0
Cd x
for
(z,w) E U1 x
U2,
E Rd — F.
(5)
is holomorphic in z E Cd, an application of Morea's theorem
Since
and Fubini's theorem shows that F2(z, w) is holomorphic in z for (z, w) E U x U2. Next, we show that F1 (z, w) is holomorphic in z E Cd for (z, w) e + n where > 0 have yet to be chosen. F1 WI and { } Let w1 = x (J2} n M. Points in the set w1 + F1 are the form (z, u')
(x + i(h(w) + v), w)
where
(x+ih(w),w)Ewi
and
vEF1.
From part (b) of Lemma 3 from the previous section with N = d + 2 and
j=
1, we have
w,
D
is a uniform constant
that is independent of z = x + i(h(w) + v) E Uj' with V E F1 and w E U2. Using (3), this estimate becomes
ID{(If)(z, w, for
E r and V E F1.
+1
The hypoanalyac wave front set
= Let we have lvi
241
>0. For (z,w) = (x+ih(w),w)+(iv,0) 0. Now the idea is to deform the domain of integration in If using Cauchy's theorem and then estimate the resulting integrand. To carry out the details, we 36/4 and choose g V(Cd) such that if ivl 0 such that
PROOF
in
I(If)(z,w,e)l for all (z,w) EU and E F. with {(O,w);w Cn_d} and No(M) with be the extrinsic Levi form of Mat 0.
As usual, we identify {(O,y,O);y E W'}. Let We have
4(w) =
Wj'tt'k
j,k=I
k
for
w = (w1,. .
.
,wnd) E
The hypoanalytic wave front set and the Levi form
By definition, the cone F0 is the convex Since F, is closed and F° =
cone,
245
(in Rd) of the image of The I', there must exist a vector v0 F0
hull
with
v that lie in the image of then the same inequality holds for all v in the convex hull of the image of Therefore, we may assume (1) holds for some vector v0 that lies in the image of By a complex linear change of coordinates in the w-variables, we may assume v
V0
= A(ei)
where
=
(1,0,... ,0) E Ca"
Now we examine the second-order Taylor expansion of h about the origin. We may assume there are no second-order pure terms in this expansion (Theorem 2
in Section 7.2). Therefore
h(w) = £o(w) + 0(3) where 0(3) involves terms that are third order in w and th. Let V)1 = (w2,.. ,Wn..d) e We have
=
h(w)
= Since
a>
0
0 small enough so that (—a/2)r2 + 4C2r4 0 small (depending on r) so that
+ 4C2r4 + [C6r2 + C63r3 + S + Denote the number on the left by —f (with c > 0). We obtain for It follows that (3)
fore EF and (z,w) E with IzI 0 are chosen suitably small as above. Since G(f)(z, w, C) is analytic in (z, w), the maximum principle implies that the above inequality also holds for IwiI < r, and Izi < 6, w'I < 6r. This together with another application of the lemma yields the estimate
forCE F and (z,w) E with lzI 0,
zECd.
is an entire function of
The first change needed for the nonrigid case is to rewrite FE so that it involves
an integral over a carefully chosen contour in Cd. For z = x + iy E Cd and w
(Cm_d let
Since If(z, w,
be the contour parameterized by the map
is an entire function of
we
can use Cauchy's theorem to
show
FE(z,w) =
J
is close to the copy of Rd Note that since Oh(0, 0)/Ox = 0, the contour = 0}. given by For the same reason, we have
for •
11
.Oh(x,w) Ox
provided
_i\t )
(x, w) is sufficiently close to the origin. The presence of the term
= r} appearing in means that the mtegrals over the side contours the change of contour process disappear as R oo. The reason we need this change of contour will become clear at a later time. Note in the rigid case, nothing has changed (since Oh(x, w)/Ox = 0). Now, we make the analogous change of contour as in the proof of Theorem 1 in Section 16.1. That is, we set
where we have used the notation
The Fourier integral approach in the nonrigid case
253
is well defined and holomorphic in E Cd for provided The expression > After dropping the hat and changing the contour of integration, as done in the proof of Theorem 1 in Section 16.1, we obtain
F€(z,w)
If(z, w,
= (27r)_d
= + presence of the term
=
f
f g(z')f(z'
—
z',
z'), we can restrict z and z' so that The means that the integrals over the side contours appearing in the change of contour process disappear ascc. 11 R} {Im As mentioned earlier, FE(z, w) is holomorphic in z E Cd and f on M as Since
—
I
0.
The definition of the hypoanalytic wave front set given in Definition 1 in Section 16.2 is unchanged for the nonrigid case except that the cone F must be a conical neighborhood of in Cd rather than an Rd.conical neighborhood as stated. The reason for this is that If(z, w, involves an integral over which is contained in Cd rather than an integral over Rd as in the rigid case. Theorem 1 in Section 16.2 holds without change in the nonrigid case. The basic ideas of the proof are similar to the ideas given in the rigid case except one must use the new transform If(z, w, defined above. For example, let us outline the key steps in the proof of the (a) implies (b) part of this theorem for the nonrigid case. Here, we are assuming that the hypoanalytic wave front set of a given CR function f is contained in a cone F and we are to show that f holomorphically extends to a set of the form w+{Fi } where F1 is a smaller subcone of the polar of F. As in the proof for the rigid case, the key idea is to estimate the real part of the exponent q(z, z', appearing in the definition of If(z, w, where z' = x' + ih(x', w) z = x + i(h(x, w) + v) (where v belongs to F1) and = ([1+ To carry out E this estimate, we shall require that there are no second-order pure terms in the Taylor expansion of h. In particular, we need for
We have z—
=x
—
x' + i(h(x, w) — h(x', w) + v)
= I+z.Oh(x,w)]j(x—x)+o(Ix—xI'2 )+iv. /
The notation o(t') for j 0 indicates terms that are small in absolute value
Further Results
254
Taking the Euclidean dot product of z — z' with
when compared to
([I + i(ah(x,
=
we obtain
(C) E
+o(Ix_x'l2)ICI+O(Ixl+ lwI)IvIICl+O(lvl2lCI). Therefore
Re{i(z — z')
—
—
0) about any given point p M such that for each f E
f
with OMf = 0 on WA there is an element u with OMU = on In all three classes of complexes, the answer to either the global or the local question does not automatically yield the answer to the other. For example, we cannot use the local solvability of 0M together with a partition of unity argument to obtain the global solvability of '9M because the product of a 0M -closed form with a smooth cutoff function is typically no longer t9M -closed. For the same reason, we cannot deduce local solvability from global solvability. For the exterior derivative and the Cauchy—Riemann operators, the answer to the solvability question is well understood. Both complexes can always be solved locally. The global solvability of the exterior derivative depends on global topological conditions on the manifold (the cohomology groups). From the theory of several complex variables, the global solvability of the Cauchy—Riemann
263
264
Solvability of the Tangential Cauchy—Riemann Complex
operator depends on the holomorphic convexity of the complex manifold. Much less is known about the global and local solvability of the tangential Cauchy—Riemann operator on a CR manifold Al. However, the theory is pretty well understood if M is a strictly pseudoconvex hypersurface in and this is the subject of much of Part IV of this book. We show that the tangential Cauchy— Riemann complex is both locally and globally solvable except at top degree. For 1 q < n — 1. the equation 8M = f for f E ( M) is overdetermined (more equations than unknown functions), and the condition = 0 is the correct compatibility condition required for both local and global solvability. If q = n — (top degree), the equation u = is no longer overdetermined.
