An Introduction to Involutive Structures

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An Introduction to Involutive Structures Detailing the main methods in the theory of involutive systems of complex vector fields, this book examines the major results from the last 25 years in the subject. One of the key tools of the subject – the Baouendi–Treves approximation theorem – is proved for many function spaces. This in turn is applied to questions in partial differential equations and several complex variables. Many basic problems such as regularity, unique continuation and boundary behavior of the solutions are explored. The local solvability of systems of partial differential equations is studied in some detail. The book provides a solid background for beginners in the field and also contains a treatment of many recent results which will be of interest to researchers in the subject. Shi feraw Berha n u is a Professor of Mathematics at Temple University in the US. Pau lo D. Corda r o is a Professor of Mathematics in the Institute of Mathematics and Statistics at the University of São Paulo in Brazil. Jorge Hounie is a Professor of Mathematics at the Federal University of São Carlos in Brazil.

NEW MATHEMATICAL MONOGRAPHS Editorial Board Béla Bollobás William Fulton Frances Kirwan Peter Sarnak Barry Simon Burt Totaro For information about Cambridge University Press mathematics publications visit http://www.cambridge.org/mathematics

An Introduction to Involutive Structures SHIFERAW BERHANU Temple University PAULO D. CORDARO University of São Paulo JORGE HOUNIE Federal University of São Carlos

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521878579 © S. Berhanu, P. Cordaro and J. Hounie 2008 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2008

ISBN-13 978-0-511-38814-9

eBook (NetLibrary)

ISBN-13

hardback

978-0-521-87857-9

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface I

page ix

Locally integrable structures I.1 Complex vector fields I.2 The algebraic structure of X I.3 Formally integrable structures I.4 Differential forms I.5 The Frobenius theorem I.6 Analytic structures I.7 The characteristic set I.8 Some special structures I.9 Locally integrable structures I.10 Local generators I.11 Local generators in analytic structures I.12 Integrability of complex and elliptic structures I.13 Elliptic structures in the real plane I.14 Compatible submanifolds I.15 Locally integrable CR structures I.16 A CR structure that is not locally integrable I.17 The Levi form on a formally integrable structure Appendix: Proof of the Newlander–Nirenberg theorem Notes

II The Baouendi–Treves approximation formula II.1 The approximation theorem II.2 Distribution solutions II.3 Convergence in standard functional spaces II.4 Applications Notes v

1 1 4 5 7 11 14 15 15 19 21 25 26 28 32 36 38 42 47 51 52 52 63 69 83 99

vi III

Contents Sussmann’s orbits and unique continuation III.1 Sussmann’s orbits III.2 Propagation of support and global unique continuation III.3 The strong uniqueness property for locally integrable solutions III.4 Proof of Theorem III.3.15 III.5 Uniqueness for approximate solutions III.6 Real-analytic structures in the plane III.7 Further applications of Sussmann’s orbits Notes

IV Local IV.1 IV.2 IV.3 IV.4

solvability of vector fields Planar vector fields Solvability in C  Vector fields in several variables Necessary conditions for local solvability Notes

V The FBI transform and some applications V.1 Certain submanifolds of hypoanalytic manifolds V.2 Microlocal analyticity and the FBI transform V.3 Microlocal smoothness V.4 Microlocal hypoanalyticity and the FBI transform V.5 Application of the FBI transform to the C  wave front set of solutions of nonlinear PDEs V.6 Applications to edge-of-the-wedge theory V.7 Application to the F. and M. Riesz theorem Notes VI Some VI.1 VI.2 VI.3 VI.4

boundary properties of solutions Existence of a boundary value Pointwise convergence to the boundary value One-sided local solvability in the plane The H p property for vector fields Notes

VII The differential complex associated with a formally integrable structure VII.1 The exterior derivative VII.2 The local representation of the exterior derivative VII.3 The Poincaré Lemma VII.4 The differential complex associated with a formally integrable structure

101 101 108 120 126 132 140 146 147 149 150 176 184 199 214 218 218 226 236 239 243 254 263 270 271 271 281 286 289 307 308 308 309 311 311

Contents VII.5 VII.6 VII.7 VII.8 VII.9 VII.10 VII.11 VII.12 VII.13

Localization Germ solvability  -cohomology and local solvability The Approximate Poincaré Lemma One-sided solvability Localization near a point at the boundary One-sided approximation A Mayer–Vietoris argument Local solvability versus local integrability Notes

vii 312 313 315 316 319 321 322 323 328 330

VIII Local solvability in locally integrable structures VIII.1 Local solvability in essentially real structures VIII.2 Local solvability in the analytic category VIII.3 Elliptic structures VIII.4 The Box operator associated with D VIII.5 The intersection number VIII.6 The intersection number under certain geometrical assumptions VIII.7 A necessary condition for one-sided solvability VIII.8 The sufficiency of condition 0 VIII.9 Proof of Proposition VIII.8.2 VIII.10 Solvability for corank one analytic structures Notes

331 332 332 333 337 340

Epilogue 1 The similarity principle and applications 2 Mizohata structures 3 Hypoanalytic structures 4 The local model for a hypoanalytic manifold 5 The sheaf of hyperfunction solutions on a hypoanalytic manifold

361 361 364 370 371

Appendix A Hardy space lemmas A.1 Multipliers in h1 A.2 Commutators A.3 Change of variables

374 374 376 378

Bibliography

381

Index

390

343 346 348 350 354 357

372

Preface

Since the first systematic exposition of the theory of involutive systems of vector fields ([T5]) was published almost 15 years ago, the subject has undergone considerable development and many new applications have been found. Systems of vector fields arise as a local basis of an involutive sub-bundle  of the complexified tangent bundle CT . Involutivity of  means that the commutation bracket of two smooth sections of  must also be a section of  . Examples of involutive structures    include foliations, complex structures, and CR structures. In these examples,  ∩  has constant rank. However, in recent work on integral geometry, natural examples of involutive structures have arisen for which the rank of  ∩  changes from point to point ([BE], [BEGM], and [EG1]). In the works [BE] and [BEGM], the cohomology of involutive structures is a key ingredient. Examples of involutive structures where the rank of  ∩  is not constant also arise naturally, for instance, on the tangent bundle of symmetric spaces (see [Sz] and the references therein) or in the study of the generalized similarity principle for the equation Lu = Au + Bu where L is a planar complex vector field not necessarily elliptic, which is intimately linked to the study of infinitesimal deformations of surfaces in R3 with non-negative curvature (see [Me3], [Me4], and the references therein). This book introduces the reader to a number of results on systems of vector fields with complex-valued coefficients defined on a smooth manifold . Most of the time, it will be assumed that the involutive structure    is locally integrable. The latter means that the orthogonal of  , which is a subbundle T  of the complexified cotangent bundle CT ∗ , is locally generated by exact differentials. When    is locally integrable, each point has a neighborhood U such that if L1      Ln are n smooth vector fields that form ix

x

Preface

a basis of  over U , then we can find m = dim  −n smooth, complex-valued functions Z1      Zm which are solutions of the equations Lj h = 0

1≤j≤n

(1)

and whose differentials are linearly independent over C at each point of U . The m functions Z = Z1      Zm  are sometimes referred to as a complete set of first integrals in the neighborhood U . In 1981, in [BT1], Baouendi and Treves proved that in a locally integrable structure, each solution of (1) can be locally approximated by a sequence Pk Z where the Pk are holomorphic polynomials of m variables and Z = Z1      Zm  is a complete set of first integrals. This approximation theorem has enabled several researchers to use the methods of complex analysis, harmonic analysis, and partial differential equations to study many problems on locally integrable structures. These problems include: the local and microlocal regularity of the solutions of (1); the determination of sets of uniqueness for solutions of (1); the solvability of the differential complex associated with the structure   ; and many other properties of the solutions of (1). This book attempts to present a systematic treatment of some of these results in a way that is accessible to graduate students with a background in real analysis, one complex variable, and basic introductions to several complex variables and linear PDEs including the theory of distributions. Chapter I introduces the basic concepts in the theory of involutive and locally integrable structures. Special classes of involutive structures such as complex structures, CR structures, elliptic structures, and real analytic structures are identified and examples are provided. Useful local representations both for general involutive and locally integrable structures are also discussed. A proof of the Newlander–Nirenberg theorem is presented in the appendix to Chapter I. Chapter II is devoted to the approximation theorem of Baouendi and Treves. It is shown that the approximation is valid in many function spaces used in analysis: the Lebesgue spaces Lp  1 ≤ p < ; Sobolev spaces; Hölder spaces; and localizable Hardy spaces hp  0 < p < . Applications to uniqueness in the Cauchy problem and extendability of CR functions are also included. Chapter III presents a variety of results on unique continuation for solutions and approximate solutions in a locally integrable structure   . The orbits of Sussmann associated with the real parts RL of the smooth sections of  play a crucial role in many problems, including the study of unique continuation and the chapter includes a discussion of some of the properties of these orbits. Chapter IV provides a detailed treatment of locally solvable vector fields. In the first part of the chapter, where the focus is on

Preface

xi

planar vector fields, the solvability condition  of Nirenberg and Treves is discussed and a priori estimates are proved in Lp and in a mixed norm that involves the Hardy space h1 R. A duality argument is then used to derive local solvability results in Lp  1 < p <  and in L R bmoR . The chapter also includes sections on the sufficiency and necessity of condition  for local solvability in higher dimensions. The first part of Chapter V introduces certain submanifolds in an involutive structure    which are important in the study of solutions. These submanifolds are generalizations of the totally real and generic CR submanifolds encountered in CR manifolds. The second part of the chapter introduces the FBI transform first in Rn and then in a locally integrable structure. The FBI transform is then applied to derive edge-of-thewedge type results. It is also applied to study the microlocal singularities of the solutions of a first-order nonlinear PDE and a generalization of the F. and M. Riesz theorem. Chapter VI studies some boundary properties of the solutions of locally integrable vector fields. These properties include the existence of a trace at the boundary, pointwise convergence of solutions to their boundary values, and the validity of Hardy space-like properties. Chapter VII describes the differential complex attached to a general involutive structure. An invariant definition of this complex is followed by a useful representation in appropriate coordinates. An approximate Poincaré Lemma for locally integrable structures is also proved in the chapter. Chapter VIII deals with the local solvability theory of the undetermined systems of partial differential equations naturally associated with a locally integrable structure, that is, the cohomology theory of its differential complex. Necessary and sufficient conditions are studied in some detail when the structure is analytic, or elliptic, or has corank one. Concerning the latter class, a thorough exposition of the geometric characterization of local solvability in degree one for real analytic structures is presented. Finally we conclude with an epilogue which summarizes some of the results obtained in recent years on diverse areas such as the similarity principle, Mizohata structures, and hyperfunction solutions in hypoanalytic manifolds. Two applications of the similarity principle are described. The first application concerns uniqueness in the Cauchy problem for a class of semilinear equations. The second application involves the theory of bending of surfaces. There are numerous interesting results on complex vector fields and involutive structures that have been obtained since the publication of [BT1] and which are not covered in this book. The authors have selected the material with which they have had first-hand experience. In the notes at the end of each chapter, we indicate some related works and provide additional references.

xii

Preface

The reader is referred to [BER] for a further reference on CR manifolds and to [T5] for additional topics on involutive structures. We are grateful to Elisandra Bär, Sagun Chanillo, Nicholas Hanges, Gustavo Hoepfner, and Gerson Petronilho for reading parts of the manuscript, pointing out errors and suggesting improvements. We are also grateful to Peter Thompson of Cambridge University Press for the expedience with which our book has been handled.

I Locally integrable structures

In this chapter we introduce the main concepts which will be studied throughout the book. In order to do so we recall some standard notions such as differentiable manifolds, vector fields, differential forms, etc., with the purpose mainly of laying down the basis for the presentation and to establish the notations. Nevertheless, we assume from the reader some familiarity with these concepts. In particular, we freely use some standard results on complex vector fields and complex differential forms on RN .

I.1 Complex vector fields Let  be a Hausdorff topological space, with a countable basis of open sets. A differentiable structure over  of dimension N is a collection of pairs  = U x , where U ⊂  is a nonempty open set, x U −→ RN is a homeomorphism onto an open subset xU of RN and the following properties are satisfied: (1)

 Ux∈

U = ; x x−1

(2) xU ∩ U   −→ x U ∩ U   is C  for each pair U x, U   x  ∈  with U ∩ U  = ∅; (3)  is maximal with respect to (1) and (2), that is, if ∅ = V ⊂  is open and y V −→ yV is a homeomorphism over an open subset of RN such y x−1

that, for any U x ∈  with U ∩ V = ∅, the composition xU ∩ V −→ yU ∩ V is C  , then V y ∈  . 1

2

Locally integrable structures

It is easy to see that given any family  ∗ = U x as above satisfying (1) and (2) there is a unique differentiable structure  over , of dimension N , such that  ∗ ⊂  . Definition I.1.1. A differentiable manifold (or smooth manifold) of dimension N is a Hausdorff topological space , with a countable basis equipped with a differentiable structure of dimension N . If, in the above definitions, we replace C  by real-analytic we obtain the concept of a real-analytic manifold of dimension N . We give some examples: (1)  = RN ,  ∗ = RN  identity map . (2) Let  be a differentiable manifold of dimension N and let W ⊂  be open. Then over W is defined a natural differentiable structure of dimension N , which is given by W = W ∩ U x W ∩U  U x ∈   W ∩ U = ∅  (3) Let f RN +1 → R be a C  function. Let  = x ∈ RN +1 fx = 0 and suppose that dfx = 0, ∀x ∈ . Then a natural differentiable structure of dimension N is defined over  (as a consequence of the implicit function theorem). Notation. An element U x ∈  will be refered to as a local chart or as a local system of coordinates. If we write x = x1      xN  then for p ∈ U its local coordinates (with respect to this given local chart) are given by x1 p     xN p. From now on, unless otherwise stated, we shall fix a differentiable manifold  (of dimension N ). We shall say that a function f  → C is smooth if for every U x ∈  the composition f x−1 is C  on xU.1 We shall denote by C   the set of all smooth functions on . We observe that C  is an algebra over C which contains, as an R-subalgebra, the set C   R of all smooth functions on  which are real-valued. Definition I.1.2. A (smooth) complex vector field over  is a C-linear map L C   −→ C   1

More generally, we say that a function f  → C is C k (k ≥ 0) if for every U x ∈  the composition f x−1 is C k on xU.

I.1 Complex vector fields

3

which satisfies the Leibniz rule Lfg = fLg + gLf

f g ∈ C  

(I.1)

We shall denote by X the set of all complex vector fields over . Proposition I.1.3. If L ∈ X and if f is constant then Lf = 0. We also have supp Lf ⊂ supp f

∀f ∈ C   L ∈ X

(I.2)

Proof. For the first statement it suffices to show that L1 = 0 and this follows from (I.1) together with the fact that 12 = 1. We shall now prove (I.2); we must show that if f vanishes on an open set V ⊂  then the same is true for Lf . Let p ∈ V be arbitrary. We select a local chart U x with p ∈ U ⊂ V and take  ∈ Cc xU such that xp = 1. Then the function g  → R defined by the rule  xq if q ∈ U gq = 0 if q ∈ U belongs to C   R and vanishes on \V . In particular, f = 1 − gf and then Lfp = 1 − gpLfp + fpL1 − gp = 0 since gp = 1. A consequence of the preceding result is the possibility of defining the restriction of an element L ∈ X to an open subset W of . More precisely, there is a C-linear map X  L −→ LW ∈ XW which turns the diagram L

C   −→ C   ↓ ↓ LW

C  W −→ C  W commutative (the vertical arrows denote the restriction map). Indeed, if p ∈ W and f ∈ C  W we set LW fp = Lf˜ p

4

Locally integrable structures

where f˜ is any element in C   which coincides with f in a neighborhood of p. Such a definition is meaningful according to Proposition I.1.3 and it is very easy to check that LW defines an element in XW. As usual we shall write L instead of LW , since the meaning will always be clear from the context.

I.2 The algebraic structure of X Given g ∈ C   and L ∈ X we can define gL ∈ X by gLf = g · Lf

f ∈ C  

Such external multiplication gives X the structure of a C  -module. A very important (internal) operation in X is the so-called Lie bracket (or commutator) between two vector fields. Given L M ∈ X we define

L M f  = L Mf  − M Lf  

f ∈ C  

(I.3)

It is a simple verification to check that L M ∈ X. This bracket operation turns X into a Lie algebra2 over C. Let U x be a local chart in  and let also L ∈ XU. We fix p ∈ U and write as before xq = x1 q     xN q

q ∈ U

Next we take V ⊂ U open such that xV is an open ball centered at xp = a = a1      aN . Given f ∈ C  U, write f ∗ = f x−1 . If x1      xN  ∈ xV, the Fundamental Theorem of Calculus applied to the function t → f ∗ a1 + tx1 − a1      aN + txN − aN  gives f ∗ x1      xN  = f ∗ a1      aN  +

N 

hj x1      xN xj − aj 

j=1

where hj ∈ C  xV and hj a = f ∗ /xj a. If we further set gj = hj x ∈ C  V, we obtain fq = fp +

N 

gj qxj q − xj p

q ∈ V

(I.4)

j=1

2

Recall that a Lie algebra over C is a C-vector space E over which is defined a bilinear form E × E  v w → v w which satisfies

u u = 0

u v w + v w u + w u v = 0

u v w ∈ E

I.3 Formally integrable structures

5

and consequently the Leibniz rule gives Lf p =

N 

  gj p Lxj p

(I.5)

j=1

Definition I.2.1. The C-linear map C  U → C  U given by f →

f ∗

x xj

defines an element in XU, which will be denoted by

 . xj

Returning to the preceding argument and notation we can write   f ∗  f p xp = gj p = hj xp = xj xj Inserting this in (I.5) gives Lfp =

N  

 Lxj p

j=1



 xj

 f p

since p was an arbitrary point taken in U we obtain the representation of L in the local coordinates x1      xN : L=

N  

Lxj

j=1

   xj

(I.6)

In particular this representation shows that the C  U-module XU is free, with basis /x1      /xN . Observe that if M ∈ XU then the representation of L M in the local coordinates x1      xN  is given by

L M =

N 

  xj

LMxj  − MLxj 

j=1

(I.7)

I.3 Formally integrable structures Denote by p the set of all pairs V f, where V is an open neighborhood of p and f ∈ C  V. In p we introduce the following equivalence relation: V1  f1  ∼ V2  f2  if there is an open neighborhood V of p, V ⊂ V1 ∩ V2 , such that f1 and f2 agree on V .  A germ of a C  function at p is an element in the quotient space C  p = p / ∼. We observe that C  p is also a C-algebra. Given a C  function f

6

Locally integrable structures

defined in an open neighborhood of p, the germ at p defined by f will be denoted by f . Notice that there is a natural C-algebra homomorphism C  p → C defined by f → fp. Definition I.3.1. A complex tangent vector (to ) at p is a C-linear map v C  p −→ C satisfying vf g = fpvg + gpvf 

f  g ∈ C  p

(I.8)

The set of all complex tangent vectors at p, denoted by CTp , has a structure of a C-vector space and is called the complex tangent space to  at p. If L ∈ X then Lp C  p → C defined by Lp f  = Lfp

f ∈ C  p

belongs to CTp . Conversely, suppose that for each p ∈  an element vp ∈ CTp  is given such that p → vp f  ∈ C  

∀f ∈ C  

Then there is L ∈ X such that Lp = vp for all p ∈ . Suppose now that p ∈ U and that U x is a local chart. If v ∈ CTp  then, according to (I.5),   N N    f  f ∈ C  p vf  = gj pvxj  = vxj  x j p j=1 j=1 In particular we conclude that  x p j = 1     N is a basis of CTp . j The complexified tangent bundle of  is defined as the disjoint union  CTp  CT = p∈

We shall also need the notion of a complex vector sub-bundle of CT of rank n and corank N − n. By this we mean a disjoint union  = p ⊂ CT p∈

satisfying the following conditions: (a) For each p ∈ , p is a vector subspace of CTp  of dimension n. (b) Given p0 ∈  there are an open set U0 containing p0 and vector fields L1      Ln ∈ XU0  such that L1p      Lnp span p for every p ∈ U0 . The vector space p is called the fiber of  at p.

I.4 Differential forms

7

Given a complex vector sub-bundle  of CT and an open subset W of , a section of  over W is an element L of XW such that Lp ∈ p for all p ∈ W . We are now in a position to introduce our main object of study: Definition I.3.2. A formally integrable structure over  is a complex vector sub-bundle  of CT satisfying the involutive (or Frobenius) condition: • If W ⊂  is open and L M ∈ XW are sections of  over W then L M is also a section of  over W . The rank (resp. corank) of  will be referred to as the rank (resp. corank) of the formally integrable structure  . Let  be a formally integrable structure over  and fix p ∈ . There is a local chart U x with p ∈ U and vector fields L1      Ln ∈ XU such that L1q      Lnq is a basis of q for every q ∈ U . If we write x = x1      xN  and Lj =

N 

ajk x

k=1

 xk

then the matrix ajk  has rank equal to n at every point; moreover, there are  cjk ∈ C  U, j k  = 1     n, such that

Lj  Lk =

n 

 cjk L 

j k = 1     n

=1

Definition I.3.3. A (classical) solution for the formally integrable structure  over  is a C 1 -function u on  such that Lu = 0 for every section L of  defined in an open subset of . More generally, we can consider the concept of (weak) solutions for the formally integrable structure  over : it suffices to consider u, in the preceding definition, belonging to the space of distributions on  (we refer to [H2] for the theory of distributions on manifolds).

I.4 Differential forms We shall denote by N the dual of the C  -module X and shall refer to its elements as differential forms over  of degree one (or one-forms for short). In other words, a one-form on  is a C  -linear map  X → C  

8

Locally integrable structures

Let  ∈ N, L ∈ X and suppose that L vanishes on an open subset V ⊂ . Then L also vanishes on V . Indeed, let p ∈ V and let g ∈ C   R be equal to one at p and vanish on \V . Then L = 1 − gL and consequently L = 1 − gL vanishes at p. In fact, we have a more precise result: Lemma I.4.1. Let  ∈ N, L ∈ X and suppose that Lp = 0. Then Lp = 0. Proof. By the preceding discussion it is clear that we can restrict a one-form on  to an open set W ⊂ , that is, given  ∈ N there is  W ∈ NW which makes the diagram 

X −→ C   ↓ ↓  W

XW −→ C  W commutative (the vertical arrows denote restriction homomorphisms). Let then U x be a local chart with p ∈ U . Then, if x = x1      xN  we have by (I.6)   N   p = 0 Lp = U LU p = Lxj pU xj j=1 The proof of Lemma I.4.1 is complete. If we then define  CTp∗  = dual of CTp  to each  ∈ N we can associate an element p ∈ CTp∗  by the formula p v = Lp where L ∈ X is such that Lp = v. As in the case for vector fields, we have a converse: if for every p ∈  an element p ∈ CTp∗  is given such that p → p Lp  ∈ C  

∀ L ∈ X

then there is  ∈ N such that p = p , for every p ∈ . Proposition I.4.2. CTp∗  = p  ∈ N . Proof. Let U x be a local chart with p ∈ U . Formula (I.6) allows one to define dxj ∈ NU, j = 1     N , by the rule

I.4 Differential forms  dxj

 xk

9

 = jk 

j k = 1     N

Hence, if  ∈ NU we have =

N 

 

j=1

 xj

 dxj 

(I.9)

where /xj  ∈ C  U. If we now observe that dxj p ⊂ CTp∗  is the dual basis of /xj p ⊂ CTp  then the conclusion will follow easily. Definition I.4.3. Given f ∈ C   we define df ∈ N by the formula dfL = Lf

L ∈ X

(I.10)

From (I.9) we obtain the usual representation in local coordinates   N N   f  dxj = dxj  df = df x x j j j=1 j=1 We now introduce the complexified cotangent bundle of  as being the disjoint union   CT ∗  = CTp∗  p∈

As before we can also introduce the notion of a complex vector sub-bundle of CT ∗  of rank m as being a disjoint union  p  = p∈

where each p is a vector subspace of CTp∗  of dimension m, satisfying the following property: • Given p0 ∈  there are an open set U0 containing p0 and one-forms 1      m ∈ NU0  such that 1p      mp span p for every p ∈ U0 . As before we shall refer to the space p as the fiber of  at the point p. Proposition I.4.4. Let  = ∪p∈ p be a complex vector sub-bundle of CT and set, for each p ∈ ,  p⊥ =  ∈ CTp∗   = 0 on p   Then  ⊥ = ∪p∈ p⊥ is a complex vector sub-bundle of CT ∗ .

10

Locally integrable structures

Proof. Given p0 ∈  there is a local chart x = x1      xN 

U0  x

with p0 ∈ U0 , and vector fields on U0 Lj =

N 

ajk

k=1

  xk

j = 1     n

such that L1p      Lnp spans p for all p ∈ U0 . After a contraction of U0 around p0 and a relabeling of the indices we can assume that the matrix ajk jk=1n is invertible in U0 . Let bjk jk=1n be its inverse and set L#j =

n 

bj L 

j = 1     n

=1

Then L#1p      L#np also spans p for all p ∈ U0 . Moreover, we have L#j =

m     + cjk xj k=1 xn+k

j = 1     n

where cjk are smooth in U0 and m = N − n. Set  = dxn+ −

n 

c dx 

 = 1     m

=1

Then 1p      mp are linearly independent for all p ∈ U0 and furthermore  L#j  = dxn+ L#j  − cj = 0 Hence 1p      mp is a basis for p⊥ for each p ∈ U0 . Remark I.4.5. It is clear that the preceding argument can be reversed. If  ⊥ is a vector sub-bundle of CT ∗  then it follows that  is a vector sub-bundle of  . When  is a formally integrable structure over  of dimension N we shall always denote the sub-bundle  ⊥ by T  . We shall also always denote by n the rank of  and by m the rank of T  . In particular, n + m = N . We shall also use the standard notation:  Tp  = v ∈ CTp  v is real   Tp∗  =  ∈ CTp∗   is real 

I.5 The Frobenius theorem

11

  T = Tp  p∈

  ∗ T ∗ = Tp  p∈

Given L ∈ X its (complex)-conjugate is the vector field L ∈ X defined by Lf = Lf 

f ∈ C  

In particular we shall say that L is a real vector field if L = L, that is, if LC   R ⊂ C   R. In the same way we can define the (complex)conjugate of an element in CTp . Given a subspace p ⊂ CTp  we define   p = v v ∈ p  It is clear from the definitions that if  is a complex vector sub-bundle of  CT then the same is true for  = ∪p∈  p . We shall refer to  as the (complex)-conjugate of the sub-bundle  . Analogous definitions and results can be introduced and obtained for CT ∗  and its fibers CTp∗ . It is also important to mention the equality ⊥

 =  ⊥ which is valid for every complex vector sub-bundle  of CT.

I.5 The Frobenius theorem We start by considering a real vector field L=

N  j=1

aj x

 xj

defined in a neighborhood of the origin in RN . Assume that L = 0. Then it is possible to find local coordinates y1  y2      yN , defined near the origin, such that  L=  y1 The proof of this result is very simple and will be recalled here. We assume that a1 0 = 0 and solve, in some neighborhood of the origin, the following Cauchy problem: ⎧ ⎨ xj /y1 = aj x1      xN  j = 1     N x 0 y2      yN  = 0 ⎩ 1 xj 0 y2      yN  = yj j = 2     N

12

Locally integrable structures

The fact that a1 0 = 0 implies that y1      yN  → x1      xN  is a smooth diffeomorphism at the origin and a simple computation shows our claim. The generalization of this result to a larger number of vector fields is the classical Frobenius theorem: Theorem I.5.1. Let L1      Ln be linearly independent, real vector fields defined in a neighborhood V of the origin in RN . Assume that the sub-bundle  of CTV generated by L1      Ln is a formally integrable structure. Then there are local coordinates y1  y2      yN , defined near the origin, such that  is generated by /y1 ,…, /yn . Proof. We shall proceed by induction on N . The case N = 1 is trivial. We then suppose that the result was proved for values < N . Applying the procedure described at the beginning of this section we can make a change of variables and assume that the given vector fields have the form: L1 =

N     Lj = ajk  j = 2     n x1 xk k=1

We then introduce a new set of generators for the bundle  : L#1 = L1  L#j = Lj − aj1 L1  j = 2     n Notice that when j ≥ 2 the vector field L#j does not involve differentiation in the x1 -variable. Thus, in a neighborhood of the origin, we have

L#j  L#k =

n 

 # Cjk L 

j k = 2     n

=2

If we then consider, in a neighborhood W of the origin in RN −1 , the vector fields N   ajk 0 x2      xN   j = 2     n Mj = x k k=2 as well as the sub-bundle   of CTW defined by them, we conclude the existence of a coordinate system y2      yN defined near the origin in RN −1 for which   is spanned by /y2      /yn  This argument has the following consequence: returning to the original coordinates x1      xN , the induction hypothesis allows us to assume from the beginning that ajk 0 x2      xN  = 0

j = 2     n k > n

I.5 The Frobenius theorem

13

Now, the coefficient of /x in the commutator L#1  L#j is equal to aj /x1 . On the other hand,

L#1  L#j =

n 

 # 1 C1j L = C1j

=1

N n      + C1j ak  x1 =2 x k k=2

and thus n aj   = C1j a  x1 =2

 = 2     N j = 2     n

Hence for each fixed  the vector a2      an  satisfies a linear system of ordinary differential equations with trivial initial condition. By the uniqueness theorem for such systems we conclude that aj = 0 if j = 2     n and  > n. Thus we have n   L#j = ajk  j = 2     n x k k=2 which concludes the proof. We now discuss the holomorphic version of the Frobenius theorem. Write the complex coordinates in C as z1      z , where zj = xj + iyj , and identify C  R2 by z = z1      z  → x1  y1      x  y  Given an open set  ⊂ C denote by  the algebra of holomorphic functions on . An element L ∈ X is said to be a holomorphic vector field if given any f ∈  we have Lf ∈  and Lf = 0. Introducing the standard notation      1   1     = −i = +i zj 2 xj yj zj 2 xj yj it is clear that every vector field L ∈ X can be written as       aj  + bj L= zj zj j=1

(I.11)

where aj  bj ∈ C  ; (I.11) is then a holomorphic vector field if and only if bj = 0 and aj ∈ , j = 1     . We now state the holomorphic version of the Frobenius theorem, whose proof is the same as that of Theorem I.5.1, working now in the holomorphic category.

14

Locally integrable structures

Theorem I.5.2. Let L1      Ln be linearly independent, holomorphic vector fields defined in a neighborhood V of the origin in C . Assume that the subbundle  of CTV generated by L1      Ln is a formally integrable structure. Then there are local holomorphic coordinates w1  w2      w , defined near the origin in C such that  is generated by /w1 ,…, /wn .

I.6 Analytic structures Let  be a real-analytic manifold, defined by the differentiable (real-analytic) structure  = V x . A function f  → C is real-analytic if for every V x ∈  the composition f x−1 is real-analytic on xV. Given U ⊂  an open set, we shall denote by U the space of real-analytic functions on U . An element L ∈ X is said to be a real-analytic vector field on  if L U ⊂ U

∀U ⊂  open

If L is given in local coordinates as in (I.6) then L is real-analytic if and only if its coefficients Lxj , j = 1     N , are real-analytic functions. Analogously, we shall say that  ∈ N is a real-analytic one-form on  if L ∈ U for every U ⊂  open and every real-analytic vector field L. From such definitions it is clear that one can introduce the notions of complex analytic vector sub-bundles of CT and of CT ∗ ; in particular we can refer to the notion of an analytic formally integrable structure over . Remark I.6.1. Suppose that  is now an open subset of RN and let L ∈ X be real-analytic. Write N   L = aj x  x j j=1 Let also u ∈  and take an open set C ⊂ CN , where the holomorphic coordinates are written as z1      zN , such that • C ∩ RN = ; • u, aj extend as holomorphic functions u˜ , a˜ j on C . Then ˜ u   Lu = L˜ ˜ is the holomorphic vector field where L ˜= L

N  j=1

a˜ j z

  zj

(I.12)

1.8 Some special structures

15

I.7 The characteristic set Let  ⊂ CT be a formally integrable structure over . The characteristic set of  is the subset of T ∗  defined by  (I.13) T 0 = T  ∩ T ∗  We shall also write Tp0 = Tp ∩ Tp∗  if p ∈ . If we recall that the symbol of a vector field L ∈ X is the function L T ∗  → C

L = Lp 

if  ∈ Tp∗ 

then we see that  ∈ Tp0 if and only if L = 0 for every section L of  . Let U x, x = x1      xN  be a local chart on . Take p ∈ U and  ∈ Tp∗ .   If we write  = Nj=1 j dxjp (j ∈ R) and L = Nj=1 aj /xj  then L =

N 

aj pj 

j=1

 Thus, if Lj = Nk=1 ajk /xk  are n linearly independent sections of  over U we can describe T 0 ∩ T ∗ U by the system of equations N 

ajk pk = 0

p ∈ U k ∈ R j = 1     n

k=1

Example I.7.1 (The Mizohata operator). If we write the coordinates in  = R2 as x t then    (I.14) M = − it ∈ XR2  t x is called the Mizohata vector field or Mizohata operator. We now describe the characteristic set of the formally integrable structure defined by M. From the equation  − it = 0 we get  0 if t = 0 0 Txt = dxp  ∈ R if p = x 0. This example in particular shows that T 0 is not, in general, a vector sub-bundle of T ∗ .

I.8 Some special structures Let  be a formally integrable structure over . We shall say that  defines • an elliptic structure if Tp0 = 0, ∀p ∈ ; • a complex structure if p ⊕  p = CTp , ∀p ∈ ;

16

Locally integrable structures

• a Cauchy–Riemann (CR) structure if p ∩  p = 0, ∀p ∈ ; • an essentially real structure if p =  p , ∀p ∈ . Before we proceed further we state some easy consequences of the preceding definitions. Proposition I.8.1. Every essentially real structure is locally generated by real vector fields. Proof. Given p0 ∈  we take vector fields L1      Ln which generate  in a neighborhood of p0 . By hypothesis the real vector fields Lj , Lj are also sections of  . Moreover,     span  Lj p  Lj p j = 1     n = p0 0

0

    and consequently n of the tangent vectors Lj p  Lj p are linearly 0 0 independent. Since this remains true in a neighborhood of p0 the result is proved. Next we recall a very elementary but useful result. Lemma I.8.2. If V is a vector subspace of CN = RN + iRN and if V 0 = V ∩ RN then V 0 ⊗R C  V 0 + iV 0 = V ∩ V . Proof. We only verify the equality. If x y ∈ V 0 then x ± iy ∈ V and so V 0 + iV 0 ⊂ V ∩ V . For the reverse inclusion take z ∈ V ∩ V . Then z=

i 1 z + z − iz − iz ∈ V 0 + iV 0  2 2

As a consequence, given any formally integrable structure  over  we have 

Tp0 ⊗R C  Tp ∩ T p 

∀p ∈ 

Since for a complex structure we also have 

Tp ⊕ T p = CTp∗  we obtain:

∀p ∈ 

(I.15)

I.8 Some special structures

17

Corollary I.8.3. Every complex structure is elliptic. Unlike what happens with Mizohata structures we have: Proposition I.8.4. If  defines a CR structure over  then T 0 is a vector  sub-bundle of T ∗  of rank d = N − 2n. Proof. If p ∩  p = 0, for all p ∈ , then     =  ⊕ = p ⊕  p p∈

is a vector sub-bundle of CT (of rank 2n) which defines an essentially real structure over . By Proposition I.4.4,  ⊥ is a vector sub-bundle of ⊥ CT ∗  of rank d which of course satisfies p⊥ =  p for all p ∈ . The same argument used in the proof of Proposition I.8.1 shows that  ⊥ has local real generators. Since these generators span T 0 the proof is complete. In order to obtain appropriate local generators for a formally integrable structure we shall need an elementary result: Lemma I.8.5. Let V be a complex subspace of CN of dimension m. Let   V0 = V ∩ RN , d = dimR V0 ,  = m − d. Let also V1 ⊂ CN be a subspace such that V0 ⊕ iV0  ⊕ V1 = V and take: 1       basis for V1 

+1      m real basis for V0 

If we write j = j + ij , j = 1     , then: 1        +1      m is a basis for V ;

(I.16)

1      m  1       is linearly independent over R

(I.17)

 + m ≤ N

(I.18)

Proof. Notice that (I.16) is trivial since +1      m is also a basis for V0 ⊕ iV0 . Next we notice that V ∩ V 1 = 0. Indeed, let z ∈ V ∩ V 1 . Then z ∈ V1 ⊂ V and consequently z z ∈ V0 , which gives z ∈ V0 ⊕ iV0  ∩ V1 = 0. Hence 1         1         +1      m is linearly independent. In particular, 2 +d =  +m ≤ N and (I.17) holds. Given a formally integrable structure  over  and fixing p ∈  we shall apply Lemma I.8.5 with the choices V = Tp 

V0 = Tp0 

18

Locally integrable structures

If 1        +1      m is the basis given in (I.16) we first take a system of local coordinates x1      x  y1      y  s1      sd  t1      tn vanishing at p such that, writing zj = xj + iyj we have dzj p = j  dsk p = +k 

j = 1      k = 1     d

Afterwards we take one-forms 1        1      d which spanT  in a neighborhood of p and such that j p = dzj p 

k p = dsk p 

j = 1      k = 1     d

If L is a complex vector field on  defined near p we can write it in the form L=



Aj

j

       + Bj + Ck + D  zj zj sk t j k 

If, furthermore, L is a section of  we necessarily must have Aj = Ck = 0 at p for all j and k. Since  + n = n, it follows that after a linear substitution we can find a set of local generators of the sub-bundle  in a neighborhood of p of the form Lj = ˜ = L

 d      + ajj  + bjk  zj j  =1 zj  k=1 sk

 d      + a˜ j  + b˜ k  t j  =1 zj  k=1 sk

j = 1     

(I.19)

 = 1     n 

(I.20)

where the coefficients ajj  , a˜ j  , bjk , b˜ k all vanish at p. We notice that the elliptic case corresponds to the situation when d = 0, the complex case to the one when d = n = 0, and the CR case to the one when n = 0. Next we introduce a generalization of the structure defined by the Mizohata operator (cf. Example I.7.1). Definition I.8.6. We shall say that a formally integrable structure  over  is a generalized Mizohata structure at p0 ∈  if p0 = p0 . Thus in the case of generalized Mizohata structures the coordinates vanishing at p0 can be taken as s1      sm  t1      tn  [d = m, n = n in this case] and  is spanned by the vector fields L =

d    + b˜ k s t  t k=1 sk

 = 1     n

I.9 Locally integrable structures

19

where bk = 0 at the origin for every  k. Finally we recall the classical notion of the so-called CR functions: Definition I.8.7. Given a CR formally integrable structure  over , any classical solution (for the formally integrable structure  ) is called a CR function. Needless to add, we can also introduce the concept of CR distributions, etc.

I.9 Locally integrable structures A complex vector sub-bundle  of CT, of rank n, is said to define a locally integrable structure if given an arbitrary point p0 ∈  there are an open neighborhood U0 of p0 and functions Z1      Zm ∈ C  U0 , with m = N − n, such that span dZ1p      dZmp = p⊥ 

∀p ∈ U0 

(I.21)

If one observes that the differential of a smooth function g is a section of  ⊥ if and only if Lg = 0 for every section of  , it follows easily that every locally integrable structure satisfies the Frobenius condition. Hence, every locally integrable structure defines a formally integrable structure. We have: • The formally integrable structure  is locally integrable if and only if, given p0 ∈  and vector fields L1      Ln which span  in an open neighborhood U0 of p0 , there are an open neighborhood V0 ⊂ U0 of p0 and smooth functions Z1      Zm ∈ C  V0  such that: dZ1 ∧    ∧ dZm = 0 Lj Zk = 0

in

V0 

j = 1     n k = 1     m

Thus, checking local integrability is equivalent to looking for a maximal number of nontrivial solutions to the (in general overdetermined) homogeneous system defined by a fixed set of independent sections of  . Theorem I.9.1. Every essentially real structure is locally integrable. Proof. By Frobenius Theorem I.5.1, in conjunction with Proposition I.8.1, given p ∈  we can find a local chart U x, x = x1      xN , with p ∈ U , such that   j = 1     n xj

20

Locally integrable structures

are sections of  over U . It suffices to take k = 1     m

Zk = xk+n 

Theorem I.9.2. Every analytic formally integrable structure is locally integrable. Proof. We shall prove that if L1      Ln are linearly independent, realanalytic vector fields in an open ball B centered at the origin in RN such that n 

L  L =

 C L 

=1

where ∈ B, then we can find real-analytic functions Z1      Zm defined in a neighborhood of the origin and satisfying  C

Lj Z = 0

j = 1     n  = 1     m dZ1 ∧    ∧ dZm = 0

We write Lj =

N 

ajk

k=1

 xk

and take an open, connected set U ⊂ CN such that U ∩ RN = B and such that  there are a˜ jk , C˜  ∈ U satisfying   a˜ jk = ajk  C˜  = C

in B

Consider then the holomorphic vector fields in U : ˜j = L

N 

a˜ jk

k=1

  zk

By analytic continuation the coefficients of the holomorphic vector fields ˜  − ˜  L

L

n 

 ˜ L C˜ 

=1

must vanish identically in U since they vanish on B and the former is connected. By the holomorphic version of the Frobenius theorem we can find holomorphic functions W1      Wm defined in an open neighborhood V ⊂ U of the origin in CN such that ˜ j W = 0 L

j = 1     n  = 1     m

I.10 Local generators

21

dW1 ∧    ∧ dWm = 0  It suffices then to set Zk = Wk V ∩B in order to obtain the desired solutions (cf. (I.12)). Example I.9.3. For the Mizohata vector field (I.14) we have MZ = 0 in R2 , where Zx t = x + it2 /2. Notice that dZ = 0 everywhere.

I.10 Local generators In this section we shall construct appropriate local coordinates and local generators of the sub-bundle T  when the structure  is locally integrable. Once more we shall apply Lemma I.8.5. Let p ∈  and let also G1      Gm be smooth functions defined in a neighborhood of p such that dG1      dGm span T  . As in Section I.8 we make the choices: V = Tp , V0 = Tp0 . If 1        +1      m is the basis given in (I.16) then we can find cjk  ∈ GLm C such that m 

cjk dGk p = j 

j = 1     

k=1 m 

cjk dGk p = j 

j =  + 1     m

k=1

We then set Zj =

m 

cjk Gk − Gk p 

j = 1     

k=1

W =

m 

c+k Gk − Gk p 

 = 1     d

k=1

It is clear that dZ1      dZ  dW1      dWd also span T  in a neighborhood of p. If we further set xj = Zj  yj = Zj  s = W then (I.17) gives that dx1      dx  dy1      dy  ds1      dsd are linearly independent at p. We are now ready to state and prove the following important result:

22

Locally integrable structures

Theorem I.10.1. Let  be a locally integrable structure defined on a manifold . Let p ∈  and d be the real dimension of Tp0 . Then there is a coordinate system vanishing at p, x1      x  y1      y  s1      sd  t1      tn and smooth, real-valued functions 1      d defined in a neighborhood of the origin and satisfying k 0 = 0 dk 0 = 0

k = 1     d

such that the differentials of the functions  Zj x y = zj = xj + iyj 

j = 1     

Wk x y s t = sk + ik z s t

(I.22)

k = 1     d

(I.23)

span T  in a neighborhood of the origin. In particular, we have  + d = m,  + n = n and also Tp0 = span ds1 0      dsd 0 

(I.24)

Proof. The proof follows almost immediately from the preceding discussion: it suffices to take smooth, real-valued functions t1      tn defined near p and vanishing at p such that dx1      dx  dy1      dy  ds1      dsd  dt1     dtn are linearly independent. Notice that dWk p = +k is real, from which we derive that dk = 0 at the origin. Since we have Wk 0 0 0 = kk  sk

k k = 1     d 

we can introduce, in a neighborhood of the origin in R2+d+n , the vector fields d   kk z s t  k = 1     d (I.25) Mk = s  k k =1 characterized by the relations Mk Wk = kk 

(I.26)

Consequently the vector fields Lj =

d  k  −i z s tMk  zj k=1 zj

j = 1     

(I.27)

I.10 Local generators

˜ = L

d   k −i z s tMk  t k=1 t

 = 1     n

23

(I.28)

are linearly independent and satisfy ˜  Zj  = L j Wk = L ˜  Wk = 0 L j Zj  = L for all j j  = 1     ,  = 1     n , and k = 1     d. Hence ˜ 1     L ˜ n span  in a neighborhood of the origin. L1      L  L

(I.29)

Notice that the one-forms dz1      dz  dz1      dz  dW1      dWd  dt1      dtn

(I.30)

span CT ∗  near the origin. Moreover, the dual basis of (I.30) is given by ˜ 1     L ˜ n  L1      L  L1      L  M1      Md  L

(I.31)

where Lj =

d  k  −i z s tMk  zj k=1 zj

j = 1     

(I.32)

the vector fields (I.31) are pairwise commuting.