f
1
E e°"'(M) then the equation
In fact, if u E
I
= consists of only one first-order partial differential equation. Based upon one's experience with d and 0, one would expect that the tangential Cauchy—Riemann
complex should at least be locally solvable at top degree. In fact, if Al and f are real analytic then local solvability follows from the Cauchy—Kowalevsky theorem. The surprise (provided by Hans Lewy's nonsolvability example [L2]) is that local solvability does not necessarily hold if f is only smooth. Adding the condition OM = 0 does not help matters since this condition always holds if f has top degree. Instead, there are other criterion for both local and global solvability at top degree. which are discussed toward the end of Part IV. We employ the integral kernel approach of Henkin [Hell, [He21 for the so-
f
lution of the tangential Cauchy—Riemann complex. We shall not discuss the £2 technique of Horrnander, Kohn et al. because there are ample references for this approach (see Folland and Kohn's book [FK]). Aside from the aesthetic appeal of exhibiting an explicit integral kernel formula for the solution to the tangential Cauchy—Riemann equations, the kernel approach makes estimating the solution rather easy (at least in the global case).
Part IV is organized as follows. In Chapter 18, we introduce the calculus of kernels and we define the concept of a fundamental solution for d, 3, and Various fundamental solutions for d and 3 are discussed in Chapter 19. As an application we prove Bochner's global CR extension theorem In Chapter 20, [Boc] for the boundary of a smooth bounded domain in we introduce Henkin's kernels, which along with the fundamental solution for O form the building blocks for the integral kernel solution to the tangential Cauchy—Riemann operator. Instead of using Henkin's notation, we use a more streamlined notation due to Harvey and Polking [HP]. Two global fundamenare introduced in Chapter 21 along with a criterion for tal solutions to global solvability at top degree. In Chapter 22, one of these global fundamental
solutions is modified to yield a solution to the local problem. Hans Lewy's local nonsolvability example is given in Chapter. along with a more general criterion of Henkin's [He2) for local solvability.
18 Kernel Calculus
Our presentation in this chapter closely follows Section 1 in [HP].
18.1
Definitions
To motivate our discussion of kernels, let K be a smooth form on RN' x RN
N+N'). We can view K as an operator from VP(R")
of degree q (O q
by defining
to
f
XERN.
yEW"
Here, it is understood that the only contributing term to this integral is the containing dy = dy1 A ... A dyN'. The dx's component of K(x, y) A appearing in this component are moved to the right (outside the integral) and then the y and dy are integrated. Since the degree of K(y, x) A is p + q, (Kcb)(x) is a differential form in x of degree q + p — N'. Recall (see Chapter 6) that as a current on RN, the form K(q5) acts on elements (with s = N + N' — (q +p)) by integration. So if E D3(RN), then
of
=
j(j VERN'
= The term (K,
0
XRN
is
well defined for any current K E x We can also replace RN and RN'
RN) and any q5 E VP(RN'), with oriented manifolds X and V of dimensions N and N', respectively. In this case, the manifold Y x X has an orientation induced on it by Y and X.
265
Kernel Calculus
266
= (xii... ,xN): U —+ RN are orientation-preserving coordinate charts for X and Y, respectively, then dy1 A...AdYN' Ads1 A...AdXN determines an orientation for Y xX in the RN' and
That is, if Y = (yi,... ,yN'): V
sense described in Section 2.5. The above discussion motivates the following definition.
(Y x X) is called a kernel of degree q on A current K in 1yq+p.- N (X) Y x X. This kernel can be regarded as an operator K: VP(Y)
DEFINITION I by
for
E VP(Y) and
Note that if
E 1)8(X) with 8 = N + N' — (q + p).
—p
in Vt(YxX); therefore,
in
This shows that
is a well-defined current
on X. If q = N' + r, then we say that K is a kernel of type r. In this case, K is an operator from VP(Y) to 7YP+T(X). For our applications r will usually be 0,—I, or —2.
Example I As already mentioned, any smooth form K on Y x X defines a kernel. Many of the theorems about kernels are motivated by considering this class of kernels.
Another closely related class of kernels is the space of forms on Y x X with locally integrable coefficients.
1]
Example 2
Y be an oriented smooth manifold of dimension N. Let = Let X given by in{(x, x); x E X} be the diagonal of X x X. The current is N. tegration over the diagonal is a kernel of type 0 since the degree of D'P(X) represents the identity map. For if VP(X) As an operator, [s]: E D"(X), then
A parameterization for
is given by x '.—. (x, x), x E X. We have
f
rEX
=
Definitions
267
= as claimed. This is analogous to the fact that the operation of convolution with the delta function is the identity operator (see Chapter 5). This analogy is made even clearer by writing the current as a form with distribution coefficients So,
=6o(x—y)d(xi —yl)A...Ad(XN Example 3 Let X and Y be oriented manifolds and suppose f: X Let Gr{f} be the graph of I in Yx X, i.e.,
—YN).
—,
I]
Y is a smooth map.
Gr{f} = {(y,x); y = f(x)}.
Gr{f} given by F(x) = (f(x), x) is a global parameterization for Gr{f}. Let us orient Gr{f} by pushing the orientation on X to Gr{f} via F. That is, the collection C T{Qr{f}} is said to be
The map F: X —÷
. .. , positively oriented if and only if is positively oriented on X. With this orientation on Gr{f}, the map F is orientation preserving. The current (or kernel) [Gr{f}] on Y x X has dimension N and therefore degree N' (where N = dimX and N' = dimY). As an operator from DP(Y) to D'P(X), we claim [Cr{f}] represents the pull back map f. To see this, let E DP(Y) and E then
= = Gr{f} Since
F: X
Gr{f}
is orientation preserving, we have
([Gr{f}](Ø),
=
f
®
= Therefore, as claimed. Note that if X = Y and f(x) = = U x E X, then this example reduces to Example 2 above.
Now let us discuss the adjoin: of a kernel K denoted by K'. If K is an operator from V(Y) to V"(X), then K' is an operator from V(X) to V"(Y) and it is defined by
=
E V(Y),
V(X).
268
Kernel Calculus
If K is of type r then K' is of type N' — N + r. We wish to find a convenient way of computing K' from K. To motivate the calculation, we first suppose that K is a smooth form on Y x X. If E D*(Y) and E V*(X), then
= =
=
f J K(y, x) A
A
xEX y€Y
± J 0(y) A ( yEY
f
K(y, x) A
vEX
J K(y,x) A
=±
xEX
'yEY
The sign in front depends on the degrees of the forms involved and it will be resolved later. From the above calculation, we have
=
±
f K(y,x) A
x that the variable of integration in K(ç5)(x)). If X = Y, then switching x with y shows that K'(y, x) = ±K(x, y). Switching x and y means that dx3 is also switched with dy3.
To generalize this calculation for more general kernels, we introduce the switch map s: X x Y Y x X, s(x, y) = (y, x). Formally, the pull back s is the identity map on forms, i.e., for e V(Y) = 0(u) A and E D(X). However in local coordinates (x,y) for X x Y, we have
Det(Ds) = (_1)NN' changes the orientation by the factor (_l)NN'. For X), we obtain
Hence,
f
YxX
=
E
f
XxY
This formula extends to currents on Y x X. Since S is a diffeomorphism, s*K is well defined for K TYt(Y x X) (see Definition 2 in Section 6.2). In addition, 84 = for 4 V*(Y xX) in view of the remark after
Definitions
Lemma
269
in Section 6.2. We obtain
2
= (s_I os*K,4,)yxx = —i
\S
—
sr.-'
WIXxY.
This equation is the analogue of (1) for the pairing between currents and smooth forms. Now we compute the kernel of
K' and keep careful track of the minus signs. VP(Y) and x X) is a kernel of type r. Let 0
Suppose K
We have
= =
Now, s(0® and N
—
p
—
(by (2)).
A In addition, the degree of = r, respectively. Commuting 0(u) and yields
= The degree of and
and
isp
Ø)v
is N'
p and the degree of
—
is p. Commuting
yields
= (_
l)NN'+P(N+N'_r)(Ø,
So we have
= To simplify the notation, we introduce the map c:
cT = Since the degree of We obtain
for is
N' —p,
T
—
E
we have
= (_I)NN'CN+N'_rS*K Define the transpose of K, Kt E
x Y), by
Kt = We have proved the following.
LEMMA I
Suppose X and Y are smooth, oriented manifolds of dimensions N and N', x X)). respectively. Suppose K is a current of type r (i.e., K E Then
K'
CN+Nr{Kt}.
Kernel Calculus
270
Kt is easy to compute, this lemma gives a convenient formula for the adjoint, K'. It may happen that a kernel sends smooth, compactly supported forms to smooth forms (rather than to currents with nonsmooth coefficients). We single out this special class of kernels. Since
DEFINITION 2
Suppose K E DIN' +r (Y x X).
K is called a
(a)
If K is a continuous map from 1Y(Y) to EP-'-"(X),
(b)
regular kernel. JfK extends to a continuous operator from E'P(Y) to is called an extendable kernel.
(c)
If both K and K' are regular kernels, then K is called biregular.
then
then
K
LEMMA 2
if a kernel K E V1N'.Ft.(Y PROOF
x X) is regular, then K' is extendable.
If T E E'(X), then define
(Ø,K'(T))y = (K(Ø),T)x,
for
E D'(Y).
E e(X)). If The right side is well defined because K is regular (so K(Ø) in e'(X); therefore in D(Y), then by hypothesis T)x. l'his shows that K'(T) is a well-defined D'T)x —' T in current on Y. In a similar manner, the reader can show that if ef*(X) then K' is extendable, as desired. K'(T) I
The lemma implies that if K is biregular, then both K and K' are extendable.
An important class of biregular kernels is the class of kernels of convolution RN be defined by type, which we now describe. Let r: RN x
r(y,x) = x — y. (RN) The linear map r has maximal rank and therefore the pull back r*: D' (RN x IRN) is well defined (see Definition 2 and Lemma 5 in Section 6.2). x RN) is said to be of convolution type A kernel K V' if there exists a current k DII(RN) with K = rk.
DEFINITION 3
then the kernel K(y,x) = So, for example, if k = udy' with u E is a kernel of convolution type. y)d(x (rk)(y, x) = u(x — — y)' LEMMA 3
A kernel of convolution type is biregular.
Definitions
271
(RN), and
(Rh'),
Suppose k the definitions, we have
PROOF
(r*k(th),
E
(q +p)
= (r*k, th ® ?,L')RN xRN
=
(3)
Suppose k = udx1 with u E 2)' (1W"), and suppose 6 = 9, E V(RN). From Lemma I in Section 6.2, we obtain
=
(RN) From
f
g(y)h(x +
=
with
A d(x +
where the only nontrivial contribution to this integral comes from the terms involving dy = dy1 A .. dyN. Depending on the multiindices J and J', the coefficient of the form is either 0 or the following function of x (up ® to a + or — sign) .
f g(y)h(x + y)dy
=
f
g(y —
= = g(—t). Together with (3), this means that 0 or it is the term where
V5)aN is either
=
up to a ± or — sign. So the coefficient function of (rk)(4) is either 0 or g, up to a + or — sign which depends only on the indices I, J, and J'. If u E 1Y(RN), then the operator g u * g for g E D(JW') is a continuous linear map from D(RN) to (see Lemma 1 in Section 5.2). Therefore,
K=
T*k is regular.
If K = T*k, then we leave it to the reader to show that Kt = where k v*k and v: RN is defined by v(x) —x. So Kt is also a kernel of convolution type. It follows that Kt is regular, and so K is biregular, as desired. I The above definitions of kernels, type, regularity, etc., also apply to the case
when X and Y are complex manifolds or CR manifolds. The only difference is that we must keep track of bidegrees rather than just degrees. For example, if X and Y are complex manifolds of complex dimension n. and m respectively, a
current ofbidegree (m+r,m+s) onYx Xis said to be a kernel of type (r, s) and K can be regarded as an operator from VPQ(Y) to Since a complex manifold has even real dimension, many of the minus signs in the above discussion disappear. For example, Kt = S*K in the complex manifold setting.
Kernel Calculus
272
For CR manifolds, the bidegree counting is a little more complicated. Suppose
M and M' are CR manifolds. Suppose dimR M = 2m + d and dimR M' = 2m' + d' where d and d' are the CR codimensions of M and M', respectively. A form of top degree on M' has bidegree (m' + d', m') (see Section 8.1 or 8.2).
Suppose K is a kernel of bidegree (m' + d' + r, m' + s) on M' >< M. Then K is said to have type (r, s) and K can be regarded as an operator from For most of our applications, M = M' will be a hypersurface in to which means m = m' = n — and d = d' = 1. In addition, r will usually 1
beOandswillbeOor—1.
18.2
A homotopy formula
In this section, we develop a homotopy formula for the exterior derivative, the Cauchy—Riemann operator and the tangential Cauchy—Riemann operator. Suppose X is a smooth manifold and let K E V1N+T(X x X). Consider the
current dK where d is the exterior derivative on X x X. Viewed as a kernel The homotopy formula operator, dK is a map from DP(X) to given in the next theorem relates this operator with K and the exterior derivative on X. The 0 and 0M versions are also given for the case of a complex manifold and CR manifold, respectively. THEOREM I HOMOTOPY FORMULA of real dimension N. JfK (a) Suppose X is a smooth
DtN+r(X x
X), then
as operators on D*(X). (b)
Suppose X is a complex manifold of dimension n. JfK E X), then
0 such that for any multiindices 1 and J B(çT, z) A
A
= B((, z) A
A
+A1
A dz'
A
z) +
A
A and A2 are differential forms which are smooth for ( there is a constant C > 0 such that
0 and an integer N > 0 such that
C
sup
IaIN
CE supp T
for
z
converges
By
examining the formula for B((,
to zero as z —' 00. Thus,
z), we
see that the right side
B(T) (z) —' 0 as z)
—' oo,
as desired.