(I.33)

Finally we observe that

Indeed it suffices to notice that if P Q are any two of the vector fields (I.31) and if F is any one of the functions Zj  Zj  Wk  t , the fact that (I.30) is dual to (I.31) gives dF  P Q  = P Q F = 0 from which we obtain that P Q = 0. In many cases we do not need the precise information provided by Theorem I.10.1 and the following particular case is enough: Corollary I.10.2. Same hypotheses as in Theorem I.10.1. Then there is a coordinate system vanishing at p, x1      xm  t1      tn and smooth, real-valued 1      m defined in a neighborhood of the origin and satisfying k 0 0 = 0

dx k 0 0 = 0

k = 1     m

such that the differentials of the functions Zk x t = xk + ik x t 

span T in a neighborhood of the origin.

k = 1     m

(I.34)

24

Locally integrable structures

If we write Zx t = Z1 x t     Zm x t then Zx 0 0 equals the identity m × m matrix. Hence we can introduce, in a neighborhood of the origin in RN , the vector fields Mk =

m 

k x t

=1

  x

k = 1     m

(I.35)

characterized by the relations Mk Z = k 

(I.36)

Consequently the vector fields Lj =

m  k  −i x tMk  tj k=1 tj

j = 1     n

(I.37)

are linearly independent and satisfy Lj Zk = 0, for j = 1     n, k = 1     m. The same argument as before gives: L1      Ln span  in a neighborhood of the origin; L1      Ln  M1      Mm are pairwise commuting and span CT RN in a neighborhood of the origin in RN .

(I.38) (I.39)

Let U be an open set of Rn and assume, given a smooth function ! U → Rm , !t = 1 t     m t. We shall call a tube structure on Rm × U the locally integrable structure  on Rm × U for which T  is spanned by the differentials of the functions Zk = xk + ik t

k = 1     m

A tube structure  has remarkably simple global generators. Indeed if we set, as usual, Z = Z1      Zm  we have Zx x t = I, the identity m × m matrix, for every x t ∈ Rm × U . This gives Mk = /xk and consequently the vector fields (I.37) take the form Lj =

m  k   −i t tj xk k=1 tj

j = 1     n

Observe that these vector fields span  on Rm × U .

(I.37 )

I.11 Local generators in analytic structures

25

I.11 Local generators in analytic structures When  is real-analytic then the functions k in Corollary I.10.2 can be taken real-analytic. We keep the notation established in the preceding section and consider the equation Zx t − z = 0 for x t z ∈ Cm × Cn × Cm in a neighborhood of the origin. Since Z 0 0 = I x we can find, by the implicit function theorem, a holomorphic function x = Hz t = H1 z t     Hm z t defined in a neighborhood of the origin in Cm × Cn satisfying H0 0 = 0

HZx t t = x

We set  Zk# x t = Hk Zx t 0

k = 1     m

Then we also have Lj Zk# = 0 j = 1     n k = 1     m dZ1# ∧    ∧ dZm# = 0 Moreover, Zk# x 0 = xk for every k. Hence, if we consider the real-analytic diffeomorphism x t → X T = Z# x t t in these new variables we can write Zk# X T = Xk + i!k# X T where now we have !k# X 0 = 0 for every k. Summing up we can state: Corollary I.11.1. Let  be a locally integrable real-analytic structure defined on a real-analytic manifold . Let p ∈ . Then there is a realanalytic coordinate system vanishing at p, x1      xm  t1      tn and real-analytic, real-valued 1      m defined in a neighborhood of the origin and satisfying k x 0 = 0

k = 1     m

such that the differentials of the functions Zk = xk + ik x t span T  in a neighborhood of the origin.

k = 1     m

(I.40)

26

Locally integrable structures

Remark I.11.2. We point out that in the coordinates x t given by Corollary I.11.1 it is elementary to find the unique analytic solution u to the Cauchy problem:  Lj u = 0 j = 1     n (I.41) ux 0 = hx where h is real-analytic. Indeed, ux t = hZ1# x t     Zm# x t solves (I.41) and in order to see that this is the unique analytic solution it suffices to notice that if v is analytic, if vx 0 = 0, and if Lj v = 0 for every j then v must vanish identically since all its derivatives vanish at the origin. Uniqueness for the distribution solutions of (I.41) holds when the structure is only C  . This, though, is a much deeper result and its discussion will be postponed to Chapter II.

I.12 Integrability of complex and elliptic structures The celebrated theorem of Newlander and Nirenberg ([NN]) states that every complex structure is locally integrable. We shall postpone the proof of this result to the appendix of this chapter and now we will apply it to prove the more general statement that in fact every elliptic structure is locally integrable. This result is due to L. Nirenberg. Theorem I.12.1. Let  be an elliptic structure over a smooth manifold . Then  is locally integrable. 

Proof. By (I.15) we have Tp ∩ T p = 0 for every p ∈  and then       Tp ⊕ T p T ⊕T = p∈

is a vector sub-bundle of CT ∗  of rank 2m. In particular, if n is the dimension of , we obtain that 2m ≤ n. Thus      ⊥    ∩ = Tp ⊕ T p p ∩  p = p∈

p∈

is a vector sub-bundle of CT. By the argument that led to the proof of Proposition I.8.1 we see then that     ∩ T = p ∩ Tp  p∈

I.12 Integrability of complex and elliptic structures is a vector sub-bundle of T. Notice that    n = dimR p ∩ Tp  = n − 2m

27

p ∈ 

Let p0 ∈  be fixed. By the Frobenius Theorem I.5.1 we can find a coordinate system x1      x2m  t1      tn  around p0 such that  ∩ T is generated near p0 by the vector fields   j = 1     n  tj Next we select m complex vector fields Lk =

2m 

ak x t

=1

 x

in such a way that L1      Lm  /t1      /tn span  in a neighborhood of p0 . After a linear substitution (as in the proof of Proposition I.4.4) we can assume that the vector fields Lk take the form Lk =

2m    + b x t  xk =m+1 k x

k = 1     m

Since  is a formally integrable structure, we know /t  Lk must be a linear combination of L1      Lm , /t1      /tn . Due to the special form of the vector fields Lk these brackets must vanish identically, that is: 2m  bk  x t = 0 x =m+1 t

∀ k

Consequently, the functions bk do not depend on t1      tn in a full neighborhood of p0 . Since, moreover, L1      Lm  L1      Lm 

   t1 tn

span CT it follows that L1      Lm  L1      Ln are linearly independent. We conclude then that L1      Lm define a complex structure (in the x-space) in a neighborhood of p0 . By the Newlander–Nirenberg theorem there are Z1 x     Zm x with linearly independent differentials such that Lk Z = 0

k  = 1     m

Since, moreover, Z = 0 tj the proof is complete.

 = 1     m j = 1     n

28

Locally integrable structures

Theorem I.10.1 gives a particularly simple local representation for an elliptic structure. Let  and  be as in Theorem I.12.1 and fix p ∈ . With the notation as in Theorem I.10.1 we have d = 0,  = m and thus there is a coordinate system x1      xm  y1      ym  t1     tn  vanishing at p such that, setting zj = xj + iyj , the differentials dzj span T  near p, and the vector fields /zk , /tj span  near p. Notice also that n = 0 corresponds to the case when  defines a complex structure.

I.13 Elliptic structures in the real plane In this section we depart a bit from the spirit we have adopted in the exposition up to now and make use of some standard results on Fourier analysis and pseudo-differential operators in order to study elliptic structures in twodimensional manifolds. The results contained here are not necessary for the comprehension of the remaining parts of the chapter and the section can be avoided in a first reading. If  is an open subset of R2 any sub-bundle  of CT of rank one defines a formally integrable structure over , for the involutive condition is automatically satisfied. Suppose that  contains the origin and let L be a complex vector field that spans  in a neighborhoord of 0. After division by a nonvanishing smooth factor it can be assumed that, in suitable coordinates x1  x2 , we can write L=

  + ax1  x2   x2 x1

As at the beginning of Section I.5 we can find a smooth diffeomorphism x t → x1  x2 , x2 = t, which reduces L to /t. Since also /x1 is a multiple of /x in these new variables, L can be written as a nonvanishing multiple of   L• = + ibx t  (I.42) t x where b is smooth and real-valued. Since both L and L• span  in a neighborhood of the origin of R2 , there is no loss of generality in assuming that our original L takes the form (I.42). The structure  is elliptic if and only if L and L are linearly independent at every point. This is equivalent to saying that the function b in (I.42) never vanishes (in the p.d.e. terminology, L is an elliptic operator). We shall now

I.13 Elliptic structures in the real plane

29

recall the standard elliptic estimates satisfied by L and its transpose t L in a neighborhood of the origin. Let L0 =

  + ib0 0  t x

(I.43)

If  ∈ Cc R2  then taking Fourier transforms gives  L0   = i − b0 0   Since b0 0 = 0 we have

  2 +  2 ≤ max 1

 1

i − b0 0 2 b0 02

and thus by Parseval’s formula we obtain, in Sobolev norms, 1 ≤ C L0 0 + 0    ∈ Cc R2 

(I.44)

where for any real s we denote by s the norm in the Sobolev space L2s R2  (see Section II.3.2 for the definition of Sobolev norms). We select an open neighborhood of the origin U ⊂  such that bx t − b0 0 ≤ 1/2C for x t ∈ U . If  ∈ Cc U then by (I.44) 1 ≤ C L0 +  L − L0  0 + 0  = C L0 +  bx t − b0 0 x 0 + 0  1 ≤ C L0 + 0  + 1  2 and thus 1 ≤ 2C L0 + 0    ∈ Cc U

(I.45)

Let now V ⊂⊂ U be an open set and let also  ∈ Cc U be identically equal to one in V . We denote by " the operator ‘multiplication by ’ and by # the operator 1 − $1/2 . For a real number s and for  ∈ Cc V, we obtain s+1 = #s "1 ≤ "#s 1 + C1 s since the commutator between #s and " has order s − 1. If we now apply (I.45) we obtain s+1 ≤ C2 L"#s 0 + "#s 0 + s ≤ C3 "#s L0 + s since both "#s and its commutator with L have order s. We then obtain: • For every V ⊂⊂ U open and every s ∈ R there is C • > 0 such that s+1 ≤ C • Ls + s   ∈ Cc V

(I.46)

30

Locally integrable structures

2s+1 Proposition I.13.1. If u ∈  U and Lu ∈ L2s loc U then u ∈ Lloc U. In particular, if u ∈  U and Lu ∈ C  U then u ∈ C  U.

Proof. Let W ⊂⊂ V ⊂⊂ U be open sets and let  ∈ Cc V be identically equal to one in W . Since there is  ≤ s such that u ∈ L2 loc V it will suffice to show that u ∈ L2+1 , for iteration of the argument will give the result. Let B% = &% ∗ ·, where &% is the usual family of mollifiers in R2 . We have B% u → u in L2 as % → 0 and also %→0

LB% u = B% Lu + L B% u −→ Lu in L2 by Friedrich’s lemma, since Lu ∈ L2 . Thus, if take %n → 0 and if we apply (I.46) for s =  and  = B%m u − B%n u we conclude that B%n u is a Cauchy sequence in L2+1 . Hence u ∈ L2+1 and the proof is complete. We shall now derive from (I.45) an estimate for the transpose of L which will lead us to a solvability result. If we notice that t L = −L − ibx x t then from (I.45) we obtain, for some constant C  > 0, 1 ≤ C  t L0 + 0    ∈ Cc U

(I.47)

Now, it is elementary that 0 ≤ 2t 0 

 ∈ Cc U

where  = sup t x t ∈ U . Consequently, if we further contract U about the origin in order to achieve 2C  ≤ 1/2, from (I.47) we finally obtain 0 ≤ 2C  t L0   ∈ Cc U

(I.48)

Proposition I.13.2. For every f ∈ L2 U there is u ∈ L2 U such that Lu = f in U . Proof. Given f ∈ L2 U consider the functional  t L → fx tx tdxdt

(I.49)

defined on t L  ∈ Cc U , where the latter is considered as a subspace of L2 U. By (I.48) it follows that (I.49) is well-defined and continuous. By the Hahn–Banach theorem we extend (I.49) to a continuous functional  on L2 U and by the Riesz representation theorem we find u ∈ L2 U such that  g = gx tux tdxdt g ∈ L2 U

I.13 Elliptic structures in the real plane In particular, if  ∈ Cc U t L =

31

 fx tx tdxdt

which is precisely the meaning of the equality Lu = f in the weak sense. Corollary I.13.3. Let D ⊂⊂ U be an open disk centered at the origin. Then L C  D = C  D

(I.50)

Proof. Given f ∈ C  D we extend it to an element f˜ ∈ Cc U and by Proposition I.13.2 we find u ∈ L2 U solving Lu = f˜ in U . Finally, by Proposition I.13.1, we have u ∈ C  U and thus its restriction to D belongs to C  D. Still under the assumption that L is elliptic we apply (I.50) in order to find v ∈ C  D such that Lv = −ibx  If we set ux t =



x

(I.51)



evx t dx

0

we get Lux t =



x

0





vt x  tevx t dx + ibx tevxt

x



−ibvx − ibx  x  tevx t dx + ibx tevxt  x = −i x bev x  tdx + ibx tevxt =

0

0

= ib0 tev0t  Then if we set Zx t = ux t − i



t



b0 t ev0t  dt

(I.52)

0

we obtain LZ = 0

Zx = ev = 0

(I.53)

that is, our original elliptic structure  is locally integrable. We have thus obtained a proof of the Newlander–Nirenberg theorem in the particular case when N = 2. We emphasize for this situation the conclusion that we have reached at the end of Section I.12:

32

Locally integrable structures

Corollary I.13.4. If L is an elliptic operator in an open subset  ⊂ R2 and if p ∈  then we can find local coordinates x y vanishing at p such that L can be written, in a neighborhood of p, as     +i  L = gx y x y where g never vanishes. Remark I.13.5. Our discussion indeed leads to a general criterion that characterizes when a rank one formally integrable structure  ⊂ CT,  ⊂ R2 open, is locally integrable. Suppose that  is spanned, in a neighborhood of the origin, by the vector field (I.42). Proposition I.13.6. The following properties are equivalent: † there is Z ∈ C  near the origin solving LZ = 0, Zx = 0; ‡ there is v ∈ C  near the origin solving (I.51). Proof. We have already presented the argument that ‡ ⇒ †. For the reverse implication we notice that 0 = LZx = LZx  + ibx Zx and consequently L logZx  = Zx−1 LZx  = −ibx 

I.14 Compatible submanifolds Let  be a smooth manifold. A subset  of  is called an embedded submanifold (or submanifold for short) of  if there is r ∈ 0 1     N for which the following is true: • Given p0 ∈  arbitrary there is a local chart U0  x, with p0 ∈ U0 and x = x1      xN , such that U0 ∩  = q ∈ U0 xr+1 q = xr+1 p0      xN q = xN p0   When p0 runs over  the pairs U0  x0 , where x0 = x1 U0 ∩      xr U0 ∩  make up a family  ∗ that satisfies properties (1) and (2) of Section I.1 Hence  is a smooth manifold of dimension r. We shall refer to the number N − r as the codimension of  (in ).

I.14 Compatible submanifolds

33

 Let p ∈  and denote by C p the space of germs of smooth functions on  at p. It is clear that the restriction to  defines a surjective homomorphism  p which gives us then a natural injection o f C-algebras C  p → C

'p CTp   CTp 

(I.54)

By transposition we thus obtain a surjection 'p ∗ CTp∗  −→ CTp∗ 

(I.55)

whose kernel will be denoted by CNp∗ . We shall sometimes refer to the disjoint union   CNp∗  (I.56) CN ∗  = p∈

as the complex conormal bundle of  in . Let now U ⊂  be open and let  ∈ NU. Given L ∈ XU ∩  the map   p → '∗p p  Lp  is easily seen to be smooth on U ∩ . By the discussion that precedes Proposition I.4.2, there is a form • ∈ NU ∩  such that •p = 'p ∗ p  for every p ∈ U ∩ . We shall denote • by '∗  and shall refer to it as the pullback of  to U ∩ . It is clear that '∗ is a homomorphism which is moreover surjective when U ∩  is closed in U . Observe also that '∗ df = df U ∩ 

f ∈ C  U

(I.57)

Let now  be a formally integrable structure over , with T  =  ⊥ , and let  ⊂  be a submanifold. If p ∈  we set     p  (I.58)  p = p ∩ CTp    = p∈

With orthogonal now taken in the duality CTp  CTp∗  we have 'p ∗ Tp  =  ⊥ p

(I.59)

since the left-hand side is the image of the composition 'p ∗

Tp  CTp∗  −→ CTp∗  and consequently is equal to the orthogonal to the kernel of the composition CTp   CTp  −→ CTp /p 

34

Locally integrable structures

Definition I.14.1. We shall say that  is compatible with the formally integrable structure  if   defines a formally integrable structure over . When  is compatible with  then, according to our previous notation,  ∗  T  p =  ⊥ p = 'p  Tp  (cf. (I.59)). The next result gives a very useful criterion: Proposition I.14.2. The submanifold  is compatible with  if (and only if) p → dim  p is constant on .

(I.60)

Proof. We must prove that (I.60) implies that   is a vector sub-bundle of  which satisfies the Frobenius condition. First we observe that (I.60) and (I.59) give the existence of  such that dim'p ∗ Tp  =  ∀p ∈ 

(I.61)

Let p0 ∈  and take 1      m ∈ NU0 , where U0 is an open subset of  that contains p0 , such that 1 q      m q span Tq for every q ∈ U0 . Select j1      j such that

∗ ' j1 p0      '∗ j p0 form a basis for 'p0 ∗ Tp 0 . Then

∗ ' j1 p      '∗ j p will still be linearly independent when p belongs to an open neighborhood V0 of p0 in  and consequently, thanks to (I.61), will form a basis to 'p ∗ Tp  for all such p. By the remark that follows Proposition I.4.4 we conclude that   is a vector sub-bundle of  . To conclude the argument it suffices to observe that if U is an open subset of  and if L M ∈ XU are such that Lp  Mp ∈ CTp  for every p ∈ U ∩  then L M p ∈ CTp  also for every p ∈ U ∩ . This property will easily imply that   satisfies the Frobenius condition. Proposition I.14.3. If  is a locally integrable structure over  and if  is a submanifold of  which is compatible with  then   is a locally integrable structure over . Proof. It follows from the proof of Proposition I.14.2 in conjunction with (I.57).

I.14 Compatible submanifolds

35

Example I.14.4. Generic submanifolds of complex space. As in Section I.5 we shall write the complex coordinates in C as z1      z , where zj = xj +iyj . If f is a smooth function on an open subset of C we shall write, as usual, f =

  f dzj  z j j=1

(I.62)

f =

  f dzj  z j j=1

(I.63)

Definition I.14.5. Let  be a submanifold of C of codimension d. We shall say that  is generic if given p0 ∈  there are an open neighborhood U0 of p0 in C and real-valued functions &1      &d ∈ C  U0  such that  ∩ U0 = z ∈ U &k z = 0 k = 1     d and &1      &d are linearly independent at each point of  ∩ U0 . Notice that every one-codimensional submanifold of C is automatically generic. Denote by  01 the sub-bundle of CT C which defines the complex structure on C , that is, the sub-bundle spanned by the vector fields /zj , j = 1     . Proposition I.14.6. If  is a generic submanifold of C of codimension d then  is compatible with  01 . Moreover,  01  is a locally integrable, CR structure for which n and m satisfy: dim  = 2n + d

m =  = n + d



The sub-bundle T  is spanned by the differentials of the restriction to  of the complex coordinate functions on C .  Proof. Let p ∈ . A vector j=1 aj /zj p belongs to CTp  ∩ p01 if and only if   & aj k p = 0 k = 1     d zj j=1 Since  is generic it follows that   dimC CTp  ∩ p01 =  − d

∀p ∈ 

By Propositions I.14.2 and I.14.3 we conclude that  is compatible with  01 and that  01  is locally integrable. Moreover, since  01 p ∩  01 p = 0 for every p ∈ C we obtain

36

Locally integrable structures  01 p ∩  01 p = 0

∀p ∈ 

which shows that  01  defines a CR structure over . Finally, we  have n = rank  01  =  − d and thus dim  = 2 − d = 2n + d and m = dim  − n = n + d. The last statement follows immediately from the proof of Proposition I.14.2.

I.15 Locally integrable CR structures When  defines a locally integrable CR structure over  then, according to  Proposition I.8.4, d = dim Tp0 = N − 2n, for all p ∈ . Using Theorem I.10.1 we obtain m = N − n = n + d,  = m − d = n and n = N − 2 − d = 0. We summarize: • Given p ∈  there is a coordinate system vanishing at p, x1      xn  y1      yn  s1      sd and smooth, real-valued functions 1      d defined in a neighborhood of the origin and satisfying k 0 = 0 dk 0 = 0

k = 1     d

(I.64)

such that the differentials of the functions Zj = xj + iyj 

j = 1     n

Wk = sk + ik z s

k = 1     d

(I.65) (I.66)



span T in a neighborhood of the origin. Notice that  is spanned, in a neighborhood of the origin, by the pairwise commuting vector fields (I.27), where  = n and there is no t-variable. Suppose that  = 1      d  is defined in a neighborhood U of the origin in Cn × Rd . Then the map F U → Cn+d 

Fz s = z s + iz s

(I.67)

has rank 2n + d and consequently FU is an embedded submanifold of Cn+d of dimension 2n + d (and of codimension d). Now we write the coordinates in Cn+d as z1      zn  w1      wd  where w = s + it, wj = sj + itj . Then FU is defined by the equations  &k z w = k z s − tk = 0 k = 1     d

I.15 Locally integrable CR structures Since

37

  &k 1 k + ik  k  = 1     d = w 2 s

we conclude, taking into account (I.64), that FU is generic if U is taken small enough so that       z s ≤ 1  z s ∈ U (I.68)  s  2 By Proposition I.14.6 the complex structure  01 on Cn+d defines a locally integrable CR structure  FU on FU for which the sub-bundle T  FU is spanned by the differentials of the restrictions of the functions z1      zn , w1      wd to FU. Since in the local coordinates z s we have zj F U = Zj z s

wk F U = Wk z s

(cf. (I.65), (I.66)), we can state: Proposition I.15.1. Every locally integrable CR structure can be locally realized as the CR structure induced by the complex structure on a generic submanifold of the complex space. Remark I.15.2. Let  be a tube structure on Rm × U (cf. Section I.10). Thus U is an open subset of Rn and we assume given smooth, real-valued functions 1      m on U such that T  is spanned by the differential of the functions Zk = xk + ik t, k = 1     m. Recall that  is then spanned on Rm × U by the vector fields (I.37 ). Let us now assume that  is also a CR structure. Let d = m − n be the rank of the characteristic set T 0 (cf. Proposition I.8.4).  ∗ Since  being CR demands that Txt + T  xt = CTxt Rm × U for every m x t ∈ R × U , we must then have rank ! t = n

∀t ∈ U

 where ! = 1      m . This implies that  = !U is an embedded submanifold of Rm of dimension n and it is clear that  can be realized as the CR structure induced by the complex structure on the generic submanifold Rm + i of Rm + iRm = Cm . One very important model of a CR structure is the Hans Lewy structure. We take as  the space C × R, where the coordinates are written as z = x + iy and s, and consider the formally integrable structure  spanned by the Hans Lewy vector field (or operator) L=

  − iz  z s

(I.69)

38

Locally integrable structures

Since L and L are linearly independent at every point it follows that  defines a CR structure which is furthermore locally integrable, since the differential of the functions z and W = s + i z 2 span T  on C × R. Notice also that the Hans Lewy structure can be globally realized as the CR structure induced on the hyperquadric  Q = z w ∈ C2 w = s + it t = z 2 (I.70) by the complex structure on C2 . More generally, given %j ∈ −1 1 , j = 1     n, we can consider the CR structure  on Cn × R spanned by the pairwise commuting vector fields Lj =

  − i%j zj  s zj

j = 1     n

(I.71)

Such a structure is also locally integrable for the differential of the functions z1      zn and W = s + iz, with z =

n 

%j zj 2 

j=1

span T  on Cn × R.

I.16 A CR structure that is not locally integrable In this section we shall prove the following quite involved result: Proposition I.16.1. Let %1 = 1

%j = −1 j = 2     n

(I.72)

There is a smooth function gz s defined in an open neighborhood  of the origin in Cn × R and vanishing to infinite order at z1 = 0, such that if we set L#j =

  − i%j zj 1 + gz s  zj s

j = 1     n

(I.73)

then the following is true: (a) the vector fields L#j are pairwise commuting; (b) if h is a C 1 function near the origin satisfying L#j h = 0 j = 1     n then h/s0 0 = 0. Before we embark on the proof we shall state and prove the important consequence of this result:

I.16 A CR structure that is not locally integrable

39

Corollary I.16.2. The vector fields (I.73) span a CR structure which is not locally integrable in any neighborhood of the origin. Indeed, first we notice that L#1      L#n  L#1      L#n are linearly independent over  which together with property (a) shows that (I.73) define a CR structure over . Now, given any smooth solution h to the system L#j h = 0

j = 1     n

(I.74)

we necessarily have h/zj 0 0 = 0 for all j = 1     n. By property (b)  we then obtain dh = nj=1 aj dzj at the origin and hence any set h1      hn+1 of smooth solutions to (I.74) must have linearly dependent differentials at the origin. In particular, the CR structure defined by the vector fields (I.73) cannot be locally integrable. Proof of Proposition I.16.1. The first step in the proof is the construction of the function g. In the complex plane we denote the variable by w = s + it and consider a sequence of closed, disjoint disks Dj , all of them contained in the sector w s < t and such that Dj → 0 as j → . Let F ∈ C  C R have support contained in the union of the disks Dj and satisfy Fw > 0 ∀w ∈ int Dj  ∀j

(I.75)

As before we shall write Wz s = s + iz, with z = z1 2 − z2 2 −    − zn 2 

(I.76)

Lemma I.16.3. The function F W vanishes to infinite order at z1 = 0. Proof. Denote by H the Heaviside function. For every  ∈ Z+ there is C > 0 such that

Fw ≤ C tHt  Then

FWz s ≤ C zHz  Since moreover zHz ≤ z1 2 , the lemma is proved. We then set  gz s =

FWz s  z1 − FWz s

(I.77)

40

Locally integrable structures

Since gz s =

FWz s 1 z1 1 − FWz s/z1

it follows from Lemma I.16.3 that g is smooth in an open neighborhood of the origin in Cn × R and that g vanishes to infinite order at z1 = 0. We shall now proceed to the proof of (a). We shall write L#j = Lj − i%j zj gz s

  s

(cf. (I.76), (I.73)). Since Lj  Lk = 0 and Lj zk = 0 for all j and k we obtain



L#j  L#k = −i %k zk Lj g − %j zj Lk g  s

(I.78)

Now Lj g =

z1 L F W z1 − F W2 j

and an easy computation making use of the chain rule gives Lj FWz s = −2i%j zj

F Wz s w

Hence from (I.78) we obtain −iz1 z1 − F W2 −2z1 = z1 − F W2

L#j  L#k =

 %k zk Lj F W − %j zj Lk F W s

 = 0 %k zk %j zj − %j zj %k zk s

We now start to prove (b). For this we set (z s = hz 0     0 s and will show that (/s0 0 = 0. We assume that ( is C 1 in a set of the form V = z s ∈ C × R z < r s <  and observe that L( − izfz s

( = 0 s

where L is the Hans Lewy operator given in (I.79) and fz s =

Fs + i z 2  z − Fs + i z 2 

is smooth in V (contracting V if necessary).

(I.79)

I.16 A CR structure that is not locally integrable

41

 Let U = w = s + it ∈ C s <  0 < t < r 2 and assume that Dj ⊂ U for all j. Define  w ∈ U (I.80) Iw = √ (z sdz

z = t

By Stokes’ theorem we have Iw =

 √

z ≤ t

 2)  ( z sdz ∧ dz = 2i z 0 0

√ t

( &ei  s&d&d z

from where we obtain   2) ( √ I 1 ( w = i  tei  sd = z sdz √ t z 0

z = t z z Consequently,

I i 1 w = L(z sdz √ w 2 z = t z

(I.81)

(cf. (I.79)). From (I.79), (I.81) and from the fact that F is supported in the union of the disks Dj we conclude that I is a holomorphic function of w in the connected open set U \ ∪j Dj . Since, moreover, Iw → 0 when t → 0+ the Schwarz reflection principle implies that I vanishes identically in U \ ∪j Dj . In particular, I ≡0

on

Dj 

∀j.

(I.82)

Next we consider, for each j, the map Dj × S 1 −→ R3 

√ w  →  tei  s

(I.83)

whose image defines a torus Tj ⊂ V . If we set  u = ( dz ∧ dW  (I.84)  where W z s = s + i z 2 , we have Tj u = 0 for all j, as a consequence of (I.82). Consequently,  du = 0 ∀j (I.85) Sj

where Sj is the solid torus whose boundary is equal to Tj . We shall now exploit property (I.85). Since dz dz dW are linearly independent we can write d( = Adz + Bdz + CdW 

(I.86)

42

Locally integrable structures

where A, B and C are continuous functions. If we apply both sides of (I.86) to L we obtain that B = L(, since Lz = LW = 0. Hence, from (I.84) we obtain du = L(dz ∧ dz ∧ dW ( = izfz s dz ∧ dz ∧ ds s ( = −2zfz s dx ∧ dy ∧ ds s which in conjunction with (I.85) gives  zfz s Sj

( dxdyds = 0 s

∀j

(I.87)

Now we observe that zfz s = Fs + i z 2 *z s, where * is smooth and satisfies *0 0 = 1. From (I.87) we conclude the existence of points Pj  Qj ∈ Sj such that     ( (  *Pj  Pj  =  *Qj  Qj  = 0 s s for all j. It suffices to let j →  to obtain that (/s0 0 = 0 and hence to conclude the proof of the proposition.

I.17 The Levi form on a formally integrable structure Let  be a formally integrable structure over a smooth manifold  and let  ∈ Tp0 ,  = 0 be fixed (recall that in particular  ∈ Tp∗  ⊂ CTp∗ ). We start with the following result: Lemma I.17.1. Let L and M be sections of  in a neighborhood of p. If   either Lp = 0 or Mp = 0 then  L M p = 0. Proof. We take complex vector fields L1      Ln which span  at each point in a neighborhood of p. Assume for instance that Mp = 0 (for the other case the argument is analogous). Then we can write M=

n  j=1

gj Lj 

I.17 The Levi form on a formally integrable structure

43

where gj are smooth functions and gj p = 0 for all j = 1     n. We have

L M =

n  

 Lgj Lj + gj L Lj j=1

  and thus  L M p = 0 since Lj p  = 0 (because  is real) and gj p = 0. From Lemma I.17.1 it follows that the following definition is meaningful: Definition I.17.2. The Levi form of the formally integrable structure  at the characteristic point  ∈ Tp0 ,  = 0 is the hermitian form on p defined by Lp v w =

 1   L M p  2i

(I.88)

where L and M are smooth sections of  defined in a neighborhood of p and satisfying Lp = v, Mp = w. Given a hermitian form H on a finite-dimensional complex vector space V , its main invariants are the subspaces V + , V − and V ⊥ of V , which give a decomposition V = V+ ⊕V− ⊕V⊥ and are characterized by: • v → Hv v is positive definite on V + ; • v → Hv v is negative definite on V − ; • V ⊥ = v ∈ V Hv w = 0 ∀w ∈ V . Thus H is itself positive definite (resp. positive negative) if V = V + (resp. V = V − ). More generally, H is said to be positive (resp. negative) if V − = 0 (resp. V + = 0 ). Also, H is said to be nondegenerate if V ⊥ = 0 . Finally, we recall that it is common to call the positive integer dim V + − dim V − the signature of H. Notice that the signature does not change after multiplication of H by a nonzero real number. A formally integrable structure  over  is nondegenerate if given any  ∈ Tp0 ,  = 0 the Levi form Lp is a nondegenerate hermitian form. We now describe the Levi form for a formally integrable CR structure over . Let p ∈ ,  ∈ Tp0 ,  = 0. According to the results described in Section I.8 we can find a system of coordinates x1      xn  y1      yn  s1      sd 

44

Locally integrable structures

vanishing at p and vector fields of the form Lj =

d d      + ajj  z s + bjk z s  zj  k=1 sk zj j  =1

j = 1     n

with ajj  0 0 = bjk 0 0 = 0 for all j j   k, which span  in a neighborhood of the origin in R2n+d . Notice, moreover, that Tp0 is equal to the span of ds1 0      dsd 0 . of the Levi Write  = 1 ds1 0 +    + d dsd 0 and denote by Ajj   the matrix

form Lp with respectto the basis /z1 p      /zn p of p . Thus, by definition, Ajj  = Lp /zj p  /zj  p and then 1  ds +    + d dsd 0  Lj  Lj  p  2i 1 1 0 d

1   L b  − Lj  bjk 0 0 = 2i k=1 k j j k

Ajj  =

that is

  d bjk bj  k 1   0 0 − 0 0  Ajj  = zj  2i k=1 k zj

(I.89)

As an example, let us consider the CR structure defined by the vector fields L#j given by (I.73). In this case d = 1 and we take  = ds 0 . We also have bj = −i%j zj 1 + gz s, where g vanishes to infinite order at z1 = 0. Then bj  0 0 = −i%j jj  zj and (I.89) gives Ajj   = diag %1      %n  Thus, Corollary I.16.2 has provided an example of a nondegenerate CR structure, defined in a neighborhood of the origin in Cn × R, for which the signature of the Levi form at ds 0 ∈ T00 ,  = 0 is equal to n − 1. In connection with this example we mention the following deep result which gives a positive answer to the problem of local integrability (or local realizability, as we have seen in Proposition I.15.1) for certain classes of CR structures. It shows that the value of the signature of the Levi form plays a crucial role. Recall that by Proposition I.8.4 the characteristic set of a CR structure is a sub-bundle of the cotangent bundle. Theorem I.17.3. Let  be a nondegenerate CR structure over a smooth manifold  and assume that its characteristic set has rank equal to one. Let

I.17 The Levi form on a formally integrable structure

45

n denote the rank of  (and thus the dimension of  is equal to 2n + 1). Suppose that for some p ∈  the signature of the Levi form at  ∈ Tp0 ,  = 0, is equal to n. If n ≥ 3 then  is locally integrable in a neighborhood of p. Finally, we shall compute the expression of the matrix Ajj   of the Levi form when  is locally integrable and CR. Invoking the local coordinates described at the beginning of Section I.15, and in particular the functions (I.66) satisfying (I.64), we see that we can take the vector fields Lj in the form (cf. (I.27)) Lj =

d  k  z sMk  −i zj k=1 zj

j = 1     n

where Mk =

d 

kk z s

k =1

  sk

k = 1     d

characterized by the relations Mk sk + ik = kk . In particular, (I.64) gives kk 0 0 = kk 

(I.90)

According to our previous notation, we have ajj  ≡ 0 for all j j  and bjk = −i

d  k   zj k k k =1

Again by (I.64) and by (I.90) we have bjk  2 k 0 0 = −i 0 0 zj  zj  zj and then by (I.89) we obtain Ajj  =

d  k=1

k

 2 k 0 0 zj  zj

(I.91)

Example I.17.4. The following discussion justifies our terminology and makes a connection with the theory of several complex variables. Let U be an open subset of Cn+1 with a smooth boundary. Let & ∈  C Cn+1  R be such that U = z &z < 0 and that d& = 0 on U = z &z = 0 . We say that U satisfies the Levi condition at the point p ∈ U if the restriction of the hermitian form  → to the space Tp =  ∈ Cn+1

n+1 

2 & pj k jk=1 zj zk

n+1

j=1 &/zj pj

= 0 is positive.

46

Locally integrable structures

The Levi condition is independent of the choice of the defining function &: it is also a holomorphic invariant. After a translation and a C-linear tranformation we can assume that 0 ∈  and that the tangent space to  at the origin is given by the real-hyperplane w = 0, where now we are writing the complex coordinates as z1      zn  w. We can also assume that the exterior normal to  at the origin is the vector 0     0 −i ∈ Cn+1 . By the implicit function theorem we conclude the existence of a smooth, real-valued function  satisfying 0 0 = 0, d0 0 = 0 such that & can be written, near the origin and in these new complex variables, as &z w = z w − w

(I.92)

Since then T0 = n+1 = 0 , the Levi condition at the origin can be written as: n  2  0 0j k ≥ 0 ∀ ∈ Cn  (I.93) z z j k jk=1 The boundary of U is a one-codimensional submanifold of Cn+1 and consequently it is generic. The complex structure  01 of Cn+1 induces on U a CR structure  01 U and, according to the discussion in Section I.15, the differentials of the functions Z j = zj 

j = 1     n Wz s = s + iz s

span T  U near the origin [we are writing s = w and considering z s as local coordinates in U ]. From (I.91) we obtain the following equivalent statement to (I.93): the Levi form of the CR structure  01 U at the characteristic point ds 0 is positive. 10 To obtain an invariant  01 ⊥ statement let us first denote by T the orthog. Given an open set U with a smooth boundary U onal sub-bundle  as above, and given p ∈ U , the map '∗p CTp∗ Cn+1 → CTp∗ U induces an isomorphism ∼

p Tp10 −→ Tp U Let  ∈ Tp0 U,  = 0. We shall say that  is inward pointing if    p−1  v > 0 for every v ∈ Tp Cn+1 which is inward pointing toward U . In the preceding set-up, when p = 0 and & is given by (I.92), then 0−1 ds 0  = dw 0 and then  = ds 0 is inward pointing if and only if  > 0. Summing up we can state:

Appendix: Proof of the Newlander–Nirenberg theorem

47

Proposition I.17.5. Let U ⊂ Cn+1 be an open set with a smooth boundary. Then U satisfies the Levi condition at p ∈ U if and only if the Levi form associated with the CR structure  U is positive at every  ∈ Tp0 U,  = 0 which is inward pointing.

Appendix: Proof of the Newlander–Nirenberg theorem In this appendix we shall present an argument due to B. Malgrange ([Mal]) which leads to the proof of the Newlander–Nirenberg theorem. We start by recalling some of the results we need from the theory of nonlinear elliptic equations. Let us consider then an overdetermined system of nonlinear partial differential equations   ! x u"  x1 u"       u"     = 0  ≤ M (I.94) where x varies in an open subset  of RN , u" = u1      uq  ∈ C M  Rq  ! = 1      p  is smooth and real-valued and q ≤ p. The system (I.94) is elliptic at u" 0 ∈ C M  Rq  in  if the linear differential operator v" →

 d  ! x u" 0 + "v x1 "u0 + "v      "u0 + "v    =0 d

(I.95)

is elliptic in the following sense: if   × RN \0  → LRq  Rp  denotes the principal symbol of (I.95) then rank x  = q

∀x  ∈  × RN \0 

We call I95 the linearization of (I.94) at u" 0 . Here is an important remark that will be quite important in what follows: if x0 ∈  and if v" →

 d  ! x0  u" 0 x0  + "v x1 "u0 x0  + x1 "v    =0 d

(I.96)

is an elliptic linear system (with constant coefficients!) then (I.94) is elliptic at u" 0 in a neighborhood of x0 . Accordingly, we shall call (I.96) the linearization of (I.94) at u" 0 at the point x0 .

48

Locally integrable structures

The two main results that are essential for Malgrange’s argument are: • If u" is a C M -solution of (I.94), if (I.94) is elliptic at u" in the sense just defined, and if the function ! is real-analytic then u" is real-analytic. • Now assume that q = p and that (I.94) is elliptic at u" 0 ∈ C M  Rq . Let x0 ∈  be such that   ! x0  u" 0 x0  x1 u" 0 x0       u" 0 x0     = 0 Then there are %0 > 0, C > 0 and 0 <  < 1 such that for every 0 < % ≤ %0 there is a smooth solution u" % to (I.94) on x − x0 < % satisfying the bounds  

 u" % x − u" 0 x ≤ C%M−  +  x − x0 < %  ≤ M We now embark on the proof of the Newlander–Nirenberg theorem. The starting point is the description of the special generators presented after Lemma I.8.5, particularly the vector fields given by (I.19), taking into account that when the structure is complex then d = n = 0. In other words, we can assume that our (complex) formally integrable structure is defined, in an open neighborhood of the origin in Cm , by the pairwise commuting vector fields Lj =

m    + ajk z  zj k=1 zk

j = 1     m

(I.97)

where ajk = 0 at the origin. For technical reasons, which are going to be clear in the argument, it is convenient to assume that ajk z = O z 2 , and this property can be achieved after performing a local diffeomorphism of the form z = z + Qz z, where Q is a homogeneous polynomial of degree two in z1      zm  z1      zm  chosen suitably. We leave the details of this (simple) computation to the reader. Malgrange’s key idea is to show the existence of a local diffeomorphism w = Hz, defined near the origin in Cm , such that, in the new variables w1      wm , the structure has a set of generators which have real-analytic coefficients. This implies the sought-for conclusion thanks to Theorem I.9.2. In order to shorten the notation and make the computations more apparent, we shall describe all the systems involved in vector and matrix notation. Thus we set ⎡  ⎤ ⎡  ⎤ ⎡ ⎤ L1 z1 ⎢  ⎥ ⎢ z1 ⎥   ⎢  ⎥ ⎥ ⎢ "  L = ⎣  ⎦  = =⎢  ⎥ z ⎣  ⎦ z ⎣  ⎦ Lm z z m

m

Appendix: Proof of the Newlander–Nirenberg theorem

49

and rewrite the system (I.97) as "= L

  + Az  z z

where Az denotes the matrix ajk z . Let w = Hz be a local diffeomorphism near the origin in Cm satisfying Hz 0 is invertible.

(I.98)

Since       + tHz  = t Hz + tHz  = t Hz w w z w w z a new set of generators for the structure is defined, in the new variables w1      wm , by the system "• = L

  + Bw  w w

(I.99)

where Bw =



t

Hz + A tHz

−1

 t Hz + A t Hz  z=H −1 w 

(I.100)

" • then a fortiori we must have If L•1      L•m denote the components of L

L•j  L•k = 0

∀j k = 1     m j < k

Writing B = bjk this property is equivalent to  m  bj bk bk bj  b w = 0 ∀j k  j < k (I.101) − − − bjr w wj wk r=1 kr wr wr We emphasize: given any local diffeomorphism H satisfying (I.98) then equations (I.101) are satisfied by B = bjk defined by (I.100). The system (I.101) together with the additional equations m  bjk j=1

wj

= 0

k = 1     m

(I.102)

make up a system of quasi-linear partial differential equations in the unknowns 2 bjk . Let us write V" = b11  b12      bmm−1  bmm  ∈ R2m . Then systems (I.101) and (I.102) can be written as  V" + +V"  ," V"  = 0

(I.103)

where  is an elliptic linear operator with constant coefficients and + is a bilinear form in its arguments. It then follows that there is a small number  > 0 such that if B0 ≤  then (I.101), (I.102) is elliptic at B in an open

50

Locally integrable structures

neighborhood of the origin. Hence any such B is a real-analytic function of w and the argument will be complete if we can show that a diffeomorphism H satisfying (I.98) can be chosen in such a way that B, defined by (I.100), is a solution of (I.102) satisfying B0 ≤ . We are left to solve the determined system m !  −1  t H z + A t H z  t Hz + A t Hz  = 0 k = 1     m (I.104) jk j=1 wj whose unknown is Hz Hz (we look at (I.104) as a determined system of 2m real equations). It is important to emphasize that these equations are now being considered in the z1      zm variables. Since Az = O z 2  it is easily seen that H0 z = z satisfies (I.104) at the origin. Furthermore, taking Hz = H0 z + Gz then for  ∈ R,  small we have  −1  t H z + A t H z  t Hz + A t Hz  = A +  t Gz + A F + O2  for some F smooth. Furthermore, since    = t Hz−1 − t Hz−1 t H z w z w we obtain   = + O  wj zj Hence, using once more the fact that Az = O z 2 , we can easily conclude that the linearization of (I.104) at H0 at the origin can be identified, in a natural way, with the complex operator # " m m    t   t  Gz j1      Gz jm G → j=1 zj j=1 zj # " m m    2 G1  2 Gm   = j=1 zj zj j=1 zj zj which is clearly elliptic (in the usual sense). We conclude that there are %0 > 0, C > 0 and  < 1 such that for every 0 < % ≤ %0 there is a smooth solution H% to (I.104) satisfying H% − H0 C 2 z

z ≤% ≤ C%2+ 

% ≤ %0 

In particular, if % > 0 is small enough we can ensure that:

(I.105)

Notes

51

• H% is a local diffeomorphism near the origin satisfying (I.98); • B defined by (I.100) satisfies B0 ≤ . The proof is complete.

Notes The first treatment of formally and locally integrable structures as presented here appeared in [T4], the main point for this being the discovery of the Approximation Formula by M. S. Baouendi and F. Treves in 1981 ([BT1]); such structures were then studied extensively in [T5]. The pioneering work though seems to be the article by Andreotti-Hill ([AH1]), where the concept of what we now call a real-analytic locally integrable structure was introduced in its full generality. This introductory chapter contains mainly results that have already been presented in standard textbooks. We mention, for instance, the Frobenius theorem, whose proof was taken from L. Hörmander’s book [H4] and the integrability of elliptic vector fields in the plane, of which we give an almost self-contained proof, depending only on very simple facts concerning commutators of certain pseudo-differential operators that can be found, for instance, in [Fo]. As mentioned in the text, Theorem I.12.1 is due to L. Nirenberg ([N2]) and the proof we present was taken from [T5]. Proposition I.16.1 is a particular case of a more general result due to H. Jacobowitz and F. Treves ([JT1]). We also refer to [JT2] where the same authors study, via a category argument, the set of all formally integrable CR structures of rank n on an open subset of R2n+1 whose Levi form has, at each nonzero characteristic point, signature n − 1. Theorem I.17.3 was originally due to M. Kuranishi ([Ku1], [Ku2]) in the case n ≥ 4. Later, T. Akahori ([Ak]) presented an improvement to Kuranishi’s argument which allowed him to prove Theorem I.17.3 also for the case n = 3. The case n = 2 is still an open problem, whereas when n = 1 the conclusion is false, according to [N3] (see also Theorem I.12.1). A proof of Theorem I.17.3 can also be found in [W3]. Finally, Malgrange’s proof of the Newlander–Nirenberg theorem that we presented in the appendix was taken from [N1], where the use of a solvability result on elliptic determined systems of nonlinear partial differential equations makes the argument a bit simpler.

II The Baouendi–Treves approximation formula

In this chapter we prove what is probably the most important single result in the theory of locally integrable structures. It states that in a small neighborhood of a given point of the domain of a locally integrable structure , any solution of the equation u = 0 may be approximated by polynomials in a set of a finite number of homogeneous solutions as soon as the solutions in that set are chosen with linearly independent differentials and the number of them is equal to the corank of . Such a set is called a complete set of first integrals of the locally integrable structure. The proof is relatively simple for classical solutions and depends on the construction of a suitable approximation of the identity modeled on the kernel of the heat equation as shown in Section II.1. The extension to distribution solutions is carried out in Section II.2. Section II.3 studies the convergence of the formula in some of the standard spaces used in analysis: Lebesgue spaces Lp , 1 ≤ p < ; Sobolev spaces; Hölder spaces; and (localizable) Hardy spaces hp , 0 < p < . The last section is devoted to applications.

II.1 The approximation theorem Since the approximation formula is of a local nature it will be enough to restrict our attention to a locally integrable structure  defined in an open subset  of RN over which ⊥ is spanned by the differentials dZ1      dZm of m smooth functions Zj ∈ C  , j = 1     m, at every point of . Thus, if n is the rank of , we recall that N = n + m. Given a distribution u ∈   we say that u is a homogeneous solution of  and write u = 0 if Lu = 0 52

on U

II.1 The approximation theorem

53

for every local section L of  defined on an open subset U ⊂ . Simple examples of homogeneous solutions of  are the constant functions and also the functions Z1      Zm , since LZj = #dZj  L$ = 0 because dZj ∈ ⊥ , j = 1     m. By the Leibniz rule, any product of smooth homogeneous solutions is again a homogeneous solution, so a polynomial with constant coefficients in the m functions Zj , i.e., a function of the form  PZ = c Z   = 1      m  ∈ Zm  c ∈ C (II.1)

 ≤d

is also a homogeneous solution. The approximation theorem states that any distribution solution u of u = 0 is the weak limit of polynomial solutions such as (II.1). Theorem II.1.1. Let  be a locally integrable structure on  and assume that dZ1      dZm span ⊥ at every point of . Then, for any p ∈ , there exist two open sets U and W , with p ∈ U ⊂ U ⊂ W ⊂ , such that (i) every u ∈  W that satisfies u = 0 on W is the limit in  U of a sequence of polynomial solutions Pj Z1      Zm : u = lim Pj Z in  U j→

(ii) if u ∈ C k W the convergence holds in the topology of C k U, k = 0 1 2     . Some well-known approximation results in analysis are particular cases of Theorem II.1.1. Example II.1.2. Let  be the locally integrable structure generated over an open set  ⊂ C by the Cauchy–Riemann vector field   1   = +i  z = x + iy 2 x y Then a distribution solution of u = 0 is just a holomorphic function and the theorem simply states that any holomorphic function can be locally approximated by polynomials in the complex variable z. Later we will give several applications of the approximation theorem but we wish to point out already one interesting consequence. Assume that two points p q ∈ U are such that Zp = Zq and let u ∈ C 0  satisfy u = 0. Then P Zp = P Zq for any polynomial P in m variables and, by the uniform approximation of u on U by polynomials in Z, it follows that up = uq. The fibers of Z in U are, by definition, the equivalence classes

54

The Baouendi–Treves approximation formula

of the equivalence relation defined by ‘p ∼ q if and only if Zp = Zq’. Thus, every solution u ∈ C 0  of u = 0 is constant on the fibers of Z. In particular, if the differentials of Z1#      Zm# span ⊥ over  it follows that Z# = Z1#      Zm#  is constant on the fibers of Z in U . Applying the theorem with Z# in the place of Z we may as well find a neighborhood U # ⊂ U of p such that Z is constant on the fibers of Z# in U # , which shows that the fibers of Z and the fibers of Z# on U # are identical. Thus, in the sense of germs of sets at p, the equivalence classes defined by Z and those defined by any other Z# = Z1#      Zm#  such that dZ1#      dZm# generates ⊥ coincide. This independence of the particular choice of Z allows us to talk about the germs at p of the fibers of  which are invariants of the structure. The fact that u is constant on the fibers of Z in U when u = 0, u ∈ C 0 , may be expressed by saying that there exists a function $ u ∈ C 0 ZU such that u =$ u Z. Thus, any continuous solution of u = 0 can be factored as the composition with Z of a continuous function defined on a subset of Cm . In general, the set ZU may be irregular but if it happens to be a submanifold of Cm , then $ u will satisfy in the weak sense the induced Cauchy–Riemann equations on ZU. Hence, at a conceptual level, the theorem links the study of solutions of u = 0 to solutions of the induced Cauchy–Riemann equations on certain sets of Cm . We will prove Theorem II.1.1 in several steps. The first step consists of taking convenient local coordinates in a neighborhood of p. Applying Corollary I.10.2, there exists a local coordinate system vanishing at p, x1      xm  t1      tn and smooth, real-valued functions 1      m defined in a neighborhood of the origin and satisfying k 0 0 = 0

dx k 0 0 = 0

k = 1     m

such that the functions Zk , k = 1     m, may be written as Zk x t = xk + ik x t

k = 1     m

(II.2)

on a neighborhood of the origin. To do so we need to assume that the real parts of dZ1      dZm are linearly independent, for which we might have to replace Zj by iZj for some of the indexes j ∈ 1     m . Notice that this will not change the conclusion of the theorem. Thus, we may choose a number R such that if V = q

xq < R tq < R

II.1 The approximation theorem then (II.2) holds in a neighborhood of V and we may assume that    j x t  1  <  x t ∈ V    2  xk

55

(II.3)

where the double bar indicates the norm of the matrix x x t = j x t/xk  as a linear operator in Rm . Modifying the functions k ’s off a neighborhood of V may assume without loss of generality that the functions k x t, k = 1     m, are defined throughout RN , have compact support and satisfy (II.3) everywhere, that is    j x t  1  <  x t ∈ RN   (II.3 )  2  xk Modifying also  off a neighborhood of V we may assume as well that the differentials dZj , j = 1     m, given by (II.2), span ⊥ over RN . Of course, the new structure  and the old one coincide on V so any conclusion we draw about the new  on V will hold as well for the original . We will make use of the vector fields Lj , j = 1     n and Mk , k = 1     m entirely analogous to those introduced in Chapter I after Corollary I.10.2, with the only difference that here they are defined throughout RN . We recall from Chapter I that the vector fields Mk =

m 

k x t

=1

  x

k = 1     m

are characterized by the relations Mk Z = k 

k  = 1     m

and that the vector fields m  k  −i x tMk  Lj = tj k=1 tj

j = 1     n

are linearly independent and satisfy Lj Zk = 0, for j = 1     n, k = 1     m. Hence, L1      Ln span  at every point while the N = n + m vector fields L1      Ln  M1     Mm are pairwise commuting and span CTp RN , p ∈ RN . Since dZ1      dZm  dt1      dtn

span CT ∗ RN

the differential dw of a C 1 function wx t may be expressed in this basis. In fact, we have n m   (II.4) dw = Lj w dtj + Mk w dZk j=1

k=1

56

The Baouendi–Treves approximation formula

which may be checked by observing that Lj Zk = 0 and Mk tj = 0 for 1 ≤ j ≤ n and 1 ≤ k ≤ m, while Lj tk = jk for 1 ≤ j k ≤ n and Mk Zj = jk for 1 ≤ j k ≤ m (jk = Kronecker delta). We now choose the open set W as any fixed neighborhood of V in . In proving the theorem we will assume initially that u is a smooth homogeneous solution of u = 0 defined in W with continuous derivatives of all orders, i.e., u ∈ C  W satisfies on W the overdetermined system of equations ⎧ ⎪ L1 u = 0 ⎪ ⎪ ⎪ ⎨L u = 0 2 ⎪ ········· ⎪ ⎪ ⎪ ⎩ Ln u = 0

(II.5)

Given such u we define a family of functions E u that depend on a real parameter , 0 <  < , by means of the formula E ux t = /)m/2

 Rm



e− Zxt−Zx 0 ux  0hx  det Zx x  0 dx 2

which we now discuss. For  = 1      m  ∈ Cm we will use the notation

 2 = 12 + · · · + m2 , which explains the meaning of Zx t − Zx  0 2 in the formula. The function hx ∈ Cc Rm  satisfies hx = 0 for x ≥ R and hx = 1 in a neighborhood of x ≤ R/2 (recall that R was introduced right before (II.3) in the definition of the set V ). Note that since u is assumed to be defined in a neighborhood of V , the product ux  0hx  is well-defined on Rm , compactly supported, and of class C  . Since Z has m components we may regard Zx as the m × m matrix Zj /xk  and denote by det Zx its determinant. Furthermore, since the exponential in the integrand is an entire function of Z1      Zm , the chain rule shows that it satisfies the homogenous system of equations (II.5) and the same holds for E ux t by differentiation under the integral sign. The second step of the proof will be to show that E ux t → ux t as  →  uniformly for x < R/4 and t < T < R if T is conveniently small. Once this is proved we may approximate in the 2 C  topology the exponential e−  (for fixed large ) by the partial sum of degree k, Pk , of its Taylor series on a fixed polydisk that contains the set √  Zx t − Zx  0 x  x < R t < R , so replacing the exponential in the definition of E by Pk Zx t − Zx  0 we will find polynomials in Zx t that approximate E ux t in the C  topology for x < R/4 and

t < T when k is large. Hence, from now on we fix our attention on the

II.1 The approximation theorem

57

convergence of E u → u. We consider the following modification of the operator E :   2 G ux t = /)m/2 e− Zxt−Zx t ux  thx  det Zx x  t dx  Rm

Notice that in the trivial case in which the functions k , k = 1     m, vanish identically so Zx t = x and det Zx = 1, G is just the convolution of ux 0hx with a Gaussian in Rm , which is a well-known approximation of the identity as  → . In general, the functions k do not vanish but they are relatively small because they vanish at the origin and (II.3 ) holds, so G is still an approximation of the identity. The idea is then to prove that G u → u and then estimate the difference R u = G u − E u using the fact that u = 0. Lemma II.1.3. Let B be an m × m matrix with real coefficients and norm B < 1 and set A = I + iB where I is the identity matrix. Then  2 det A e− Ax dx = ) m/2  Rm

Proof. We may write Ax 2 = t AAx · x (the dot indicates the standard inner 2 product in Rm and also its extension as a C-bilinear form to Cm ) so e− Ax = e−Cx·x where the matrix C = t AA has positive definite real part C = I − t BB because B < 1. It is then known that (see, e.g., [H2, page 85])  e−Cx·x dx = ) m/2 det C−1/2 Rm

where the branch of the square root is chosen so det C1/2 > 0 when C is real. Since det C = det A2 the proof is complete. Set hxux t det Zx x t = vx t. For x t fixed, the matrix Zx x t = I + ix x t satisfies the hypotheses of the lemma in view of (II.3 ). Thus, we may write   2 hxux t = ) −m/2 e− Zx xtx vx t dx  Rm

Introducing the change of variables x → x + −1/2 x in the integral that defines G u we get  −1/2  2 G ux t = ) −m/2 e− Zxt−Zx+ x t vx +  −1/2 x  t dx  Rm

Then G ux t − hxux t = I + J 

58

The Baouendi–Treves approximation formula

where I x t = ) −m/2



 2

Rm

e− Zx xtx vx +  −1/2 x  t − vx t dx

and 

J x t = ) −m/2



Rm

e− Zxt−Zx+

−1/2 x t 2

  2 − e− Zx xtx vx +  −1/2 x  t dx   2

 2

 2

 2

To estimate I we observe that e− Zx xtx = e− x + x xtx ≤ e−3 x /4 in view of (II.3 ). We also observe that ,x vx t is bounded in Rm ×  t ≤ R because v vanishes for large x, so the mean value theorem gives   2 e−3 x /4 x dx ≤ C   −1/2 

I x t ≤ C −1/2 Rm

showing that I x t → 0 as  →  uniformly on Rm × t ≤ R . To estimate −1/2  2  2  2 J we first observe that e− Zxt−Zx+ x t − e− Zx xtx ≤ 2e−3 x /4 , so  −1/2  2  2

e− Zxt−Zx+ x t − e− Zx xtx dx

J x t ≤ C

x R and invoking Stokes’ theorem, we have  d G ux t − E ux t = Rm × 0t

where 0 t denotes the segment joining the origin of Rn to the point t ∈ Rn . To compute d we will take advantage of expression (II.4). We have d = dv ∧ dZ so the only terms in (II.4) that matter here are those that do not  contain dZj , j = 1     m , i.e., d = nj=1 Lj v dtj ∧ dZ. Since the exponential factor in v is an entire function of Z1      Zn , and thus satisfies (II.5) as well as u, we obtain n     2 R ux t = /)m/2 e− Zxt−Zx t  ux  t Lj hx  dtj ∧ dZx  t  m j=1 R × 0t

Assume now that x ≤ R/4 and t ≤ T , where T will be chosen momentarily. We wish to estimate the exponential factor 





 2 − x−x 2 

e− Zxt−Zx t  = e xt−x t  2



We have

x t − x  t  ≤ x t − x  t + x  t − x  t  1 ≤ x − x + C t − t 2 1 ≤ x − x + CT 2 because t ∈ 0 t and t ≤ T . Hence, 1

x t − x  t  2 ≤ x − x 2 + 2T 2 2 and

& − Zxt−Zx t  2 & &e & = e2T 2 − x−x 2 /2 

where  is a bound that depends only on  and does not depend on u. Since Lj h vanishes for x ≤ R/2 we have that x ≥ R/2 in all integrands in the expression of R , so x − x ≥ R/4 and

R ux t ≤ Ce2T

2 −R2 /32



60

The Baouendi–Treves approximation formula

We may now choose T small enough so as to achieve R ux t ≤ Ce−R /33 . This proves that R ux t → 0 uniformly on U =  x ≤ R/4 ×  t ≤ T . Summing up, we have found a neighborhood of the origin U such that for any C  -solution u of (II.5) defined in W , E u → u uniformly on U , which partially proves part (i) of the theorem for very regular distributions. The third step is to prove part (ii) of the theorem for k =  (the cases 1 ≤ k <  will be proved later). The main tool is the use of commutation formulas for the vector fields Mk with G . 2

Lemma II.1.4. For u ∈ C 1 W and k = 1     m the following identity holds: Mk G ux t − G Mk ux t = Mk G ux t   2 = /)m/2 e− Zxt−Zx t ux  tMk hx  det Zx x  t dx  Rm

(II.6)

Proof. By the symmetry in the variables x and x of the expression Zj x t − Zj x  t we have

j = 1     m

 jk = Mk x t Dx Zj x t − Zj x  t

 = −Mk x  t Dx Zj x t − Zj x  t 

Thus, if F is an entire holomorphic function and we set fx x  t = FZx t − Zx  t we also have, by the chain rule, Mk x t Dx fx t t  = −Mk x  t Dx fx t t  Applying this to F = e−  we get, after differentiation under the integral sign that 2

Mk G ux t = −/)m/2   2 Mk x  t Dx e− Zxt−Zx t ux  thx  dZx  t Rm

where we have used the fact that the pullback to any slice t = const. of the m-form dZ1 ∧ · · · ∧ dZn is given by det Zx x  t dx . Next, using the ‘integration by parts’ formula   Mk v w dZ = − v Mk w dZ (II.7) Rm

Rm

II.1 The approximation theorem

61

which is valid if v and w are of class C 1 and one of them has compact support, we get Mk G ux t = /)m/2    2 e− Zxt−Zx t Mk ux  thx  + ux  tMk hx  dZx  t Rm

which proves (II.6). To complete the proof we show that (II.7) holds. Consider the exact m-form defined by ' k ∧ · · · ∧ dZm  k = duv dZ1 ∧ · · · ∧ dZ ' k ∧ · · · ∧ dZm = duv ∧ dZ1 ∧ · · · ∧ dZ where the hat indicates that the factor dZk has been omitted. The pullback of k to the slice t × Rm is exact, so  k = 0 (II.8) t ×Rm

Using (II.4) to compute duv and observing that the pullback to the slice of terms that contain a factor dtj vanish, we get k t ×Rm = −1k+1 vMk u + uMk v dZ t ×Rm  so (II.8) implies (II.7). Next we prove for the Lj commutation formulas analogous to (II.6). We write Lj = =

m  k  −i x tMk  tj k=1 tj m    + jk  tj k=1 xk

j = 1     n

We start with a technical lemma. Lemma II.1.5. m jk det Zx   det Zx  + ≡ 0 tj xk k=1

j = 1     n

(II.9)

Proof. Note that (II.9) says that the vector field det Zx Lj is divergence free, i.e., div det Zx Lj  = 0, or that t Lj det Zx  = 0 where t Lj is the transpose

62

The Baouendi–Treves approximation formula

of Lj . Take a test function vx t and consider the compactly supported exact form   $ j ∧ · · · ∧ dtn j = d v dZ ∧ dt1 ∧ · · · ∧ dt $ j ∧ · · · ∧ dtn = dv ∧ dZ ∧ dt1 ∧ · · · ∧ dt = −1m+j−1 Lj v dZ ∧ dt = −1m+j−1 Lj v det Zx  dx ∧ dt whose integral over RN vanishes, that is,   Lj vdet Zx  dxdt = v t Lj det Zx  dxdt = 0 RN

RN

Since v is arbitrary, Lj det Zx  ≡ 0 and (II.9) is proved. t

If g˜  t is a smooth function on Cm × Rn that is holomorphic with respect to  and we set gx t = g˜ Zx t t we have, by the chain rule, that Lj gx t =

g˜ Zx t t tj

because Lj Zk = 0, k = 1     m. To take advantage of this fact we may write ˜  uZx t t, where G ux t = /)m/2 G   2 ˜  u t = G e− −Zx t ux  thx  det Zx x  t dx  Rm

so Lj G ux t = /)m/2

˜ u G Zx t t tj

To compute the right-hand side of the last identity we write e  x  t =  2 e− −Zx t , differentiate with respect to tj under the integral sign, and observe that det Zx  e uh det Zx  e uh = det Zx + e uh tj tj tj m   = det Zx Lj e uh − det Zx jk e uh xk  k=1 + e uh

det Zx   tj

Note that the integral over Rm of the second term of the right-hand side may be written, after integration by parts, as  m   jk det Zx  dx e uh x k k=1

II.2 Distribution solutions

63

so the integral of the second and third terms together yields " #  m det Zx    + jk det Zx  dx = 0 e uh tj k=1 xk in view of (II.9). Since Lj e  = 0, we also have that det Zx Lj e uh = det Zx e Lj uh + det Zx e uLj h. This shows that   ˜ ˜  Lj u t + e  x  tuLj h det Zx x  t dx  G u t = G tj When  = Zx t we obtain Lemma II.1.6. For u ∈ C 1 W and j = 1     m the following identity holds: Lj G ux t − G Lj ux t = Lj G ux t   2 e− Zxt−Zx t ux  tLj hx  det Zx x  t dx  = /)m/2 Rm

(II.10)

Let us assume now that u ∈ C  W satisfies u = 0 and we wish to prove that E ux t → ux t in C  U. We have already proved that G u → hu uniformly in  t ≤ T × Rm . Since Lj Mk u = Mk Lj u = 0, 1 ≤ j ≤ n, 1 ≤ k ≤ m, Mk u is a smooth solution of the system, so we also have that G Mk u → hMk u uniformly on  t ≤ T ×Rm . Now, the expression (II.6) of Mk  G u is almost identical to that of G , the only difference being that h has been replaced by Mk h, so Mk  G u → Mk hu. Restricting our attention to U where h = 1 and Mk h = 0, we conclude that Mk G u = G Mk u + Mk  G u → Mk u uniformly on U as  → . A similar conclusion can be obtained for Lj G u using (II.10) instead of (II.6), that is, Lj G u → Lj u uniformly on U . Since any first-order derivative D may expressed as a linear combination with smooth coefficients of the Mk ’s and the Lj ’s, we see that DG u → Du uniformly on U . This shows that G u → u in C 1 U. Of course, the argument can be iterated for higher-order derivatives to conclude that G u → u in C  U.

II.2 Distribution solutions We continue the proof of Theorem II.1.1, keeping the notations of Section II.1. In order to extend the arguments of the previous section to a distribution u ∈  W such that u = 0—which is the fourth step of the proof of Theorem II.1.1—it is enough to check the following facts: (a) E u is well-defined for u ∈  W; (b) G u is well-defined for u ∈  W;

64

The Baouendi–Treves approximation formula

(c) G u → u in  U as  →  for u ∈  W; (d) R u = G u − E u → 0 in  U as  →  for u ∈  W. We start by observing that since u satisfies the system of equations (II.5) on a neighborhood of V , the wave front set WFu of u is contained in the characteristic set of  and therefore does not intersect the set

x  t < R   = 0 

x t 0  ∈ RN × RN 

for some R > R. Thus, WFhu is contained in the same set and, in particular, the restriction of u to W belongs to C   t ≤ R    x < R  On the connection between wave front sets and restrictions of distributions, we refer to [H2, chapter VIII]. Moreover, since V =  x < R ×  t < R is relatively compact in W we may assume that t → u· t is a continuous function with values in the L2 based local Sobolev space L2s loc BR  of order s, for all t ≤ R and some real s, where BR denotes the ball of radius R centered at the origin of Rm (for the definition of local Sobolev spaces see Section II.3.2 below). Thus, for any t ≤ R, the trace u· t is well-defined and belongs to L2s loc BR . Then, E ux t (resp. G ux t) is well-defined if we interpret the integral as duality between the distribution u· 0 and the test function  2 /)m/2 e− Zxt−Zx 0 hx  det Zx x  0 (resp. u· t and the test function  2 /)m/2 e− Zxt−Zx t hx  det Zx x  t). This takes care of (a) and (b). To prove (d), it is convenient to express R u by a reinterpretation of the formula obtained for smooth u using Stokes’ theorem. We point out that the formula could also have been written as   n rj x t t   dtj  (II.11) R ux t =

0t j=1

where rj x t t   = /)m/2





Rm



e− Zxt−Zx t  ux  t Lj hx  det Zx x  t  dx 2

(II.12) and 0 t denotes the straight segment joining 0 to t. In other words, by integrating first in x we may express the integral of an m + 1-form over the cell Rm × 0 t as the integral of a 1-form over the segment 0 t . In this form, Stokes’ theorem is just a restatement of the fundamental theorem of calculus for a 1-form. To prove this claim, write for fixed  and     2 ˜  u t  = e− −Zx t  ux  t hx  det Zx x  t  dx  gt  = G Rm

II.2 Distribution solutions Then, gt − g0 =



65

n  g   t  dtj  

0t j=1 tj

(∗) 

To compute the derivatives of g we write e  x  t = e− −Zx t , differentiate with respect to tj under the integral sign, and recall that 2

det Zx  e uh det Zx  e uh = det Zx + e uh   tj tj tj m   e uh = det Zx Lj e uh − det Zx jk xk  k=1 + e uh

det Zx   tj

a fact we already used in the proof of (II.10). Once again, the integral over Rm of the second term of the right-hand side may be written, after integrating by parts, as  m   jk det Zx  dx e uh k=1 xk so the integral of the second and third terms together yields " #  m det Zx    + jk det Zx  dx = 0 e uh tj k=1 xk in view of (II.9). Since Lj e u = 0, we also have that det Zx Lj e uh = det Zx e uLj h. This shows that g  t  = r˜j  t   tj where r˜j  t   =





Rm



(∗∗)

e− −Zx t  ux  t Lj hx  det Zx x  t  dx  2

Hence, (∗) for  = Zx t gives an alternative proof of the fact that R u = G u − E u as given by (II.11) and (II.12). Notice that (II.12) makes sense if u ∈ C   t ≤ R    x < R  as soon as we change the integral symbol by the duality pairing between the distribution u· t  and the appropriate test function; furthermore, R u = G u − E u is still given by (II.11) and (II.12) in the case of distribution solutions since (∗∗) is easily seen to remain valid in this case. Note also that R ux t is a smooth function of x t. We will prove a stronger form of (d).

66

The Baouendi–Treves approximation formula

Proposition II.2.1. Let u ∈  W satisfy the system (II.5). Then, R ux t → 0 in C  U

(II.13)

Proof. We already saw that the exponential in (II.12) may be majorized by e−c for some positive constant c > 0 when x < R/4, x ≥ R/2, t < T and t ∈ 0 T . Let $x denote the Laplacian in Rm . For k ∈ Z+ we may write  Lj hx  ux  t  det Zx x  t  =(x 1 − $x k 1 − $x −k Lj hx   ux  t  det Zx x  t   where (x  is a cut-off function that vanishes for x ≤ R/4 such that (x Lj hx  = Lj hx . Let us write vj x  t  = 1 − $x −k Lj hx  ux  t  det Zx x  t   It follows that vj ∈ C 0 V for an appropriate choice of k and we may write, after an integration by parts,    2 rj x t t   = /)m/2 vj x  t 1 − $x k (x e− Zxt−Zx t  dx  Indeed, the convolution operator 1 − $−k fx =

1  ix· f  d e 1 +  2 −k$ 2)m

f ∈ Rm 

maps continuously L2s Rm  onto L2s+2k Rm  and the latter is contained in L Rm  ∩ C 0 Rm  if s + 2k > m/2 by Sobolev’s embedding theorem. Hence, rj x t t   is continuous with respect to t and converges to 0 uniformly for x ≤ R/2 t ≤ t ≤ T , as  → , since the derivatives in 1 − $x k produce powers of  that are dominated by the exponential e−c . Hence, R ux t → 0 uniformly as  →  and it is easy to see, by differentiating (II.11), that the same holds for the derivatives of any order with respect to x and t of R ux t, as we wished to prove.  Finally, it is enough to prove that (c) holds assuming that u ∈ C 0  t ≤  Let us start with the case k = 0. We assume R  L2k loc BR for some integer k.  2 0 that u ∈ C  t ≤ R  Lloc BR  (with R slightly larger that R) and we wish to prove that 

G ux t − ux t 2 dx → 0 uniformly in t ≤ T 

x ≤R/4

II.2 Distribution solutions

67

which certainly implies (c) in this case. Redefining u by zero off BR × Rn we may assume that ux t ∈ L2 Rm  for each fixed t, t ≤ T . Using once more (II.3 ), we see that for any x x ∈ Rm and t ∈ Rn

Zx t − Zx  t 2 = x − x 2 − x t − x  t 2 ≥ 3/4 x − x 2  so the exponential inside the integral that defines G u has a bound   2  2

e− Zxt−Zx t  ≤ e−3 x−x /4 . If we set F x =  m/2 e−3 x /4  2

0 <  < 

we easily conclude for fixed t ≤ R that

G ux t ≤ C F ∗ u  x t where the convolution is performed in the x variable and t plays the role of a parameter. Since F L1 = F1 L1 = C, Young’s inequality for convolution implies sup G u· tL2 Rm  ≤ C sup u· tL2 Rm  

t ≤T

(II.14)

t ≤T

On the other hand, we proved in Section II.1 that if u ∈ Cc V then G u → u uniformly in U = BR/4 ×  t < T , which implies convergence in the mixed norm space C 0  t ≤ T  L2 BR/4  . So the operator G U convergesto the restriction operator u → u U , as  → , on a dense subset of C 0  t ≤ because of T  L2 BR  and the family of operators G U is equicontinuous   (II.14). Thus, G u U → u U in the whole space C 0  t ≤ T  L2 BR  . a cut-off Assume now that u ∈ C 0  t ≤ T L21 BR  , R > R. Introducing  function we may assume that u ∈ C 0  t ≤ T  L21 Rm  without modifying u for x < R. Thus, for t ≤ T fixed, we see that u, u/xk  and u/tj  are in L2 Rm  for 1 ≤ k ≤ m, 1 ≤ j ≤ n. Since we are assuming that x t is compactly supported, the coefficients of Lj and Mk are bounded, with bounded derivatives. In particular, Lj u and Mk u are in L2 Rm  for 1 ≤ k ≤ m, 1 ≤ j ≤ n, uniformly in t ≤ T . To obtain the convergence result for k = 1 we will be able to reason as with the case k = 0 as soon as we prove an estimate analogous to (II.14) for the L21 norm, i.e., sup G u· tL21 Rm  ≤ C sup u· tL21 Rm  

t ≤T

(II.15)

t

≤T

Any first-order derivative with respect to x is a linear combination with bounded coefficients of the Mk ’s, so it is enough to prove for t ≤ T , 1 ≤ k ≤ m, 1 ≤ j ≤ n, that

68

The Baouendi–Treves approximation formula Mk G u· tL2 Rm  ≤ C sup u· tL21 Rm  

(II.16)

t ≤T

Writing Mk G = Mk  G + G Mk we are led to estimate G Mk uL2 and  Mk  G uL2 . By (II.14) we have G Mk uL2 ≤ CMk uL2 ≤ C  uL21 . Notice that an estimate like (II.14) holds as well with Mk  G in the place of G because G and Mk  G have very similar kernels, as (II.10) shows. Thus,  Mk  G uL2 ≤ CuL21 , which proves (II.16) and gives (II.15). This process can be continued to prove sup G u· tL2k Rm  ≤ Ck sup u· tL2k Rm  

t ≤T

t

≤T

k = 1 2   

(II.17)

To deal with the case in which k is a negative integer, i.e., k = − k = −k, we consider a slight modification of G , namely, G ux = hxG ux. Of course, G u U = G u U because hx = 1 for x ≤ R/2, so this change will not affect our conclusions for x ≤ R/4. The advantage of considering G is that for fixed t it becomes a formally symmetric operator in the xvariables, as soon as we use the pairing given by the complex measure dZx t = det Zx x t dx. More precisely, for fixed t and v w ∈ Cc Rm  we have #G v w$ = #v G w$ where we are using the notation #a b$ =  axbx det Zx x t dx, when a b ∈ C  Rm  and one of them has compact support. Thus, G u· tL2k Rm  ≤ C =C ≤C

sup

#G u· t w$

sup

#u· t G w$

sup

u· tL2k G wL2k

w∈Cc Rm  wL2k ≤1 w∈Cc Rm  wL2k ≤1 w∈Cc Rm  wL2k ≤1

(II.18)

≤ Cu· tL2k  where we have used (II.17) for the positive integer k in the last inequality. This extends (II.17) to all integers k ∈ Z, proving the equicontinuity of G   in all spaces C 0  t ≤ T  L2k BR  , k ∈ Z, which together with the convergence of G u U to u U for the space of test functions Cc BR ×  t ≤ T    which is dense in any C 0  t ≤ T  L2k BR  proves that G u → u in     C 0  t ≤ T  L2k BR/4  for any u ∈ C 0  t ≤ T  L2k BR  . This proves (c) and concludes the proof of part (i) of Theorem II.1.1.

II.3 Convergence in standard functional spaces

69

To prove part (ii) of the theorem—this is the fifth and final step of the proof—using the same method of proof, it will be enough to prove the equicontinuity of G on the spaces   C j  t ≤ T  Cbk Rm   j k = 0 1 2     where Cbk Rm  is the space of functions on Rm possessing continuous bounded derivatives of order ≤ k. For j k = 0 this is easily achieved by noting that

G ux t ≤ C F ∗ u  x t ≤ C  uC 0  t ≤T C 0 Rm   b

For j k ≤ 1 one expresses the derivatives in terms of the vector fields Lj and   Mk and reduces the equicontinuity for the norms of C j  t ≤ T  Cbk Rm  to the case j = k = 0 by introduction of the commutators G  Lj and G  Mk , as was done before for Sobolev norms; iteration of this process gives the result for k = 2 3    This concludes the proof of Theorem II.1.1.

II.3 Convergence in standard functional spaces As proved in Proposition II.2.1, R u = G u − E u → 0 in C  U, for any distribution u satisfying u = 0 in a larger open set V . This reduces the problem of the convergence E u → u in any space with coarser topology than C  -topology to the convergence of G u → u in the same space. Now, as the reader probably noticed in the proof of Theorem II.1.1, the operator G is very close to convolution with a Gaussian in the x-variables with t playing the role of a parameter, and as such it is a very well-behaved approximation of the identity. Hence, loosely speaking, we may expect that the convergence G u → u on U holds in the topology of many functional spaces used in analysis, provided that u belongs to that space over the larger set V . In this section we deal with this question and the approach will always be the same: to prove convergence in a given space of distributions XU we will first prove the equicontinuity of G in the space XRN  and then try to apply the standard fact that under the hypotheses of equicontinuity it is enough to check the convergence on a convenient dense subset of XV. Usually the dense subset will be the space of test functions * ∈ Cc V, for which we know that G * → * in C  U . Thus, this approach works if (i) XV is a normal space of distributions (i.e., Cc V is dense in XV), and (ii) C  U  ⊂ XU with continuous inclusion. We have already applied this principle in the proof   of Theorem II.1.1 with XV = C 0  t ≤ R  L2k BR  .

70

The Baouendi–Treves approximation formula

II.3.1 Convergence in L p The main result of this subsection is: Theorem II.3.1. Let  be a locally integrable structure on  and assume that dZ1      dZm span ⊥ at every point of . Then, for any z ∈ , there exist two open sets U and W , with z ∈ U ⊂ U ⊂ W ⊂ , such that for any u ∈ Lploc W, 1 ≤ p ≤ , satisfying u = 0, E ux t −→ ux t a.e. in U as  → .

(II.19)

In case p is finite, i.e., 1 ≤ p < , we also have E ux t −→ ux t in Lp U as  → .

(II.20)

In (II.19) and (II.20) we may replace the operator E by a convenient sequence of polynomials in Z, P Z1      Zm . In the proof of Theorem II.3.1 we may assume from the start by shrinking W that u ∈ Lp W and we will do so. We are also tacitly assuming that we are using special coordinates x t adapted to a given set of local generators dZ1      dZm of ⊥ with linearly independent real parts so that Z = x + ix t, where x t is smooth, real, has compact support and satisfies (II.3 ). Once the special coordinates x t are fixed, the operator E referred to in (II.19) and (II.20) is defined precisely as in the proof of Theorem II.1.1. We will also prove below theorems similar to Theorem II.3.1 for different norms and in all of them the first step will be to choose special local coordinates where Z has this special form where the operators E and G are defined and have good convergence properties. To avoid repetitions we will always assume that this step has already been carried out, even if not mentioned explicitly. According to the considerations made at the beginning of the section, we need only prove that G u −→ hu

in

Lp W

 −→ 

u ∈ Lp W

(II.21)

For 1 ≤ p < , the space Cc0 W is dense in Lp W and (II.20) will be a consequence of G u −→ hu

uniformly

 −→ 

u ∈ Cc0 W

(which we already know by Theorem II.1.1) and the uniform bound that we will prove later: G up ≤ Cup  where  p denotes the Lp -norm.

u ∈ Lp RN 

 > 0

(II.22)

II.3 Convergence in standard functional spaces

71

Let us set W = Bx × Bt , where Bx =  x < R and Bt =  t < R . Let u ∈ Lp W and set ut x = ux t. Fubini’s theorem guarantees that ut is defined for a.e. t, it is measurable, and it belongs to Lp Bx . If, moreover, u satisfies u = 0, we know that u has a trace Tt u and Bt  t → Tt u ∈  Bx  is a smooth function. It will be useful to compare both types of restrictions of u to the slices t = const. Lemma II.3.2. If u ∈ Lp W 1 ≤ p ≤ , and u is a solution of the system (II.5) then Tt u = ut for a.e. t ∈ Bt . In particular, Tt u ∈ Lp Bx  for a.e. t ∈ Bt . Proof. We take functions  ∈ Cc Bx  and * ∈ Cc Bt . We know that t → #Tt u $ is a C  -function defined in Bt , t → #ut  $ belongs to Lp Bt  and     ux tx dx *t dt #Tt u $*t dt = (II.23)  = #ut  $*t dt  If we take *t = (j t −t0 , (j t = j n (jt, 0 ≤ ( ∈ Cc  t ≤ 1 , (dt = 1, and let j → , the left-hand side of (II.23) converges for every t ∈ Bt to #Tt u $ while the right-hand side converges a.e. to #ut  $. Hence, there is a null set N ⊂ Bt such that  ∈ Cc 

#Tt u $ = #ut  $

t % N

If we apply the last identity to a dense sequence k ⊂ Cc Bx  and set  N = Nn  we obtain that Tt u = ut as elements of  Bx  when t is not in the null set N . Remark II.3.3. One cannot expect in general that, under the conditions of Lemma II.3.2, Tt u ∈ Lp for all t. For instance, if  = −1 1 × −1 1 ⊂ R2 , Z = x + it2 /2, L = t − itx is the Mizohata operator and ux t = 1/Zx t, it is simple to verify that u ∈ Lp  for 1 ≤ p < 3/2, Lu = 0 in the sense of distributions and Tt u ∈ C   −1 1  ⊂ L −1 1 ⊂ Lp −1 1 for t = 0 but for t = 0 we have T0 u = pv1/x − i)x % Lp −1 1. We now prove Theorem II.3.1. Consider the maximal operator associated with G u: G∗ ux t = sup G ux t  ≥1

We claim that, for u ∈ L1 , there exists a constant C > 0 such that G∗ ux t ≤ CMhxTt ux

(II.24)

72

The Baouendi–Treves approximation formula

for any t such that Tt u ∈ L1 Bx . Here Mfx t = sup r>0

 1

fx  t dx

Bx r Bxr

is the Hardy–Littlewood maximal operator acting in the x-variable, Bx r is the ball of radius r centered at x, and Bx r denotes its Lebesgue measure. In fact, G ux t can be estimated by   2  2 /)m/2 e− x−x − xt−x t  Tt ux 

hx 

detZx x  t dx Rm

and this expression can be dominated by the maximal operator   2 sup F ∗ h Tt u det Zx = C sup  m/2 e−3 x−x /4 ≥1

Rm

≥1 

Tt ux 

hx 

det Zx x  t dx where F x = C m/2 e−3 x

2 /4

and C is a constant. Hence, G∗ ux t ≤ sup F ∗ h Tt u detZx ≤ CMh·Tt u·x ≥1

The last inequality follows from the fact that F1 x = Ce−3 x /4 is radial decreasing and belongs to L1 Rm  (see, for instance, [S1, page 62]). Thus, (II.24) is proved. If u ∈ Cc0 W, we know that G ux t → hxux t  →  uniformly. The standard properties of the maximal operator allow us to conclude that for any t ∈ Bt such that Tt ux ∈ L1 Bx  there exists a subset Nt ⊂ Bx with

Nt = 0 such that 2

G ux t → hxux t

x % Nt 

Hence, if we choose x t ∈ U such that Tt u ∈ L1 Bx  and x % Nt , we get (recalling that R u → 0 uniformly in U ) E ux t → hxux t = ux t ae in U and therefore E ux t → ux t a.e. in U as we wished to prove. We now prove (II.22). We observe that

G ux t ≤ F ∗ h Tt u det Zx and then Young’s inequality for convolution implies G u· tLp dx ≤ F 1 h Tt u det Zx Lp dx ≤ CTt uLp dx 

II.3 Convergence in standard functional spaces

73

since the L1 norm of F does not depend on  and h det Zx is bounded. Raising this inequality to the pth power and integrating with respect to t we obtain (II.22). Since G u → hu uniformly in W as  →  when u is continuous, the usual density argument shows that (II.21) holds for 1 ≤ p < . Thus, (II.19) and (II.20) have been proved. Finally, since E u can be approximated in C  U by polynomials in Z for fixed , the proof is complete. It is obvious that (II.20) is, in general, false for p =  because the uniform limit of a sequence of continuous functions, such as E ux t, is continuous. A simple consequence of Theorem II.3.1 is: Corollary II.3.4. Let  be a locally integrable structure over a C  manifold U and let u ∈ Lploc U, 1 ≤ p ≤ , v ∈ Lqloc U, 1/p + 1/q = 1, be solutions of the system (II.5). Then the product w = uv ∈ L1loc U also satisfies (II.5). Proof. By localization we may assume that U is the neighborhood where the conclusions of Theorem II.3.1 hold. Set u = E u, w = u v. Leibniz’s rule shows that w = 0, as u ∈ C  U. By Theorem II.3.1 and Hölder’s inequality w → w in L1loc U,  → , showing that w = 0 in the sense of distributions.