I Part (b) of the lemma is useful since the Cauchy kernel and the Bochner— Martinelli kernel each has properties not a priori possessed by the other. The Cauchy kernel has a nice support property described in part (a) which is not apparent for the Bochner—Martinelli kernel. On the other hand, the Bochner— Martinelli kernel has nice regularity properties since B((, z) has locally in-
tegrable coefficients. Part (b) states that if T is a 0-closed compactly sup-
ported (0,1)-current, then C(T) and B(T) both enjoy these properties since B(T) = C(T). We now state and prove Bochner's global CR extension theorem. THEOREM I
Suppose D is a bounded open set in C'2 (n 2) with smooth boundary. Suppose f is a CR function on OD of class C'. Then there is a holomorphic function F on D which has a continuous nontangential extension to OD from D such that FIOD = f. Moreover, F is given by either of the following integral formulas:
F = —B([OD]°"f) = —C([OD]°"f).
Bochner's global CR extension theorem
293
Note there are no convexity assumptions on OD for this global theorem.
Let M = OD. 1ff is aCR on M, then from Lemma 5 in Section 8.1, (MJ°" f is a 8-closed, compactly supported current of bidegree (0,1) on C's.
PROOF
Let
F = —B([M]°"f). By part (b) of Lemma 1, we have
F= From Theorem 3 in Section 19.2, From part (a) of Lemma 1, F 0 on nontangential extension to M from D. In addition, the F has a continuous — D) is equal to f. Since boundaxy value jump of F across M (from D to
F is holomorphic on D. Since
0,
Theorem 2 in Section 18.2 yields
OF = —[M]°"f which has support in M. Therefore, F is holomorphic on particular on D), as desired.
—
M (and in
I
Note that Bochner's theorem does not hold for domains in C'. This is because every function on a closed contour in C is a CR function (there are no tangential Cauchy—Riemann equations). The above proof breaks down in n = 1 because part (a) of Lemma I does not hold for n 1. For a simple closed contour in C, the condition that replaces 8M = 0 for Bochner's theorem is the moment condition, which means
f
J
for
n=0,l,2
Note by Cauchy's theorem, this condition is necessary for f to be the boundary values of a holomorphic function defined on the inside of y. To see that this is a sufficient condition, let
F(z) = —
1
[
2rriJ (—z <E.y
Theorem 3 in Section 19.2 still holds (regardless of whether or not f satisfies
the moment condition). In particular, the boundary value jump of F across 'y (from the inside to the outside of y) is equal to f. From a series expansion of 1/(( — z) in powers of (together with the moment condition, we see that F(z) = 0 for z outside 'y. It follows that the boundary values of F from the inside of -y agree with f.
20 The Kernels of Henkin
In Chapter 19, we constructed two fundamental solutions for 0 on
As
mentioned in Chapter 18, if K is a fundamental solution for 0 and if f
with Of
= K(f). with Of
0, then the equation Ou =
f
can be solved on
by setting Now suppose D is a bounded domain in CT' and suppose f E 0 on D. In this case, we cannot directly apply K to I without first
extending f and then multiplying by suitable cutoff function so that f has compact support. This process produces an extended f which is no longer 0closed. Another way to cut off f is to multiply f by the characteristic function on D (denoted If Of = 0 on D, then
= —[OD}°'
A 1.
As mentioned in Section 18.2 (see Theorem 2), we have
9{K(xDf)}
—
A 1) =
Therefore, the term A f) is the obstruction to solving the equation Ou = on D. In this chapter, we define a general class of kernels due to Henkin which allows us to solve the equation thj1 = K([OD]°" Af) on a strictly convex domain D and so in this case, u = K(XDI) — m solves the equation Ou = on D. We should also mention that a slightly different kernel approach to the solution of the equation Ou = f was discovered by Rarnirez IRa].
f
f
20.1
A general class of kernels
The kernels we are about to define are due to Henkin; however we shall employ the more streamlined notation of Harvey and Policing [HP].
294
A general class of kernels
Let V u3: V
be
295
1<j
x Ctm. For each an open subset of is a smooth map. We write
< N, suppose
u3((,z) = and
we use the notation
. (ç -
z)
- zk)
=
.d((-z)
.d((
-zk)
- z) =
Ad((k - zk).
Here, Ô refers to the Cauchy—Riemann operator on
x
(i.e., in both (
and
z).
For 1 <j t = the proof of part (a). For part (b), we first note that for some 0 the real hessian of p at (is positive definite provided —c < p(() < c. Therefore, part (b) follows from a second-order Taylor expansion of p about the point (. I ((— z)
0
C" x C"
Now we define the functions
C" for j = 1, 2, 3, which will
generate our desired kernels. Let
(ôp(()
Op(()
— Op(z) — (Op(z)
ap(z)
—
u'((,z) — —
u2((,z)
O(
—
u3((,z) = (— z.
—
—
*9z1
The solution to the Cauchy-Riemann equations on a convex domain
301
Note that u1 only depends on ( and u2 only depends on z, but we wish to think of both u1 andu2 as being defined on Ctm x
Using (L1,U2,U3, we define the kernels E1 E(u'), E2 = E(u2), E3 = E(u1, u3), E23 = E(u2, u3), and E123 = E(u3), E12 = E(&, u2), E13 u2, u3) as in Section 20.1. For historical reasons, we assign the following labels to these kernels:
L=
E1
(after Leray)
H=E13 (afterHenkin). As already mentioned, the Bochner—Martinelli kernel is given by
B=E3. By definition, the transpose of a kernel is obtained by switching ( with z. Since u3(z,() = —u3((,z) and u2(z,() u'((,z), we obtain
Ht((,z)
= E23((,z)
Lt((,z)
= E2((,z)
and
Bt=B. Recall that up to a sign, the transpose of a kernel is equal to its adjoint as an operator on forms (see Section 18.1). The L and H kernels are smoothly defined on the set
{
((,z)
E
x Ctm;O
((-z)
In view of part (a) of Lemma 1, this set includes the set {((,z) E M x D}. Reversing the roles of ( and z, we see that Lt and Ht are smoothly defined on the set {((, z); ( E M and 0 < p(z) < }. From Theorem I in Section 20.2, we have
OH=L-B onMxD. on{((,z);(EMandO 0 (independent
C
IH([ODJ°" A
sup
Kernels of Henkin
of f E D (K)) such that sup
zEDflK PROOF Since the forms
H and L have smooth coefficients on the set {((, z) E M x D},
H(fM]°" A f)(z)
A
=
f)(z) =
J(EM [H((, z) A - J(EM
[L((,z) A Af) and Lt([M]o.t A
depend smoothly on z E D. Similarly, the forms
We shall defer the proof of the estimate given in the lemma until the next section where we shall prove a more general result. I Another key fact is that L and Lt act nontrivially only on forms of highest
and lowest degree, respectively, as the next lemma shows. LEMMA 3
Suppose f E
Then
L([M]°" A
A f) = 0 unless q = n — 1.
f)
0 unless
q
0.
In addition,
depends only on (, and examination of the formula for L((,z) shows that the degree of L in d( is n — 1. Therefore, PROOF
Since u1
z) =
L([M]°" A f)(z) =
f has degree 0 in
-J
IL(C, z) A
—
Similarly, the degree of Lt((, z) in d( is 0 and so Lt([M}°" A 1) = 0 unless the degree of f in d( is n — I. I
Now we present Henkin's integral kernel solution to the Cauchy—Riemann
equations on a strictly convex domain. His solution has an L°° estimate which was one of the motivations for the construction of integral kernel solutions to the Cauchy—Riemann equations.