II.3.2 Convergence in Sobolev spaces In this subsection we prove Theorem II.3.5. Let  be a locally integrable structure with first integrals Z1      Zm , defined in a neighborhood of the closure of W = Bx × Bt . There exists a neighborhood U ⊂ W of the origin such that for any u ∈ Lps loc W, 1 < p < , s ∈ R, satisfying u = 0, E ux t −→ ux t in Lps loc U

 −→ 

(II.25)

As usual, we may replace the operator E in (II.25) by a convenient sequence of polynomials in Z, P Z1      Zm . We recall that for 1 ≤ p ≤  s ∈ R,  Lps RN  = f ∈  RN  f ps = #s f p <  where #s fx =  −1 1 +  2 s/2  f x and  denotes the Fourier transform in RN (#s is the Bessel potential and  denotes the space of tempered distributions). For k ∈ Z+ and p in the range 1 < p <  the space Lpk RN  is exactly the subspace of the functions in Lp RN  whose derivatives of

74

The Baouendi–Treves approximation formula

order ≤k in the sense of distributions belong to Lp RN . This space is equivalently normed by ([S1]) uLpk =



D up 

(II.26)

 ≤k

 The space Lps loc  is the subspace of  of the distributions u such that p N *u ∈ Ls R  for all test functions * ∈ Cc , equipped with the locally convex topology given by the seminorms u → *ups , * ∈ Cc . Fix p ∈ 1 , s ∈ R and choose the open sets U and W as in Theorem II.1.1. The theorem will be proved if we show that

lim G v = h v

→

in

Lps W

∀v ∈ Cc W

(II.27)

and there exists a positive constant C such that G wps ≤ Cwps

∀w ∈ Lps RN 

(II.28)

Indeed, (II.27) and (II.28) imply as usual, by density and triangular approximation, that G w − hwps → 0 as  →  for any w ∈ Lps RN  ∩   W— where   W denotes the space of distributions compactly supported in W —which implies that G w → w in the topology of Lps loc U. We know that for u ∈ Cc U, G u → u in C  U, thus (II.27) is clearly true and we need only worry about proving (II.28), which we prove first for a positive integer s = k ∈ Z+ . The vector fields Lj and Mk form a basis of CT Rn and we may express the derivatives D in (II.26) in terms of the vector fields Lj , j = 1     n, Mk , k = 1     m. This gives G wLpk ≤ C



M 1 L2 G wp 

(II.29)

1 + 2 ≤k

We write Lj G w = G Lj w + Lj  G w Mk G w = G Mk w + Mk  G w As shown in Lemmas II.1.4 and II.1.6, the operators Lj  G and Mk  G are given by the same expression as G with hx replaced respectively by Lj hx and Mk hx. Hence, the proof of Theorem II.3.1 gives bounds in Lp for the commutators that may be written as  Lj  G vp +  Mk  G vp ≤ Cvp 

v ∈ Lp RN 

(II.30)

II.3 Convergence in standard functional spaces

75

Thus, for 1 ≤ j ≤ n, 1 ≤ k ≤ m, Lj G wp + Mk G wp ≤ CLj wp + Mk wp + wp  ≤ Cwp1 + wp  ≤ Cwp1 

(II.31)

where we have used (II.22) to estimate G Lj w and G Mk w in the first inequality. Thus, combining (II.26) for u = G w and k = 1 with (II.31) we get (II.28) for k = 1. This reasoning can be iterated for any s = k ∈ Z+ and the theorem is proved for s ∈ Z+ . To prove (II.28) for nonintegral s > 0, we use interpolation of Sobolev spaces (on the subject of interpolation see, for instance, [C1] and [C2]). First we take k ∈ Z+ such that 0 < s < k. The operator G is of type p p 0 0 and also of type p p k k k ∈ Z+ , that is, it verifies G wp ≤ Cwp 

w ∈ Cc RN 

and G wpk ≤ Cwpk 

w ∈ Cc RN 

By complex interpolation we obtain that G is of type p p s s; that is, (II.28) holds for 0 < s < k and w ∈ Cc RN  and by density it also holds for w ∈ Lps RN . Finally, to prove (II.28) for s < 0, we invoke a slight variation of the duality argument that was used to extend (II.18) from positive integers to negative integers: we consider the modification of G , G ux = hxG ux which is formally symmetric in the x-variables for fixed t for the pairing given by integration with respect to dZx t = det Zx x t dx and thus also symmetric in both variables x and t for the pairing given by integration with respect to dZx t ∧ dt = det Zx x t dxdt. Since this is a nonsingular continuous pairing for the spaces Lps RN  and Lq−s RN , 1/p + 1/q = 1, it extends (II.28) to s < 0 as follows: G wLps RN  ≤ C ≤C ≤C

sup *∈Cc RN  *Lq ≤1 −s

sup *∈Cc RN  *Lq ≤1 −s

sup *∈Cc RN  *Lq ≤1 −s

#G w· t *$

#w G *$ wLps G *Lq−s

≤ Cs wLps RN  

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The Baouendi–Treves approximation formula

where in the last inequality we used (II.28) with q in the place of p and −s > 0 in the place of s. Thus, (II.28) is completely proved and the proof of Theorem II.3.5 is complete.

II.3.3 Convergence in Hölder spaces Let  ⊂ R be an open, bounded, convex set. The Hölder space C   is defined as N

C   = u ∈ C k  u <  where u = u  + u 0 

u 0 = sup ux  x∈

ux − uy 

x − y  xy∈

u  = sup

0 <  ≤ 1

x=y

u  =



D u −k 

k <  ≤ k + 1

k ∈ Z+

u ∈ C k 

≤k 

The spaces C RN  are defined similarly. The approximation theorem is: Theorem II.3.6. Let  be a locally integrable structure with first integrals Z1      Zm , defined in a neighborhood of the closure of W = Bx × Bt . There exists a convex neighborhood U ⊂  of the origin such that for any u ∈ C  W,  > 0 satisfying u = 0 in a neighborhood of W and any 0 ≤  <  E ux t −→ ux t in C  U

 −→ 

(II.32)

As usual, we may replace the operator E in (II.32) by a convenient sequence of polynomials in Z, P Z1      Zm . Proof. As always, since Cc W is dense in Cc W for the C  norm, we need only prove G u −→ u

in

C  W

u ∈ Cc W

and the inequality G u ≤ Cu 

u ∈ Cc W

It is obvious that G u − u → 0 when  → , u ∈ Cc W, because by Theorem II.1.1 G u → u,  →  in C k W for every positive integer k. We may assume without loss of generality, as we always do, that Zx t = x + ix t is defined and satisfies (II.3 ) throughout RN and reduces to

II.3 Convergence in standard functional spaces

77

Zx t ≡ x for x t outside a compact set. We shall then prove G u ≤ Cu 

u ∈ Cc RN 

(II.33)

We assume first that 0 <  < 1. It will be useful to use the following well-known characterization of C  RN  ([S2, page 256]): Lemma II.3.7. A function u belongs to C  RN , 0 <  < 1, if and only if there exist a sequence of functions uk  ∈ C 1 RN , bounded and with bounded gradients, such that (i) uk L ≤ K 2−k , k = 0 1     (ii) ,uk L ≤ K 21−k , k = 0 1      N (iii) uz =  k=0 uk z, z ∈ R . It also follows that the best constant K in (i) and (ii) above is proportional to u . Such a sequence is usually called a sequence of best approximation for  u. We start by writing u = uk with uk  a sequence of best approximation  for u. Then, G u = G uk and we need to estimate the essential supremum of G uk and ,G uk . Taking account of (II.22) with p =  and (i) of Lemma II.3.7 we derive G uk L ≤ Cuk L ≤ CK2−k 

k ∈ Z+ 

(II.34)

In order to estimate ,G uk it is convenient to express any partial derivative in terms of the vector fields Lj and M , 1 ≤ j ≤ n, 1 ≤  ≤ m. Then, we are led to estimate Lj G uk , j = 1     n and M G uk ,  = 1     m. We may write Lj G uk = G Lj uk + Lj  G uk and recall that  Lj  G uk L ≤ Cuk L  which follows from (II.30) with p = . We get Lj G uk L ≤ CLj uk L + uk L  ≤ C,uk L + uk L 

j = 1     n

k = 1 2   

Similar estimates are true for M G uk ,  = 1     m, k ∈ Z+ and we obtain ,G uk L ≤ Cuk L + ,uk L  ≤ C  K21−k 

k ∈ Z+ 

(II.35)

Thus, (II.34), (II.35) and Lemma II.3.7 imply that (II.33) holds for 0 <  < 1. Let us assume next that there is a positive integer k such that  = k + , 0 <  < 1 and we wish to estimate   D G u ≤ C M 1 L2 G u  G u ∼

 ≤k

1 + 2 ≤k

78

The Baouendi–Treves approximation formula

Using the commutation formulas of Lemmas II.1.4 and II.1.6 it is easy to prove (II.33) by induction on k, adapting the reasonings we used to deal with Sobolev norms of integral order in Section II.3.2; we leave the details to the reader. Finally, to prove (II.33) for  = k = 1 2    , we observe that in this case u = uk ∼ uLk so (II.33) is a variation of the estimates already considered for Sobolev norms. This completes the proof of Theorem II.3.6.

It is not possible to take  =  in Theorem II.3.6, as we will see next. Example II.3.8. Consider in R2 , where we denote the coordinates by x t, the structure  spanned by t with first integral Zx t = x and let 0 <  ≤ 1. Consider a function ux ∈ Cc R2  independent of t (so it satisfies u = 0) such that ux = x  for x ≤ 1. If wx t is of class C 1 in a neighborhood of the origin, we have for 0 < - < 1 sufficiently small,

u − w  ≥

u- − w- 0 − u0 − w0 0 ≥ 1 − C-1− -

and the left-hand side is ≥ 1/2 for - small, showing that u cannot be approximated by continuously differentiable functions in the C  topology.

II.3.4 Convergence in Hardy spaces We recall that the real Hardy space H p RN , 0 < p < , introduced by Stein and Weiss ([SW]), is equal to Lp RN  for p > 1, is properly contained in L1 RN  for p = 1, and is a space of not necessarily locally integrable distributions for 0 < p < 1. For p ≤ 1, H p RN  is a substitute for Lp RN  ([S2]), as the latter is not a space of distributions and has trivial dual if p < 1; even for p = 1, L1 RN  does not behave as well as Lp RN , 1 < p < , for example on questions concerning the continuity of pseudo-differential  operators. Let us choose a function ! ∈ RN , with !dz = 0 and write !- z = -−N !z/-, z ∈ RN , and M! fz = sup !- ∗ fz  0 0 such that for all hp atoms az  (II.39) G aphp = m! G azp dz ≤ C  ≥ 1 Indeed, 

" m ! G



#p  k ak

dz ≤

k

≤



#p

"





k m! G ak

k

k p



dz

m! G ak p dz

k

because p ≤ 1. We assume without loss of generality that ! ≥ 0 is supported in the unit ball (in fact, changing the function ! by any other function in RN 

II.3 Convergence in standard functional spaces

81

with nonvanishing integral will produce an equivalent ‘norm’ in H p RN ). 2 We set Fx = e−3 x /4 , x ∈ Rm , F x =  −m Fs/ and we check that by the estimates of Section II.3.1 (see (II.24)): x

!- ∗ G ax t ≤ C !- ∗ F ∗ ax t x

= C !- ∗ a ∗ F x t 

 =  −1/2 

x

where the symbol ∗ denotes convolution in the x-variable. Let Q = Q1 × Q2 , Q1 ⊂ Cm , Q2 ⊂ Cn , be a cube containing the support of a. Thus, invoking (i), we get m! G ax t ≤ C Q −1/p (Q2 t

(II.40)

Here and in the sequel, (A will denote the characteristic function of a measurm able set A. Let Q∗1 (resp. Q∗∗ 1 ) be the cube in R concentric with Q1 having twice (resp. four times) the side length. Then (II.40) shows that 

m! G ax t p dx dt ≤ C (II.41) n Q∗∗ 1 ×R

with C > 0 independent of 0 < - ≤ 1,  ≥ 1, az an atom. Thus, (II.39) will be proved as soon as we obtain  x sup !- ∗ F ∗ ax t p dx dt ≤ C 0 <  ≤ 1 (II.42) n Rm \Q∗∗ 1 ×R 0 m, we obtain for a large integer d = L − m

!-1 ∗ a ∗ F x t ≤ C(Q2 t Q −1/p x − x0 −d 

82

The Baouendi–Treves approximation formula

n Convolving with !-2 t gives, for x % Q∗∗ 1 and t ∈ R , x

t

!- ∗ F ∗ ax t ≤ C Q −1/p x − x0 −d !-2 ∗(Q2 t ≤ C Q −1/p x − x0 −d (Q∗2 t Choose d = m + 1. If we take the supremum in 0 < - ≤ 1, raise both sides to n the pth power and integrate in Rm \Q∗∗ 1  × R , we obtain (II.42), under the assumption r ≥ 1. Let us assume now that r < 1, so az satisfies the moment  conditions (ii). It is clear that these properties are inherited by a- z, i.e., z a- z dz = 0,

 ≤ N1/p − 1. We start by writing Fx as a convergent series in Rm ,  Fx = k F k x with F 0 supported in the unit ball B = B0 1 and each F k supported in some ball of radius 1. We aim at proving (II.42) with F k in the place of F . Using the vanishing of the moments of a  x a- ∗ Fk x t = (Q∗2 t ay t Gk - x − y dy  = ay t Gk (II.43) - x − y − qx- y dy 1 k where Gk - = !- ∗ F and qx- y is the Taylor polynomial of degree d of the function y → Gk - x − y expanded about x0 and d is the integral part of N1/p − 1. The usual estimates for the remainder of the Taylor expansion imply that the integrand in (II.43) is ≤ C Q −1/p  −d+1+m r d+1 . We assume first that k = 0 so F 0 is supported in the unit ball. Since x − x0 ≤ C x − y 0 when y ∈ Q∗1 and x % Q∗∗ 1 , x − y ≤  on the support of F x − y, and a is supported in the cube Q∗1 of measure 2rm it follows that for any 0 < % ≤ 1 and 0 <  ≤ 1 d+m+1p  r

a- ∗ F0 x t p ≤ C0 (Q2 t  x % Q∗∗ 1 

x − x0

which after integration gives  sup !- ∗ F0 ∗ ax t p dx dt ≤ C0  n Rm \Q∗∗ 1 ×R 0 0 a.e. in  x − x0 < % .  This shows that & = 0 ∩  j xj bj = 0 has measure zero. In view of Lemma IV.3.6, L ( = 0 so to obtain (IV.53) it is enough to prove separately the inequalities (Lp Rn+1  ≤ CT L(Lp Rn+1  

 ∈ Cc T 

1 − (Lp Rn+1  ≤ CT L1 − (Lp Rn+1  

∈

Cc T 

(IV.55) (IV.56)

The third step. We prove inequality (IV.55). The proof of (IV.55) is easy because L( = (L = (t , so  t L(x s ds (xx t = −T

Hence, (·· tLp Rn  ≤



t

−T

L(· sLp Rn  ds 

≤ 2T1/p L1 − (Lp Rn+1   with p −1 + p−1 = 1. Raising both sides to the power p and integrating with respect to t between −T and T we obtain (IV.55) with C = 2. The fourth step. We introduce a partition of unity that reduces the proof of inequality (IV.56) to the proof of local estimates for test functions. Note that the function 1 − ( is not even continuous which, of course, is a source

IV.3 Vector fields in several variables

189

of trouble. The main idea to overcome this difficulty is to write 1 − ( as a series of convenient test functions supported in \N. We start by proving some lemmas. Lemma IV.3.7. Let &x and N be as defined above. (1) The function &x is Lipschitz and " L  ,&L ≤ ,x b

(IV.57)

(2) Outside N the vector v"x is locally Lipschitz and satisfies

, v"x ≤

" L 2,x b &x

for x % N

(IV.58)

" t . Then Proof. Let x y ∈ Rn and let t ∈ −1 1 such that &x = bx " t ≤ by " t + by " t − bx " t &x = bx " L x − y  ≤ &y + ,x b " L x − y and interchanging x and y This shows that &x − &y ≤ ,x b " we are led to &x − &y ≤ ,x bL x − y for all x y ∈ Rn . This implies (IV.57). " 0  t > 0. Then Next, given x0 % N select t ≤ 1 such that &x0  = bx " t is positive and differentiable in a neighborhood of x0 , so

bx & & & & & & , b" & b" " L " && 2, b

b , & & x & x x0  t& ≤ & + b" ⊗

, v"x0  = &,x &≤ & & b & b &x " "2 &

b " ≤ ,x b . This proves (IV.58). where we have used that ,x b

In the sequel, cube will mean a closed cube in Rn , with sides parallel to the axes. Two such cubes will be said to be disjoint if their interiors are disjoint. If Q is a cube with side length  and  > 0 is a positive number, Q will denote the cube with the same center as Q and side length equal to . Lemma IV.3.8. Let f Rn −→ R+ be a Lipschitz continuous function with Lipschitz constant 0 <  ≤ 1, i.e., fx − fy ≤  x − y , x y ∈ Rn . Assume that F = f −1 0 is not empty and set  = x ∈ Rn fx > 0 . There exists a collection of cubes  = Q1  Q2     such that  Qj =  = Rn \F ; (1) j

(2) the Qj ∈  are mutually disjoint; (3) diam Qj  ≤ inf fx ≤ sup fx ≤ 5 diam Qj . Qj

Qj

190

Local solvability of vector fields

Proof. Let 0 denote the family of cubes with side length one and vertices with integral coordinates. For every integer k we define k = 2−k Q

Q ∈ 0

so the cubes in k form a mesh of cubes of side length 2−k and diameter √ −k n2 . Each cube ∈ k gives rise to 2n cubes ∈ k+1 by bisecting the sides. Set for any integer k √ √ k = x ∈ Rn 2 n2−k < fx ≤ 4 n2−k   Note that k ⊂  and  = k k . We now define  0 = Q ∈ k Q ∩ k = ∅  k

√ √ Let Q ∈ 0 ∩ k . There exists x ∈ Q such that 2 n2−k < fx ≤ 4 n2−k . Given y ∈ Q we have fx −  y − x ≤ fy ≤ fx +  y − x  √ so using that  ≤ 1 and y − x ≤ n2−k = diam Q we get diam Q ≤ inf fx ≤ sup fx ≤ 5 diam Q Q

Q

Since fy > 0 on Q it follows that Q ⊂ . Also, given y ∈  there exists a unique k such that y ∈ k and y also belongs to some Q ∈ k because   Q ∈ k = Rn , so y ∈ Q and Q ∈ 0 , which shows that Q ∈ 0 = . Thus, the cubes of 0 satisfy (1) and (3) although they may not be disjoint. To obtain the required collection  we must discard from 0 the superfluous cubes, which is easy because if two distinct cubes in 0 are not disjoint one contains the other. Namely, if Q1  Q2 ∈ 0 are not disjoint, then Q1 ∈ k1 and Q2 ∈ k2 with k1 = k2 , so if, say, k1 > k2 it turns out that Q1 ⊂ Q2 . Hence, if Q ∈ 0 is contained in some other cube Q ∈ 0 we discard Q and apply the same procedure to Q , discarding it if it is contained in a bigger cube of 0 and keeping it in the opposite case. For a fixed cube Q, this process stops after a finite number of steps, otherwise the cubes Q ⊂ Q ⊂ Q ⊂ · · · would fill Rn , contradicting that F = ∅. Thus, each cube Q ∈ 0 is contained in a maximal cube of 0 and the collection  of those cubes of 0 which are maximal satisfies (1), (2), and (3). We now need a more detailed discussion of the family  defined in the previous lemma. Although two distinct cubes Q1 and Q2 ∈  are always disjoint in the sense that they have disjoint interior their intersection may be

IV.3 Vector fields in several variables

191

nonempty, as they could share a vertex, an edge, or some k-dimensional face, k < n. In this case we say that Q1 and Q2 touch. Proposition IV.3.9. If two cubes Q1  Q2 ∈  touch, then 1 diam Q2  ≤ diam Q1  ≤ 4 diam Q2  4 Proof. Let Q1 and Q2 ∈  have a common point x in their boundaries and assume without loss of generality that diam Q1  ≥ diam Q2 , so their respective sides 1 and 2 are related by 2 = 2−k 1 for some integer k ≥ 0. If z ∈ Q2 we have √ √ √ fz ≤ fx +  n2 ≤ n1 5 + 2−k  ≤ 6 n1  where we have used that Q1 satisfies (3) of Lemma IV.3.8 to estimate fx. Now, (3) applied to Q2 gives diam Q2  ≤ supz∈Q2 fz ≤ 6 diam Q1 . Since the quotient diam Q2 /diam Q1  is a power of 2, the latter estimate implies that diam Q2 /diam Q1  ≤ 4. Proposition IV.3.10. If Q ∈  , less than 12n cubes of  touch Q. Proof. Let Q ∈  have side  = 2−k . There are exactly 3n − 1 cubes in k that touch Q and each one of them contains at most 4n−1 cubes that belong to k+2 and touch Q. Since by Proposition IV.3.9 the cubes of  that touch Q may only have the side lengths , /2, or /4 it is easily seen that the total number of cubes of  that touch Q is ≤ 3n − 14n−1 < 12n . The family  that disjointly fills up  with closed cubes gives rise to a cover by open cubes that has the bounded intersection property. We fix 0 < - < 1/4 and for any Q ∈  denote by Q∗ the cube with the same center as Q but with side dilated by the factor 1 + -. Let Q1 and Q2 ∈  do not touch. We claim that Q∗1 and Q2 cannot intersect. Indeed, the union of Q1 with all the cubes of  that touch Q1 (among which Q2 is not) contains, by Proposition IV.3.9, the cube 5/4Q1 whose interior contains Q∗1 . This shows that Q∗1 ∩ Q2 = ∅. Consider now a point x ∈  and select Q ∈  such that x ∈ Q. If x ∈ Q∗j for some Qj ∈  then Q ∩ Q∗j = ∅, which implies that Q and Qj touch. Then Proposition IV.3.10 shows that x belongs to at most 12n cubes Q∗j . If z ∈ Q∗ then fz ≥ inf Q f − - diam Q ≥ 3/4diam Q ≥ 3/5diam Q∗ . Similarly, fz ≤ 5 diam Q + - diam Q ≤ 5 diam Q∗ . Thus, for every Q ∈  we have 1 diam Q∗  ≤ inf∗ fx ≤ sup fx ≤ 5 diam Q∗  Q 2 Q∗

(IV.59)

192

Local solvability of vector fields

This estimate implies that Q∗ ⊂  and since the interior Int Q∗  ⊃ Q we see that Int Q∗  is an open cover of  with the bounded intersection property. Lemma IV.3.11. Let N ⊂ Rn be the closed set defined in (IV.54) and let 0 <  ≤ 1, 1 < p < . There exists a covering of Rn \N by open cubes with sides parallel to the coordinate axes Int Q∗j  , j = 1 2    , such that the intersection of 12n cubes of the family is always empty and for any j = 1 2     we have the estimate: 1 diam Q∗j  ≤  inf∗ &x ≤  sup &x ≤ 5 diam Q∗j  Qj 2 Q∗j

(IV.60)

Furthermore, there are functions j ∈ Cc Rn \N such that pj is a partition of unity in Rn \N subordinated to the covering Int Q∗j  and for a certain constant C > 0, ,j L ≤

C  diam Q∗j 

j = 1 2   

(IV.61)

" L ≤ 1. Proof. From now on we assume without loss of generality that ,x b We apply Lemma IV.3.8 with fx = &x so F = N. The hypotheses are satisfied because the Lipschitz constant of &x is 1 by (IV.57) and the complement of N is bounded so N = ∅. Thus we obtain the collection  of disjoint cubes Qj which, dilated by the factor 1 + -, yields the associated collection Q∗j of cubes whose interiors cover Rn \N, have the bounded intersection property, and satisfy (IV.59). This proves (IV.60). Fix a function 0 ≤ * ∈ Cc Rn  supported in x < 1 + -/2 such that * p x is smooth and *x = 1 if x ≤ 1/2 (such a function is easily constructed). If Qj ∈  , denote by xj its center and by j its side length. Then *j x = *x −xj /j  ∈ Cc Int Q∗j  and *j x = 1 on Qj . We have ,*j L ≤

,*L C  ≤ j diam Q∗j 

(IV.62)

 Note that . = j *jp is smooth and ≥ 1 in Rn \N. Let us estimate ,.x on the support of *j . If x ∈ Q∗j and *k x = 0 for some k ∈ Z+ it follows that Q∗j ∩ Q∗k = ∅. We know that Q∗j is contained in the union of Qj with those cubes of  which touch it and the same can be said about Qk . This implies that there are cubes Qj  and Qk in  such that (1) Qj  touches Qj ; (2) Qk touches Qk ; (3) Qj  ∩ Qk = ∅ so they either coincide or touch.

IV.3 Vector fields in several variables

193

Applying Proposition IV.3.9 three times we obtain that diam Qk  ≥ 4−3 diam Qj  and Proposition IV.3.10 tells us that there are less than N = 123n integers k such that Q∗j ∩ Q∗k = ∅. This shows that at most N terms *kp x of the infinite sum that defines .x are not zero if x ∈ supp *j . Thus, using the analogue for *kp of (IV.62) we obtain sup ,.x ≤ Q∗j

 k

sup ,*kp x ≤



Q∗j

k

C 43 NC ≤  ∗ diam Qk  diam Q∗j 

(IV.63)

Since

,. −1/p x ≤

1 . −1−1/p L ,.x ≤ ,.x  p

because . ≥ 1, (IV.63) implies sup ,. −1/p x ≤ Q∗j

C  diam Q∗j 

(IV.64)

Set j x =

*j x  . 1/p x

Then, pj is a partition of unity in Rn \N with the required properties. Indeed, to prove (IV.61) we use the Leibniz rule and invoke (IV.62) and (IV.64). The fifth step. We prove estimate (IV.56) when x t is supported in Q∗j × −T T, Qj ∈  . Assume that  is supported in Q∗k × −T T for a certain cube ∈  ; the value of T < 1 will be chosen momentarily. Since we are " L ≤ 1, (IV.58) yields assuming that ,x b

, v"x ≤

2 &x

for x % N

This shows, in view of (IV.60), that , v"x ≤ 4/diam Q∗j  on Q∗j . Furthermore, Rn \N is bounded so diam Q∗j  ≤ C, j ∈ Z. Hence, v"x is approximately constant on Q∗j if  is small; this allows us to rectify its flow as follows. Since v" is a unit vector, we may assume without loss of generality that at the center √ √ xj of Q∗j we have v1 xj  ≥ 1/ n. Then, v1 xj  − v1 x ≤ 4  < 1/2 n for  fixed once for all, small but independent of j, and we may assume that √ v1 x ≥ 1/2 n on Q∗k . Solving the differential equations dxj vj x =  dy1 v1 x

xj 0 = yj 

j = 2     n

(IV.65)

we obtain a change of variables on a neighborhood of Q∗k given by x1 = y1 , xj = xj y1  y2      yn , 1 < j ≤ n, where the right-hand side denotes the

194

Local solvability of vector fields

solution of (IV.65). In the new coordinates v"xy = v1 xy/y1 and L assumes the form   − ib1 xy t t y1 " with b1 > 0, since b1 xy t 0     0 = bxy t implies b1 xy t = "

bxy t . Set By t = b1 xy t. Then, by the chain rule, ,y BL ≤ C,x b1 L ≤ C  because the Lipschitz constant of the change of variables y → xy is bounded by a constant independent of j, as follows from the fact that the right-hand side of the ODE (IV.65) is bounded by C. Now we apply Theorem IV.1.9 with p = q to the vector field L1 =

  − iBy t t y1

(IV.66)

that we regard as a vector field in two variables depending on a parameter y = y2      yn . For some constants C and T0 whose size only depends on ,x b1 L we get for any 0 < T ≤ T0 · · y pLp ≤ CT L1 · · y pLp 

 ∈ Cc Q†j × −T T

where the Lp norms are taken in the variables y1  t and the map y → xy takes Q†j onto Q∗j . Integrating this estimate with respect to y we get  ∈ Cc Q†j × −T T

pLp ≤ CT L1 pLp 

Observing that the absolute value of the Jacobian determinant of y → xy is close to 1 uniformly in j ∈ Z+ , the latter estimate implies in the original variables x t pLp ≤ CT LpLp 

 ∈ Cc Q∗j × −T T

(IV.67)

which may be regarded as estimate (IV.56) for  ∈ Cc Q∗j × −T T. The sixth step. We prove (IV.56) in general. Let  ∈ T and set j = j  where j is the collection of functions described by Lemma IV.3.11. We have  1 − (x x t p = j 1 − (xx t p  j

Integrating this identity and taking account of (IV.67),   1 − (pLp = j 1 − (pLp ≤ CT Lj pLp j

j

IV.3 Vector fields in several variables ≤ CT 1 − (LpLp + CT



195

Lj 1 − (pLp

j

 where we have used the Leibniz rule and the fact that j pj = 1. The second term on the right-hand side is dominated by CT 1 − (pLp . Indeed, " t · ,x j x ≤ sup b " ,x j ≤

Lj x = bx Q∗j

C ≤ C1 

in view of the definition of &, (IV.60) and (IV.61). Hence, Lj x p ≤ C and since Lj x p = 0 except for at most 12n values of j we also have  p j Lj x ≤ C. Thus, 1 − (pLp ≤ CT 1 − (LpLp + CT 1 − (pLp and the last term can be absorbed as soon as CT < 1/2. This proves (IV.56). We have already seen in steps 1 and 2 that (IV.53) follows in general once (IV.55) and (IV.56) are proved for L of the form (IV.51), so the proof of Theorem IV.3.5 is now complete for L and we may also replace L by −L + cx t in (IV.53) if cx t is any bounded function provided we shrink the neighborhood U of the origin, in particular, we may replace L by the " transpose operator t L = −L − idivx b. As usual, we obtain by duality Corollary IV.3.12. Let L given by (IV.49) satisfy (IV.50) and condition  in a neighborhood of the origin and fix 1 < p < . Then, there exist R0 and C > 0 such that for every 0 < R < R0 and f ∈ Lp Rn+1  there exists u ∈ Lp Rn+1  with norm uLp Rn+1  ≤ C Rf Lp Rn+1  that satisfies the equation Lu = f

for x 2 + t2 < R2 

(IV.68)

Moreover, the constants C and R0 depend only on p and the L norms of the derivatives of order at most two of the coefficients of L. Let us assume now that we are dealing with a locally integrable vector field L in an open set of Rn+1 that contains the origin. After an appropriate local change of coordinates x t we may assume that there are functions Zj x t, j = 1     n defined on a neighborhood of the origin of the form Zj x t = xj + ij x t

j = 1     n

196

Local solvability of vector fields

with j x t smooth and real satisfying j 0 0 = ,x j 0 0 = 0

j = 1     n

such that LZj = 0

j = 1     n

We denote by Z the function Z = Z1      Zn  with values in Cn and similarly write  = 1      n , so Zx t = x + ix t. The n × n matrix ⎞ ⎛ ··· 1 /xn 1 /x1 ⎟ ⎜    x = ⎝ ⎠    ···

n /x1

n /xn

vanishes at the origin and after modification of L outside a neighborhood of the origin we may assume that the functions j x t are defined throughout Rn+1 , have bounded derivatives of all orders, and satisfy 1 x x t ≤  x t ∈ Rn+1  2 This implies that the matrix Zx = I + ix is everywhere invertible and we write Zx−1 x t = jk x t. Then the vector fields Mj =

n 

jk x t

k=1

  xk

j = 1     n

(IV.69)

commute pairwise and the vector field L1 =

n    − k x t t k=1 xk

commutes with M1      Mn and is proportional to L if k x t = −i

n 

kj x t

j=1

j x t t

Furthermore, M1      Mn  L are linearly independent at every point and generate T Rn+1 . Multiplying L by a nonvanishing factor we may assume that L = L1 . We now extend Theorem IV.2.3 to several variables. Theorem IV.3.13. Assume that L is a smooth vector field defined in an open subset  ⊂ Rn+1 and let cx t ∈ C  . If L satisfies  in  and is locally integrable then every point p ∈  has a neighborhood U such that the equation Lu + cu = f

f ∈ Cc U

IV.3 Vector fields in several variables

197

may be solved with u ∈ C  U. Conversely, if L is locally solvable in C  then L is locally integrable. Proof. The construction of smooth solutions is a straightforward extension of the two-dimensional case. We write D = −L2 − M12 + · · · + Mn2  where M1      Mn are given by (IV.69) and  > 0 is a large parameter. Since L = L1 and Mj commute, j = 1     n, it follows that L and D commute. If x t   denotes the symbol of L, mj x t  denotes the symbol of Mj and dx t   = −2 + m21 + · · · + m2n x t   is the principal symbol of D, we have  d x t   = 0

x t   ∈ R2n+1 

For large  > 0, D is a uniformly elliptic second-order differential operator. Consider, for fixed s ∈ R, the pseudo-differential operator  1 B% ux t = eix·+t px t  $ u  dd 2)n+1 RN +1 with symbol b% x t   =

(t1 + dx t  s/2 1 + % dx t  1/2

where (t ∈ Cc −T T and (t = 1 for t ≤ 3/4T . Here we choose T so that the estimate ux tL2 Rn+1  ≤ CLux tLn+1 R2 

(IV.70)

holds for every u ∈ Cc Rn × −T T for some C > 0, as guaranteed by the proof of Theorem IV.3.5. The estimate can be extended to any u ∈ L2c Rn −T T × −T T such that Lu ∈ L2c Rn −T T × −T T by Friedrich’s s lemma. It follows that b% → b = (1 + ds/2 in the symbol space S10 and that us ∼ BuL2 if B is the pseudo-differential with symbol b and u ∈ Hcs Rn × −T/2 T/2. Furthermore, L B has order s − 1 on R × −T/2 t/2. If u ∈ Hcs−1 Rn × −T/2 T/2 is such that Lu ∈ Hcs Rn × −T/2 T/2 we may apply (IV.70) to B% u. Letting % → 0 we obtain BuL2 Rn+1  ≤ CBLuL2 Rn+1  +  L B uL2 Rn+1   which implies that u ∈ H s Rn+1  and uH s Rn+1  ≤ Cs LuH s Rn+1  + uH s−1 Rn+1  

(IV.71)

198

Local solvability of vector fields

Once (IV.71) is known, general arguments lead to an a priori estimate uH s Rn+1  ≤ Cs LuH s Rn+1  

(IV.72)

if u ∈ Hcs−1 Rn × −T/2 T/2 is such that Lu ∈ Hcs R × −T/2 T/2 and to the existence of local smooth solutions, as described in the proof of Theorem IV.2.3. We leave details to the reader. While the method to obtain smooth solutions starting from the existence of L2 solutions is essentially the same independently of the number of variables, the proof that smooth local solvability implies local integrability is rather different if n = 1 or n > 2. In the proof of Theorem IV.2.3 it was shown that, for n = 1, solving Lu = f for a specific f obtained from the coefficients of L was enough to produce locally a smooth Z such that LZ = 0 and dZ = 0. Nothing like this is available if n > 1 and we must proceed indirectly. Assume that L given by (IV.51) is locally solvable in C  and we wish to find n first integrals with linearly independent differentials defined in a neighborhood of a given point p that we may as well assume to be the origin. The first step is to find a complete set of approximate first integrals, namely, n smooth functions Zj# , j = 1     n, such that LZj# = fj vanishes to infinite order at the origin—i.e., fj x = O x k , k = 1 2    —and dZ1# 0     dZn# 0 are linearly independent. To find Zj# we solve first the noncharacteristic Cauchy problem  LUj = 0 Uj x 0 = xj  in the sense of formal power series. The coefficients of the formal series Uj corresponding to monomials that do not contain t are determined by the initial condition Uj x 0, i.e., they are all zero with the exception of the coefficient of xj which is 1. The coefficients of monomials of the form t x are determined from LUj = 0 inductively on . Once the formal series Uj has been found we take as Zj# any smooth function that has Uj as its Taylor series at the origin (the existence of such a function is usually called Borel’s lemma). By their very definition Z1#      Zn# are approximate first integrals. To obtain exact first integrals by correction of Z1#      Zn# we must solve the equations Luj = fj , j = 1     n, in a neighborhood of the origin and then define Zj = Zj# − uj . Clearly, LZj = 0, so the problem is now to verify that dZ1 0     dZn 0 are linearly independent. This will be guaranteed if we can make sure that duj 0 is small. Let K be a ball centered at the origin such that LC  K = C  K and let  denote the subspace of C  K of the (equivalence classes of) functions h such that Lh = 0. Then L defines a continuous linear map from C  K/ onto C  K which, by the open

IV.4 Necessary conditions for local solvability

199

mapping theorem for Fréchet spaces, has a continuous inverse. This means, in particular, that given % > 0 there exists  > 0 and m ∈ Z+ such that for every f ∈ C  K such that D f L K <  for all  ≤ k there exist u ∈ C  K such that Lu = f and duL K < %. Let (x t ∈ Cc Rn+1  be equal to 1 for

x 2 + t2 < 1 and set fj& x t = fj x t(&x &t. Since fj vanishes to infinite order at the origin we see that, choosing & big enough, D fj& L <  for all  ≤ k. Choose now uj such that Luj = fj& and duj L K < %. Since fj& = fj for x 2 +t2 < 1/& we see that the functions Zj = Zj# −uj , j = 1     n form a complete set of first integrals in a neighborhood of the origin if % is taken small enough.

IV.4 Necessary conditions for local solvability In this section we discuss the necessity of condition  for the local solvability of a locally integrable vector field. Assume that L defined in  ⊂ Rn+1 by (IV.49) is locally solvable in the sense of Definition IV.1.2. We will show that L must satisfy condition  in . In doing so, due to the local nature of the problem, we may assume that L is given by (IV.51) and that  = B × −T T where B ⊂ Rn is a ball centered at the origin. We may also assume that there is a vector-valued function Zx t = Z1 x t     Zn x t defined in a neighborhood U of  such that LZj = 0, j = 1     n and I − Zx  < 1/2 in , where I denotes the identity matrix. In particular, the form dZ1 ∧ · · · ∧ dZn does not vanish in  and the pairing  Cc  × Cc   f v → fv detZx  dxdt is nondegenerate. The formula   f Lv detZx  dxdt = − Lf v detZx  dxdt

v f ∈ Cc 

means that L and −L are each other’s formal transpose with respect to this pairing. The formula is also valid by continuity if v ∈   provided that we replace the integration by the standard duality between distributions and test function, i.e., #Lv f detZx $ = −#v Lf detZx $

f ∈ Cc  v ∈  

(IV.73)

One of the basic tools in the study of necessary conditions for local solvability is Hörmander’s lemma ([H6]), of which we give the following version.

200

Local solvability of vector fields

Lemma IV.4.1. Let L be as described above and suppose that for every f ∈ Cc  there exists u ∈   such that Lu = f . Then, for any compact set K ⊂  there exist constants C > 0, M ∈ Z+ such that & &   & &   Dxt f L Dxt LvL (IV.74) & fv detZx  dxdt& ≤ C

 ≤M

 ≤M

for all f v ∈ Cc K. Proof. Let K ⊂⊂  with nonempty interior be given and consider the bilinear form (IV.73) restricted to pairs f v ∈ Cc K × Cc K. Endow the first  factor with the topology defined by the seminorms Dxt f L —so it becomes a Fréchet space—and the second factor with the countable family of semi norms Dxt LvL . Our solvability hypothesis implies that the latter topology is Hausdorff, indeed, if v ∈ Cc K is such that Lv = 0 we may choose for any f ∈ Cc K a distribution u ∈   such that Lu = f , so we have #f v detZx $ = #Lu v detZx $ = −#u Lv detZx $ = 0 for any f ∈ Cc K, which implies that v = 0. For fixed v, the bilinear form clearly depends continuously on f . The solvability hypothesis implies that the dependence on v is also continuous for f fixed. Indeed, we may assume that f = Lu for some u ∈  . Hence  fv detZx  dxdt = #Lu f detZx $ = −#detZx u Lf $ in view of (IV.73), which shows the continuity with respect to f for fixed v. A bilinear form defined on the product of a Fréchet space and a metrizable space which is separately continuous is continuous in both variables. This proves (IV.74). The last lemma shows that in order to prove that L is not solvable it is enough to violate the a priori inequality (IV.74). We now describe a method to violate (IV.74) provided we find a solution h of the homogenous equation Lh = 0 with certain geometric property. Let g ∈ C 0  be a real function and K ⊂⊂  be compact. We say that g assumes a local minimum over K if there exists a ∈ R and V open, K ⊂ V ⊂  such that (1) g ≡ a on K; (2) g > a on V \K. Note that we may always replace the open set V with one of its open subsets with compact closure that contains K. In this case, still denoting the new set

IV.4 Necessary conditions for local solvability

201

by V we have inf g = a1 > a V

Then, taking a < b < a1 we see that the set W = g < b ∩ V has compact closure contained in V and g ≥ b > a on V \W . The proof of the next lemma shows how (IV.74) may be violated. Lemma IV.4.2. Assume that there exists h ∈ C   such that (i) Lh = 0; (ii) h assumes a local minimum over some K1 ⊂⊂ . Then there exists f ∈ Cc  such that Lu = f for all u ∈  . Proof. By Lemma IV.4.1 it will be enough to show that for a convenient choice of K ⊂⊂ , (IV.74) cannot hold for all f v ∈ Cc K whatever the choice of M ∈ Z+ and C > 0. By hypothesis h assumes a local minimum over K1 ⊂⊂  for some homogeneous solution h. Subtracting a constant we may assume that h = 0 on K1 and h ≥ - > 0 on V \W for some open sets  V ⊃ W ⊃ K1 such that K = V ⊂⊂ . Select  ∈ Cc K, 0 ≤  ≤ 1, such that  = 1 on W and set, for a large parameter & > 0, v& x t = x te−&hxt  Since e−&hxt is a homogeneous solution, Lv& = e−&h L. Furthermore, L is supported in K\W so it follows that   Dxt Lv& L ≤ C&M e−-&  (IV.75)

 ≤M

∈ Cc V,

0 ≤ * ≤ 1, such that * = 1 on K1 and hx t < -/2 Next, choose * on the support of *. Define f& x t = Then



*x t e&hxt  detZx x t

 Dxt f& L ≤ C&M e-&/2 

(IV.76)

 ≤M

On the other hand, since  and * are positive in a neighborhood of K1 ,   f& v& detZx  dxdt = x t*x t dxdt = c > 0 which together with (IV.75) and (IV.76) shows that (IV.74) cannot hold for the pair f&  v&  ∈ Cc K × Cc K if & is large enough.

202

Local solvability of vector fields

Our next task is to produce solutions of the homogeneous equation Lh = 0 whose real part assumes a local minimum over a compact set assuming that condition  does not hold. We will first discuss this in the case n = 1, which is technically simpler and the geometric ideas involved are easier to spot. Suppose n = 1, L = t − Zt /Zx x , Z = x + ix t, x t ∈ R2 . We know by Lemma IV.2.2 that if  does not hold then t → x0  t is not monotone for some x0 , or equivalently that t → t x0  t takes opposite signs and, in particular, vanishes for some t0 . The simplest situation occurs when   t x0  t0  = 0 and tt x0  t0  = 0. If tt x0  t0  = A > 0, xt x0  t0  = B and  xx x0  t0  = C set, for  > 0 to be chosen later, wx t =

x − x0 + ix t − x0  t0  1 + ix x0  t0 

hx t = w2 x t − iwx t Note that wx0  t0  = 0, wt x0  t0  = 0, wx x0  t0  = 1—which implies that  wx x0  t0  = 0—and it is also clear that Lh = 0. Let us write ux t = hx t, so ux t = wx t2 − wx t2 +  wx t and it follows that ux0  t0  = ut x0  t0  = ux x0  t0  = 0. Then, uxx x0  t0  = 2 wx x0  t0 2 + c C = 2 + c C utt x0  t0  =  wtt x0  t0  = c tt x0  t0  = c A > 0 uxt x0  t0  = c xt x0  t0  = c B where c = 1 + x2 x0  t0 −1 > 0, which shows that the Hessian of u at x0  t0  is positive definite if  > 0 is small enough. Then h has a strict local minimum at x0  t0 , i.e., the hypotheses of Lemma IV.4.2 are satisfied if we choose K1 = x0  t0  . If tt x0  t0  = A < 0 we reason similarly, taking  < 0 and small. The previous discussion shows that when looking for a homogeneous solution h whose real part assumes a local minimum over a compact set we may work under the assumption that t x t = 0 &⇒ tt x t = 0

(IV.77)

Assume that condition  does not hold in any square centered at 0 0. Then given % > 0 we may find points x∗  t1 , x∗  t2  in the cube Q centered at the origin with side length % such that, say, t1 < t2 , t x∗  t1  < 0, and t x∗  t2  > 0. We consider homogeneous solutions of the form hx t x0  = Zx t − Zx0  02 − i

Zx t − Zx0  0 Zx x0  0

IV.4 Necessary conditions for local solvability

203

and the difficulty is to show under assumption (IV.77) that for an appropriate choice of  ≤ 1 and x0 ≤ 1 our function h assumes a local minimum over a compact set. Writing h in terms of its real and imaginary parts, hx t x0  = ux0 x t + ivx0 x t we obtain ux0 x t = x − x0 2 − x t − x0  0 2 +  c x t − x0  0 − x x0  0x − x0 

(IV.78)

where c = 1 + x2 x0  0−1 > 0. A straightforward computation shows that x ux0 x0  0 = uxx0 x0  0 = 0. Since uxxx0 0 0 = 2+xx 0 0 we may assume, taking  small but fixed and shrinking Q, that uxxx0 > 0 on Q. Then the connected component x0 that contains the point x0  0 of the level set x t

uxx0 x t = 0

is a smooth curve that intersects transversally the x-axis at x0  0. Hence, the curves x0 foliate a neighborhood of the origin and shrinking % > 0 if  necessary we may assume x0 ≤% x0 ⊃ Q. From now on we will assume that

x0 ≤ %. Note that the vector field x

uxt0   (IV.79) − x0 t uxx x is tangent to the curve  x0 along  x0 so this curve may be realized as the graph of a function x = xx0 t, t < %0 . Let us take a closer look at the behavior of ux0 on the curve x0 . For any x  t  ∈ x0 we have that uxx0 x  t  = 0 and uxxx0 x  t  > 0 so x → ux0 x t  attains a strict minimum precisely at x = x (geometrically, the graph of x → ux0 x t  looks like a parabola pointing upwards with vertex at x ). Hence, there is a tubular neighborhood V of x0 such that =

min ux0 x t = min ux0 x t V

x0

Thus, if we can find points x1  t1 , x0  t0 , x2  t2  in x0 such that t1 < t0 < t2 and ux0 x1  t1  > ux0 x0  t0  ux0 x2  t2  > ux0 x0  t0  it follows that there is a compact set K ⊂  x0 such that ux0 x t assumes a local minimum over K. To study the variation of ux0 along x0 we consider the parameterization x0 s = xx0 s s and differentiate ux0 xx0 s s

204

Local solvability of vector fields

with respect to s. Since uxx0 xx0 s s ≡ 0, we obtain d x0 x u x0 s = ut 0 x0 s = t c − 2 − x0  02  x0 s ds Shrinking % < %0 we may assume that 2 x t − x0  0 2 < c  /2. Thus, ux0 is monotone along x0 if and only if t does not change sign on x0 . Hence, if for some curve x0 we find points x1  t1 , x2  t2  in x0 such that t1 < t2 , t x1  t1  < 0, t x2  t2  > 0, then for  > 0 and small the curve x0 will contain a compact subset K over which ux0 assumes a local minimum; if, instead, t x1  t1  > 0 and t x2  t2  < 0 we take  < 0 in the definition of h to achieve the desired homogeneous solution. To see that such x0 exists, consider the quadrilateral Q having as horizontal sides the segments t = ±% and as ‘vertical’ sides the curves x0 with x0 = ±%. Then Q is the union of the curves x0 , −% < x0 < %. Assume by contradiction that t does not change sign along any of these curves. We may decompose Q into three disjoint sets: the union Q+ of the curves x0 that contain at least one point on which t > 0, the union Q− of the curves x0 that contain at least one point on which t < 0, and the union Q0 of the curves x0 on which t vanishes identically. Observe that Q+ and Q− are open sets and neither Q+ nor Q− can be empty, for this would imply that t does not change sign on some square containing the origin and condition  would be satisfied in that square, contradicting our assumptions. Since Q+ and Q \Q+ are invariant sets (i.e., they are a union of the curves x0 that intersect them) so is the boundary of Q+ . Let p be a boundary point of Q+ and let x0 be the curve passing through p. We claim that x0 is a vertical segment. Indeed, x0 ⊂ Q0 since it cannot meet Q+ ∪ Q− . So t vanishes identically on x0 and also does tt because of x (IV.77). Let q ∈ x0 . If xt0 q = 0 the set S = x = 0 is a smooth curve in a neighborhood of q and since tt = 0 on S we conclude that the intersection of S with a neighborhood of q must be a vertical segment, in particular, the x tangent to x0 at q is vertical. Assume now that xt0 q = 0. Differentiating twice (IV.78), first with respect to x, then with respect to t and evaluating x the result at q we get uxt0 q = 0 because t q = xt q = 0. Then the vector field  given by (IV.79) reduces to t at q. Thus the velocity vector of x0 is always vertical and x0 is itself the vertical segment x0 × −% %. Let us return to the points x∗  t1 , x∗  t2  in the cube Q centered at the origin with side length % such that t1 < t2 , t x∗  t1  < 0 and t x∗  t2  > 0. Then trivially x∗  t1  ∈ Q− and x∗  t2  ∈ Q+ so there exists a point x∗  t0  ∈ Q+ such that t1 < t0 < t2 . But, as we have seen, this implies that x∗ = x∗ × −% % and t x∗  t = 0 for t < %, which is a contradiction. Thus, for some x0 < %, t assumes opposite signs on x0 , ux0 is not monotone

IV.4 Necessary conditions for local solvability

205

on x0 , and hx t x0  is a homogeneous solution whose real part assumes a local minimum over a compact set. Essentially the same approach works in a higher number of variables although the proofs are technically more involved. The following elementary lemma about real quadratic forms in R2 will be useful: Lemma IV.4.3. Assume that the real quadratic form q1 x y = Ax2 + 2Bxy + Cy2 

x y ∈ R2  A B C ∈ R

has positive trace A + C > 0 and set )  ( C −A 2 C −A + iB x + iy2 = x − y2  − 2Bxy q2 x y =  2 2 Then q1 x y + q2 x y =

A+C 2 x + y2  2

is diagonal and positive definite. Proof. The assertion is self-evident. We consider a vector field L given by (IV.51) defined on  = B × −T T ⊂ Rn × R

B = x ∈ Rn

x < 

and assume that there exist n first integrals Z1      Zn , LZj = 0, j = 1     n, with dZ1      dZn linearly independent in . We write Z = Z1      Zn  and further assume that detZx  = 0 in , Z0 0 = 0 and Zx 0 0 = I. We also use the notation   " t = b1 x t     bn x t  bx Lemma IV.4.4. Assume that there exists x0  t0  ∈  and  ∈ Rn such that " 0  t0  ·  = 0; (i) bx " (ii) bt x0  t0  ·  = 0. Then there exists f ∈ Cc  such that Lu = f for all u ∈  . Proof. By Lemma IV.4.2 we need only show that there exists a solution h of Lh = 0 such that h assumes a local minimum at p = x0  t0 . Set Z = Z1      Zn  = Zx−1 x0  t0  Zx t − Zx0  t0  . Then LZj = 0, j = 1     n, Z p = 0, Zx p = I. Then, the change of coordinates x = x − x0 , t = t − t0 ,

206

Local solvability of vector fields

shows that there is no loss of generality in assuming from the start that x0  t0  = 0 0. Write !j x t = Zj x t − xj , so !j 0 0 = x !j 0 0 = 0, j = 1     n. Set Wx t = Zx t ·  =

n 

j Zj x t

j=1

Then LW = 0 and in view of (i) we get   n  !j 0 0 + ibj 0 0 = i!t 0 0 ·  0 = LW0 0 = j t j=1 where ! = !1      !n . Hence, !0 0 = !t 0 0 = !x 0 0 = 0. We distinguish two cases. " 0 = 0. Differentiating with respect to t the equation LW = 0 Case 1. b0 we obtain !tt 0 0 ·  + ib"t 0 0 ·  = 0 and using (ii) we derive !tt 0 0 ·  = 0 Set hx t = Z12 x t + · · · + Zn2 x t − i Wx t  ux t = hx t = x + ! 2 − !x t 2 +  !x t ·  Thus, u0 0 = 0, ut 0 0 = 0, ,x u0 0 = 0 and if we choose  with the same sign as  = !tt 0 0 ·  it follows that the Taylor series of u at the origin is n  ux y = x12 + · · · + xn2 +  t2 +  cj xj t + · · · j=1

where the dots indicate terms of order > 2. Thus, the Hessian of u at the origin with respect to x t is positive definite and u has a strict local minimum at the origin for  small. Case 2. b"j 0 0 = 0 for some 1 ≤ j ≤ n. After a linear change in the x-variables we may assume that ⎧ ⎨ b1 0 0 = 1 b 0 0 = 0 j = 2     n ⎩ j  = 0 2      n  Since (ii) implies that  = 0 this case can only occur if n ≥ 2. Set Wx t = iZx t ·  = i

n  j=2

j Zj x t

IV.4 Necessary conditions for local solvability

207

Proceeding as in Case 1 we obtain Wt 0 0 = Wxj 0 0 = 0, for j = 1     n. Differentiating the equation LW = 0 with respect to t we obtain 2 tt2 W0 0 + itx W0 0 = b"t 0 0 ·  = 0 1

while differentiation with respect to x1 gives 2 tx W0 0 + ix21 x1 W0 0 = 0 1 2 Using both equations to eliminate the term tx W0 0 and replacing  by − 1 if necessary we obtain  tt2 W0 0 + x21 x1 W0 0 = 2 > 0

Applying Lemma IV.4.3 to the quadratic form q1 x1  t = tt2 W0 0t2 + 2x21 x1 W0 0tx1 + x21 x1 W0 0x12 we find a complex number  such that q1 x1  t +  x1 + it2 is positive definite. Since x1 Z1 0 0 = 1 and it follows from LZ1 0 = 0 that t Z1 0 0 = i the Taylor expansion in the variables x1  t of Z12 is Z12 x1  0     0 t =  x1 + it2 + · · · Thus, W + Z12 x1  0     0 t = t2 + x12  + · · · If we now set 2 hx t = Z12 x t + · · · + Zn−1 x t +  Wx t + Z12   ux t = hx t

we may check as in case (i) that Lh = 0 and that for  > 0 small u = h has a positive definite Hessian at the origin. Remark IV.4.5. Lemma IV.4.4 has the following geometric interpretation. " and X Y = Writing L = X +iY with X and Y real we have that X = t , Y = b, b"t . Then conditions (i) and (ii) at p = x0  t0  mean that X Y p, Xp, and Yp are not linearly dependent. Indeed, if AXp + BYp + C X Y p = 0, the obvious fact b" · X = b"t · X = 0 implies that A = 0 so X Y p and Yp would be collinear, contradicting (i) and (ii). This implies that the orbit 0 of the pair of vectors X Y that passes through p cannot have dimension ≤ 2. In fact, the three vectors X Y p, Xp, and Yp belong to Tp 0 so dim 0 ≤ 2 would force a linear relationship between them. Hence, (i) and (ii) of Lemma IV.4.4 imply that dim 0 ≥ 3, which violates (1) of condition  in Definition IV.3.2.

208

Local solvability of vector fields

In order to find a solution h of Lh = 0 with the property that its real part assumes a local minimum over a compact set we need only worry about those cases not covered by Lemma IV.4.4, i.e., we may always assume that t x t ·  = 0 &⇒ tt x t ·  = 0

x t ∈ 

 ∈ Rn 

(IV.80)

Let us assume that L does not satisfy condition  in any cube centered on the origin and let us try to produce the required homogeneous solution h. As in the case of two variables we will look for solutions h = u + iv such that the Hessian matrix uxx is everywhere positive definite and the critical points of x → ux t are located on a certain curve  so that when looking for a local minimum of u we only need to direct our attention to the restriction u  . Then, assuming by contradiction that u is monotone on  and that this happens for all the functions u of this type, we must conclude that L is forced to satisfy  in some neighborhood of the origin. The first step is then to show the abundance of solutions of this type, which is taken care of by the next lemma that describes a family of solutions depending on two parameters, x0 ∈ B and  ∈ Rn . The general form of these solutions is based on the function h introduced in case (i) of Lemma IV.4.4. Lemma IV.4.6. If T and  are small enough there exists a smooth function h ∈ C   × B × Rn , hx t x0   = ux t x0   + ivx t x0   with u and v real such that (i) Lh = 0 in  for all x0   ∈ B × Rn ; (ii) ux x0  0 x0   = 0 and vx x0  0 x0   = ; (iii) uxx x t x0   is positive definite at all points x t ∈  for all x0   ∈ B × Rn . Proof. Set hx t x0   =  1 +  2 1/2

k 

Zx t − Zx0  02

j=1

+ i · Zx−1 x0  0 Zx t − Zx0  0  Since h is a polynomial in Z1      Zn it is apparent that (i) holds. Differentiating h with respect to x and evaluating the result at x t = x0  0 we get hx x0  0 x0   = i which shows (ii). Finally, write 1 +  2 −1/2 u = F = f +g. Then f is independent of  and fxx 0 0 0  = 2I, I = identity matrix, so fxx has n eigenvalues ≥  > 0 on  × B × Rn if T and  are chosen small.

IV.4 Necessary conditions for local solvability

209

Since gxx is uniformly bounded in  × B × Rn , taking  large we obtain that Fxx is positive definite in  × B × Rn , which implies the positivity of uxx . We regard the function h defined in Lemma IV.4.6 primarily as a function in the variables x t that depends on the parameters x0  , whose geometric meaning is furnished by (ii). To the function h we associate the real vector field V defined for x t x0    ∈  × B × Rn × Rn by V=

   − Ax t x0   · + Bx t x0   · t x 

where A = A1      An , B = B1      Bn  are defined by A = u−1 xx uxt  B = vtx − vxx A Note that the jth component of Vux is Vux j = utxj −uxx Aj = utxj −utxj = 0, j = 1     n so ux is constant along the integral curves of V . A similar computation shows that V −vx j = 0, j = 1     n so  −vx is also constant along the integral curves of V . It follows that V is tangent to the submanifold of  × B × Rn × Rn of dimension 2n + 1 0 = x t x0   

ux x t x0   = 0

 = vx x t x0   

Since x0  0 x0    ∈ 0 by (ii) of Lemma IV.4.3 the partial derivative of x t x0    → ux x t x0    − vx x t x0   with respect to x0   at 0 0 0 0 0 0 is the identity. Thus, 0 may be parameterized by x t  for x < 1 t < T1 ,  < 1 as the graph of a smooth map x t  → x0 x t  x t  with values in  x0 < 2 ×   < 2 . We may assume, if  and T are further shrunken, that the image of x < 1 t < T1 ,  < 1 by the map x t  → x0 x t  t x t  covers  × B ×   <  . Thus, the vector field V∗ =

   − x t  · + x t  · t x 

where x t  = Ax t x0 x t  x t  x t  = Bx t x0 x t  x t 

(IV.81)

210

Local solvability of vector fields

agrees with V on 0—in particular, V∗ is tangent to 0—and its coefficients do not depend on x0 and . Fix x0 ∈ B and  <  and consider the function of ux t x0   as a function of x t. By (iii) of Lemma IV.4.6 the roots of the equations ux x t x0   = 0,  − vx x t x0   = 0 determine a smooth curve ˜ x0  in x t -space contained in 0 that passes through the point x0  0 . The curves ˜ x0  may be parameterized as ˜ x0  s = xs x0   s s x0   and they foliate 0 as x0 ,  vary. The vector field V∗ is tangent to ˜ x0  at every point of ˜ x0  so we may parameterize ˜ x0  so that its velocity vector is V∗ . The projection of ˜ x0  on x t-space gives a curve x0  passing through x0  0 on which ux vanishes and uxx is positive definite. Hence, there is a tubular neighborhood V of x0  such that min ux t x0   = min ux t x0   x0 

V

Thus, if the restriction of u to x0  assumes a local minimum over a compact subarc K of x0  we will also have that u itself assumes a local minimum over K. In order for the restriction of u to x0  to assume a local minimum over a compact subarc K we must find points t1 < t2 such that d

uxs x0   s t1  < 0 ds

and

d

uxs x0   s t2  > 0 ds

Now, writing xs x0   = xs and s x0   = s to simplify the notation, d d uxs s = ux xs s x0   xs + ut xs s x0   ds ds = ut xs s x0   " = bxs s · vx xs s x0   " = bxs s · s Note that the identity ux = b" ·vx is just the real part of the equation Lh = 0. This reduced the problem of finding a homogeneous solution h = u + iv whose real part assumes a local minimum over a compact set for an appropriate choice of x0   to the problem of finding a curve ˜ x0  such that the function " t· changes from negative to positive along ˜ x  . Thus, from qx t  = bx 0

the fact that  is not satisfied in any neighborhood of the origin—which amounts to saying that any cube centered at the origin contains an integral curve of X = t along which qx t  changes sign—we must derive that there exists an integral curve of V∗ along which qx t  changes sign. The tool to compare the changes of sign of a function along the integral curves of two different vector fields is provided by

IV.4 Necessary conditions for local solvability

211

Lemma IV.4.7. Let U ⊂ RN be an open set, X and V∗ Lipschitz vector fields in U and q ∈ C 1 U a real function such that (1) qx = 0 implies Xqx ≤ 0; (2) qx = 0 and dqx = 0 imply that Xx = V∗ x. Assume that the integral curves  of V∗ have the following property: • if qx < 0 for some x ∈  then qy ≤ 0 for all points y ∈  that lie ahead of x in the order determined by the flow. Then, the integral curves of X also satisfy property •. We postpone the proof of Lemma IV.4.7 and continue our reasoning. We apply the lemma with U given by x < 1 , t < T1 ,  < 1 , x0 < , " t · .

 < , N = 4n + 1, X = t , V∗ given by (IV.81) and qx t  = bx Let us check that hypotheses (1) and (2) in the lemma are satisfied. From (IV.80) we get (1). Assume now that qx t  = dqx t  = 0 at some point x t . Since q is independent of x0   we may say q and dq vanish at p = x t x0    ∈ 0, x0 = x0 x t ,  = x t  and since V∗ = V on 0 and X and V∗ do not depend on x0   we need only prove that Vp = Xp. " t = 0, b"t x t ·  = 0, From qx t  = dqx t  = 0 we derive that bx "bx x t ·  = 0, j = 1     n. The real part of Lh = 0 is ut = b" · ,v which, j differentiated with respect to xj , gives utxj x t  = 0, so the coefficient Aj of /xj in V satisfies Aj x t x0   = 0. Similarly, differentiating vt + b" ·,u = 0 " = 0 at x t  so V∗ p = we get that B = vxt − vxx A = vxt = −b"x · ux − uxx b Vp = t = Xp which proves (2). Since L does not satisfy  there is an integral curve of X contained in U on which q changes sign from minus to plus. Then, by Lemma IV.4.7, V∗ cannot possess property • showing the existence of a curve ˜ x0  along which b" ·  changes sign from minus to plus as required to show that ux t x0   assumes a local minimum over a compact set of , which, by Lemma IV.4.2 implies that L is not solvable in . Summing up, Theorem IV.4.8. Assume that L, given by (IV.49), is locally solvable in . Then every point p ∈  has a neighborhood U such that L satisfies condition  in U . To complete the proof of the theorem we must prove Lemma IV.4.7. We start by recalling that if f a b → R is a continuous function we define fx + - − fx D+ fx = lim sup %(0

212

Local solvability of vector fields

which may vary in the range −  . The mean value inequality states that if f ∈ C 0 a b there exists c ∈ a b such that fb − fa ≤ D+ fca − b. If fa = fb it is enough to choose c ∈ a b so that fc = inf fx and the general case is reduced to this one by subtracting the affine function fa + x − afb − fa/b − a. It follows that if D+ fx ≤ 0, x ∈ a b, then fx is monotone nonincreasing. Let V be a Lipschitz vector field in U ⊂ RN , that is, Vx − Vy ≤ K x − y , x y ∈ U . We denote by !t x, the forward flow of V stemming from x, i.e., the solution !t x defined in a maximal interval 0 ≤ t < Tx of the ODE  d ! x = V!t x dt t !t 0 = x Let F ⊂ U be a closed set. We say that F is positively V -invariant, or just V -invariant for brevity, if x ∈ F &⇒ !t x ∈ F

for all

t ∈ 0 Tx

The characterization of V -invariant sets given below is due to Brézis ([Br]). The following properties are equivalent: (i) F is positively V -invariant; dist x + -Vx F (ii) ∀x ∈ F lim = 0. -(0 Indeed, assume (i). Then dist x + -Vx F x + -Vx − !- x ≤ & & & & ! - x − x & & = &Vx − & and the right-hand side converges to 0 as - ( 0. Conversely, assume that (ii) holds. To prove (i) it is enough to show that the Lipschitz continuous function f 0 Tx → 0  defined by ft = dist !t x F vanishes identically. This will follow if we prove that e−At ft is nonincreasing for some A > 0, since f0 = 0. Thus, it is enough to show that Dt+ e−At ft ≤ e−At Dt+ ft − Aft ≤ 0 which in turn is implied by Dt+ ft ≤ Aft. Fix t ∈ 0 Tx and choose zt ∈ F such that ft = !t x − zt . For small - > 0 we have ft + - = dist !t+- x F

IV.4 Necessary conditions for local solvability

213

≤ !t+- x − !- zt  + !- zt  − zt − -Vzt  + dist zt + -Vzt  F Now !t+- x − !- zt  = !- !t x − !- zt  , so by Gronwall’s inequality,

!- !t x − !- zt  ≤ eK- !t x − zt = eK- ft for - > 0 small, where K is the Lipschitz constant of V . Thus, ft + - − ft eK- − 1ft ≤ & & & ! z  − zt & dist zt + -Vzt  F + && - t − Vzt && + and letting - ( 0 we get Dt+ ft ≤ Kft, since the right-hand side’s middle term obviously → 0 and the last one also does because we are assuming that (ii) holds. This shows that e−Kt ft is nonincreasing and proves (i). We now prove Lemma IV.4.7. Proof. Let U − be the V∗ -flow out of the set x ∈ U qx < 0 , i.e., a point x ∈ U − if x = !t y for some y ∈ U with qy < 0 and 0 ≤ t < Ty, where !t is the flow of V∗ . Hence, U − is an open set and qx < 0 ⊂ U − ⊂ qx ≤ 0 because of •. By its very definition, U − is positively V∗ -invariant and so is its closure F = U − . Indeed, if x ∈ F there exist a sequence xj  ⊂ U − such that xj → x. If 0 < t < Tx, then 0 < t < Txj  for large j because s → Tx is lower semicontinuous. Then !t xj  ∈ U − by the V∗ -invariance of U − and !t x = limj !t xj  ∈ U − . To prove the lemma we will show that F is X-invariant, which clearly implies that X has property • because F ⊂ qx ≤ 0 . We must show that lim -(0

dist x + -Xx F = 0 -

x ∈ F

(IV.82)

If qx < 0 this is trivially true, since x + -Xx ∈ F for small - > 0. If qx = dqx = 0, (2) implies that Xx = V∗ x so dist x + -Xx F dist x + -V∗ x F = and the right-hand side → 0 as - ( 0 because F is V∗ -invariant. If qx = 0 and dqx = 0, the set qy = 0 ∩ W is a C 1 manifold where W is a convenient ball centered at x. It is easy to find a smooth unit vector field Ny that meets qy = 0 transversally and points toward qy < 0 , so  Nq < 0 on W ∩qy = 0 . Let .t y  denote the flow of the vector X  = X + N ,  ≥ 0. Then, (1) implies that X  qy < 0 on W ∩qy = 0 for any  > 0.

214

Local solvability of vector fields

Note that no integral curve of X  ,  > 0, that stems from a point in W ∩qy < 0 can cross W ∩ qy = 0 (this would amount to traveling against the flow at W ∩ qy = 0 ) and this implies that q.t x  < 0 for  > 0, t > 0 small, in particular .t x  ∈ U − . Hence, .t x 0 = lim(0 .t x  ∈ F where the limit holds by the continuous dependence on the parameter . Thus, the flow .t x 0 of X does not exit F for small values of t > 0, which easily implies (IV.82), as in the proof of ‘(i) &⇒ (ii)’ of the characterization of flow-invariant sets.

Notes A few years after the publication of Hans Lewy’s example [L1], Hörmander ([H6], [H7]) shed new light on the nonsolvability phenomenon explaining it in a novel way. Although his results are set in the framework of general order operators of principal type we will describe its consequences for vector fields. He proved that if a (nonvanishing) vector field L is locally solvable in  then the principal symbol of the commutator L L between L and its conjugate must vanish at every zero of the principal symbol x  of L. A vector field with this property is said to satisfy condition  . For the Lewy operator condition   is violated at every point. If the coefficients of L are real or constant L L vanishes identically. This was a most remarkable advance because it explained a phenomenon that had appeared as an isolated example in terms of very general geometric properties of the symbol, an invariantly defined object. However, it turns out that condition   does not tell apart the solvable vector fields from the nonsolvable ones among some examples considered by Mizohata ([M]), which we now describe. Let k be a positive integer and consider the vector field in R2 defined by Mk =

  − iyk  y x

If k = 1 condition   is violated at all points of the x-axis so, in particular, M1 is not locally solvable at the origin. For k ≥ 2 condition   is satisfied everywhere. On the other hand, it follows from relatively simple arguments that Mk is locally solvable at the origin if and only if k is even ([Gr], [Ga]). The principal symbol of Mk is m1 = −i − iyk . The crucial difference between k odd and k even is that in the first case the function yk changes sign and in the second case it doesn’t. Nirenberg and Treves ([NT]) elaborated these examples and identified a property that turned out to be the right condition for local solvability of vector fields, i.e., condition . When L

Notes

215

satisfies  the arguments in [NT] allow Lu = f to be solved locally with u in the Sobolev space L2−1 for f ∈ L2 . This result was improved by Treves ([T2]) to L2 solvability, i.e., u can be taken in L2 . Concerning the regularity of the coefficients, it was shown in [Ho1] that if L is in the canonical form n  u u + i bj x t  (a) Lu = t x j j=1 with bj real-valued and Lipschitz and satisfies  then it is locally solvable in L2 . Since there is loss of one derivative in the process of obtaining coordinates in which L has this form one must require, in general, that derivatives up to order one of the coefficients of L be Lipschitz. However, in two variables (i.e., when n = 1) it is possible to prove L2 solvability directly without assuming that L is in the special form (a) ([HM1]). Hence, planar vector fields with Lipschitz coefficients that satisfy  are locally solvable in L2 . This result is essentially sharp in the sense that there are counterexamples to L2 solvability and to the existence of L2 a priori estimates if the coefficients are only restricted to belong to the Hölder class C  for any 0 <  < 1 ([J1], [HM1], [HM2]). Whether any vector field with Lipschitz coefficients that satisfies  in three or more variables is locally solvable in L2 is an open problem at the time of this writing. It is a characteristic feature of locally solvable operators of order one that the L2 a priori estimates that they satisfy can be extended to Lp estimates for 1 < p < , a fact that turns out to be false for second-order operators in three or more variables (for results in that direction see [Li], [K], [KT1], [KT2], [Gu], [Ch1]). Solvability in Lp for vector fields was first considered in [HP], where the method involved pseudo-differential operators and demanded smooth coefficients. On the other hand, using the method of H. Smith ([Sm]), Lp a priori estimates in the range 1 < p <  can be proved in one stroke under the same regularity hypothesis on the coefficients initially known to guarantee just L2 estimates ([HM2]). This is the point of view used in the presentation of a priori estimates in this book, although for simplicity we have not included the proof that in two variables Lp estimates for vector fields with Lipschitz coefficients are valid without assuming they are in the canonical form (a) ([HM2]). The proof of a priori estimates in several variables is reduced, thanks to the geometry of  that prevents the existence of orbits of dimension higher than 2, to two-dimensional a priori estimates that are glued by a partition of unity associated with a convenient Whitney decomposition in cubes. The presentation in this chapter owes much to the discussion in [S1] about decomposition of open sets in cubes.

216

Local solvability of vector fields

While it is true that for any locally solvable vector field L and 1 < p <  the equation Lg = f can locally be solved in Lp if f is in Lp , this is false, in general, for p =  as we saw in the example after Remark IV.1.12 that was taken from [HT2]. This difficulty can be dealt with by introducing the space X = L Rt  bmoRx  of measurable functions ux t such that, for almost every t ∈ R, x → ux t ∈ bmoR and ut ·bmo ≤ C <  for a.e. t ∈ R, where bmoR is a space of bounded mean oscillation functions, dual to the semilocal Hardy space h1 R of Goldberg. This was first observed in [BHS], where it is proved that for a substantial subclass of the class of locally solvable vector fields L, the equation Lu = f can be locally solved with u ∈ X if f ∈ L . This result was later improved by showing that for any locally solvable vector field L the equation Lu = f can be locally solved with u ∈ X for any f ∈ X ([daS], [HdaS]) which can be regarded as an ersatz for p =  of the Lp local solvability valid for 1 < p < . The presentation in Section IV.1.2 follows closely [HdaS] but replaces lemma 4.5 of that paper—which is true but incorrectly proved—by Lemma IV.1.17 which is sharper. A priori estimates in L2 easily give a priori estimates in L2s for any s ∈ R but the absorption of lower-order terms requires shrinking of the neighborhood in which the estimate holds in a way that makes its diameter tend to zero when s → . Therefore, the technique of a priori estimates gives solutions of arbitrary high but finite regularity for smooth right-hand sides. Using a different approach, Hörmander ([H9]) proved solvability for differential operators of arbitrary order that satisfy  by studying the propagation of singularities of the equation Pu = 0 mod C  , showing the existence of semiglobal solutions, i.e., solutions defined on a full compact set under the geometric assumption that bicharacteristics do not get trapped in the given compact set. Furthermore, the solutions can be taken smooth if f is smooth. In Sections IV.2 and IV.3 of this chapter, the construction of smooth solutions is simplified by the assumption that the vector fields are locally integrable. Since vector fields that satisfy  are indeed locally integrable, the local integrability hypothesis is superfluous, however this fact depends on the difficult and long theorems on smooth solvability by Hörmander ([H9], [H5]). Thus, it would be interesting to have a shorter ad hoc proof of the local existence of smooth solutions for vector fields that satisfy  without invoking local integrability. Concerning the necessity of , Nirenberg and Treves had shown in their seminal paper [NT] that local solvability implies  for vector fields with real-analytic coefficients and conjectured the same implication should hold for smooth coefficients. This state of affairs remained unchanged for 15 years

Notes

217

until Moyer ([Mo]) removed in 1978 the analyticity hypothesis for operators in two variables in a never published manuscript. His ideas, however, were applied by Hörmander [H4] to extend the result for operators in any number of variables with smooth coefficients. The discussion of the necessity of  in Section IV.4 of this chapter is again simplified by the assumption of local integrability and follows the presentation in [T3] (see also [T5] and [CorH2]).

V The FBI transform and some applications

This chapter begins with a discussion of certain submanifolds in CR and hypoanalytic manifolds. We then introduce the FBI transform which is a nonlinear Fourier transform that characterizes analyticity. We also present a more general version of this transform which characterizes hypoanalyticity. We will discuss several applications of the FBI transform to the study of the regularity of solutions in locally integrable structures.

V.1 Certain submanifolds of hypoanalytic manifolds This section discusses certain submanifolds of hypoanalytic manifolds. We begin with a discussion of CR manifolds in CN . CR manifolds are good models for hypoanalytic manifolds. Later in the chapter we will see that a hypoanalytic structure can be locally embedded in a CR structure. This can sometimes be useful in reducing problems about general vector fields in hypoanalytic structures to CR vector fields. We will first recall the concept of a complex linear structure on a real vector space and apply it to the real tangent bundle of real submanifolds in CN . Let V be a vector space over R and suppose J V −→ V is a linear map such that J 2 = −Id (where Id = the identity). Clearly J is an isomorphism and dimV is even since det J2 = det−Id = −1dimV . The map J is called a complex structure on V. Indeed, with such a J , V becomes a complex vector space by defining a + ibv = av + bJv for a b ∈ R v ∈ V. Conversely, if V is a complex vector space, it is also a vector space over R and the map Jv = iv is an R-linear map with J 2 = −Id. If v1   vN is a basis of V over C, then v1   vN  Jv1   JvN is a basis of V over R. 218

V.1 Certain submanifolds of hypoanalytic manifolds

219

Example V.1.1. In CN let zj = xj + iyj  1 ≤ j ≤ N , denote the coordinates. We will identify CN with R2N by means of the map z1   zN  −→ x1  y1   xN  yN  Multiplication by i in CN then induces a map J R2N −→ R2N given by Jx1  y1      xN  yN  = −y1  x1      −yN  xN  Note that J 2 = −Id and so J is a complex structure on R2N , called the standard complex structure on R2N . Example V.1.2. With notation as in the previous example, for p ∈ CN , a basis of the real tangent space Tp CN is given by x p  y p      x p  y p . This 1 1 N N basis can be used to identify Tp CN with R2N by choosing the usual basis e1 = 1 0     0     e2N = 0     0 1

of

R2N 

This leads to a complex structure J Tp CN −→ Tp CN given by & &       &&  &&  J

= and J

=−  j = 1     N xj p yj &p yj p xj &p This complex structure is independent of the choice of the holomorphic coordinates z1      zN . To see this, suppose w = Fz is a biholomorphic map defined near 0 with F0 = 0 where we are assuming as we may that p = 0. Write F = U + iV and let wj = uj + ivj  j = 1     N . We need to show that dF0 J = J dF0 . We have:       Ul   Vl    dF0 J = dF0 = + xj yj l yj ul l yj vl and

 J dF0



 xj



" #  Ul   Vl  =J + l xj ul l xj vl =

 Ul   Vl  −  l xj vl l xj ul

where everything is to be evaluated at 0. Thus an application of the Cauchy– Riemann equations to the Uj and Vj shows that dFJ x  = JdF x . The j

j

equality also holds in the same fashion for y . Thus, J is independent of the j holomorphic coordinates. This also means that J can be defined on the real tangent space of any complex manifold. Note that J extends to a C-linear map from CTp CN into itself and the extension still satisfies J 2 = −Id. We will also denote this extension by J . The fact that J 2 = −Id implies that

220

The FBI transform and some applications

J CTp CN −→ CTp CN has only two eigenvalues: i and −i. Define Tp10 CN to be equal to the eigenspace associated with i, and Tp01 CN will be the eigenspace associated with −i. We get corresponding vector bundles T 10 and T 01 . Observe that T 10 is generated by z      z . Hence T 10 is the bundle 1 N of holomorphic vector fields introduced in Chapter I (see the discussion preceding Theorem I.5.1). Likewise, T 01 is generated by z      z . 1

N

Definition V.1.3. Let  be a real submanifold of CN . For p ∈ , define p  = CTp  ∩ Tp01 CN  Definition V.1.4. Let  be a real submanifold of CN and p ∈ . The complex tangent space of  at p denoted Tpc  is defined by Tpc  = Tp  ∩ JTp  It is easy to see that Tpc  = v ∈ Tp  Jv ∈ Tp  . Observe that J Tpc  −→ Tpc  and so Tpc  is equipped with a complex vector space structure. It is also evident that J CTpc  −→ CTpc . Example V.1.5. Let  be a hypersurface in CN through the point 0. Let & be a defining function for  near 0. Since d&0 = 0 and & is real-valued, &0 = 0. After a complex linear change of coordinates, we may assume that & 0 = 0     0 1 z That is, we have coordinates z w z = x + iy ∈ CN −1  w = s + it ∈ C, such & that x 0 = y& 0 = 0 j = 1     N − 1, & 0 = 0 and & 0 = 0. These s t j j conditions on the partial derivatives of & allow us to apply the implicit function theorem and conclude that near 0 the submanifold  is given by  = z s + iz s  where  is real-valued, 0 0 = 0, and d0 0 = 0. Hence T0  = span at 0 of         j = 1     N − 1  xj yj s while 0  = the C-span at 0 of   

j = 1     N − 1 zj and T0c  = the R-span at 0 of     

j = 1     N − 1  xj yj

V.1 Certain submanifolds of hypoanalytic manifolds

221

The spaces Tpc  and p  are related. To see this, we recall the following result from [BER] where p  denotes the real parts of elements of p : Proposition V.1.6. For p ∈ , (a) p  = Tpc ; (b) CTpc  = p  ⊕ p ; (c) p  = x + iJx x ∈ Tpc  . Proof. Observe first that for any x ∈ Tp CN  x +iJx ∈ Tp01 CN . Let x ∈ Tpc . Then x and Jx ∈ Tp  and so x + iJx ∈ p . Thus x ∈ p . Conversely, if x ∈ p , then there is y ∈ Tp CN such that x+iy ∈ p  ⊆ CTp  implying that x ∈ Tpc  since y = Jx and y ∈ Tp . We have thus proved (a) and (b) follows from (a) trivially. The proof of (c) is also contained in that of (a). From Proposition V.1.6 we see that dim Tpc  = 2 dimC p  Definition V.1.7. A submanifold  of CN is called CR (for Cauchy– Riemann) if dimC p  is constant as p varies in . In this case, dimC p  is called the CR dimension of . Definition V.1.8. A CR submanifold  of CN is called totally real if its CR dimension is 0. Example V.1.9. The copy of RN in CN given by

x + iy ∈ CN y = 0 is a totally real submanifold. Example V.1.10. Let k and N be positive integers, 1 ≤ k ≤ N . Write the coordinates in CN as z w, z = x + iy ∈ Ck and w = u + iv ∈ CN −k . Let  Rk → Rk and * Rk → CN −k be smooth functions with 0 = 0, d0 = 0, *0 = 0, and d*0 = 0. Then the submanifold

 = x + ix *x x ∈ Rk is totally real near the point 0 ∈ . Conversely, if  is any totally real submanifold of CN , then near each of its points, there are holomorphic coordinates in which  takes the form of  above (see proposition 1.3.8 in [BER]). If  is also real-analytic, holomorphic coordinates can be found so that  ≡ 0 and * ≡ 0.

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The FBI transform and some applications

Lemma V.1.11. Suppose  is a submanifold of CN of real codimension d. Then 2N − 2d ≤ dim Tpc  ≤ 2N − d Proof. Since Tpc  ⊆ Tp , dim Tpc  ≤ dim Tp  = 2N − d On the other hand, Tp  + JTp  ⊆ Tp CN and so dim Tpc  = 2 dimTp  − dimTp  + JTp  ≥ 2N − 2d Example V.1.12. A hypersurface  ⊆ CN is a CR submanifold of CR dimension N − 1. Indeed, this follows from the lemma since Tpc  always has even real dimension, which when p ∈  has to equal 2N − 2. Example V.1.13. Let  be a complex submanifold of CN of complex dimension n. Then  is a CR submanifold of CR dimension n. This follows from the J -invariance of Tp . To see this, let X ∈ Tp . If fj = uj + ivj 1 ≤ j ≤ N − n are local holomorphic defining functions near p ∈ , then by the Cauchy–Riemann equations we have JXuj = JXvj = 0 for all j. Hence JX ∈ Tp . Suppose  ⊆ CN has codimension d and is locally defined by &j = 0 j =   l 1     d. Then a vector v = Nj=1 vj z ∈ p  if and only if Nj=1 vj & =0 zj j for all l. Hence dimp  = N − r where r = the dimension of the C-span of &1 p     &d p . Example V.1.14. Let  be the two-dimensional submanifold of C2 defined by &1 = x2 − x12 = 0 and &2 = y2 − y12 = 0. Then by calculating &1 p and &2 p, we easily see that dimp  = 0 or 1 depending on whether p ∈  ∩ x1 = y1 . Hence  is not a CR manifold. If  ⊆ CN has codimension d, since 2 dimC p  = dim Tpc , Lemma V.1.11 tells us that the minimum possible value of dimC p  = N − d. This minimum value is attained precisely when the forms &1 p     &d p are linearly independent. The CR submanifolds for which dimp  has such minimal value are the generic ones introduced in Chapter I. It will be convenient to present here an equivalent definition. Definition V.1.15. A CR submanifold  ⊆ CN of codimension d is called generic if for p ∈ , dimC p  = N − d. Example V.1.16. A hypersurface of CN is a generic CR submanifold.

V.1 Certain submanifolds of hypoanalytic manifolds

223

Example V.1.17. A complex submanifold of CN that is not an open subset is a nongeneric CR submanifold. Example V.1.18. Let z w denote the coordinates of Cn+d where z = x+iy ∈ Cn and w = s + it ∈ Cd . Let 1 z s     d z s be smooth, real-valued functions. Then  = z w tj − j z s = 0 1 ≤ j ≤ d is a generic CR manifold. This is easily checked by noting that &j =

wj − w j − j z s 2i

are defining functions with &1 p     &d p linearly independent at each point. Remark V.1.19. Conversely, as we saw in Chapter I, given any generic CR submanifold , in appropriate holomorphic coordinates,  takes the form in the example. Let  be a CR submanifold of CN of codimension d that is not generic and assume 0 ∈ . We will show that in a certain sense, near the point 0,  can be viewed as a generic CR submanifold of CL for some L < N . Define = T0  + JT0  Let Y be a subspace of T0  such that T0  = T0c  ⊕ Y Note that JY ∩ T0  = 0. Let v1      vn be a C-basis of the complex space T0c . Then: v1      vn  Jv1      Jvn  is an R-basis of T0c . Complete this to a basis v1      vn  Jv1      Jvn  u1      ur of T0  where 2n + r + d = 2N . Then since JY ∩ T0  = 0, it follows that  = v1      vn  Jv1      Jvn  u1      ur  Ju1      Jur  is a basis of . Extend  to a basis  =  ∪ u1      ul  Ju1      Jul  of T0 CN , N = n + r + l. Split the coordinates in CN = Cn+r+l as z w p where z = x + iy ∈ Cn , w = s + it ∈ Cr , and p = s + it ∈ Cl . Define the map A T0 CN → T0 CN by Avi  = x  AJvi  = y  1 ≤ i ≤ n; Auk  = i i   AJuk  = t  1 ≤ k ≤ r; Auj  = s  AJuj  = t  1 ≤ j ≤ l. Note s k

k

j

j

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The FBI transform and some applications

that the map A commutes with J and hence after a complex linear map (see Remark V.1.20 below) we are in coordinates z w p ∈ Cn+r+l where       = span of   

1 ≤ j ≤ n 1 ≤ k ≤ r xj yj sk tk and T0  = span of  x  y  s . It follows that near 0,  can be expressed j j k as a graph of the form:  = x + iy s + ifx y s gx y s where f is valued in Rr and g is valued in Cl . The components of the functions s + ifx y s and gx y s are CR functions. Observe that the projection ) Cn+r+l → Cn+r , )z w p = z w is a diffeomorphism of  onto the generic CR submanifold ) of Cn+r . Remark V.1.20. Recall the identification of CN with R2N of Example V.1.1 given by z1      zN  → x1  y1      xN  yN . With this identification, it is easy to see that a real linear map A R2N → R2N induces a C-linear map on CN if and only if A commutes with the operator J . Proposition V.1.21. If  is a totally real submanifold of CN of codimension d, then d ≥ N and hence dim  ≤ N . If  is also generic, then d = N . Thus, a totally real submanifold of maximal dimension has dimension = N . Proof. Let p ∈  and &1   &d be defining functions of  near p. Since p  = 0 , we must have: spanC &1  · · ·  &d = spanC dz1      dzN at the point p. Hence d ≥ N . If  is also generic, then &1   &d are linearly independent and so d = N . The map J can be used to characterize CR, generic CR, and totally real submanifolds. Proposition V.1.22. Let  be a submanifold of CN . Then (i)  is CR if and only if dimTp ∩JTp  is constant as p varies in . (ii)  is totally real if and only if Tp ∩JTp  = 0 for all p ∈ . (iii)  is a generic CR submanifold if and only if Tp  + JTp  = Tp CN

for all p ∈ .