Boundary value results for Henkin 's kernels
303
THEOREM 1
(See fHe2].) Suppose D is abounded strictly convex domain in
boundary M. Let f E
q
I
n with t9f =
0
with smooth
on D. Then the ftrm
+ H(jMj°' A f)
=
f
is a solution to the equation = on D. Moreover, there is a constant C > 0 which is independent of f such that D
D
We apply part (b) of Theorem 2 in Section 18.2 to the fundamental and since 9f = 0 on D, we solution B with T = xof. Since PROOF
obtain
ô{B(XDf)} — Using the equation OH = L A
—
A
A f) ± L([M1°' A
and
(1)
B on M x D, we obtain
f)
Since f E
1) = XDf
f)
on D.
q I, the second term on the right vanishes by
Lemma 2. Using the homotopy formula in part (b) of Theorem I in Section 18.2, the first term on the right equals —O{H([M}°'
Since a{fM}°' A 1)
f) + H(O{[MJ°' A f})}.
0, we obtain
B([M1°'
A
f) =
—O{H([M]°1
A
f)}
on D.
Inserting this into (1) yields our solution n. The estimate on Iul follows from the estimate given in Lemma 2 and from the fact that the Bochner—Martinelli kernel B((, z) is uniformly integrable in
(EDforzED.
20.4
I
Boundary value results for Henkin's kernels
In this section, we shall prove the estimate given in Lemma 2 of the previous section. We shall also examine the smoothness of these kernel operators up to the boundary of our given convex domain. The boundary values of these kernel operators are the key ingredients for the construction of one of the fundamental solutions for the tangential Cauchy—Riemann complex on the boundary. Let D = {z E C'; p(z) < 0}, let = {z E C?z; p(z) > 0}, and let
M be the boundary of D. From Lemma 2 in the previous section, the forms
304
The Kernels of Henkin
A f) and L([M}°' A f) are smooth on D and A f) and Lt([M]oi A f) axe smooth on the set fl U where U is a neighborhood of M in We shall examine the regularity of the boundary values (on M from D) of H([M}°" Af) and Af) along with their tangential derivatives. Likewise, we shall examine the regularity of the boundary values (on M from of Ht([M}°"Af) and Lt([MJ°"Af) along with their tangential derivatives. Recall that a vector field X on is considered tangential to M if Xp = 0 near M where p is the defining function for M. Our main boundary value results for Henkin's kernels are contained in the following two theorems.
THEOREM I
Suppose D is a strictly convex domain with smooth boundary M. There is a neighborhood U of M such that the following holds. Let N be any nonnegative integer and suppose X1, .. , are tangential vector fields to M. If f is a compactly supported (p, q)-form with coefficients of class CN on lvi, then .
X1
. .
.
X.v{H(EM)°' A f))ID- and X,
A
. . .
continuous extensions to M. Moreover, for any compact set K in a constant C > 0 which is independent of f such that X1
D-nK sup
have
there is
.. XN{H([M]°" A f)}I
X1
.
.
. .
A f)}I
{ D+ riUnK }
for any smooth form f on M with support in K. Here, If ICN(M) is the usual
of f on Zt'i.
THEOREM 2
Suppose D is a strictly convex domain with smooth boundary M. There is a neighborhood U of M such that the following holds. If f is any compactly supported (p, q)-form with coefficients that are of class CN+I on M, then any derivative of order N of have A f)ID- and of V([M)°' A continuous extensions to M.
If D is bounded, then the above two theorems also apply to forms without compact support. We remark that the case N = 0 (i.e., no derivatives) is given in [He2] (see also [He3]). Some refinements of these boundary value results due to Harvey and Polking are given in Chapter 24.
In Theorem 2, we are allowed to take normal derivatives of L([M]°t A f) and Lt([M1oI A f), whereas in Theorem 1, we are only allowed to take tangential derivatives of H(1M]°' A f) and Ht([jll}o.l A f). There is no loss of differentiability in Theorem 1. That is, the form f is assumed to be of class CN and we obtain a This contrasts with Theorem 2 where there
Boundaiy value results for Henkin 's kernels
305
is a loss of differentiability. As we shall see from the proofs, this is due to the
fact that L((, z) and Lt((, z) are not integrable in ( E M for fixed z E M, whereas both H(C, z) and Ht((, z) are integrable in E M for fixed z M. The rest of this section is devoted to the proofs of these theorems. The basic idea of the proofs is to localize and then make a change of variables that flattens out it'!. The strict convexity allows us to estimate the resulting kernels. We then show that tangential derivatives do not worsen these estimates.
We shall prove these theorems for H and L. The proofs for Ht and Lt similar. By examining the L and H kernels, we see that we must analyze a term of the form
K(f)(z)
=
J(EM K((, z)f(()
where k((,
K'\
p+q—n, p, q_>
where f is a function of class CN on M and where dc(() form on M. For II, k is a smooth function satisfying
denotes
the volume
z) = O(I( — z()
both p and q are at least 1. For L, k is simply a smooth function without any estimate and q = n, p = 0. We fix a point z0 E M and examine the regularity of K(f)(z) as z approaches M from D near zo. Fix (o E M. If (o z0 then by Lemma 1, ((Op(C)/O(). — z)) 0 for x z) in some neighborhood Ui x U2 C of ((h, zn). If 0 is a smooth compactly supported function in U1, then K(Øf) is smooth on U2. In particular, K(Of) is smooth up to M fl U2 from U2 fl D. By a partition of unity argument, it suffices to assume Co z0. That is, we may assume that f has compact support in a set of the form U fl M where U is an open neighborhood of Zo in (to be chosen later). We must show that K (f) has the desired regularity on U fl D. We need the change of variables given in the following lemma. and
LEMMA 1
For each z0 e M,
'I': U x U (a)
(b)
there is a neighborhood U of zo in
with
and a smooth map
the following properties:
(p(z),0,. ..,0). If we write W((,z) = (w1((,z),.. W(z,z) =
.
E
then
.(C_z)}.
306
(c)
The Kernels of Henkin
Foreach z EU, the map
= W((,z) is a
given by
: U '—+
from U to PROOF
We let
.((_z)} as required by (b). We have = dp(zo) + ilm{Op(zo)}
((,
= dp(zo)
—
ZJ*d()
where the last equation uses the fact that Op = 1/2(dp
—
iJdp)
(see Lemma 5
in Section 3.3). The real vectors dp(zo) and J*dp(zo) span (over R) the 1complex dimensional subspace generated by Op(z0). Since Op 0 on M, we can find vectors w2, .. . , so that {(Op(zo)/e9z), w2,.. . , form a basis for C'2 over C. We let
=
. .
where w1 ((, z) is defined above and
w3((,z) =
to3
((— z)
for 2 j n.
The real (-derivative of 'I'((, zo) at (=
zO is nonsingular. So property (c) follows from the inverse function theorem. Property (a) follows from the definition ofW. I
From
now on, we require f to have support in U fl M where U is a neigh-
borhood of
which is small enough to satisfy Lemma 1. We shall also require
U to be small enough so that the following estimate holds (from part (b) of Lemma 1 in Section 20.3) for(,zEU.