V.1 Certain submanifolds of hypoanalytic manifolds

225

Proof. (i) follows from the definition of Tpc  and Proposition V.1.6. (ii) also follows from Proposition V.1.6. To prove (iii), if  is generic and &1   &d are local defining functions, then the linear independence of &1   &d is equivalent to: dimC p  = N − d Hence, by Proposition V.1.6, dim Tp  ∩ JTp  = 2N − d. But then dim Tp  + JTp  = 2N , implying that Tp  + JTp  = Tp CN for all p ∈ . Conversely, if Tp  + JTp  = Tp CN , then dimTp  ∩ JTp  = 2N − d and so by Proposition V.1.6, dimC p  = N − d showing that  is generic. We will next describe certain submanifolds in hypoanalytic structures that play important roles in the analysis of the solutions of the sections of the associated vector bundle. Let    be a hypoanalytic structure.  is a smooth manifold of dimension N and  is an involutive sub-bundle of CT  of fiber dimension n whose orthogonal bundle T  in CT ∗  is locally generated by the differentials of m = N − n smooth functions. Recall from  0 Chapter I that T 0 = Tp denotes the characteristic set of the structure   .

p∈

Definition V.1.23. A submanifold  is called noncharacteristic if Tp  = Tp  + p 

∀p ∈ 

Definition V.1.24. A submanifold  is called strongly noncharacteristic if CTp  = CTp  + p

∀p ∈  

Definition V.1.25. A submanifold  of  is called maximally real if CTp  = CTp  ⊕ p

∀p ∈  

Clearly, a maximally real submanifold is strongly noncharacteristic. If  is strongly noncharacteristic, then dim  ≥ m, while if  is maximally real, dim  = m. A strongly noncharacteristic submanifold is a noncharacteristic submanifold. A noncharacteristic hypersurface in  is strongly noncharacteristic. Example V.1.26. Denote the coordinates in R3 by x y t and consider the structure generated by L = t + i y . The orthogonal of L is generated by dZ1 and dZ2 where Z1 = x and Z2 = t + iy. If S = x 0 0 , then S is a noncharacteristic submanifold that is not strongly noncharacteristic.

226

The FBI transform and some applications

A CR submanifold of CN is strongly noncharacteristic if and only if it is generic. It is maximally real precisely when it is totally real of maximal dimension. The proofs of the following propositions are left to the reader. Proposition V.1.27. A submanifold  of  is noncharacteristic if and only if the natural map T ∗   −→ T ∗  maps T 0 injectively into T ∗ . Proposition V.1.28. A submanifold  of  is strongly noncharacteristic if and only if the natural map CT ∗   −→ CT ∗  maps T   injectively into CT ∗  . Proposition V.1.29. A submanifold  of  is maximally real if and only if the natural map CT ∗   −→ CT ∗  induces a bijection of T   onto CT ∗  . Distribution solutions have traces on a noncharacteristic submanifold of  (see proposition 1.4.3 in [T5]). In particular, a solution can always be restricted to a maximally real manifold. The local and microlocal regularity of solutions are often studied by analyzing their restrictions to maximally real submanifolds. Instances of this will occur later in this chapter.

V.2 Microlocal analyticity and the FBI transform The Paley–Wiener Theorem (see Theorem V.3.1 in the next section) characterizes the smoothness of a tempered distribution u in terms of the rapid decay of its Fourier transform uˆ . This characterization is very useful in studying the local and microlocal regularity of solutions of partial differential equations with smooth coefficients. There is also a characterization of analyticity in terms of the Fourier transform ([H8]). However, the latter is based on estimates using a sequence of cut-off functions making it more difficult in applications. The FBI transform is a nonlinear Fourier transform which characterizes analyticity (see Theorem V.2.4 below). Definition V.2.1. Let u ∈   Rm . Define the FBI transform of u by  2 Fu x  = e ix−y·− 

x−y uy dy (V.1) for x  ∈ Rm × Rm , where x − y ·  =

m  j=1

xj − yj j 

V.2 Microlocal analyticity and the FBI transform

227

The integral is to be understood in the duality sense. Theorem V.2.2. (Inversion with the FBI.) Let u ∈ Cc Rm . Then  1 m 2 ux = lim+ Fu t eix−t·−%   2 dtd m 3 %→0 4)  2 where the convergence is uniform. Remark V.2.3. If u ∈   Rm , the theorem also holds where convergence is understood in the distribution sense. Proof. From the Fourier transform of the Gaussian, we have:   )  m2 − x−y 2 2 eix−y·−%  d = e 4%  % Rm Hence  − x−y 2 1 1  ix−y·−%  2 uy ddy = e e 4% uy dy m 2)m 2m )% 2 √ 1  −t2 = e ux − 2 %t dt m 2 ) → ux uniformly on Rm since u ∈ Cc Rm . Thus 1  ix−y·−%  2 uy dyd ux = lim e %→0 2)m  1 m 2 2 = eix−y·− 

t−y −%   2 uy dtdyd m lim 3 4)  2 %→0  1 m 2 = Fu t eix−t·−%   2 dtd m lim 3 4)  2 %→0 The following characterization of analyticity by means of an exponential decay of the FBI transform may be viewed as an analogue of the Paley–Wiener Theorem. Theorem V.2.4. Let u ∈   Rm . The following are equivalent: (i) u is real-analytic at x0 ∈ Rm . (ii) There exist a neighborhood V of x0 in Rm and constants c1 , c2 > 0 such that

Fu x  ≤ c1 e−c2  for x  ∈ V × Rm 

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The FBI transform and some applications

Proof. We will assume that u is continuous and leave the general case for the reader. i ⇒ ii Suppose u is real-analytic at x0 . Let 0 ≤  ≤ 1  ∈ C0 Rm   ≡ 1 near x0 , and supp ⊆ x u is analytic at x . The integrand in Fu x  has a holomorphic extension in a neighborhood of y = x0 in Cm . We will denote by u the holomorphic extension of u near x0 . In the integration defining Fu x , we deform the contour from Rm to the image of Rm under the map  where s is chosen small enough so that u is defined y → y = y − isy  m on the image R . We then have  (V.2) eQxy uy det  y dy Fu x  = Rm

where Qx y  = ix − y ·  −  x − y2 . Observe that Qx y  = −s  y 1 − sy − 

x − y 2 

(V.3)

Let  > 0 such that y ≡ 1 when y − x0 ≤ . Choose s = With these choices, (V.2) and (V.3), we get:   eQxy dy + c eQxy dy

Fu x  ≤ c  . 4

y−x0 ≤

y∈suppu y−x0 ≥

= I1 x  + I2 x  Note then that I1 x  ≤ c



≤ ce

y−x0 ≤

e−s  y 1−sy dy

− 8 

for any , and for x − x0 ≤ 2 . Moreover,  2 e− 

x−y dy I2 x  ≤ c

y−x0 ≥

≤ ce Hence, for x − x0 ≤

 2

− 2 2 



and any  ∈ Rm ,

Fu x  ≤ c1 e−c2  for some c1  c2 > 0 (ii)⇒(i) Assume without loss of generality that x0 = 0. Suppose then that for some c1  c2 > 0,

Fu x  ≤ c1 e−c2  for all  ∈ Rm , and for all x near 0. We will use the inversion given by Theorem V.2.2. Write  m 2 Fu t eix−t·−%   2 dtd = I1% x + I2% x + I3% x + I4% x

V.2 Microlocal analyticity and the FBI transform

229

where for some A1  A2  B to be chosen later, I1% x = the integral over

t  t ≤ A1   ∈ Rm 

I2% x = the integral over t  A1 ≤ t ≤ A2   ≤ B  I3% x = the integral over t  t ≥ A2   ∈ Rm  I4% x = the integral over t  A1 ≤ t ≤ A2   ≥ B  Our goal is to show that there is a neighborhood of the origin in Cm to which the Ij% extend as holomorphic functions and for each j, Ij% z converges uniformly on this neighborhood as % → 0. Consider first I1% . Recall that x0 = 0. Choose A1 > 0 so that

Fu x  ≤ c1 e−c2 

for x ≤ A1 

If we complexify x to z = x + iy in the integrand of I1% , we see that the integrand is bounded by a constant multiple of

 2 e−c2 + y   m

which therefore has an integrable majorant for y ≤ c22 . Hence, as % → 0, the entire functions I1% z converge uniformly on a neighborhood of 0 to a holomorphic function. The functions I2% easily extend as entire functions of z and converge uniformly on compact subsets to an entire function as % → 0. Next choose A2 so that   A suppu ⊆ y y ≤ 2  4 Then note that when t ≥ A2 ,

Fu t  ≤ ce ≤ ce

2  A −  t − 42  − 

t 2 4

A2

+ 162





Using the latter we see that after integrating in t, the integrand in I3% is uniformly bounded by a constant multiple of A22

e− 16   This allows us to complexify as in I1% to conclude that I3% z converges uniformly to a holomorphic function in a neighborhood of 0. Write  m 2 2 I4% x = eix−y·− 

t−y −%   2 uy dydtd R

where R = y t   ≥ B A1 ≤ t ≤ A2  y ∈ supp u 

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The FBI transform and some applications

Note that the function  →  has a holomorphic extension #$ in the region

 <  , where # 21 " m  j2 #$ = j=1

and an appropriate branch of the square root is taken. We change the contour in the  integration from Rm to its image under the map  =  +is  x −y for s small, s > 0. The number s is chosen to be small enough to ensure that for  = 0  <  . We then have, modulo entire functions that converge uniformly to an entire function,  m ePxyt% #$ 2 uy dydtd I4% x = where Px y t  % = ix − y ·  − s x − y 2  − #$ t − y 2 − %2  Note that for s small, 2 ≥  2 and #$ ≥ exponential term can be bounded as follows: 2

2

2  − t−y 2

ePxyt% ≤ e−s x−y

 . 2

 − 2%  2

Hence the crucial



In particular, when x = 0, since t ≥ A1 , there is a constant c > 0 so that

eP0yt% ≤ e−c 

for all 

This gives us enough freedom to complexify x to z and vary z near 0 to conclude that I4% z converges uniformly to a holomorphic function near 0. We consider now the boundary values of holomorphic functions defined on wedges with flat edges, that is, edges that are open subsets of Rm . Let + ⊆ Rm \0 be an open convex cone with vertex at the origin, V ⊆ Rm open. For  > 0, let + = + ∩ v v <   If +  is another cone, we write +  ⊂⊂ + if +  ∩ S m−1 ⊂ + ∩ S m−1 where S m−1 denotes the unit sphere in Rm . Definition V.2.5. A holomorphic function f ∈ V + i+  is said to be of tempered growth if there is an integer k and a constant c such that

fx + iy ≤

c 

y k

V.2 Microlocal analyticity and the FBI transform

231

For f ∈ V + i+   ∈ C0 V, and v ∈ +, set  #fv  $ = fx + ivxdx

Theorem V.2.6. Suppose f ∈ V + i+  is of tempered growth and k is as in the definition above. Then bf = lim  fv v→0v∈+



exists in V and is of order k + 1. Proof. Assume that

fx + iy ≤

c 

y k

We may assume that + = y = y1      ym  y < C1 ym for some C1 > 0.

y0 . If y ∈ +0 , we have Fix y0 ∈ +. Let 0 = 2C 1

(a) ym0 ≥ (b)

y0

C1

ym0 − ym

≥ 2 y ≥ 2ym and ≥ ym0 − ym ≥

y0 C1

Fix  ∈ C0 V. For y ∈ + , let hy =

− y >



y0 . 2C1

fx + iyx dx

Using the growth condition on f and the fact that f is holomorphic, we can integrate by parts and arrive at

D hy ≤

CC

y k

for all 

where C = sup D  

Let  = k. We will estimate D hy on +0 . Assume first that k ≥ 2. For y ∈ +0 , & 1 & & &  0  0 0  &

D hy − D hy  = & DD hty + 1 − ty  · y − y  dt&& 0 " #  1  1 ≤ C C dt

ty + 1 − ty0 k 0

 =k+1 " #  1  1 ≤ C C dt 0 k 0 tym + 1 − tym 

 =k+1 " #    1 1 1 ≤ C C 0 − ym − ym ymk−1 ym0 k−1

 =k+1

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The FBI transform and some applications " ≤ C

#



C

 =k+1

1 

y k−1

We have used (a) and (b) and the fact that since + is convex, ty +1−ty0 ∈ +. Thus there is C > 0 such that for all ,  = k, # "  1  C whenever y ∈ +0 

D hy ≤ C

y k−1

 =k+1 Continuing this way, we get k−2  C > 0 such that # "  1 2 C whenever y ∈ +k−2 

D hy ≤ C

y

 =k+1 Note that this inequality also holds when k = 1. Fix yk−2 ∈ +k−2 . Let k−1 =

yk−2 . 2C1

Using the preceding inequality, for y ∈ +k−1 , we can easily get: " # 

Dhy ≤ C C log y

for some C > 0

 =k+1 

Let now y y ∈ +k−1 . We have: & & 1 & &

hy − hy  ≤ && Dhty + 1 − ty  · y − y  dt&& 0 " #& & & 1 &   & ≤ C C & log ty + 1 − ty dt&& y − y

 =k+1

" ≤ C



#" C

0



1

#

1

tym + 1 − tym 2 " #" #  ym + ym ≤ C C + √ ym + ym

 =k+1 " #  ≤ C C  y + y 

 =k+1

0

1

dt y − y

 =k+1

Hence lim+y→0 hy exists and as +  y → 0, 

D 

L

hy ≤ C

 ≤k+1

with C independent of . Remark V.2.7. We note here that when m = 1, the theorem above says that if a holomorphic function f defined on a rectangle Q = −a a×0 b satisfies

V.2 Microlocal analyticity and the FBI transform

233

the growth condition fx + iy ≤ yck , then the traces f + iy converge in

 −a a to a distribution of order k + 1. 1 Example V.2.8. Consider fx y = x+iy which is holomorphic and of tempered growth in the upper half-plane y > 0. By the theorem, f has a boundary value bf ∈  R. It is not hard to show that in fact,   1 bf = pv − i)0 x

where pv denotes the Cauchy principal value. Distributions which are boundary values of holomorphic functions of tempered growth arise quite naturally. Indeed, we have:  Theorem V.2.9. Any u ∈   Rm  can be expressed as a finite sum nj=1 bfj where each fj ∈ Rm + i+j  for some cones +j ⊆ Rm , and the fj are of tempered growth. Proof. Let u ∈   Rm . There exist an integer N and a constant c > 0 such that the Fourier transform $ u satisfies the estimate $ u ≤ c1 +  N . Let j  1 ≤ j ≤ k be open, acute cones such that Rm =

k 

j

j=1

and j ∩ l has measure zero when j = l. Define the cones +j = v ∈ Rm v ·  > 0

∀ ∈ j 

For each j = 1     k, define fj x + iy =

1  ix+iy· $ e ud 2)m j

Note that fj is holomorphic on Rm + i+j . Let +j be a cone, +j ⊂⊂ +j . Then there exists c > 0 such that y · ≥ c y

 ∀y ∈ +j  ∀ ∈ j . Hence for x +iy ∈ Rm + i+j ,  e−c y

 $ u d

fj x + iy ≤  e−c y

 1 +  N d ≤ c Rm

≤

cj 

y m+N

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The FBI transform and some applications

Thus each fj is of tempered growth on Rm + i+j and so by Theorem V.2.6 the  fj have boundary values bfj ∈  Rm . To prove u = kj=1 bfj , let  ∈ C0 V. Then  #bfj  $ = lim  fj x + iyxdx y→0y∈+j Rm

=



lim

y→0y∈+j Rm

 j

eix+iy·

x$ u

d dx 2)m

1  −y· $ u$ −d e y→0y∈+j 2)m j 1  $ u$ −d = 2)m j

=

lim

Hence #u $ =

k 

#bfj  $

j=1

Example V.2.10. Let f1 x y = Then it is not hard to show that

1 x+iy

1 for y > 0 and f2 x y = − x+iy for y < 0.

−2)i0 = bf1 + bf2  Granted this, since u ∗ 0 = u for any u ∈   R, we get an explicit decomposition of u as a sum of two distributions each of which is the boundary value of a tempered holomorphic function on a half-plane. Definition V.2.11. Let u ∈  Rm  x0 ∈ Rm   0 ∈ Rm \0 . We say that u is microlocally analytic at x0   0  if there exist a neighborhood V of x0 , cones + 1      + N in Rm \0 , and holomorphic functions fj ∈ V + i+j  (for some   > 0) of tempered growth such that u = Nj=1 bfj near x0 and  0 · + j < 0 ∀j. Remark V.2.12. When m = 1, if we take x0 = 0 and  0 = −1, then u is microlocally analytic at 0 −1 if there is a tempered holomorphic f on some rectangle −a a × 0 b such that u = bf on −a a. Definition V.2.13. The analytic wave front set of a distribution u, denoted WFa u, is defined by WFa u = x  u is not microlocally analytic at x   Observe that from Definition V.2.13 it can easily be shown that the analytic wave front set is invariant under an analytic diffeomorphism, and hence microlocal analyticity can be defined on any real-analytic manifold. The following theorem provides a very useful criterion for microlocal analyticity in terms of the FBI transform:

V.2 Microlocal analyticity and the FBI transform

235

Theorem V.2.14. Let u ∈  Rm  x0 ∈ Rm   0 ∈ Rm \0 . Then x0   0  % WFa u if and only if there is a neighborhood V of x0 in Rm , an open cone + ⊂ Rm \0  0 ∈ + and constants c1  c2 > 0 such that

Fu x  ≤ c1 e−c2  ∀x  ∈ V × + The proof uses the inversion formula of Theorem V.2.2 and ideas similar to those in the proof of Theorem V.2.4 (see also Theorem V.3.7). The reader is referred to [Sj1] for the proof of this theorem. Corollary V.2.15. A distribution u is analytic near x0 if and only if for every  0 ∈ Rm \0  x0   0  %WFa u. Corollary V.2.16. (The edge-of-the-wedge theorem.) Let V ⊂ Rm be a neighborhood of the point p, and + +  + − be cones such that + − = −+ + . Suppose for some  > 0, f + ∈ V + i++ , f − ∈ V + i+−  are both of tempered growth and bf + = bf − . Then there exists a holomorphic function f defined in a neighborhood of p that extends both f + and f − . In particular, bf + is analytic at p. Example V.2.17. Let

 ux =

3

x 2  3 2

i x 

x≥0 x ≤ 0 3

Then ux = bfx where fx y = x + iy 2 for y > 0 and we take the principal branch of the fractional power. Since f is holomorphic for y > 0, it follows that 0 −1 % WFa u. On the other hand, since u is not analytic (it is not even C 2 ), by Corollary V.2.15, 0 1 ∈ WFa u. Example V.2.18. Let x t denote the variables in Rm+n , x ∈ Rm and t ∈ Rn . Let t = 1 t     m t be real-analytic functions near the origin and consider the associated tube structure generated by Lk =

m  j   −i  tk j=1 tk xj

k = 1     n

It was shown in [BT5] that this system is analytic hypoelliptic at 0, i.e., every solution u of Lk u = 0 k = 1     n is analytic at 0 if and only if, for every  ∈ Rm , the function t → t ·  does not have a local minimum at 0. This result was proved using the FBI transform. The authors also proved a microlocal version of this result.

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When a distribution u is a solution of a partial differential equation with analytic coefficients, the analyticity or microlocal analyticity of the solution can sometimes be established by using the FBI transform. Sections V.4 and V.5 contain results in this direction. The notes at the end of this chapter contain several references to such applications of the FBI transform.

V.3 Microlocal smoothness In this section we introduce the concept of the C  wave front set which is a refined way of describing the singularities of distributions. It is well known that a distribution u of compact support is C  if and only if its Fourier transform uˆ  decays rapidly as  → . More precisely, we recall Paley–Wiener’s Theorem: Theorem V.3.1. (Theorem 7.3.1 in [H2].) A distribution u with support in the ball x ∈ Rm x ≤ R is C  if and only if uˆ  is entire on Cm and for each positive integer k there is Ck such that

ˆu ≤ Ck

eR  1 +  k

∀ ∈ Cm 

Definition V.3.2. Let u ∈    ⊆ Rm open, x0 ∈ , and  0 ∈ Rm \0 . We say u is microlocally smooth at x0  0  if there exists  ∈ C0 ,  ≡ 1 near x0 and a conic neighborhood + ⊆ Rm \0 of  0 such that for all k = 1 2   and for all  ∈ +, '

u ≤

Ck on + 1 +  k

Definition V.3.3. The C  wave front set of a distribution u denoted WFu is defined by WFu = x  u is not microlocally smooth at x   It is easy to see that a distribution u is C  if and only if WFu = ∅. When a distribution u is a solution of a linear partial differential equation with smooth coefficients, its wave front set is constrained. We quote here a basic result along this line:  Theorem V.3.4. (Theorem 8.3.1 in [H2].) Let P =  ≤k a xD be a smooth linear partial differential operator on an open set  ⊂ Rm and suppose u ∈  . Then WFu ⊂ char P ∪ WFPu

V.3 Microlocal smoothness where the characteristic set  char P = x  ∈  × R \0 m



237

 a x = 0  

 =k

In particular, if Pu is smooth, then WFu ⊂ char P. If Pu is smooth, and P is elliptic, then u has to be smooth. In Section V.5 we will consider an analogous result for solutions of first-order nonlinear partial differential equations. 2 is a Definition V.3.5. Let f ∈ C  ,  ⊆ Rm open, and suppose  m  2 neighborhood of  in C . A function f˜ x y ∈ C  is called an almost analytic extension of fx if f˜ x 0 = fx ∀x ∈  and for each j = 1     m, f˜ x y = O y k  for k = 1 2    zj Remark V.3.6. Lemma V.5.1 in Section V.5 shows that each smooth function of one real variable has an almost analytic extension. Such extensions also exist in higher dimensions (see [GG]). The following theorem characterizes microlocal smoothness in terms of almost analytic extendability in certain wedges. Theorem V.3.7. Let u ∈  Rm . Then x0   0  % WFu if and only if there exist a neighborhood V of x0 , open acute cones + 1      + N in Rm \0 , and almost analytic functions fj on V + i+j (for some  > 0) of tempered growth  such that u = Nj bfj near x0 and  0 · + j < 0 for all j. Proof. Suppose x0   0  % WFu. Let  ∈ C0 Rm ,  ≡ 1 near x0 such that ' u decays rapidly in a conic neighborhood of  0 . By the Fourier inversion formula, 1  ix· ' u = e u d 2)m where the formula is understood in the duality sense, that is, for * ∈ C0 Rm ,  1   ix· ' *x dx u d #u *$ = e 2)m Let j  1 ≤ j ≤ N be open, acute cones such that Rm =

N 

j

j=1

and j ∩ k has measure zero when j = k. We may assume that  0 ∈ 1 and  0 %  j for j ≥ 2. This implies that we can get acute, open cones + j  2 ≤ j ≤ N

238

The FBI transform and some applications

and a constant c > 0 such that 0 · + j < 0

and y ·  ≥ c y

 ∀y ∈ + j  ∀ ∈ j 

For each j = 2     N , define fj x + iy = and set g1 x =

1  ix+iy· ' u d e 2)m j 1  ix· ' e u d 2)m 1

For j ≥ 2, fj is holomorphic on Rm + i+ j and as we saw in the proof of Theorem V.2.9, it is of tempered growth and hence has a boundary value  decays bfj ∈  Rm . Since x0   0  % WFu, we may assume that u  m rapidly in the cone 1 . It is then easy to see that g1 is C on R . By Remark V.3.6, the function g1 has an almost analytic extension f1 which is smooth  on Cm . It follows that u = Nj bfj near x0 with the fj ’s as asserted. For the converse, we may assume that on some neighborhood V of x0 , u = bf where f is almost analytic and of tempered growth on V + i+, + is an open cone, and  0 · + < 0. Let  ∈ C0 V,  ≡ 1 near x0 . We have  ' u = #u xe−ix· $ = lim fx + iye−ix· x dx +y→0 Rm

Let !x y be an almost analytic extension of x. Fix y0 ∈ + and let D = x + ity0 ∈ Cm x ∈ V 0 ≤ t ≤ 1  Consider the m-form fx + iye−ix+iy· !x y dz1 ∧ · · · ∧ dzm where each zj = xj + iyj , 1 ≤ j ≤ m. By Stokes’ theorem, ' u −

 V

fx + iy0 e−ix+iy0 · !x y0  dx =

m   j=1 D

 f!e−ix+iy· zj

dzj ∧ dz1 ∧ · · · ∧ dzm  After contracting + if necessary, we may assume that for some c > 0, y0 ·  ≤ −c  for all  ∈ +. This latter inequality, together with the almost analyticity of f and !, and the tempered growth of f , imply that on D, for any integer k ≥ 0, we can find a constant Ck such that & & & && &  & f!x + ity0 & &e−ix+ity0 · & ≤ C  ty0 k ety0 · ≤ Ck  k & & z

 k j

V.4 Microlocal hypoanalyticity and the FBI transform

239

Observe also that the inequality y0 ·  ≤ −c  ( ∈ +) implies that the integral  fx + iy0 e−ix+iy0 · !x y0  dx V

decays rapidly in +. It follows that x0   0  % WFu. Corollary V.3.8. Let u ∈  Rm . If x0   0  ∈ WFu, then x0   0  ∈ WFa u.

V.4 Microlocal hypoanalyticity and the FBI transform A hypoanalytic structure (or manifold) is an involutive structure    with charts U  Z  where the U form an open covering of , and the Z = Z1      Zm  are a complete set of first integrals on U that are determined on the overlaps up to a local biholomorphism of Cm . A basic example is a generic CR submanifold  of Cm . A function f on a hypoanalytic manifold is said to be hypoanalytic if in a neighborhood of each point p it is of the form f = hZ1      Zm  for some holomorphic function h defined in a neighborhood of Z1 p     Zm p in Cm . In the case of generic CR submanifolds of Cm , the hypoanalytic functions are the restrictions to  of holomorphic functions defined in a neighborhood of . Hypoanalytic structures will be discussed some more in the epilogue. For more details on hypoanalytic structures, the reader is referred to [BCT] and [T5]. In this section we will briefly discuss the notion of the hypoanalytic wave front set. This notion is a generalization of the concept of microlocal analyticity we discussed in Section V.2 and the reader is referred to the work [BCT] for more details. We begin with the concept of a wedge in CN whose edge is a generic CR manifold. Let  be a generic CR manifold in CN of codimension d. Then dim  = 2n + d m = n + d = N and the bundle T  = T   is generated by the differentials of the restrictions to  of the N complex coordinates in CN . Fix p ∈  and let h = h1      hd  be smooth defining functions of  in a neighborhood U of p in CN . Definition V.4.1. For + an open convex cone with vertex at 0 ∈ Rd , the set U h + = z ∈ U hz ∈ + is called a wedge with edge . The wedge is said to be centered at p and to point in the direction of +. Observe that U h + is an open set in CN and  ∩ U lies in its boundary. When  is a hypersurface, + = 0  or − 0 and so a wedge with

240

The FBI transform and some applications

edge  in this case is simply a side of . Although the definition of a wedge involves the defining functions, the following proposition shows some independence from the defining functions. Proposition V.4.2. (Proposition 7.1.2 in [BER].) Assume that h = h1   hd  and g = g1   gd  are two defining functions for  near p. Then there is a d × d real invertible matrix B such that for every U and + as above, the following holds: for any open convex cone +1 ⊆ Rd with B+1 ∩ S d−1 relatively compact in + ∩ S d−1 (S d−1 denotes the unit sphere in Rd ), there exists a neighborhood U1 of p in CN such that  U1  g +1  ⊆  U h + The reader is referred to [BER] for the proof of this proposition. We mention that if az is a d × d smooth invertible matrix satisfying g = ah near p, then the matrix B = ap −1 . Definition V.4.3. A holomorphic function f defined on a wedge  = U h + is said to be of tempered growth if there exists a constant c > 0 and an integer k such that c ∀z ∈   (V.4)

fz ≤

hz k By using a diffeomorphism that flattens  near p, it is easy to see that the growth condition (V.4) is equivalent to

fz ≤

c distz k

∀z ∈  

Recall from Chapter I that for the generic  we can find complex coordinates z1      zn  w1      wd  vanishing at p ∈ , z = x + iy ∈ Cn , w = s + it ∈ Cd , and smooth real-valued functions 1      d defined near 0 0 in z s space with k 0 = 0 dk 0 = 0 1 ≤ k ≤ d such that near 0,  is given by &k z w = k z s − tk = 0

1 ≤ k ≤ d

That is, near 0,  = z s + iz s . By Proposition V.4.2, there exist % > 0 and a convex open cone +  ⊆ Rd such that if   = z s + iz s + iv z < % s < % v < % v ∈ +  then   ⊆ U h +. The description of   makes it clear what a wedge with edge  means. Observe also that a holomorphic function fz w on   is of tempered growth if and only if it satisfies an estimate of the form c

fz s + iz s + iv ≤ k

v

V.4 Microlocal hypoanalyticity and the FBI transform

241

for v ∈ +  small and z s ∈ Cn × Rd near 0 0. Holomorphic functions of tempered growth in a wedge have distributional boundary values on the edge of the wedge. We have: Theorem V.4.4. (Theorem 7.2.6 in [BER].) Let fz w be a holomorphic function of tempered growth in a wedge   as above. Then there exists a CR distribution u = bf defined in a neighborhood of 0 in  by  fz s + iz s + iv*x y sdxdyds #u *$ = lim +v−→0 R2n+d

for any smooth function * of sufficiently small compact support near the origin in R2n+d . Proof. The proof will use arguments similar to those used in the proof of Theorem V.2.6. For *x y s smooth, supported near the origin, set  hv = fz s + iz s + iv*x y sdxdyds R2n+d



for v ∈ + , v < %. We will estimate the derivatives of h. We have  h f v = i z s + iz s + iv*x y sdxdyds 2n+d vj wj R for each j = 1     d. Observe that since

  d  f k d  fz s + iz s + iv = z s + iz s + iv km + i dsm sm k=1 wk

and the matrix I + is is invertible near the origin, there are smooth functions ajm z s such that for each k = 1     d, d  f d z s + iz s + iv = akm z s fz s + iz s + iv wk ds m m=1

It follows that d   h d v = fz s + iz s + ivajm z s*x y sdxdyds 2n+d ds vj m m=1 R

We can thus integrate by parts and iterate the procedure to conclude that for some constant C > 0 and every multi-index ,

D hv ≤

CC

v k

where C = sup D * . It then follows, as in the proof of Theorem V.2.6, that hv has a limit as +   v → 0. Set #u *$ = limv→0 hv. Note that u

242

The FBI transform and some applications

is CR since it is the distributional limit of the CR functions   z s −→ fz s + iz s + iv. The reader is referred to [BER] for an invariant formulation of Theorem V.4.4 (corollary 7.2.9 in [BER]). Suppose now X is a hypoanalytic structure of codimension 0. Such an X often arises as a maximally real submanifold in a hypoanalytic structure. The structure bundle of X is all of CT ∗ X and since  is empty, any distribution is a solution. Fix p ∈ X and let Z = Z1      Zm  be a hypoanalytic chart near p. In a neighborhood V of p in X, the map Z V −→ Cm is a diffeomorphism onto ZV. ZV is a generic submanifold of Cm which is totally real of maximal dimension. In what follows, we will identify V with ZV. 

Definition V.4.5. A distribution u ∈ X is microlocally hypoanalytic at  ∈ Tp∗ X\0 if there exist open convex cones 1      k in Tp X satisfying v < 0 ∀v ∈ j  1 ≤ j ≤ k and wedges 1      k in Cm with edge ZV centered at p and pointing in the directions of +1      +k respectively such that J j ⊆ +j and for each j, there is a holomorphic function of tempered  growth uj on j such that u = kj=1 buj in V . Definition V.4.6. The hypoanalytic wave front set of u, denoted WFha u is defined by WFha u =  ∈ T ∗ X\0 u is not microlocally hypoanalytic at   The hypoanalytic wave front set for solutions in structures of positive codimension is defined by restriction to a maximally real submanifold as follows ([BCT]). Let    be a hypoanalytic structure and u a distribution solution near p ∈ . Select a maximally real submanifold  through p. We recall that the restriction u  is well-defined and by Proposition V.1.29  inherits a hypoanalytic structure of codimension 0. Hence the hypoanalytic wave front set WFha u   is defined and lives in T ∗  \0 . Since  is maximally real, by Propositions V.1.27 and V.1.29, the inclusion i  →  induces an injection i∗ T 0  → T ∗ . We will say a covector  ∈ Tp0 \0 is in the hypoanalytic wave front set of u if i∗  ∈ WFha u  . This set will be denoted  by WFhap u This definition is independent of the choice of the maximally real submanifold  through p (see [BCT] for the proof) and thus for any  such , we have a bijection i∗ WFhap u → WFhap u  , where WFhap denotes the hypoanalytic wave front set at p. We will next recall the FBI transform of [BCT] which gives a very useful Fourier transform criterion for microlocal hypoanalyticity. X is a hypoanalytic structure of codimension 0 as above. If p ∈ X, by the results in Chapter I (see

V.5 Application to the C  wave front set

243

for example Corollary I.10.2), we may choose local coordinates x1      xm for X vanishing at p so that locally, X becomes a neighborhood U of 0 in Rm and we may assume that a hypoanalytic chart has the form Zj = xj + ij x

1 ≤ j ≤ m

 = 1      m  real-valued. For 2 > 0 and u ∈   U, define  2 F 2 u z  = ei·z−Zy−2#$ z−Zy uydZy U

and for any  ∈ Cm with  <   #$ = where z ∈ C  w 2 21 2 1 + · · · + m  (the principal branch of the square root). m

2

= w12 +· · ·+wm2 ,

Definition V.4.7. F 2 u z  is called the FBI transform of u (with parameter 2). In [BCT] the authors characterized microlocal hypoanalyticity in terms of an exponential decay of the FBI transform. In particular, when 0 = 0 and d0 = 0, they proved: Theorem V.4.8. There is a universal constant M > 0 such that if 2 > M sup  x, the following holds: for  ∈ Rm \0 , u ∈   U V a neigh-

x∈U   =2

borhood of 0 in Cm 

+ ⊆ Cm \0 a complex conic neighborhood of , if

F 2 u z  ≤ c1 e−c2  

∀z ∈ V

∀ ∈ +

and for some c1  c2 > 0, then 0  % WFha u. Here U is a neighborhood of 0 in Rm .

V.5 Application of the FBI transform to the C  wave front set of solutions of nonlinear PDEs In this section the FBI transform will be used to prove a result on the C  wave front set of solutions of first-order nonlinear PDEs. Suppose u = ux t is a C 2 solution of a nonlinear pde ut = fx t u ux  where fx t 0   is complex-valued, C  in all the variables, and holomorphic in 0  . Here x varies in an open set in Rm , t in an interval of R, and 0   in an open set in Cm+1 . We will present Asano’s ([A]) proof of

244

The FBI transform and some applications

Chemin’s ([Che]) result that the C  wave front set of any C 2 solution is contained in the characteristic set of the linearized operator Lu =

m f    x t u ux   − xj t j=1 j

We begin with some lemmas about linear vector fields: Lemma V.5.1. Let L=

N M      + bk x t  + aj x t  xj k=1 k t j=1

where the coefficients aj and bk are C  in the variables x t ∈  × J ⊂ RN × R and holomorphic in the variable  ∈  ⊂ CM ,  open. Let fx  be a C  function defined on  ×  , holomorphic in . There exists a C  function ux t  holomorphic in  which is an approximate solution of Lu = 0 in the sense that Lux t  = Otk 

for

k = 1 2   

and such that ux 0  = fx . Proof. The conditions that u has to satisfy determine the Taylor coefficients of the formal series   ux t  = uj x tj j=0

where uj x  = Lu = Ot

j

t ux0 . j!

Set u0 x  = fx . For each j, since we want

, we must have tj−1 Lux 0  = 0. This then leads to ( N up 1  1  q a uj x  = − x  qk x 0  j p+q=j−1 q! k=1 xk t ) M  up q b x  qk x 0  + t k=1 k

j+1

for j ≥ 1. Note that the functions uj x  are C  and holomorphic in . Let ( ∈ C0 R be such that ( ≥ 0 ( ≡ 1 in − 21  21 and supp ( ⊂ −1 1  Then there exists a sequence Rj > 1 Rj / + such that the series ux t  =

  j=1

(Rj tuj x tj

V.5 Application to the C  wave front set

245

is convergent in C  . It follows that u is C  in all the variables and holomorphic in . Moreover, from the way the functions uj are defined, u is an approximate solution of Lu = 0 with the property that ux 0  = fx . In the following lemma, WF denotes the C  wave front set. Lemma V.5.2. Let X ⊂ Rm be open, U an open neighborhood of X × 0 in Rm+1 , U+ = U ∩ Rm+1 + . Let L=

m    + aj x t xj t j=1

be a C l vector field in U for some positive integer l. Assume f ∈ C 1 U+  satisfies

Lfx t = Otk 

k = 1 2   

uniformly on compact subsets of X. Suppose there exist C l functions .1 x t     .m x t on U such that Zx t = x + t.x t satisfies Zx 0 = x and LZx t = Otk 

k = 1 2   

Let ax t = a1 x t     am x t. Assume tj ax 0 = 0 ∀j < l

∀x ∈ X

and that #tl ax0  0  0 $ > 0

for some x0 ∈ X

 0 ∈ Rm 

Then x0   0  % WFfx 0. Remark V.5.3. If L is C  , then Lemma V.5.1 insures that the Zj exist and the proof below will show that in this case, we only need to assume that f ∈ C 0 T   Rm . Proof. Without loss of generality, we may assume that x0 = 0. For j =   1     m let Mj = m k=1 bjk x t x be vector fields satisfying k

Mj Zk = kj 

Mj  Mk = 0

Note that for each j,

Mj  L =

m  s=1

cjs Ms

(V.5)

246

The FBI transform and some applications

where each cjs = Otk  k = 1 2  Indeed, the latter can be seen by expressing

Mj  L in terms of the basis L M1      Mm and applying both sides to the m + 1 functions t Z1      Zm . For any C 1 function g, observe that the differential # " m m   (V.6) Mk gdZk + Lg − Mk gLZk dt dg = k=1

k=1

This is verified by evaluating each side at the basis vector fields L M1      Mm  Using (V.6) we get:

"

dgdZ1 ∧ · · · ∧ dZm  = Lg −

m 

# Mk gLZk dt ∧ dZ1 ∧ · · · ∧ dZm  (V.7)

k=1

For  ∈ Rm  s ∈ Rm , let Es  x t = i · s − Zx t −  s − Zx t 2   2 where for w ∈ Cm , we write w 2 = m j=1 wj . Let B denote a small ball centered m  at 0 in R and  ∈ C0 B,  ≡ 1 near the origin. We will apply (V.7) to the function gs  x t = xfx teEsxt where s  are parameters. We get:   m  dgdZ = Lf + fLE − Mk f + fMk ELZk eE dt ∧ dZ k=1

(V.8) where dZ = dZ1 ∧ · · · ∧ dZm . Next by Stokes’ theorem we have, for t1 > 0 small:   t1   gs  x 0dx = gs  x t1 dx Zx t1  + dgdZ (V.9) B

B

0

B

We will estimate the two integrals on the right in (V.9). Write Z = Z1      Zm  = x + t.x t and . = .1 + i.2  Since the Zj are approximate solutions of L, we have . + t.t + I + t.x  · a = Otk 

k = 1 2   

and hence tj .x 0 = 0

j < l and #tl .2 x 0  0 $ < 0

(V.10)

V.5 Application to the C  wave front set

247

for x in a neighborhood V of B (after shrinking B, if necessary). Observe that Es  x t = t · .2 x t −  s − x − t.1 2 − t2 .2 x t2  Because of (V.10), continuity and homogeneity in , we can get c1 > 0 such that Es  x t ≤ −c1  tl+1 

for x ∈ V

0 ≤ t ≤ t1

(V.11)

s ∈ Rm and  in a conic neighborhood + of  0 . Going back to the integrals in (V.9), we clearly have & & & & & gs  x t1 dx Zx t1 & ≤ e−c2   B

t  for some c2 > 0, for s ∈ Rm and  ∈ +. To estimate 0 1 B dgdZ, we use (V.8) and look at each term that appears there. We first consider the term LfeE . For any k,

LfeE ≤ Ck tlk e−c1 t  ≤ l

Ck 

 k

Moreover, for the x-integral  Lf eE dZ = #f t LeE $ B

after decreasing t1 , we can get  > 0 such that if s ≤  and  ∈ +,

#f t LeE $ ≤ Ce−c  for some constants c C > 0. In the latter, we have used the constancy of  near 0. It follows that the integral   t1 LfeE dt ∧ dZ B 0

decays rapidly in . The term fLEeE is estimated using the fact that for l any k, LE ≤ ck tk  for some constant ck and that eE ≤ e−c1 t  . This shows that   t1 fLEeE dt ∧ dZ B 0

decays rapidly in . The integrals of the terms fMk ELZk eE and Mk f LZk eE are estimated in the same fashion. Thus   t1 dgdZ B 0

248

The FBI transform and some applications

has a rapid decay in , and going back to (V.9), we have shown:  2 Fs  = ei·s−x−  s−x xfx 0dx

(V.12)

B

decays rapidly for s ≤  in Rm and  in a conic neighborhood + of  0 . The function Fs  is the standard FBI transform of the distribution xfx 0. To conclude the proof, we will exploit the inversion formula for the FBI, namely,  n 2 (V.13) xfx 0 = lim+ cm eix−s·−%  Fs   2 dsd %→0

where cm is a dimensional constant. Assume now that x is supported in the ball centered at the origin with radius M. We will study the inversion integral in (V.13) by writing it as a sum of three pieces: I1 %, I2 %, and I3 %. The first piece consists of integration over the region  s s ≥ 2M . In the second piece we integrate over  s  ≤ s < 2M , and in the third piece over  s s ≤  . For the integral I1 %, after integrating in s, one gets an exponential decay in  independent of %, and hence lim%→0+ I1 % is in fact a holomorphic function near the origin in Cm . To study the second piece, we write it as  m 2 2 I2 % = cm eix−y·−  s−y −%  yfy 0  2 dydsd ys ≤ s 0 and vj ·  0 < 0. We now write n  I3 % = Kj % j=0

V.5 Application to the C  wave front set

249

where Kj equals the integral over +j . The decay in the FBI established in (V.12) shows us that K0 is a smooth function even after setting % = 0. Each of the remaining functions Kj , after setting % = 0, is a boundary value of a tempered holomorphic function in a wedge whose inner product with  0 is negative. Hence 0  0  % WFa Kj 0+ where WFa denotes the analytic wave front set. By Corollary V.3.8, the latter implies that 0  0  % WFKj 0+ We have thus proved that 0  0  % WFfx 0

Consider now the vector field L=

m    + aj x t t j=1 xj

where the aj are C 1 on an open set  ⊂ Rm x × Rt . To L we associate vector fields  L = − e−i L s where s ∈ R is a new variable and  ∈ 0 2) is a parameter. Suppose that for each  ∈ 0 2) there exist C 1 functions .1 x t s     .m x t s defined on  × J (J ⊂ R is an open interval centered at the origin) such that Zj x t s = xj + s.j x t s

j = 1     m

are approximate solutions of L Zj = 0 in the sense that L Zj are s-flat at s =   x t s = e−i and Zm+1 x t s = t + e−i s and note that 0. Define also .m+1    = 0. If we write .  = .1      .m+1  and Z = Z1      Zm+1 , then L Zm+1     −i .0 0 e a0 0 = e−i Zs 0 0 0 = .  0 0 0 = − −i e −1 and

       .0 0 cos  − .0 0 sin    · · . 0 0 0 =  sin   =  · .0 0 cos  +  −  · .0 0 sin 

250

The FBI transform and some applications

So the condition

   · .  0 0 0 = 0 

for some  ∈ 0 2) is equivalent to saying that 0 0   is not in the characteristic set of L. Suppose now hx t is a C 1 function with the following property: there exist C 1 functions h x t s such that h x t 0 = hx t and L h is s-flat at s = 0. If 0 0  0   0  is not in the characteristic set of L, we know that there is  ∈ 0 2) such that  0  · .  0 0 0 = 0 0 By replacing  by  + ) or  − ) if necessary, we may assume that  0  · .  0 0 0 < 0 0 and we can apply what we saw in the proof of Lemma V.5.2 to an FBI in x t-space to conclude the following: there exist a conic neighborhood + of  0   0  in Rm+1 \0 and a neighborhood  of the origin in Rm+1 such that Fh 0 x  t     2     2 ei ·x −x+t −t −  #x −x$ +t −t h x t 0dxdt = B×J

= Fhx  t    is rapidly decreasing for   ∈ + and x  t  ∈ . We have thus proved: Lemma V.5.4. For each  ∈ 0 2) let L = s − e−i L and suppose there  ∈ C 1  × J such that Z = x t + s.  x t s is an exist .1      .m+1 approximate solution of L Z = 0 in the sense that L Z is s-flat at s = 0. Suppose moreover that there exist h ∈ C 1  × J such that h x t 0 = hx t and L h is s-flat at s = 0. Then WFh 0 ⊂ charL 0  The preceding linear results will next be applied to a nonlinear equation. Let  ⊂ Rm+1 be a neighborhood of the origin and suppose u ∈ C 2  is a solution of ut = fx t u ux 

(V.14)

where fx t 0   is a C  function in the variables x t ∈  and holomorphic in the variables 0   ∈  ⊂ C × CM 

a  = u0 0 ux 0 0 ∈  

V.5 Application to the C  wave front set Consider =

 m  f    x t 0   − xj t j=1 j

and Lu =

251

(V.15)

 m  f    x t u ux  −  t j=1 j xj

Let v = u ux . It is easy to check that v solves the quasi-linear system Lu v = gx t v

(V.16)

where g0 x t 0   = fx t 0   −

m  j=1

j

f x t 0   j

and gi x t 0   = fxi x t 0   − i

f x t 0   0

i = 1     m

Consider now the principal part of the holomorphic Hamiltonian of (V.16) H =  + g0

m    + gj  0 j=1 j

For .x t 0   a C  function in x t  0  and holomorphic in the variables 0   ∈  , set . v x t = .x t vx t and let p denote the vector field in  obtained by plugging px t for  0  in the coefficients of . Note that v = Lu . Equation (V.16) implies that v . v = H.v where .x t 0   is any C  function in x t ∈  and holomorphic in 0   ∈  . Let Zj x t 0   j = 1     m, and 3k x t 0   k = 0     m be t-flat solutions of H. = 0 such that Zj x 0 0   = xj  j = 1     m, ˜ ˜ and 3k x 0 0   = k  k = 0     m. Let Zz t 0   and 3z t 0  , m m z = x + iy ∈ R ⊕ iR be almost analytic extensions of Zx t 0   and ˜ ˜ 3x t 0   respectively, i.e., Zx t 0   = Zx t 0  , 3x t 0   = 3x t 0   and for all k ∈ N there exists Ck > 0 such that for j = 1     m we have & & &  & & Zz & ≤ Ck z k ˜ t    0 & z & j

252

The FBI transform and some applications

and

& & & &  ˜ & 3z t 0  && ≤ Ck z k  & z j

Since the Jacobian ˜ 3 ˜ ˜ Z ˜ 3 Z z z 0  0    is nonsingular near t = 0, we may solve  ˜ Zz t 0   ˜ 3z t    0

˜ = Z ˜ =3

with respect to z 0   in a neighborhood of 0 a  by the implicit function theorem and get  ˜ ˜ t 3 z = PZ ˜ ˜ t 3 0   = QZ with P0 0 a  = 0 and Q0 0 a  = a . We get  ˜ t QZ ˜ ˜ ˜ t 3 ˜ t 3 ˜ ZP Z = Z ˜ ˜ t QZ ˜ ˜ ˜ t 3 ˜ t 3 3P Z =3 and differentiating with respect to Z˜ we obtain ˜ ˜ 3 Z ˜ ˜ t QZ ˜ P Q Z ˜ t 3 ˜ t 3 ˜ t 3 PZ z 0   ˜ Z +

˜ ˜ 3 Z

˜ t QZ ˜ P Q Z ˜ = 0 ˜ t 3 ˜ t 3 ˜ t 3 PZ z 0   ˜ Z

If Az t 0   denotes a generic entry of the matrix ˜ ˜ 3 Z z 0  

z t 0  

then Az t 0   ≤ Ck z k for all k. It follows that for each k & & & Q & &k & & 0 ˜ ˜ && ˜ && ≤ C  &&PZ ˜ t 3 Z t 3 ∀j = 1     m & k & Z˜ & j and Q0 is holomorphic in 0  . Now consider ˜ ˜ .z t 0   = Q0 Zz t 0   0 3z t 0  

V.5 Application to the C  wave front set

253

and observe that . v x 0 = .x 0 ux 0 ux x 0 ˜ ˜ 0 ux 0 ux x 0 0 ux 0 ux x 0 0 3x = Q0 Zx = ux 0 ˜ ˜ t 0   are t-flat at t = 0. We will Observe that H Zx t 0   and H 3x next show that # " # " m m   Q0 Q0 ˜ Q0 Q0 ˜ H Z˜j + H 3k + H 3k H Z˜j + H. = ˜ Z˜j j=1 k=0  3k Z˜j 3˜ k is t-flat. Note that Px 0 0   = PZx 0 0   0 3x 0 0   ˜ ˜ 0 0   = PZx 0 0   0 3x = x This implies that for some C > 0, & & & & ˜ ˜ t 0  & ≤ C t  t 0   0 3x &PZx Hence implies

Q0 ˜ ˜ Zx t 0   0 3x t 0   is t-flat Z˜j that for all k ∈ N, there exists Ck > 0 such

at t = 0, which in turn that

H.x t 0   ≤ Ck t  k

Hence Lu . v = v . v = H.v is t-flat at t = 0, and so we have found hx t = . v x t such that Lu h is t-flat at t = 0 and hx 0 = ux 0. Now ux t is also a solution of the equation us = e−i ut − fx t u ux  which is of the same kind as (V.14), and the associated vector field  as in (V.15) is given by   = − e−i  s with  as before. Note that   v    = − e−i v = − e−i Lu =  Lu   s s It follows that there exists a C 1 function h x t s such that    u   −i u L  h = − e L h s

254

The FBI transform and some applications

is s-flat at s = 0 and h x t 0 = ux t. We apply Lemma V.5.4 and conclude that WFu 0 ⊂ char Lu 0 . By translation we may apply the same argument to all points of  and state Theorem V.5.5. Let u ∈ C 2  be a solution of (V.14). Then the C  wave front set of u is contained in the characteristic set of the linearized operator Lu .

V.6 Applications to edge-of-the-wedge theory Consider now a hypoanalytic structure   , dim  = N , fiber dimension of  = n and m = N − n. If  is a strongly noncharacteristic submanifold of , then Proposition V.1.28 shows that  induces a hypoanalytic structure on  by taking as the structure bundle in  the image of T  under the natural map CT ∗   → CT ∗   The associated bundle of vector fields will be denoted by   and we have   =  ∩ CT  . Note that for any p ∈   dimC p  = dim  − m. For p ∈  define

p = L ∈ p L ∈ Tp   Lemma V.6.1.   is a real sub-bundle of   of rank n+ dim  − m. The map  which takes the imaginary part induces an isomorphism between   /  and T   /T  . Proof. Let p ∈  . The map  p → Tp  induces a map  p → Tp /Tp  . This latter map is surjective. Indeed, given v ∈ Tp , since  is strongly noncharacteristic, we can find L ∈ p and w ∈ CTp  such that iv = L + w. Taking real and imaginary parts, we see that L ∈ p and v = L + w as desired. Since the kernel of the map  p → Tp /Tp  is p  , we get an isomorphism as asserted in the lemma. Hence, dim p = dim Tp  − dim Tp  + dimR p  = n + dim  − m for any p ∈  . Definition V.6.2. Let E be a submanifold of , dim  = r + s, dim E = r. We say a subset  is a wedge in  at p ∈ E with edge E if the following holds: there exists a diffeomorphism  of a neighborhood V of 0 in Rr+s onto a neighborhood U of p in  with 0 = p and a set B × + ⊆ V with B a ball centered at 0 ∈ Rr and + a truncated open convex cone in Rs with vertex at 0 such that B × + =  and B × 0  = E ∩ U .

V.6 Applications to edge-of-the-wedge theory

255

If E   and p ∈ E are as in the previous definition, the direction wedge +p  ⊆ Tp  is defined as the interior of c 0 c 0 1 →  smooth, c0 = p

ct ∈ 

∀t > 0 

If  is as in Definition V.6.2, +p   = dRr × v v ∈ + ,  > 0  . Note that +p  is a linear wedge in Tp  with edge equal to Tp E. Set  +p   + = p∈E∩U

Suppose now  is a strongly noncharacteristic submanifold of  and  is a wedge in  with edge  . Let u ∈    be a solution of  and let f ∈   . In a neighborhood of p ∈  we may choose coordinates x y vanishing at p such that y = 0 defines  locally and  has the form B × + with B a ball centered at 0 in x-space and + a truncated cone in y-space with vertex at 0. Since u is a solution and  is noncharacteristic, by proposition 1.4.3 in [T5], ux y is a smooth function of y ∈ + valued in  B. We say f is the boundary value of u and write bu = f if +  y → u y extends continuously to + ∪ 0 with u 0 = f , and that this is true for any p ∈  . In this case, since   =  ∩ CT  , it is readily seen that f is a solution of   , i.e., of the induced structure on  . If the codimension of  is 1, then a wedge  with edge  is simply a side of  and distribution solutions in  in this case with boundary values in  were studied in [T5]. We continue to assume that  is a wedge in  with edge  which is strongly noncharacteristic. For p ∈  , define

+p   = L ∈ p L ∈ +p    and



+pT   = L L ∈ +p   

+pT  is an open cone in p  ∩ Tp  . To see this, fix p ∈  and let L1      Ll be an R-basis for p  and complete this to an R-basis L1      Ll  V1      Vk of p . Observe that p  ∩ Tp  is spanned by L1      Ll  V1      Vk  Note also that +p   is a linear wedge in Tp  and hence is translation invariant by elements of Tp  . Therefore   l k k    T ai Li + bj Vj ai ∈ R bj ∈ R bj Vj ∈ +p   +p  = 1

1

1

256

The FBI transform and some applications

This description shows that +pT   is an open cone in p  ∩ Tp  . Lemma V.6.3. Let    be a CR structure, p ∈  and v ∈ Tp . Then there is a maximally real submanifold  ⊆  with p ∈  and v ∈ Tp  . Proof. Recall from Chapter I that there are local coordinates x1      xn  y1      yn  s1      sd  vanishing at p and smooth, real-valued 1   d defined near the origin such that the differentials of zj = xj + iyj  wk = sk + ik x y s

j = 1     n k = 1     d

span T  in a neighborhood of the origin, 0 = 0 and d0 = 0. Let v=

n  k=1

ak

n d      + bk + ck xk k=1 yk k=1 sk

be a real tangent vector at the origin, v = 0. If aj = 0 = bj for all j, we can take  = x y s y = 0 . Otherwise, assume without loss of generality that a1 + ib1 = 0. Consider the subspace S of the tangent space at the origin generated by the n + d linearly independent vectors v s   s  x   x . 1 d 2 n Let  be a submanifold of dimension m = n + d through the origin so that T0  = S (can take  to be a linear space). We claim that  is maximally  real near the origin. To see this, suppose a one-form  = nj=1 Aj dzj 0 + d k=1 Bk dsk is orthogonal to T0  . Then 8 9   =0 ∀j sj ; : and so Bj = 0 ∀j. Moreover, since  x = 0 ∀l ≥ 2, we get Aj = 0 for l j ≥ 2. Finally, note that 0 = # v$ = A1 a1 + ib1  and so since a1 + ib1 = 0 A1 = 0 showing that  = 0. Hence  is maximally real near 0. We observe that Lemma V.6.3 is not valid for a general hypoanalytic structure    which has a section L in  such that at a point p ∈  , Lp is a real vector field. Recall next Marson’s technique of locally embedding a hypoanalytic structure into a generic CR manifold ([Ma]). Suppose    is a hypoanalytic structure with the integers m and n having their usual meaning. Let d = dim Tp0 for some p ∈ . Choose a coordinate system x1      xm  y1      yn 

V.6 Applications to edge-of-the-wedge theory

257

vanishing at p and smooth, real-valued functions 1      d defined in a neighborhood U of the origin and satisfying k 0 = 0

dk 0 = 0

∀k = 1     d

such that T  over U is spanned by the differentials of zj = xj + iyj 

j = 1     

z+k = x+k + ik x y s

k = 1     d

Let U  = U × Rn− and suppose xm+1      xm+n−  are the coordinates for Rn−v . Define zm+k = xm+k + iy+k 

for k = 1     n − 

Let   be the sub-bundle of CTU  that is orthogonal to the bundle generated by dz1      dzm+n− . It is easy to see that U      is a CR structure and for any L ∈ p , L = L − i

n−  l=1

Ly+l 

 ∈ p   xm+l



Here for p ∈ U , we write p ∈ U  to be any point of the form p = p x. Moreover, the preceding association L → L is an isomorphism of p onto p . In particular, any solution of  is also a solution of   depending on fewer variables. Characteristic covectors  ∈ Tp0 U embed into characteristic covectors  0 ∈ Tp0 U  for any p = p x. If  is a strongly noncharacteristic submanifold of U , then   =  × Rn− is a strongly noncharacteristic submanifold of U  and if p ∈  and p = p x ∈   , we have: 

p = L L ∈ p where L is determined by L as above. If  is a wedge with edge  in U , then   =  × Rn− is a wedge in U  with edge   and 

+p    = L L ∈ +p    Finally, if u ∈   , it may be viewed as a distribution in   and it is easy to see that 

  WFha p u × 0 ⊆ WFha p u

We are now ready to present an application of the FBI transform to the hypoanalytic wave front set of a distribution u on a strongly noncharacteristic  which extends to a solution in a wedge. The result is due to Eastwood and Graham ([EG1]).

258

The FBI transform and some applications

Theorem V.6.4. ([EG1]) Let    be a hypoanalytic structure,  a strongly noncharacteristic submanifold, and let  be a wedge in  with edge . Suppose f ∈    is the boundary value of a solution of  on . Then WFha f ⊆ + T  0 = the polar of + T   in the duality between T  and T ∗  . Proof. Let p ∈  and  ∈ Tp∗  /0 satisfy  % +pT 0 . If we embed  near p into a CR structure as in the preceding discussion, then   =  0 %   +pT   0 , and so because of the relation between WFha p f and WFha p f, it suffices to prove the theorem under the assumption that    is CR. Since  % +pT 0 , there is L ∈ +p   such that # L$ < 0. By Lemma V.6.3, there is a maximally real submanifold  ⊆  with p ∈  and L ∈ Tp  (note that the induced structure on  is CR). Since  is maximally real and L = 0, L % Tp  . Choose a submanifold  of  such that  ⊆ , and Tp  is spanned by Tp  and L. Thus  is a hypersurface in . Since  is maximally real,  inherits a hypoanalytic structure of codimension 1 from   . This induced structure on  is CR near p, is generated by L at p, and  is a maximally real submanifold of . We may assume that near p,  divides  into two components  + ,  − where  + is the side toward which L points. Since L ∈ +p  ,  + ⊆  near p.  + may be regarded as a wedge in  with edge  . If F is the solution in  with bF = f on  , then F restricts to  + (since  + is noncharacteristic) and this restriction is a solution for the structure on . Moreover, this restriction has a boundary value equal to f  . To prove the theorem, we have to show that i∗  % WFha p f  . Note that we also have #i∗  L$ < 0. Choose local coordinates x1      xm  t on  vanishing at p so that in these coordinates  is given by t = 0 and   L = A + i t where A = m 1 Aj xj is a real vector field. We therefore need to show that if  ∈ T0∗ Rm and #A $ < 0, then  % WFha bf . This will follow from Theorem V.6.9. Corollary V.6.5. Suppose  ⊂  is a maximally real submanifold, p ∈  , and let  + and  − be wedges in  with edge  such that +p  +  = −+p  − . If f ∈    is the boundary value of a solution of  on  + and also the boundary value of a solution of  on  − , then WFhap f ⊂ i∗ Tp0 . Proof. By Theorem V.6.4, WFhap f ⊆ +pT  + 0 ∩ +pT  − 0  Note that since +p  +  = −+p  − , +pT  +  = −+pT  − . Hence if  ∈ +pT  + 0 ∩ +pT  − 0 , then # v$ = 0 for every v ∈ +pT  + . Since +pT  + 

V.6 Applications to edge-of-the-wedge theory

259

 ⊥ is an open cone in p ∩ Tp  , it follows that  ∈ p ∩ Tp  . Therefore the corollary follows from the fact that ⊥  i∗ Tp0 = p ∩ Tp   Corollary V.6.6. (Theorem V.3.1 in [BCT].) If f is defined in a full neighborhood of p and p ∈  is strongly noncharacteristic, then  ∗ WFhap f ⊂ i Tp0 

Corollary V.6.7. (The edge-of-the-wedge theorem.) If the structure  on  is an elliptic structure and f is the boundary value of solutions in two wedges  +   − with edge a maximally real  as in Corollary V.6.5, then f extends to a hypoanalytic function in a full neighborhood of p in . Corollary V.6.7 is a generalization of the classical edge-of-the-wedge theorem of several complex variables. The example of the structure in the plane generated by y for which the x-axis is maximally real shows that the corollary may not be valid when the structure is not elliptic. ∗ Remark V.6.8. Notice that in general i Tp0  ⊆ +pT 0 

We will next present a result on the hypoanalytic wave front set of the trace of a solution when the vector field in question is locally integrable. We consider a smooth vector field L = X + iY where X and Y are real vector fields defined in a neighborhood U of the origin. Let 0 be an embedded hypersurface through the origin in U dividing the set U into two regions, U + and U − , where U + denotes the region toward which X is pointing. We assume that L is noncharacteristic on 0, which means (after multiplying L by i if necessary) that X is noncharacteristic. Our considerations will be local and so after an appropriate choice of local coordinates x t and multiplication of L by a nonvanishing factor, the vector field is given by L=

m    + aj x t t j=1 xj

(V.17)

and 0 and U + are given by t = 0 and t > 0 respectively. We will need to consider the integral curve −% %  s → s of X that passes through the origin, i.e.,   s = X s, 0 = 0. It is clear that for small % > 0 and

s < %, s ∈ U + if and only if s > 0, so −% %∩U + = 0 %. To simplify the notation we will simply write  + to denote 0 %.

260

The FBI transform and some applications

  Theorem V.6.9. Let L = t + m j=1 aj x t xj be locally integrable. Suppose f ∈  U+  has a boundary value at t = 0 and Lfx t = 0

x t ∈ U + 

Assume that there is a sequence pk ∈  + , pk → 0 such that for each k = 1 2    , Xpk  and Ypk  are linearly independent. Then there exists a unit vector v such that  0 ∈ Rn 

v ·  0 > 0 &⇒ 0  0  % WFha bf

In particular, the hypoanalytic wave front set of bf at the origin is contained in a closed half-space. Proof. Let Z1      Zm be a complete set of smooth first integrals of L near the origin in U and choose new local coordinates x t in which the Zj ’s may be written as Zj x t = xj + i!j x t

k = 1     m

with !0 0 = 0 , !x 0 0 = 0, and !xx 0 0 = 0. For j = 1     m let   Mj = m k=1 bjk x t x be vector fields satisfying k

Mj Zk = kj 

Mj  Mk = 0

It is readily checked that for each j = 1     m,

Mj  L = 0

(V.18)

For any C 1 function g, the differential may be expressed as dg = Lg dt +

m 

Mk g dZk 

(V.19)

k=1

Using (V.19) we get: dgdZ1 ∧ · · · ∧ dZm  = Lg dt ∧ dZ1 ∧ · · · ∧ dZm 

(V.20)

For  ∈ Cm  z ∈ Cm , let Ez  x t = i · z − Zx t − 2#$ z − Zx t 2  Let B denote a small ball centered at 0 of radius r in Rm and  ∈ C0 B,  ≡ 1 for x ≤ r/2, the precise value of r as well as the value of the positive parameter 2 in the definition of E will be determined later. We will apply (V.20) to the function gz  x t = xfx teEzxt

V.6 Applications to edge-of-the-wedge theory

261

where z  are parameters. We get: dgdZ = fL eE dt ∧ dZ

(V.21)

where dZ = dZ1 ∧ · · · ∧ dZm . Next by Stokes’ theorem we have, for t1 > 0 small:    t1  gz  x 0 dx Zx 0 = gz  x t1  dx Zx t1  + dgdZ B

B

0

B

(V.22) We will estimate the two integrals on the right in (V.22) and our aim is to show that for z close to the origin in complex space, both decay exponentially as  →  in a conic neighborhood of  0 . Write Z = Z1      Zm  = x + i!x t

! = !1      !m 

Observe that, assuming without loss of generality that  0 = 1, E0  0  x t = !x t ·  0 − 2 x 2 − !x t 2  Our main task will be to determine convenient values of t1 , 2 and r such that for some  > 0 (i) E0  0  x t1  ≤ − for x ≤ r; (ii) E0  0  x t ≤ − for 0 ≤ t ≤ t1 and r/2 ≤ x ≤ r. In order to find the vector v mentioned in the statement of the theorem we will need Lemma V.6.10. There exists a sequence tk ( 0 such that (1) !0 tk  = 0; (2) !0 t ≤ !0 tk  for 0 ≤ t ≤ tk ; (3) lim !0 tk / !0 tk  = −v. tk →0

We will postpone the proof of Lemma V.6.10 and continue our reasoning with v given by (3) in Lemma V.6.10. The assumptions on ! allow us to write !x t = !0 t + ex t

ex t ≤ A xt + B x 2

(V.23)

for some positive constants A and B. Suppose first !t 0 0 = 0, which is the case that is needed for Theorem V.6.4. Then there is  < 0 such that !t 0 0 = v. Since !0 0 = 0 and !x 0 0 = 0, we can write !x t ·  0 = !t 0 0 ·  0 + O x 2 + t2  = v ·  0 + O x 2 + t2 

262

The FBI transform and some applications

Hence given 2 > 0, we can find t1  r and  > 0 such that (i) and (ii) above hold. We may therefore assume that !t 0 0 = 0 and so the quotient !0 t /t2 ≤ C for 0 t ∈ U + . We have !0 tk  + !0 tk  v = o !0 tk  . We recall that by hypothesis  0 · v > 0. Hence, !0 tk  ·  0 = − !0 tk  v ·  0 + o !0 tk   < − !0 tk  v ·  0 /2 = −c !0 tk   for tk small and 0 < c < 1. We now take r =  !0 tk  /tk , with  and tk small to be chosen later. Hence, for x ≤ r and 0 ≤ t ≤ tk , we can choose  small enough (depending on A, B and C but not on tk ) so that

ex t ≤ A !0 tk 

t

!0 tk  + B2

!0 tk  tk tk2

!0 tk  ≤c  2

(V.24)

This implies that on the support of x we have c −1 + c !0 tk  ≤ !x tk  ·  0 ≤ − !0 tk   2 Let 2 = %/ !0 tk  . A consequence of (V.23), (V.24) and the fact that

!0 t ≤ !0 tk  for 0 ≤ t ≤ tk is

!x t ≤ 1 + c !0 tk  

!x t 2 ≤ 1 + c2 !0 tk  2 

(V.25)

2 !x t ≤ %1 + c !0 tk  2

2

for x in the support of x and 0 ≤ t ≤ tk . Choosing % = c/41 + c2 (thus, independent of tk ), we get, on the support of x, c !x tk  ·  0 + 2 !x tk  2 ≤ − !0 tk  + %1 + c2 !0 tk  2 c ≤ − !0 tk  4 which leads to an exponential decay in the first integral on the right of (V.22) for z complex near 0 and  in a complex conic neighborhood of  0 , as soon as we replace t1 by tk . For the second integral, note that for 0 ≤ t ≤ tk and x in the support of , we may invoke again (V.25) to estimate the size of

!x t and 2 !x t 2 which gives, in view of the previous choice of %, c

!x t + 2 !x t 2 ≤ 1 + c !0 tk  + !0 tk  ≤ 1 + 2c !0 tk  4

V.7 Application to the F. and M. Riesz theorem

263

while on the support of L, x ≥ r/2 =  !0 tk  /2tk so 2 x 2 ≥ and

%2 !0 tk  4tk2

  %2 !x t ·  0 − 2 x 2 − !x t 2  ≤ 1 + 2c − 2 !0 tk   4tk

Hence, if tk is chosen sufficiently small, we also get exponential decay for the second integral on the right-hand side of (V.22) with t1 replaced by tk . We have thus shown that the function  Fz  = eEzx0 xfx 0 dx Zx 0 B

satisfies an exponential decay of the form

Fz  ≤ Ce−R  for z near 0 in Cm and  in a complex conic neighborhood of  0 in Cm . In particular, since Z0 0 = 0 and dx !0 0 = 0, by Theorem V.4.8, 0  0  % WFha bf. We now return to the proof of Lemma V.6.10; it is here that we use the fact that X and Y are linearly independent on a sequence pk ∈  + that approaches the origin. We will show that !0 t cannot vanish identically on any interval 0 % . Let us write L = t + a · x , Z = x + i!, Zx = I + i t !x and recall that t !x has small norm for x t close to 0. Now LZ = 0 leads to a = −iI + i t !x −1 !t . If !0 t vanishes identically on 0 % we will have, for those values of t, that !t 0 t = 0, a0 t = 0, and Y0 t = a0 t = 0. Furthermore, X0 t = t for 0 < t < % , showing that s = 0     0 s for 0 < s < % . Thus, Xs and Ys are linearly dependent for 0 < s < % , a contradiction. Therefore, there exists a sequence sk ( 0 such that !0 sk  > 0 and since !0 0 = 0 there is another sequence tk ( 0 satisfying (1) and (2), which in turn possesses a subsequence that satisfies (1), (2), and (3).

V.7 Application to the F. and M. Riesz theorem The classical F. and M. Riesz theorem states that a complex measure  defined on the boundary T of the unit disk $ all of whose negative Fourier coefficients vanish, i.e.,  2) $ exp−ik d = 0 k = −1 −2     (V.26) k = 0

264

The FBI transform and some applications

is absolutely continuous with respect to Lebesgue measure d. Observe that condition (V.26) is equivalent to the existence of a holomorphic function fz defined on $ whose weak boundary value is . In other words, the theorem asserts that if a holomorphic function f on $ has a weak boundary value bf that is a measure, then in fact bf ∈ L1 T. The F. and M. Riesz theorem has inspired an extensive generalization in two different directions: (i) generalized analytic function algebras, which has as a starting point the fact that (V.26) means that  is orthogonal to the algebra of continuous functions f on T that extend holomorphically to F on $ with F0 = 0; (ii) ordered groups, which emphasizes instead the role of the group structure of T in the classical result. We will next briefly describe these two directions. Let A denote the algebra of continuous functions f on T which have a holomorphic extension F into $. The map f −→ F0 is a continuous homomorphism  of A and so there is a set M of measures on T each of which represents . In this case, it is clear that the normalized Lebesgue measure d is the unique element of M . The kernel of  is the closure of the linear span A0 of expin n > 0. Hence the condition $ n = 0 for all n < 0 is equivalent to  ∈ A⊥ . Such a  decomposes as  = a + s , where a 0 (resp. s ) is absolutely continuous (resp. singular) with respect to d, that is, with respect to every measure in M . The classical F. and M. Riesz theorem ⊥ ⊥ consists of two parts:  ∈ A⊥ 0 ⇒ s ∈ A0 and s ∈ A0 ⇒ s = 0. For function algebras A on compact Hausdorff spaces X other than T, one looks at continuous homomorphisms  of A and their sets of representing measures M . It is known that any measure  on X can be decomposed as  = a + s , with a (resp. s ) absolutely continuous (resp. singular) with respect to every measure in M . Under a variety of hypotheses on A or M , ⊥ the implication  ∈ A⊥ 0 ⇒ s ∈ A0 has been proved and this kind of result turns out to be a crucial ingredient in the theory of generalized analyticity in the algebra A. For more details on this, we mention the book [BK] by Klaus Barbey and Heinz Konig. In the second direction of generalization, one starts with a locally compact $ written additively, is assumed to contain abelian group G. Its dual group G, $ Denote by ME an order, that is, a semigroup P which satisfies P ∪ −P = G. the convolution algebra of complex Borel measures on G whose Fourier $ Each measure  decomposes as transforms vanish on the subset E of G. a + s with respect to Haar measure on G. In this set-up, the implication  ∈ MP ⇒ s ∈ MP has been proved. Under some conditions on G and P, the implication  ∈ MP ⇒ s = 0 has also been proved. There are also results for compact groups (see [K1] and [K2]).

V.7 Application to the F. and M. Riesz theorem

265

Thus, although absolute continuity with respect to Lebesgue measure is a local property, the generalizations mentioned above involve global objects: function algebras and groups. In the paper [B], Brummelhuis used microlocal analysis to prove generalizations of a local version of the theorem of F. and M. Riesz. Among other things, in [B] it is shown that if a CR measure on a hypersurface of Cn is the boundary value of a holomorphic function defined on a side, then it is absolutely continuous with respect to Lebesgue measure. It is easy to use his methods to get a similar result for CR measures on CR submanifolds of any codimension whenever the measure is the boundary value of a holomorphic function defined in a wedge. Another proof of this result was given by Rosay in [Ro]. There are also results when the edge of the wedge has lower regularity ([CR2] and [BH8]). Another way of stating the F. and M. Riesz theorem is to say that if a holomorphic function fz defined on a smoothly bounded domain D of the complex plane has tempered growth at the boundary and its weak boundary value is a measure, then the measure is absolutely continuous with respect to Lebesgue measure. If we regard holomorphic functions as solutions of the homogeneous equation f = 0, it is natural to ask for which complex vector fields L it is possible to draw the same conclusion for solutions of the equation Lf = 0. We will present here an extension of the F. and M. Riesz theorem to all locally integrable, smooth complex vector fields in the plane for smooth domains at the noncharacteristic part of the boundary. We recall that a nowhere vanishing smooth vector field L = ax y

  + bx y x y

is said to be locally integrable in an open set  if each p ∈  is contained in a neighborhood which admits a smooth function Z with the properties that LZ = 0 and the differential dZ = 0. Theorem V.7.1. Suppose L = t + ax t x is smooth in a neighborhood U of the origin in the plane. Let U+ = U ∩ R2+ , and suppose f ∈ CU+  satisfies Lf = 0 in U+ and for some integer N ,

fx t = Ot−N 

as t → 0+ 

Assume that L is locally integrable in U . If the trace bf = fx 0 is a measure, then it is absolutely continuous with respect to Lebesgue measure. The existence of the trace bf = fx 0 under the assumptions on f follows from theorem 1.1 in [BH1]. In his work [B], the author gives a microlocal

266

The FBI transform and some applications

criterion for the absolute continuity of a measure analogous to (V.26) based on Uchiyama’s deep characterization of BMO Rn  [U]. Similarly, one of the main steps in the generalization of the F. and M. Riesz theorem is Theorem V.6.9, which involves the location of the hypoanalytic wave front set of the trace of a solution of a locally integrable vector field in Rn . On the other hand, while in the classical case and the generalizations in [B] the location of the wave front set of the measure under consideration always satisfies a restrictive hypothesis which leads to absolute continuity, this restriction is not fulfilled in general by the trace of a solution of an arbitrary locally integrable vector field even if the solution is smooth (an example concerning a vector field with real-analytic coefficients is shown in example 4.3 of [BH1]). Thus, we need to deal as well with points where the wave front set of the measure may contain all directions; at those points, the vector field L exhibits a behavior close to that of a real vector field (in a sense made precise in Lemma V.7.2 below) and absolute continuity may be proved directly. Lemma V.7.2. Let L=

n    + i bj x t t xj j=1

be smooth on a neighborhood U = B0 a×−T T of the origin in Rn+1 with B0 a = x ∈ Rn x < a . We will assume that the coefficients bj x t, j = 1     n are real and that all of them vanish on F × 0 T, where F ⊂ B0 a is a closed set. Assume that f ∈ CU +  satisfies Lf = 0 on U + = B0 a × 0 T, has tempered growth as t ( 0 and its boundary value bfx = fx 0 is a Radon measure . Then the restriction F of  to F defined on Borel sets X ⊂ B0 a by F X = X ∩ F is absolutely continuous with respect to Lebesgue measure. Proof. If x˜ is an arbitrary point in F we may write bj x t =

n 

xk − x˜ k jk x x˜  t

k=1

with jk x x˜  t real and smooth. The proof of theorem 1.1 in [BH1] shows that for any  ∈ C −a a we have  T  fx t Lt !k x t dxdt (V.27) # $ = fx T!k x T ds + 0

where !k x t =

k 

B0a

tj j x t  0 x t = x j! j=0

(V.28)

V.7 Application to the F. and M. Riesz theorem

and

j x t = −

267

n    % x − x˜  j x x˜  tj−1 x t j−1 x t − t s=1 xs

for j = 1     k, with k a convenient and fixed positive integer. We can write



!k x t = Ax t Dx x

(V.29)

where Ax t Dx  =  ≤k a x tDx is a linear differential operator of order k in the x variables with coefficients depending smoothly on t. The coefficients a are obtained from the coefficients bj x t of L by means of algebraic operations and differentiations with respect to x and t. Observe that given any point x˜ ∈ F , Ax t Dx  may be written as Ax t Dx  =

n 

A x x˜  t x − x˜  Dx  

(V.30)

 ≤k =1

Notice that A x x˜  t ≤ C, for x ∈ B0 a, x˜ ∈ F , t ∈ 0 T,  ≤ k, and  = 1     n because the coefficients of L have uniformly bounded derivatives on B0 a. Hence, we obtain from (V.29) and (V.30) the estimate & &   & & dx F  Dx x dx (V.31) & fx T!k x T dx& ≤ C

 ≤k+1 B0a

where dx F = inf x˜ ∈F x − x˜ . We next consider the second integral on the right in (V.27). We will first show that for any j, Lt !j  =

j+1 j t j!

(V.32)

To see this, note first that (V.32) holds for j = 0 from the definition of 1 . To proceed by induction, assume (V.32) for j ≤ m. Then   m+1 m+1 t Lt !m+1  = Lt !m  + Lt m + 1!   m+1 m m+1 m+1 = t + Lt t m! m + 1! Lt m+1  m+1 = t m + 1! m+2 m+1 t  = m + 1! This proves (V.32). Next we observe that since the coefficients bj x t vanish on F × 0 T , each j has the form  c x tDx x (V.33) j x t =

 ≤j

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The FBI transform and some applications

where the c are smooth and satisfy the estimate

c ≤ Cdx F   The form (V.33) is clearly valid for 0 = . Assume it is valid for j . Then it will also be valid for j+1 since by definition, j+1 = Lt j . If we now choose k = N + 1, (V.32) and (V.33) imply that & T  &  T & & k+1 x t k t k & & t dxdt & 0 B0a fx t L ! x t dxdt& ≤ 0 B0a fx t k!  T

k+1 x t dxdt (V.34) ≤C 0 B0a   dx F  Dx x dx ≤C

 ≤k+1 B0a

Thus the second integral on the right-hand side of (V.27) also satisfies an estimate of the kind in (V.31). Consider now a compact subset K ⊂ F with Lebesgue measure K = 0 and choose a sequence 0 ≤ % x ≤ 1 ∈ C  B0 a

% → 0

such that (i) % x = 1 for all x ∈ K; (ii) % x = 0 if dx K > %; (iii)

Dx % x ≤ C %−  . Note that % x converges pointwise to the characteristic function of K as % → 0 while D % x → 0 pointwise if  > 0. Let * ∈ C  B0 a and use (V.31) and (V.34) with  = % * keeping in mind the trivial estimate dx F ≤ dx K. By the dominated convergence theorem,  # % *$ → * d K

while

dx K  Dx % xL1 ≤ %  Dx % xL1 → 0 as % → 0 (when  = 0 one uses the fact that K = 0). Thus, (V.31) and (V.34) show that  * d = 0 * ∈ C  B0 a K

which implies that the same conclusion holds for any continuous function * on K (first extend * to a compactly supported function on B0 a and then approximate the extension by test functions). Thus the total variation  K of  on K is zero and by the regularity of  it follows that  F   = 0 whenever F  ⊂ F is a Borel set with F  = 0. This proves that F is absolutely continuous with respect to Lebesgue measure.

V.7 Application to the F. and M. Riesz theorem

269

We now consider the set F0 = x ∈ B0 a

∃% > 0 bj x t = 0 ∀t ∈ 0 %  j = 0     n

which is a countable union of the closed sets Fk = x ∈ B0 a

1 bj x t = 0 ∀ 0 ≤ t ≤  j = 0     n k

to which we can apply Lemma V.7.2 and conclude that Fk is absolutely continuous with respect to Lebesgue measure. Thus, F0 is also absolutely continuous with respect to Lebesgue measure and the Radon–Nikodym theorem implies that there exists g ∈ L1loc B0 a such that  F0 X = gx dx X ⊂ B0 a a Borel set. X

Theorem V.6.9 and Lemma V.7.2 imply Theorem V.7.1: End of the proof of Theorem V.7.1. We may assume that the vector field has the form   L = + ibx t t x where bx t is real and smooth on a neighborhood of U = B−a a × −T T of the origin in R2 . Since the trace bf is a measure, by the Radon– Nikodym theorem, we may write bf = g +  where g is a locally integrable function and  is a measure supported on a set E of Lebesgue measure zero. Suppose x0 is a point for which we can find a sequence tj converging to 0 with bx0  tj  = 0. Let Zx t be a first integral satisfying Zx0  0 = 0, and Zx x0  0 = 1. If Zt x0  0 = 0, then L will be elliptic in a neighborhood of x0  0 and so by the classical F. and M. Riesz theorem, we can conclude that bf is absolutely continuous near x0  0. Otherwise, the proof of Theorem V.6.9 shows that the FBI transform with this Z as a first integral and arbitrarily large 2 decays exponentially in a complex conic neighborhood of x0  0 , for some nonzero covector. By theorem 2.2 in [BCT], it follows that near the point x0 , modulo a smooth nonvanishing multiple, the trace bf is the weak boundary value of a holomorphic function F defined on a side of the curve x −→ Zx 0. But then, again by the classical F. and M. Riesz theorem, bf is locally integrable near x0 , that is, x0 % E. Hence the set E is contained in the set F0 = x ∈ B0 a

∃% > 0 bj x t = 0 ∀t ∈ 0 %  j = 0     n 

270

The FBI transform and some applications

But we already observed that the restriction of bf to F0 is absolutely continuous with respect to Lebesgue measure which implies that  is zero.

Notes For a more detailed account of CR manifolds the reader is referred to the books [Bog] and [BER]. The book [T5] contains a detailed discussion of hypoanalytic manifolds. The characterization of microlocal analyticity (Theorem V.2.14) was proved by Bony. Microlocal analyticity was generalized to microlocal hypoanalyticity in the work [BCT]. Several mathematicians have used the FBI transform to study the regularity of solutions in involutive structures and higher-order partial differential equations. Some of these applications can be found in the works [BCT], [BT3], [BRT], [Hi] and [HaT], [Sj1], and [EG1]. Theorem V.5.5 was proved by Chemin [Che] by using para-differential calculus. The main ideas for the proof presented here are due to Hanges and Treves ([HaT]), who proved the analytic version of Chemin’s result. Subsequently, Asano [A] used the techniques in [HaT] to give a new proof of Chemin’s result. Most of the material in Section V.6 is taken from a paper of Eastwood and Graham ([EG1]). Section V.7 is taken from [BH1]. For a generalization of the F. and M. Riesz theorem to systems of vector fields, we refer the reader to [BH7].

VI Some boundary properties of solutions

In this chapter we will explore certain boundary properties of the solutions of locally integrable vector fields. In the first section we present a growth condition that ensures the existence of a distribution boundary value for a solution of a locally integrable complex vector field in RN . This condition extends the well-known tempered growth condition for holomorphic functions which we will recall in Theorem VI.1.1 below. Section VI.2 considers the pointwise convergence of solutions of planar, locally integrable vector fields to their boundary values. Sections VI.3 and VI.4 explore the class of vector fields in the plane for which Hardy space-like properties are valid. The chapter concludes with applications to the boundary regularity of solutions. The boundary variant of the Baouendi–Treves approximation theorem, namely, Theorem II.4.12, will be crucial for the results in Sections VI.2 and VI.4.

VI.1 Existence of a boundary value Suppose L is a smooth complex vector field, L=

N 

aj x

j=1

 xj

defined on a domain  ⊆ R and u ∈ C is such that Lu = 0 in . Assume  is smooth. We would like to explore conditions on u that guarantee that u will have a distribution boundary value on . Theorem V.2.6 showed us that when u is holomorphic on a domain D ⊆ Cn , then u has a boundary value if C (VI.1)

uz ≤ distz Dk N

271

272

Some boundary properties of solutions

for some C, k > 0. Conversely, it is well known that if a holomorphic function on  has a distribution trace on D, then uz has a tempered growth as in (VI.1). For simplicity, we recall here a precise version in the planar case: Theorem VI.1.1 (Theorems 3111, 3114 [H2].). Let A, B > 0, Q = −A A ×0 B and f holomorphic on Q. (i) If for some integer N ≥ 0 and C > 0,

fx + iy ≤ Cy−N 

x + iy ∈ Q

then there exists bf ∈ D −A A of order N + 1 such that  lim+ fx + iy*xdx = #bf *$ ∀* ∈ C0N +1 −A A y→0

(ii) If limy→0+ f· + iy exists in Dk −A A, then for any 0 < A < A, and 0 < B < B, there exists C  such that

fx + iy ≤ C  y−k−1 

x + iy ∈ −A  A  × 0 B 

Because of the local equivalence of L1 and sup norms for solutions in the elliptic (Cauchy–Riemann) case, the preceding theorem asserts that a holomorphic function f on Q has a trace at y = 0 if and only if for some integer N > 0, 

fx + iy yN dxdy <  Q

It is natural to investigate generalizations of this theorem for nonelliptic vector fields. It turns out that the tempered growth condition (VI.1) is sufficient to ensure the existence of a boundary value for a general nonvanishing vector field that may not be locally integrable. Indeed, we have: Theorem VI.1.2 (Theorem 11 [BH4]). Let L be a C  complex vector field in a domain  ⊆ Rn , f ∈ C, Lf = 0 in . Suppose

fx ≤ C distx −N for some C N > 0. If 0 ⊆  is open, smooth and noncharacteristic for L, then f has a distribution boundary value on 0. The preceding result suggests that for a locally integrable vector field, in general, one should seek a growth condition that is weaker than a tempered growth expressed in terms of dist x .