Re{wi((,z)} = p((), contains a neighborhood of the origin inthecopyofR2'2' given by {w C'2;Rew1 =0} for each fixed z MflU. After pulling back the integral in K(f) to this copy of R2'2' via 'I';', we Since
obtain
K(f)(z)
=
f
{Rewi =O}
Kj(w,z)fi(w,z)dv(w)
307
Roundwy value results for Henkin 's kernels
where
K1(w,z) = (p(w,
k1(w,z) —1
(1)
(w) — zI2P
dv(w) = volume form on {Rewi
0}
k1 (w, z)dv(w) = k('Içt (w), z)W' (dc)
fi(w,z) p(w,z) =
p
(w))
(w) — z).
Since f has compact support in U, there is an > 0 such that the w-support of fj(w, z) is contained in {w E C'1; wi 0 such that (c)
and
e(iwhl2
p(w,
+ yiJ)
z)i e(iw'12 + )YI
for z EU1 and w = (y1,w') EU2. PROOF The proofs of parts (a)—(c) of this lemma are similar to the proofs of the corresponding parts of Lemma 2 in Section 20.4. As already mentioned, W is the copy of is a local diffeomorphism on an open set U C and given by {Rew1 = z for z M, there is an open OL Since neighborhood U1 of zO in M and an open neighborhood U2 of the origin in with 'P;'{U2} c Un M for each z E U1. Parts (a) and (b) now follow about the origin. from a Taylor expansion in w E For part (c), we first note from part (b) of Lemma 1 in Section 20.3 that if ( and z belong to M = {p = 0} then I
-
- z)}
Re {
With the roles of ( and z reversed, we obtain
ôp(z)
z)}
—Re {
-
for and z E M. Therefore, part (c) follows by letting (= using part (b).
Part (d) also follows from part (b) and the equation
((—z)
öp(z)
—
(Op(()
=0((_z)2.
ôp(z)
and then
A second fundamental solution to the tangential Cauchy—Riemann complex
323
The first estimate in part (e) follows from the estimate Rep(w, z) fIwI2 in part (c) together with the fact that Imp(w, z) = y'. For the second estimate in part (e), we first note from part (c) that
w
U2. From part (d), we have
IImp(w,z)I — IIm{p(w,z) _p*(w,z)}I —
C(Iw'12 + IyiI2) —
Clw'12
provided vt and 1w'! are suitably small, where C is some uniform positive I
constant. Therefore, we obtain Ip*(w,z)I max
—
CIwhl2}.
(4)
Now we use an easily established inequality (also used in a similar context in
[GL]): if a,/3,y > 0 then
max{a,j3—'y} (2+
(a+$).
(5)
a + 8 and this inequality follows by dividing If a then 2a + — through by 2+ ('y/a). On the other hand if a B — -y, then
l} to a smooth form on = I } (see Theorem 2 in Section 20.4), the above equation shows that If extends from {lwj < I } to a smooth function on M. Since If is holomorphic on {IwI < l}, the extension of If to M is a CR function. Also
M = {
J
z
note that (Lt)+(f) has real analytic coefficients near E it'! if and only if the If(VJ) is real analytic in w near ZO. This in turn is equivalent to function w If(w) for w near From Theorem I the real analyticity of the function ic in PvI extends in Section 9.1, a real analytic CR function on M near a point to a holomorphic function on a neighborhood of in C's. Therefore from Theorem 1, we obtain the following corollary. COROLLARY I The equation OM u = f can be solved on M
I } near Zo E M if and {z only if the function If extends to a holomorphic function in a neighborhood of
To construct examples of nonsolvability of the tangential Cauchy—Riemann
equations on the sphere near a given point z0, we first find a holomorphic < l} which extends smoothly to M = {IzI = l} but does function f on not holomorphically continue past we can let
For example, if ZO
= (1,0,... ,O) then
where we use the principal branch of the square root defined in the right half plane in C. Next we define for KI =
do A
f(() Thedegreeoffis(n,n—l). is the inclusion map. It follows that
do =
d). Therefore
.
on M. We obtain
If(w) =
(—21r2)
-n JgI=i
(1
—
Henkin's criterion for local solvability at the top degree
After the change of variables (i-i'
341
(n the above integral, we obtain
11(w) 1 —
= The
f
f(() A
d((
- w) A(d(( w).d(( ]
factor of (—1 )" disappears due to the change in orientation of the map
(p-. (. From the formula for L (with z replaced by w). we obtain 11(w) = L([M1°" A f)(w). Since I is holomorphic on the set {IwI < 1}, f = —L([M]°" Af) on {IwJ < 1) by Corollary 2 in Section 21.1. We obtain
If =—! on {IwI < 1). Since / does not holomorphically continue past the point = = (1,0,. . . , 0), neither does If, and so the equation OMU = cannot be
solved in any neighborhood of zo in M according to Corollary 1.
f
24 Further Results
24.1
More on the Bochner—Martinelli kernel
The boundary value result in Theorem 3 in Section 19.2 for the Bochner— Martinelli kernel can be strengthened. If we only assume that f is of class for 0 < < on M, then B([M]°" A f) is of class Ca on D and This can be established by adapting the proof of the corresponding result for 1
the Cauchy kernel in Lemma 2 in Section 15.4. Cauchy's integral formula used in the proof of that lemma must be replaced by the equation
f
= -B([M]°"f)
(EM
=xv-f
(1)
which holds for functions f which are holomorphic on D and continuous up to M (see Theorem 3 in Section 18.2). Here, XD- is the characteristic function of D whose manifold boundary is M. The minus sign in the first equality in (1) results from commuting the current [M)°" with the (2n — 1)-form B((, z). Adapting the computations in the proof of Lemma 2 in Section 9.4 for the Bochner—Martinelli kernel is somewhat tedious since the Bochner—Martinelli kernel is more complicated to unravel than the Cauchy kernel. However, the basic ideas are the same. This result for the Bochner—Martinelli kernel is due to Cirka [C], who a.lso generalized Theorem i in Section 19.3 to the case where the ambient space is a Stein manifold (instead of merely It is interesting to compare the boundary values of the Bochner—Martinelli kernel with the principal value limits of the Bochner—Martinelli kernel. The
latter is defined for f
Bbf(z) = lim EI—.O
342
by
f
{B((, z) A
z E M.
kernel
More on the
343
The following equations hold for f E
B(f) = Bb(f)
—
(2)
= Bo(f) +
(3)
Here, M is oriented by the equation dxD- = —[M}. These relationships are well known from residue theory for the Cauchy kernel. Note that subtracting these two equations gives the key boundary value jump formula B (f) = which is crucial in Chapter 21 for the construction of the fundamental solution for 8M• Equations (2) and (3) are due to Harvey and Lawson
f
[HLJ for the case where f is a function and to Harvey and Polking [HPI for the case of higher degree forms.
We shall sketch the ideas involved in the proof of (2). The proof of (3) is similar. The proof of (2) can be reduced to the case where f is a smooth function by using Lemma 2 as in the proof of Theorem 3 in Section 19.2. We assume that M is the boundary of a bounded domain D. The case where D is unbounded can be handled in a similar way by using cutoff functions as in the proof of Theorem 3 in Section 19.2. Fix Zo E M. If f is a smooth function is an integrable function of ( E M. that vanishes at zo, then IB((, exists by the dominated convergence and Bbf(zo) = So Bbf(zo) theorem. This proves (2) at zo for the case of a smooth function f that vanishes at z0. For more general f, we may write f(() = (f(() — f(zo)) + f(zo). Since we know (2) holds at zo for the function ('—' f(() — f(z0), it suffices to show (2) for the constant function I (() = 1. For z E D and 0, we have
B([M]°'t l)(z) =
-
f
z)
(EM
-
f
B((,z)
(4)
—
where we have set
M; K
< €}. Also set
—
= {(
E
Note that = [St] — [M€], where M( has the same orientation as M has the induced boundary orientation from From (1) with
and
= 0 for z E D. For
instead of D, we have B([8D7]°")(z) = z E D, we obtain
J B((, z) (EM.
=
f
B((, z)
J
B((, zo)
as z
Equation (4) becomes
= Bb(l)(Zo) — urn f B((,zo).
Further Results
344
FIGURE 24.1
To prove (2), it therefore suffices to show
urn f
2
—C.)
E
By a translation, we may assume that z0 is the origin. We may also pull back the resulting integral via the change of scale map ( — e( for = 1. The inverse image of the set under this change of scale converges to half of a unit sphere in as —f 0. Since the Bochner—Martinelli kernel is also invariant under the unitary group in C". we may assume this half sphere is the set
{( =
(,,);
CI
= I and Im(1 > 0}.
Therefore, we must show
f b(() =
(5)
345
Kernels for strictly pseudoconvex boundaries
where b(() = B((,0). To see (5), let S be the corresponding half of the unit we have sphere with Im(j < 0. In view of (I) with D as the unit ball in
1= f b(()+ f b((). (ES
is a diffeomorphism from Moreover, the conjugation map ( C(() which changes the orientation by a factor of (—1 )fl. By an easy S to the unit sphere. This uses the fact that on the computation, C'b (— We obtain sphere, 0 = d( . d(. + ( =(
fh= b.
This together with the previous equation implies that (5) holds and so the proof of (2) is complete.
24.2
Kernels for strictly pseudoconvex boundaries
For a compact convex boundary M = -, we use the complex gradient of a suitable defining function p as the generating function ?i for the kernels L, Lt, H, R, etc. The key properties possessed by u are the following.
(I) The function u((,z) is holomorphic in z e D for each fixed ( E M. (2) There are positive constants 6, and C such that if ( belongs to a 6neighborhood of and if K — zI 0. In addition, R(f) has C'°-coefficients provided the coefficients of f are compactly supported and belong to for any c> 0. Estimates on the local solution for the tangential Cauchy—Riemann complex
are more difficult than for the global solution. Indeed, there does not yet appear to be a local kernel solution for the tangential Cauchy—Riemann complex that exhibits a gain in regularity (maybe none exists). The solution given in Chapter 22 does not even preserve regularity due to the presence of the term E121([Ow.x]°" A 1). The best result available in this context is a recent result of Shaw [Sh2] who shows that a modification of the solution operator given in Chapter 22 is continuous in for 1
, given in Chapter 22 does not necessarily have continuous boundary values on 8WA due again to the presence of the term A f). In Section 5 of [He3], Henkin modifies this solution to one that is continuous up to 9WA provided f is a continuous (p, q)form with I q < n — 3. This solution operator (different than Shaw's) is also continuous in the on wX.
24.4
Weakly convex boundaries
The full strength of properties (2) and (3) in Section 24.2 for the generating function u or ü is not needed for some of the results in Chapters 20, 21, and 22 Many of these results only require property (1) in Section 24.2 (i.e., that u((, z)
Solvability of the tangential Cauchy—Riemann complex in other geometries
is holomorphic in z E
349
for (in M) and the following weaker version of
properties (2) and (3).
The function u((, z).((—z) is nonvanishing for (çz) E x Moreover, V is a neighborhood of Mx M in if ((o, zo) E V with u((o, zo) ((o — zo) = 0, then the real derivatives and z '—' u((o,z).((o—z) have of the maps respectively. maximal rank at the points (= (o and z = For a bounded convex domain that is not strictly convex, a defining function can be found so that its complex gradient satisfies this weaker property. Harvey and Polking show in [HP] that integral kernels can be constructed for the solution
of the 0-problem on domains that have generating functions that satisfy the above weaker condition. Their approach is somewhat different than Henkin's. Harvey and Policing take principal value limits across the singular sets of these kernels (i.e., the set {((, z) V; u((, z) ((— z) = 0}). As a result, they obtain new integral kernel solutions for the equation &u = on convex domains. These new solutions can be applied to more subtle problems. For example,
f
if I
smooth on D (but not necessarily up to M), then they provide an integral kernel solution u (in certain bidegrees) which is smooth on D. If f has compact support then they also obtain a solution u with compact support is
(in certain bidegrees).
The boundary value results for the L, Lt, H, and Ht kernels given in Theorems 1 and 2 in Section 20.4 also only require that the generating function u satisfy the above weaker condition. So these theorems (and the result concerning a fundamental solution for 0M given in Theorem 1 of Section 21.1) also hold for boundaries of weakly convex domains. In addition, the boundary values of one normal derivative of Af)JD- and of Af)ID-I. exist on M (Theorems I and 2 in Section 20.4 only discuss the boundary values of tangential derivatives). These and related results can be found in Sections 8 and 9 of [HP].
24.5
Solvability of the tangential Cauchy—Riemann complex in other geometries
The goal of much of Part IV is the discussion of the local and global solvability of the tangential Cauchy—Riemann complex on a strictly pseudoconvex hypersurface in In this section, we briefly discuss the local solvability of the tangential Cauchy—Riemann complex in other geometries. We fix a point zO in
a smooth real hypersurface M in C" and fix an integer q with 1 5q Sn— I. We ask the following question. What conditions on the Levi form of M at zO are needed to ensure that the equation = f can be solved near zo where f
350
Further Results
is a given smooth OM-closed (p, q)-form on M near Zqj? The answer is that 1k! must satisfy the Y(q) condition of J. J. Kohn at the point z0. The hypersurface M is said to satisfy condition Y(q) at the point z0 if the Levi form of M at ZO has either max{n — q, q + l} eigenvalues of the same sign or min{n — q, q + l} pairs of eigenvalues of opposite sign (i.e., min{n — q, q + 1 } positive eigenvalues and min{n q, q + 1 } negative eigerivalues). If M is strictly pseudoconvex at z0, then M satisfies condition Y(q) at z0 for all 1 q n — 2. More generally, suppose the Levi form of M at 21J has r-positive and s-negative eigenvalues with r + s = n — 1, then M satisfies condition Y(q) at z0 for each 0 q ii — s. Folland and Kohn use £2 techniques in their book except q = r and q
[FK] to show that if M satisfies condition Y(q) at every point zo in M, then the equation = is globally solvable where f is any smooth OM-closed
f
(p, q)-form. Local solvability under condition Y(q) was established by Andreotti and Hill [AnHi2J. We shall outline an integral kernel approach to Andreotti and Hill's result. For details, see [BS}.
Since there are several cases to consider for the condition Y(q), we shall discuss the case where the Levi form of M at ZO has (q+ 1)-positive eigenvalues with q + I n — q. The other cases of condition Y(q) are similar to this one.
We assume that zO is the origin and M is graphed over its tangent space at the origin as M = {z E p(z) = 0) where
p(z) =Imz1 Here, we have diagonalized the Levi form of M at the origin and the numbers P2,.. . , p7, are its eigenvalues. By replacing p with p + Cp2 as in the proof of Theorem i in Section 10.3, we may assume that the complex hessian of p at the origin is positive definite in the z1 -direction. Since the Levi form of M at the origin has (q + 1)-positive eigenvalues, we may assume that P2,... are positive. After a change in scale, we obtain n
+
p(z) = Imz1 +
+ O(1z13)
3=1
where
(I)
In the strictly pseudoconvex case, q + 2 n and the function used to generate the kernels in Chapters 20 and 21 is given by the complex gradient of p. Here, '—* x ui,) : be given by we let u' = . .
,
if
if
of the tangential Cauchy—Riemann complex in other geometries
351
A Taylor expansion argument together with the estimate in (1) imply that there is an open set U in containing the origin and a constant C> 0 such that
2Re{n'((,z).((—z)} p(() —p(z)+C(I(—z12) for (,z E U. Let u2((, z) = u'(z,
Reversing the roles of ( and z in the
above inequality yields
2Re{u2((,z)
— z)}
p(()
— p(z)
—
C(I( — zt2)
for (,z EU. We form the kernels L = E(u'), Lt E(u2), and R = E(u',u2) as in Chapters 20 and 21. The above estimates imply that L((, z) is smooth
for ( E Mn U, z E Un D and Lt((,z) is smooth for ( E Mn U and z E UflDt where D = {z E = {z E C'2; p(z) > 0}. p(z) —A for A > 0. Since M is strictly convex (in Chapter 22), these open sets shrink down to the origin as A '—+ 0.
in this section, M is only strictly convex in the (zi,...
,
zq+2)-directions.
Therefore, we shall obtain a local neighborhood basis of open sets by intersecting M with a trough that is flat in the . , )-directions and that "bends up" More precisely, let in the (Zq+3,. , .
.
WA =
EM; Imz1 > —A+2
1z312}.
j=q+3
The choice of the number 2 is motivated by the estimate (1) on the eigenvalues which ensures that the above trough bends up faster than M in 11q+3' .. , the directions (zq+3, ... , zn). In fact, combining the defining equation for M and the defining equation for WA, we obtain q+2
forzEwA. j=1
The estimate in (1) together with this inequality imply that the diameter of W), 0. From now on, we is proportional to VX. So WA shrinks to the origin as A restrict A so that WA is contained in U. Let
r(z) = Imz1
—2
j=q+3
The defining equation for WA is given by {z E M; r(z) > i—' C'3 by C" x
—A}.
Define
As in Chapter 22, we form the kernels E13, E23, and E123. As in the proof of part (a) of Lemma 1 in Section 20.3, the (weak) convexity of r implies that if
Solvability of the tangential Cauchy-Riemann complex in other geometries
zE
and ( A
then u3((, z) ((— z) A f),
0. So if f E
353
then
A 1) are smooth forms
On W.,.
The same arguments used for the L kernel above allow us to show that the degree of E13((, z) in d( is at least q. Therefore, E13 acts nontrivially only
on currents of bidegree (p, s) with s
n—q—
1.
Since q n — q —
1,
E13([OwAI°'1 A f) vanishes for f E As with the strictly pseudoconvex case, the term E23([Ow,j°" A 1) does not vanish purely from bidegree considerations. However, if f is a then an approximation argument similar to that at smooth (p, q)-form on
the end of Chapter 22 can be carried out to show that E23([aw.x]°" A f) vanishes. In this case, the kernel E23((, z) is (formally) holomorphic only in the So the approximation argument must be carried out in variables (it... treated as parameters. We refer the reader to these variables with (q+3, . . the end of [BS] for details. A f) both vanish if f is a smooth, — Since E13([OwA]°" A f) and the procedure in Chapter 22 can be carried out in q)-form on aM-closed is given by (3) in Chapter 22. this context so that the solution to OMU = on In the strict pseudoconvex case, we showed in Chapter 23 (with Lewy's example) that the tangential Cauchy—Riemann complex is not locally solvable at top degree (q = n — 1). More generally, if the Levi form of a hypersurface at a point zo has p-positive eigenvalues and q-negative eigenvalues, then the .
,
f
tangential Cauchy—Riemann complex is not locally solvable in degrees p and q. This is a result of Andreotti, Fredricks, and Nacinovich [AFN}. Generalizations of the results in this section to manifolds of higher codimension have been obtained by Airapetyan and Henkin [AH1], [AH2].
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Notation
The following is a partial list of notation. The page number refers to the page where the notation is first encountered or defined. space of smooth functions on — p. 2. the space of compactly supported smooth functions on — p. 2. the space of real analytic functions on — p. 4. the space of holomorphic functions on Il — p. 6. the space of Holder continuous functions with exponent — p. 215. the norm on the space — p. 216. T(X) the tangent bundle to the manifold X — p. 24. Tc (M) the complexified tangent bundle to M p. 40. 11(M) the complex tangent bundle to a CR manifold M p. 97. 11C (M) the complexification of H(M) — p. 101. X(M) the totally real tangent bundle to a CR manifold M — p. 98. the cotangent bundle of X — p. 26. (M) the complexified cotangent bundle of it'! — p. 40. the bundle of r-forms on Al — p. 27. (X) the bundle of complexified r-forms on X — p. 40. J a complex structure map — pp. 7, 41. the induced complex structure map on forms — p. 42. VLO (resp. V°") the +i (resp. —i) eigenspace of J on the vector space V — the
p.
43. the
bundle of (p,q)-forms on Al — p. 46.
see p. 123. the space of smooth r-forms on M — p. 27. 17(M) the space of smooth r-forms with compact support on M — p. 79. the space of smooth (p, q)-forms on a complex manifold M — p. 46. the space of elements of with compact support — p. 49. the space of smooth (p, q)-forms on a CR manifold Al — p. 124. the space of elements of with compact support — p. 124. the space of smooth sections of — p. 124.
359
Notation
the
space of smooth sections of
with compact support — pp. 124,
128. D'(Il) the space of distributions on Il — p. 62.
£'
the space of distributions with compact support on the
—
p. 62.
space of currents of degree r on — p. 81. the space of currents of degree r with compact support on —
p. 81.
the space of currents of bidegree (p, q) on — p. 83. the space of currents of bidegree (p, q) on a CR manifold M — p. 129. F defined on vectors (or currents) — p. 8. Ft the pull back map via F defined on forms (or currents) — p. 11. DF the matrix which represents the derivative of F — p. 7. d or dM the exterior derivative operator — p. 28. d* the £2-adjoint of the exterior derivative on RN — p. 16. o the Cauchy—Riemann operator — p. 47. 0* the £2-adjoint of 0 on CM — p. 49. °M the tangential Cauchy—Riemann operator (either extrinsic or intrinsic) — p. 124.
the tangential piece of the form f along M —
ftM dci
a
d/.SM (
,
)c2
p. 124.
volume form — p. 36. surface measure on M — p. 63. pairing between forms and vectors at the point p — p. 9. pairing between distributions and functions on — p. 62. pairing
between currents and forms on — p. 80.
the intrinsic Levi form at p — p. 156. the extrinsic Levi form at p — p. 160. the convex hull of the image of — p. 200. F1