VI.1 Existence of a boundary value

273

As a motivation, suppose Z = x + ix y is smooth in a neighborhood of the origin in R2 ,  real-valued. Then Z is a first integral for L=

iy   −  y 1 + ix x

Assume that x y > 0 when y > 0 and x 0 = 0, for all x. Then for any 1 integer N > 0, since the holomorphic function x+iy N has a boundary value + as y → 0 , it is not hard to see that uN x y =

1 Zx yN

also has the same boundary value. Note that LuN = 0 when y > 0, uN 0 y =

uN x y ≤

1

0y N

, while

1 1 = 

x y N

Zx y − Zx 0 N

Observe that  may be chosen so that uN x y is not bounded by any power of y as y → 0+ . In general, if L is locally integrable, Z is a first integral of L near the origin and Lu = 0 in the region y > 0, then the growth condition

ux y

Zx y − Zx 0 N ≤ C < 

(VI.2)

is sufficient for u to have a distribution boundary value at y = 0. When L is real-analytic, (VI.2) is also a necessary condition for the existence of a boundary trace at y = 0 (see [BH5]). Before we state the main result of this section, as a motivation for its proof, we review the classical case of holomorphic functions. Consider a holomorphic function f on the rectangle Q = −A A × −B B satisfying the growth condition

fx + iy yN ≤ C <  We wish to show that f has a boundary value at y = 0. Let * ∈ C0 −A A. Fix 0 < T < B. For each integer m ≥ 0, choose *m x y ∈ C  −A A ×

0 B  such that (i) *m x 0 = *x and (ii) *m x y ≤ Cym where C depends only on the size of the derivatives of * up to order m + 1. Indeed, if we define *m x y =

m  * k x k=0

k!

iyk 

274

Some boundary properties of solutions

then it is easy to see that (i) and (ii) hold. Note that since f is holomorphic, for any 0 < % < T , and g ∈ C01 −A A, integration by parts gives:  A  A fx + iTgx Tdx fx + i%gx % dx = −A −A  A T + 2i fx + iygx y dxdy −A %

Plugging gx y = *N x y − % in the preceding formula yields  A  A fx + i%*x dx = fx + iT*N x T − %dx −A −A  A T + 2i fx + iyex y % dxdy −A %

where ex y % ≤ C y − % . Since fx + iy yN ≤ C, as y → 0, the righthand side in the formula converges. This proves that fx + iy has a boundary value at y = 0. We will prove now the sufficiency of (VI.2) in a more general set-up. Let L be a smooth, locally integrable vector field defined near the origin in Rm+1 . In appropriate coordinates x t we may assume that L possesses m smooth first integrals of the form Zj x t = Aj x t + iBj x t j = 1     m defined on a neighborhood of the closure of the cylinder Q = Br 0 × −T T where Br 0 is a ball in x space Rm and Zx 0 0 is invertible. Thus, after multiplication by a nonvanishing factor, L may be written as N

L=

m   Zk − M t k=1 t k

(VI.3)

where the Mk are the vector fields in x space satisfying Mk Zj = kj  1 ≤ k j ≤ m. The next theorem gives, in particular, a sufficient condition for the existence of a boundary value of a continuous function f when f is a solution of Lf = 0. Theorem VI.1.3. Let L be as above and let f be continuous on Q+ = Br 0 × 0 T. Suppose (i) Lf ∈ L1 Q+ ; (ii) there exists N ∈ N such that  T

Zx t − Zx 0 N fx t dxdt <  0

Br 0

Then limt→0+ fx t = bf exists in D Br 0 and it is a distribution of order N + 1.

VI.1 Existence of a boundary value

275

Proof. Note first that by taking complex, linear combinations of the Zj ’s, we may assume that Zx 0 0 = Id, the identity matrix. This will not affect hypothesis (ii) in the theorem. Let * ∈ C0 Br 0. For each integer k ≥ 0, we will show that there exists *k x t ∈ C  Br 0 × 0 T  such that (i) *k x 0 = *x and (ii) L*k x t ≤ C Zx t − Zx 0 k where C depends only on the size of D *x for  ≤ k + 1. To get *k x t with these properties, we will use a smooth function uk = uk x y defined near 0 ∈ 0 = Zx 0 in Cm and satisfying: (a) uk Zx 0 = *x and (b)  x + i y uk x y ≤ C distx y 0k for j = 1     m. j

j

Assuming for the moment that such a uk with these properties exists, we set *k x t = uk Ax t Bx t where Ax t = A1 x t     Am x t

Bx t = B1 x t     Bm x t

Then *k x 0 = *x so that (i) above holds. To check (ii), observe that from the equations LZj  = LAj + iBj  = 0 we have L*k  =

j = 1     m

 m  m   uk u u LAj  + k LBj  = 2 LAj  k  xj yj zj j=1 j=1

(VI.4)

It follows that

L*k x t ≤ C1 uk Ax t Bx t ≤ C2 distAx t + iBx t 0k ≤ C2 Zx t − Zx 0 k  Thus if uk satisfies (a) and (b), then *k x t will satisfy (i) and (ii). We will next write a formula for the uk . Since the map x → Ax 0 is invertible, there is a smooth map G = G1      Gm  such that Zx 0 = Bx 0 = GAx 0 This and some of what follows may require decreasing the neighborhood around the origin. Note that since dB0 0 = 0, and dA0 0 = 0, dG0 0 = 0.

276

Some boundary properties of solutions

Let Vj be the vector fields satisfying Vj xs + iGs x = js  1 ≤ j s ≤ m. For each k = 1 2    define uk x y =

 i   ˜ − Gx V *xy !

 ≤k

˜ where by definition, *x = *Ax 0−1 . Clearly, uk Zx 0 = *x. We claim that for each j = 1     m,   1    u ˜ (VI.5) V *x y − Gx  2 k = ik zj ! x j

 =k In particular, the claim implies property (b) for uk . Indeed, after contracting the neighborhood of the origin, we may assume that 0 = x + iGx . Since dG0 0 = 0, it follows that

y − Gx ≤ dist x y 0 which gives (b). The claim will be proved by induction. We have: u1 ˜ x + iy = iVj *x yj and m m     ˜  *˜ u1 ˜ Gs + i x + iy = − i Vs * Vs * ys − Gs x xj xj xj s=1 s=1 xj

Next observe that m   Gs x =i Vs + Vj xj xj s=1

(VI.6)

which can be seen by applying both sides to the m linearly independent functions x1 + iG1 x     xm + iGm x. Hence m  u1   ˜  u +i 1 = i Vs * ys − Gs x xj yj s=1 xj

which proves the claim for k = 1. Assume next that (VI.5) holds for k − 1, k ≥ 1. We can write uk x y = uk−1 x y + Ek x y where Ek x y = ik

  1   ˜ y − Gx  V *x !

 =k

(VI.7)

VI.1 Existence of a boundary value For any 1 ≤ j ≤ m, by the induction assumption, we have  1     u 2 k−1 = ik−1 V *˜ y − Gx  zj ! xj

 =k−1 Observe that   1 Ek  ˜ k x y = i V * ! xj xj

 =k

277

(VI.8)

˜  y − Gx y − Gx + V * xj 





(VI.9) and

˜   V  * Ek x y = ik y − Gx  yj ! y j

 =k

Using the expression for 2

 xj

(VI.10)

from (VI.6), (VI.8) can be written as

m   uk−1 k   1 Gs =i xVs V  *˜ y − Gx zj ! xj

 =k−1 s=1  1    + ik−1 Vj V *˜ y − Gx  !

 =k−1

(VI.11)

From (VI.7), (VI.9), (VI.10) and (VI.11), we get   1    u ˜ 2 k = ik V *x y − Gx ! x zj j

 =k which establishes property (b) for uk . Hence for each k we have *k which satisfies (i) and (ii) and has the form  ˜ *k x t = P x t Dx *Ax tBx t − GAx t (VI.12)

 ≤k

where P x t Dx  is a differential operator of order  involving differentiations only in x. Observe next that if gx t is a C 1 function, the differential of the m form gx tdZ1 ∧ · · · ∧ dZm where Zj = Aj x t + iBj x t is given by dg dZ1 ∧ · · · ∧ dZm  = Lg dt ∧ dZ1 ∧ · · · ∧ dZm  This observation and integration by parts lead to:   fx %*N x %dZx % = fx T*N x T dZx T Br 0

Br 0

+ +





T

Br 0 %





Br 0 %

T

fx tL*N x t dt ∧ dZ (VI.13) Lfx t*N x t dt ∧ dZ

278

Some boundary properties of solutions

where dZ = dZ1 ∧ dZ2 ∧ · · · ∧ dZm . Now by the hypotheses on fx t and property (ii) of *N x t, fx tL*N x t ∈ L1 and so the second integral on the right in (VI.13) has a limit as % → 0. The third integrand on the right is in L1 since Lf is. Therefore,  fx %*N x % dZx % exists. (VI.14) lim %→0 Br 0

We can clearly modify *n by dropping the tilde in its definition and use (VI.14) to conclude:  lim fx %.N x % dZx % exists (VI.15) %→0 Br 0

where for any smooth function *x,  P x t Dx *Ax tBx t − GAx t  .n x t =

 ≤n

Let Px t = Bx t − GAx t. For gx t ∈ C  Br 0 × −T T whose x-support is contained in a fixed compact set independent of t, and n a non-negative integer, define  Tn gx t = P x t Dx gx tPx t  T0 gx t = gx t (VI.16)

 ≤n

Using (VI.15), we will show next that in fact,  lim fx tTN gx t dZx t exists t→0 Br 0

(VI.17)

for any g = gx t. To see this, for * = *x, we change variables y = Ax t in (VI.15) to write   fx t.N x t dZx t = fHy t tQy t Dy *y dy where Q is a differential operator (with differentiation only in y) and y → Hy t is the inverse of x → Ax t. Since  lim fHy t tQy t Dy *y dy exists t→0

it follows that

 lim t→0

fHy t tQy t Dy *y t dy

exists

for any smooth *y t with a fixed compact support in y. Going back to the x coordinates, we have shown that  lim fx tSN gx t dZx t exists (VI.18) t→0 Br 0

VI.1 Existence of a boundary value where by definition



Sn gx t =

279

P x t Dx gAx t tPx t

 ≤n

for any smooth g = gx t. Observe that the integral in (VI.18) can be written in the form  ux tgAx t t dx where this latter integral denotes the action of a distribution u t on the smooth function x → gAx t t. Now since x t → Ax t t is a diffeomorphism near the origin, any function *x t is of the form gAx t t for some g = gx t. We can therefore use (VI.18) to conclude that for any gx t,  fx tTN gx t dZx t exists (VI.19) lim t→0 Br 0

which proves (VI.17). For *x t ∈ C  Br 0 × −T T whose x-support is contained in a fixed compact set and a given multi-index  with  = N , plug gx t = *x tPx t = *x tBx t−GAx t in (VI.19). Note that we may write   h x tP  e x tP  + (VI.20) TN *P  x t = *P  + *

 =N

 >N

where the h and e are smooth functions and 

lim Dx e x t = 0 t→0

∀  

Observe that for each  with  > N ,  fx th x tPx t dZx t lim t→0 Br 0

exists.

(VI.21)

Indeed, this follows from applying the integration by parts formula (VI.13) to the m-form fx th x tPx t dZ1 ∧ · · · ∧ dZm , using the hypotheses on f , and the bound Px t ≤ Zx t − Zx 0 . From (VI.19) and (VI.21) we conclude that # "   (VI.22) fx t *P  + * e x tP  dZx t exists. lim t→0 Br 0

 =N

We can plug * for * in (VI.22) and sum over  with  = N to conclude " # " #     fx t P * + * E x t dZx t exists lim t→0 Br 0

 =N

 =N

(VI.23)

280

Some boundary properties of solutions

where all order derivatives of the E go to zero as t → 0. Observe that given *  =N as above, we can find   =N such that " # # "      P  +  E = P * 

 =N

It follows that

 =N

 lim

t→0 Br 0

fx t



 =N

* P  dZx t exists

(VI.24)

 =N

whenever the functions * x t ∈ C  Br 0 × −T T have their x-support contained in a fixed compact set independent of t. We now return to a general gx t ∈ C  Br 0 × −T T with x-support contained in a fixed compact set independent of t. From (VI.19) and (VI.24) we conclude that  fx tTN −1 gx t dZx t exists (VI.25) lim t→0 Br 0

for any gx t ∈ C  Br 0 × −T T with x-support contained in a fixed compact set independent of t. We will prove by descending induction that for any such gx t and 0 ≤ k ≤ N ,  lim fx tTk gx t dZx t exists, t→0 Br 0

which for k = 0 and gx t = *x ∈ Cc Br 0 gives us the desired limit. To proceed by induction, suppose 1 ≤ k ≤ N and assume that for any multi-index  with  = k, the limits  lim fx tP  x tgx t dZx t and t→0 Br 0 (VI.26)  fx tTk−1 gx t dZx t lim t→0 Br 0

both exist for any gx t ∈ C  Br 0 × −T T with x-support contained in a fixed compact set independent of t. We have already seen that (VI.26) is true for k = N as follows from (VI.24) and (VI.25). Fix  with  = k − 1.  Plug gx t = *x tPx t in the limit on the right in (VI.26) and observe that Tk−1 g may be written as    e x tP  + h x tP  (VI.27) Tk−1 gx t = *P  + *

 =k−1

 ≥k

where the e and h are smooth, the x-supports of the h x t are contained in a compact set that is independent of t, and all order derivatives of the e

VI.2 Pointwise convergence to the boundary value

281

go to zero as t → 0. From the existence of the two limits in (VI.26) we derive that # "    lim (VI.28) fx t *P  + * e x tP  dZx t t→0 Br 0

 =k−1

exists. We now argue as before by replacing * by * and summing over

 = k − 1 to conclude that  (VI.29) fx tPx t *x t dZx t exists lim t→0 Br 0

for all  with  = k − 1 and *x t ∈ C  Br 0 × −B B with x-support contained in a fixed compact set independent of t. Hence, taking account of (VI.26) and (VI.29) we conclude that  fx tTk−2 gx t dZx t exists. (VI.30) lim t→0 Br 0

We have thus proved that (VI.26) holds for k − 1, completing the inductive step. Therefore,  fx %*x dZx % exists (VI.31) lim %→0 Br 0

and thus bf = limt→0 f t exists. Moreover, since the functions x −→ *N x % − *x

x −→ Zx % − Zx 0

and

and all their x-derivatives converge to zero as % → 0, (VI.13), (VI.14), and (VI.31) imply the following formula for bf :  fx T*N x T dZ (VI.32) #Zx x 0bf *$ = Br 0

+



 Br 0 0

+



T

fx tL*N x t dt ∧ dZ 

Br 0 0

T

Lfx t*N x t dt ∧ dZ

This formula shows that bf is a distribution of order N + 1.

VI.2 Pointwise convergence to the boundary value Suppose L is a locally integrable vector field in a planar domain  with a smooth boundary. Let f ∈ L1loc , and assume that f has a weak trace bf which is in L1loc . In this section we will discuss the pointwise convergence

282

Some boundary properties of solutions

of f to bf . It is classical that when L is the Cauchy–Riemann operator, the holomorphic function f converges nontangentially to bfp for almost all p in . In general, this approach region cannot be relaxed. Indeed, we recall: Theorem VI.2.1. (Theorem 744 in [Zy].) Let C0 be any simply closed curve passing through z = 1 situated, except for that point, totally inside the circle

z = 1, and tangent to the circle at that point. Let C be the curve C0 rotated around z = 0 by the angle . There is a Blaschke product Bz which, for almost all 0 , doesn’t tend to any limit as z → expi0  inside C0 . This theorem shows us that for nonelliptic vector fields, we can’t expect nontangential convergence. Indeed, by the theorem, if Lk =

  − ik + 1tk t x

k = 1 2 3    

then for each k, we can get a bounded solution fk = Fk x + itk+1  of Lk with Fk holomorphic in a semidisk in the upper half-plane, bfk x = bFk x ∈ L1 −1 1, but each fk x t doesn’t converge nontangentially on a subset of −1 1 of positive measure. It suffices to take Fk holomorphic and bounded on the semidisk z z < 1 z > 0 such that on a set of full measure in −1 1, Fk has no limit in certain appropriate regions. By considering the Lk with k even, we see that nontangential convergence may fail even for vector fields that are C  and analytic hypoelliptic. Note that for each k, and for almost all p ∈ −1 1, there is an open region +k p with p ∈ + k p such that fk x t converges to bfk p in +k p. On the other hand, if we take the real vector field t , and the solution ux t ≡ bux = (, the characteristic function of a Cantor set C of positive measure in −1 1, the only sets of approach for which ux t → bux x ∈ C, are the vertical segments. Thus for a general locally integrable vector field, we cannot get approach sets for convergence larger than curves. Suppose now L = X + iY is a smooth, locally integrable vector field near the closure of a planar domain . Assume 0 ⊆  is a smooth curve that is noncharacteristic for L, f ∈ L1loc  Lf = 0 and f has a trace bf ∈ L1 0. Multiplying by i if necessary, we may assume that X is not tangent to 0 anywhere and that it points toward . For each p ∈ 0, let p be the integral curve of X through p and set p+ = p ∩ . We shall classify the points of 0 into two types: (I) A point p ∈ 0 is a type I point if the vector fields X and Y are linearly dependent on an arc p+ s 0 < s < % for some % > 0. (II) A point q ∈ 0 is a type II point if there is a sequence qk ∈ p+ converging to q such that L is elliptic at each qk .

VI.2 Pointwise convergence to the boundary value

283

Theorem VI.2.2. Let Lu = 0 in , u ∈ L1loc  bu ∈ L1 0, and 0 is noncharacteristic for L. Assume L is locally integrable in a neighborhood of 0. For each p ∈ 0, there is an approach set +p ⊆  such that: (i) (ii) (iii) (iv)

p ∈ +p and if q ∈ 0 ∩ +p, then q = p; p+ ⊆ +p; for a.e. p ∈ 0, lim+pq→p uq = up; if p is a type II point, +p is an open set, otherwise +p = p+ .

Proof. Since the problem is local, we may assume that we are in coordinates x t where  = −1 1 × 0 1, 0 = −1 1 × 0 , and Zx t = x + ix t is a first integral of L with  real, 0 0 = 0 and x 0 0 = 0. Modulo a nonvanishing factor, L= and so

 t  −i 1 + ix x t

     t x X= −  t 1 + x2 x

Y=

−t   1 + x2 x

Observe that L is elliptic, i.e., X and Y are linearly independent precisely at the points where t = 0. Assume now that 0 ∈ 0 is a type II point. Then t → 0 t can’t vanish on any interval 0 %  % > 0. Indeed, otherwise, we would conclude that L = X on 0 × 0 %—contradicting the hypothesis that 0 is a type II point. For  > 0 small, define mx = inf x t 0≤t≤

Mx = sup x t 0≤t≤

Then since m0 < M0, we may choose A > 0 so that mx < Mx for

x ≤ A. After decreasing A and , by the boundary version of the Baouendi– Treves approximation theorem in Chapter II (Theorem II.4.12), there is a sequence of entire functions Fk satisfying: (a) Fk Zx t → ux t pointwise a.e. on −A A × 0 ; (b) Fk Zx 0 → bux a.e. on −A A. Set A =  =  + i  < A m <  < M  We may assume that the sequence Fk converges uniformly on compact subsets of A to a holomorphic function F and ux t = FZx t for x t ∈ Z−1 A . Indeed, this is clearly true if ux t is continuous for t > 0. In general, we can use the fact that we can express u as Qh where h is a

284

Some boundary properties of solutions

continuous solution and Q is an elliptic differential operator that maps solutions to solutions. The operator Q can be taken to be a convenient power of the operator D defined in Section IV.2. Since 0 is a type II point, by theorem 3.1 in [BH1] and [BCT] (page 465), for some 0 < A1 < A 0 < 1 < , there is a holomorphic function G of tempered growth defined on the region 1 = Zx 0 + iZx x 0v x < A1  0 < v < 1 such that for every * ∈ C0 −A1  A1 ,  #bu *$ = lim GZx 0 + iZx x 0v*xdx v↓0

Since bu ∈ L1 , the holomorphic function Gz converges nontangentially to bux a.e. in −A1  A1 . We may assume that A1 and 1 are small enough so that 1 ⊆ A . We will show that G = F on 1 . Define the subsets of

−A1  A1 : E1 = x x t = x 0 t ∈ 0 

for some  > 0 

E2 = x x t ≥ x 0 t ∈ 0 

for some  > 0 

E3 = x x t ≤ x 0 t ∈ 0 

for some  > 0 

E4 = x for some tj → 0 sj → 0 x sj  < x 0 < x tj   Observe that −A1  A1 = E1 ∪ E2 ∪ E3 ∪ E4 . If x0 ∈ E4 , then by theorem 3.1 in [BH1], there is a holomorphic function H defined in a neighborhood of Zx0  0 such that ux t = HZx t for x t in a neighborhood of x0  0 t > 0. Hence in this case, Fz has a holomorphic extension to a neighborhood of Zx0  0 and since ux t = FZx t for t > 0, we have FZx 0 = bux = bGZx 0. Therefore, by theorem 2.2 in [Du], Fz = Gz on 1 . We may therefore assume that E4 = ∅. Each of the other three sets E1  E2 , and E3 can be written as a countable union of closed sets as follows:  1 E1 =  j=1 E1j , where E1j = x ∈ −A1  A1 x t = x 0 t ∈ 0 j ;  E2 = j=1 E2j , where E2j = x ∈ −A1  A1 x t ≥ x 0 t ∈ 0 1j ;  1 and E3 =  j=1 E3j , where E3j = x ∈ −A1  A1 x t ≤ x 0 t ∈ 0 j . Thus the interval −A1  A1 is a countable union of the closed sets Eij and hence by Baire’s Category Theorem, one of these sets contains an interval with nonempty interior. Case 1: Suppose x t = x 0 on A2  A3 × 0 T for some T > 0, A2 < A3 . Then L = t on A2  A3 × 0 T and so ux t = bux on this rectangle. This implies that Fz extends as a continuous function in 1 up to the boundary piece Zx 0 A2 < x < A3 and therefore bFZx 0 = bux for x ∈ A2  A3 . But then F ≡ G in 1 .

VI.2 Pointwise convergence to the boundary value

285

Case 2: Suppose x t ≥ x 0 on A2  A3 × 0 T , for some T > 0, A2 < A3 . For % > 0 sufficiently small, define u% x t = GZx t + i%

x t ∈ A2  A3  × 0 T

Observe that Lu% = 0. Recall that G is holomorphic on the region 1 = Zx 0 + iZx x 0v x < A1  0 < v < 1 . Let 2 = Zx 0 + iZx x 0v

x < A1  0 < v < 2 for some 0 < 2 < 1 , and for each p = Zx 0 x < A1 , define the nontangential approach region +p = z ∈ 2 z − p < 2 distz 2   Denote by G∗ x the nontangential maximal function of Gz, that is, G∗ x = sup Gz z ∈ +Zx 0  We have:

u% x t ≤ G∗ x ∈ L1 A2  A3  Let wx t = lim u% x t (the pointwise limit) %→0  Gx + ix t if x t > x 0 = bux if x t = x 0 Then u% → w in L1 A2  A3  × 0 T and so Lw = 0 in A2  A3  × 0 T. Since

Gx + ix t ≤ G∗ x

and a.e.

Gx + ix t → bux as t → 0

we conclude that wx t → bux in L1 A2  A3  as t → 0 Therefore ux t = wx t in a neighborhood of A2  A3  × 0  t > 0. In particular, since we may assume that x t ∈ A2  A3  × 0 T x t > x 0 is not empty (otherwise, we would be placed under Case 1), Fz ≡ Gz on 1 . Case 3: Suppose x t ≤ x 0 on A2  A3 × 0 T  T > 0 A2 < A3 . We may assume that there exists x0 ∈ A2  A3  and sj → 0 such that x0  sj  < x0  0. Indeed, otherwise, matters will reduce to Case 1. By theorem 3.1

286

Some boundary properties of solutions

in [BH1] and [BCT] (page 465), after decreasing A2  A3 × 0 T , we get a tempered holomorphic function G1 z defined on the region 1 = Zx 0 + iZx x 0v A2 < x < A3  −T < v < 0 such that for every * ∈ C0 A2  A3 ,  #bu *$ = lim G1 Zx 0 + iZx x 0v*xdx v→0

By the edge-of-the-wedge theorem, there is a holomorphic function vz defined in a neighborhood of Zx 0 A2 < x < A3 that extends G and G1 . Hence Fz = Gz in 1 . We have thus shown that F ≡ G on 1 . Now for almost every p ∈ −A1  A1  Gz converges nontangentially at ˜ Zp 0 (in 1 ) to bup. Pick such a point p and let +p be a nontangential −1 ˜ approach region for Gz at Zp 0. Define +p = Z +p. Then lim

+pxt→p

ux t =

lim

FZx t

+pxt→p

= lim Gz = bup ˜ +pz

We have thus shown that if p is a type II point, then there is an interval around it such that a.e. in the interval, pointwise convergence holds as asserted. Consider now a type I point x0  0. Then Zx0  t ≡ Zx0  0 for t in some interval 0 % . This implies that Fk Zx0  t ≡ Fk Zx0  0 for t ∈ 0 % , and so because of the a.e. convergence stated in (a) and (b), we conclude that for almost every type I point x, ux t → bux as t → 0.

VI.3 One-sided local solvability in the plane In Section VI.4 we will explore the boundary regularity of solutions of the inhomogeneous equation Lf = g where L = Ax t

  + Bx t t x

is a smooth, locally integrable complex vector field defined on a subdomain  of R2 . If Lf = g in , and f has a trace bf on  with a certain degree of regularity, we will investigate whether the regularity persists near  under some smoothness assumption on g. As usual, the motivation comes from what is known in the elliptic case. Suppose hz is a holomorphic function of one variable defined on the rectangle Q = −A A × 0 T with a weak trace bh

VI.3 One-sided local solvability in the plane

287

at y = 0. From the local version of the classical Hardy space (H p  theory for holomorphic functions in the unit disk, we have: (i) if bh ∈ C  −A A, then h is C  up to y = 0; (ii) if bh ∈ Lp −A A 1 ≤ p ≤ , then for any B < A, the norms of the traces h· y in Lp −B B are uniformly bounded as y → 0+ . The main results of Section VI.4 will extend (i) and (ii) above to solutions of complex vector fields that satisfy a one-sided solvability condition. In the elliptic case, property (i) follows easily from part (ii) of Theorem VI.1.1. We will show in Section VI.4 that in general, property (i) follows from property (ii) above and a boundary solvability condition. When a vector field exhibits property (ii), we will say that it has the H p property. To describe the class of vector fields with the H p property, consider a curve 0 in  such that \0 has two connected components, \0 = + ∪ − . It turns out that the local solutions of the equation Lu = 0 on + possess the H p  property at q ∈ 0 if and only if there is a neighborhood U of q such that L satisfies the solvability condition  of Nirenberg and Treves ([NT]) on U ∩ + . This leads to a one-sided version of  that we denote by  +  (or  −  if + is replaced by − ) to indicate the side where it holds. If  holds at q, then both  +  and  −  hold at q. However,  +  and  −  may hold at q ∈ 0 and yet  may not hold in a neighborhood of q. The Mizohata vector field provides an example illustrating this. Write L = X + iY with X and Y real. Let  ⊂ U be 5 a two-dimensional orbit of L in U and consider X ∧ Y ∈ C  U 2 TU. 52 TU has a global nonvanishing section e1 ∧ e2 , X ∧ Y is a real Since multiple of e1 ∧ e2 and this gives a meaning to the requirement that X ∧ Y does not change sign on any two-dimensional orbit  of X Y in U . Recall from Chapter IV that the vector field L satisfies condition  at p ∈ 0 if there is a disk U ⊆  centered at p such that X ∧ Y does not change sign on any two-dimensional orbit of L in U . Definition VI.3.1. We say that L satisfies condition  +  at p ∈ 0 if there is a disk U ⊆  centered at p such that X ∧ Y does not change sign on any two-dimensional orbit of L in U + = U ∩ + . Definition VI.3.2. We say that L is one-sided locally solvable in Lp , 1 < p <  (resp. in C  ) at q ∈ 0 if there is a neighborhood U ⊆  of q such that—after interchanging + and − if necessary—for every f ∈ Lp U (resp. f ∈ C  U ∩ +  ) there exists u ∈ Lp U (resp. u ∈ C  U ∩ + ) such that Lu = f on U + = U ∩ + .

288

Some boundary properties of solutions

Definition VI.3.3. We say that L is one-sided locally integrable at p ∈ 0 if there is a disk U ⊂  centered at p such that—after interchanging + and − if necessary—there exists Z ∈ C  U such that: (1) LZ vanishes identically on U + = U ∩ + ; (2) dZp = 0. Let us assume that L is one-sided locally integrable at p ∈ 0 and let Z satisfy (1) and (2) of Definition VI.3.3. Replacing Z by iZ if necessary and decreasing U we may choose local coordinates x t such that xp = tp = 0, Zx t = x + ix t

(VI.33)

with  real, U is the rectangle U = −a a × −T T, 0 ∩ U = x 0

x < a and U + = −a a × 0 T. Thus, modulo a nonvanishing multiple, we may assume that t x t   −i  t 1 + ix x t x    t  X = + t x2  Y =−  t 1 + x x 1 + x2 x L=

(VI.34)

and so X ∧Y =

 t x y  ∧  1 + x2 x t

The proof of the following lemma is essentially the same as the one for Lemma IV.2.2. Lemma VI.3.4. Let Zx t and L be given by (VI.33) and (VI.34) respectively. Then, L satisfies  +  at the origin if and only there exist T a > 0 such that 0 T  t → x t is monotone for every x ∈ −a a. We now recall from [BH6] the local equivalence between  +  and one-sided solvability. More precisely, Theorem VI.3.5. Let Zx t and L be given by (VI.33) and (VI.34) respectively. The following properties are equivalent: (1) L satisfies  +  ( or  − ) at the origin; (2) L is one-sided locally solvable in Lp , 1 < p < , at the origin; (3) L is one-sided locally solvable in C  at the origin. The following proposition is concerned with continuous solvability up to the boundary and will be useful in the applications to boundary regularity in Section VI.4.

VI.4 The H p property for vector fields

289

Proposition VI.3.6. Let Zx t and L be given by (VI.33) and (VI.34) respectively and assume that L satisfies  +  at the origin, i.e., for some U + = −r r × 0 T, the function 0 T  t → x t is monotone for x < 3 r. If fx t ∈ LipU there exists u ∈ 02r  u + d

−p 2r p −     r ≤ 2 + 

u d j r< p− 1/j. Similarly, if q ∈ Cj1 , q = Aj , and q = Bj then distq U = distq C1  = 1/j. Thus, every point q ∈ Cj is at a distance 1/j of U , we can always find z ∈ U such that q − z = distq U, and z is uniquely determined by q except when q = Aj or q = Bj (in which case the distance may be attained at two distinct boundary points). In particular,

VI.4 The H p property for vector fields

295

whatever the value of  > 0, q ∈ + z for all q ∈ Cj and gq ≤ g ∗ z for any function g defined on U . Given g ∈ H p U we must show that  (VI.39) sup gz p dz ≤ M <  j

We have

 Cj2

Cj

gq p dq = ≤

 

−1 C  j2 j

−1 C  j2 j

≤C



C2



gj2 z p j2 z dz 

g ∗ z p j2 z dz

g ∗ z p dz 

Similarly, using the map j1 x = x + i1/j ∈ Cj1 , we get  

gq p dq ≤ C

g ∗ z p dz  Cj1

(VI.40)

(VI.41)

C1

so adding (VI.40) and (VI.41) we obtain  

gq p dq ≤ C

g ∗ z p dz Cj

U

g

pH p .

which implies (VI.39) with M = C To prove the other inclusion we first assume that p = 2. Given f ∈ E 2 U ⊆ 1 E U it has an a.e defined boundary value f + = bf ∈ L2 U and the Cauchy integral representation 1  bf d z ∈ U fz = 2)i U  − z is valid ([Du], theorem 10.4). Furthermore,

f

Ep U  f + Lp U . Next we recall Lemma VI.4.8 that gives the estimate f ∗ z ≤ T∗ f + z + CMf + z

z ∈ U \A B 

(VI.42)

It is well known that M is bounded in L2 U. Furthermore, T∗ is also bounded in L2 U by Theorem VI.4.7. Therefore (VI.42) implies that f H 2 U = f ∗ L2 U ≤ Cf + L2 U ≤ C  f E2 U  The same technique leads to the inclusion E p U ⊂ H p U for p > 1 because T∗ and M are bounded as well in Lp U for 1 < p <  but the method breaks down for p = 1. This case will be handled in the proof of Theorem VI.4.1 using the fact that if f ∈ E p U, 1 ≤ p < , f has a canonical factorization f = FB where F has no zeros, and B ≤ 1. This is classical for the unit disk

296

Some boundary properties of solutions

$, where B is obtained as a Blaschke product and the general case is obtained from the classical result. We are now ready to present the proof of Theorem VI.4.1. We begin by defining mx = min x y 0≤y≤B

Mx = max x y 0≤y≤B

−A ≤ x ≤ A

The function Zx y takes the rectangle Q = −A A × 0 B onto ZQ =  + i

−A ≤  ≤ A

m ≤  ≤ M 

The interior of ZQ is  + i

−A <  < A

m <  < M 

We will consider three essential cases, in each of which we will show that the assertions of the theorem are valid on a half-interval 0 a . Since the same arguments also apply to the half-intervals −a 0 , the theorem will follow. Case 1: Assume that M0 = m0 and Ma = ma for some a > 0. In this case we will first assume that the solution f is smooth on Q. If Mx = mx for every x ∈ 0 a , then L would be t in 0 a and fx t = fx 0 for all t ∈ 0 B , which trivially leads to the inequality we seek on the half-interval

0 a . Hence we may assume that there is x ∈ 0 a for which mx < Mx. Then the set Z0 a × 0 B has nonempty interior. Every component of the interior of this set has the form  + i  <  <  m <  < M where   is a component of the open set x ∈ 0 a Mx > mx . Let  x ∈ 0 a Mx > mx = k  k  k

be a decomposition into components. Fix k and consider one of these components k  k . Note that mk  = Mk  and mk  = Mk . Since for each x, the function t −→ x t

is monotonic

either mx = x 0 and Mx = x B or mx = x B and Mx = x 0 on k  k . Without loss of generality, we may assume that mx = x 0 and Mx = x B for every x ∈ k  k . Let Uk = the interior of Zk  k  × 0 B. Thus Uk = x + iy k < x < k  x 0 < y < x B 

VI.4 The H p property for vector fields

297

Since the solution f is assumed smooth on Q in the case under consideration, by the Baouendi–Treves approximation theorem, there exists Fk ∈ C  Uk , holomorphic in Uk such that fx y = Fk Zx y

∀x y ∈ k  k × 0 B 

Note that Uk is a bounded, simply connected region lying between two smooth graphs and its boundary Uk is smooth except at the two end points k  Mk  and k  Mk . Note also that Uk has a rectifiable boundary of length bounded by  k <  k < 1 + x2 x B dx + 1 + x2 x 0 dx

Uk ≤ k k = ≤ 2k − k  1 + sup , 2 = Kk − k  Q

where the constant K is independent of k. For each p ∈ Uk , and p % k  Mk  k  Mk  , define the approach region +p = z ∈ Uk z − p ≤ 2 distz Uk   Define the maximal functions Fk∗ and T∗ Fk on Uk (except at the two cusps) by Fk∗ p = sup Fk  ∈+p

and

& & T∗ Fk z = sup && ∈U %>0

k

−z >%

& & 1 F  d &&   −z k

z ∈ Uk 

Recall the Hardy–Littlewood maximal function 1  + MFk z = sup

f  d  z = k + iMk  k + iMk 

I I where the sup is taken over all subarcs I ⊆ Uk that contain z and I denotes the arclength of I. Next, since each Uk is Ahlfors-regular, Lemma VI.4.8 gives the estimate Fk∗ z ≤ T∗ Fk z + CMFk z

z ∈ Uk \k + iMk  k + iMk   (VI.43) The constant C in (VI.43) is independent of k because the aperture of the +p is independent of k. Next we will show that any z ∈ Uk lies in +p for some p ∈ Uk . Let z ∈ Uk . Then for some x t ∈ k  k  × 0 B, z = x + ix t

298

Some boundary properties of solutions

and x 0 < x t < x B. Let p = x + ix B and q = x + ix 0. We claim that z ∈ +p ∪ +q . Indeed suppose first

x B − x t ≤ x t − x 0 

(VI.44)

Then for any y: 1

x + ix t − y − iy B ≥  x − y + x t − y B  2 1 ≥  x − y + x t − x B − x B − y B  2 1 1 ≥  x t − x B since x ≤ 2 2 1 = z − p  (VI.45) 2 We also have: 1

x + ix t − y − iy 0 ≥  x − y + x t − y 0  2 1 ≥  x t − x 0 2 1 ≥  x B − x t by (VI.44) 2 1 = z − p  (VI.46) 2 From (VI.45) and (VI.46) we see that if (VI.44) holds, then z ∈ +p . By a similar reasoning, if (VI.44) does not hold, then z ∈ +q . We have thus shown that  Uk ⊆ +p  (VI.47) p∈Uk

Next fix x t ∈ k  k  × 0 B. If x + ix t ∈ Uk , i.e., if x 0 < x t < x B, then by (VI.47),

Fk x + ix t ≤ Fk∗ x + ix 0 + Fk∗ x + ix B

(VI.48)

On the other hand, if x t = x 0, then since x 0 < x B, there exists t ≤ y < B such that x y = x 0 = x t and y is the maximum such. Let ym → y ym > y. Then by (VI.48),

Fk x + ix ym  ≤ Fk∗ x + ix 0 + Fk∗ x + ix B Letting m → , we get

Fk x + ix t = Fk x + ix y ≤ Fk∗ x + ix 0 + Fk∗ x + ix B

VI.4 The H p property for vector fields

299

Thus for any x t ∈ k  k  × 0 B, we have:

fx t = Fk x + ix t ≤ Fk∗ x + ix 0

(VI.49)

+ Fk∗ x + ix B From (VI.43) and (VI.49), for any x t ∈ k  k  × 0 B, we have:

fx t ≤ T∗ Fk x + ix 0 + T∗ Fk x + ix B + CMFk x + ix 0 + MFk x + ix B

(VI.50)

where we recall that the constant C is independent of k. Let 1 < p < . The cases p = 1  will be treated separately at the end. Since Uk is an Ahlforsregular domain, both T∗ and M are bounded in Lp Uk  ([D]) and so (VI.50) leads to  k 

fx t p dx ≤ C

Fk z p dz for any 0 < t < B (VI.51) k

Uk

Since fx t = Fk Zx t on k  k × 0 B , we conclude that for any 0 < t < B:     k  k k p p p

fx t dx ≤ C

fx 0 dx +

fx B dx (VI.52) k

k

k

where C is independent of k. We can write " #   0 a = k  k  S k

where S = x ∈ 0 a x 0 = x B . Observe that for x ∈ S, the function t −→ fx t is constant since L = t on x × 0 B. Hence for any 0 ≤ t ≤ B,  S

fx t p dx =

 S

fx B p dx

Using (VI.53) and summing up over k in (VI.52), we conclude:  a   a  a p p p

fx t dx ≤ C

fx 0 dx +

fx B dx 0

0

(VI.53)

(VI.54)

0

for any 0 < t < B. Finally, we use a refinement of the approximation theorem as in Theorem II.4.12 to remove the smoothness of f . Case 2: Assume that M0 = m0 and Mx > mx for every 0 < x ≤ A. We will need to use the boundary version of the Baouendi–Treves approximation formula. Let hx ∈ C0 −A A, hx ≡ 1 in a neighborhood of 0. For  > 0, define   2 E fx t = /)1/2 e− Zxt−Zx 0 fx  0hx Zx x  0 dx R

300

Some boundary properties of solutions

and G fx t = /)1/2

 R



e− Zxt−Zx t fx  thx Zx x  t dx 2

where fx  t is the distribution trace of f at t ≥ 0. Let R fx t = E fx t − G fx t The Baouendi–Treves approximation theorem asserts that after decreasing A and B, E fx t converges to fx t in the sense of distributions in the open set −A A × 0 B. However, here we need the refined boundary result in Chapter II (Theorem II.4.12) which guarantees convergence up to t = 0 in appropriate function spaces. More precisely, according to the result, there exist a b > 0 such that R fx t → 0

in C   −a a × 0 b 

Since it is clear that G fx t → fx t in Lp −a a whenever f t ∈ Lp −a a, it follows that E fx t → fx t

in Lp  −a a 

if f t ∈ Lp −a a

(VI.55)

Let F z be the entire function satisfying F Zx t = E fx t. Let Ua = the interior of Z0 a × 0 b. Recall that m0 = M0 but mx < Mx for any 0 < x ≤ A. The domain Ua is also an Ahlfors-regular domain. Therefore, we can apply the arguments in Case 1 to the smooth functions E f to arrive at:   a

E fx t p dx ≤ C

F fz p dz  (VI.56) 0

Ua

Note that this time Ua has three pieces and so (VI.56) leads to:  a  a  a

E fx t p dx ≤ C

E fx 0 p dx +

E fx b p dx 0 0 0   b p +

E fa s s a s ds  0 < t < b (VI.57) 0

We now wish to let  →  in (VI.57). From (VI.55) we know that if f 0 and f b are in Lp −a a, then  a  a

E fx 0 p dx →

fx 0 p dx and 0 0  a  a

E fx b p dx →

fx b p dx 0

0

VI.4 The H p property for vector fields

301

We thus need only compute the limit of the s integral in (VI.57). We will show that for almost all a ,  b  b (VI.58)

fa  s p s a  s ds

E fa  s p s a  s ds → 0

0

We know that Mx > mx for every 0 < x ≤ A. We may also assume that x t > x 0 for every x ∈ 0 A  t ∈ 0 b . Indeed, otherwise, we will be placed in the context of Case 1. The approximation theorem then implies that for each x > 0, f is continuous at x t for t > 0 small. Since R fx t → 0 uniformly in 0 a × 0 b , (VI.58) will follow if we show that for almost all a ,  b  b

G fa  s p s a  s ds →

fa  s p s a  s ds (VI.59) 0

0

Choose two numbers a1  a2 such that 0 < a1 < a < a2 ≤ A. By the approximation theorem, after decreasing b, since f is continuous at x t for t = tx > 0 small, there exists F continuous in Za1  a2  × 0 b, holomorphic in W = the interior of Za1  a2  × 0 b such that FZx t = fx t. Observe that W = x + iy x ∈ a1  a2  x 0 < y < x b and F has a distributional boundary value = fx 0 on the curve x+ix 0 a1 < x < a2 . For x ∈ a1  a2 , define F ∗ x = sup Fx + ix t  0 m0. Let a > 0 such that Mx > mx for every x ∈ −a a. If Wa = Z−a a × 0 B, there is a function F holomorphic on the interior of Wa such that fx y = FZx y. This time the boundary of Wa has four pieces. One can then reason as in the previous case to get the required estimate on the interval −a a. Finally, observe that estimates on the interval of the form −a 0 are also valid under Cases 1 and 2. The theorem for 1 < p <  follows from these three cases. We consider next the case when p = 1. Assume we are in the situation of Case 1 where M0 = m0 and Ma = ma for some a > 0. As before we assume first that fx t is smooth on Q+ , Fk ∈ C  Uk , holomorphic in Uk and fx y = Fk Zx y on k  k ×

0 B . Since Uk is simply connected, by a classical result (see the corollary of theorem 10.1 in [Du]), Fk has a factorization Fk = Gk Bk where each factor is

VI.4 The H p property for vector fields

303

holomorphic in Uk , Gk has no zeros, Gk ∈ E 1 Uk , Bk z ≤ 1, and Bk z = 1 on Uk . The fact that Gk ∈ E 1 Uk  implies (see theorem 10.4 in [Du]) that it has a nontangential limit bGk a.e. on Uk , and Gk equals the Cauchy transform of bGk . Observe that since Bk z = 1 on Uk , bGk z = Fk z on Uk . Since Gk has no zeros on the simply connected region Uk , it has a holomorphic square root Hk . Note that Hk ∈ E 2 Uk  = H 2 Uk  (by the discussion preceding this proof). We have Hk∗ z ≤ T∗ bHk z + CMbHk z

(VI.66)

Using (VI.66) and the equality Gk = Fk on Uk we get:  k  k

fx t dx =

Fk x + ix t dx k

k

≤ ≤



k

k



Uk

≤C

Gk x + ix t dx =



k k

Hk x + ix t 2 dx

Hk∗ z 2 dz



bHk z 2 dz by the L2 boundedness of T∗ and M     k k

fx 0 dx +

fx B dx for any 0 < t < B =C Uk

k

k

(VI.67) Summing up over k and adding the contributions from the set S = 0 a\ ∪k k  k , we get:  a   a  a

fx t dx ≤ C

fx 0 dx +

fx B dx (VI.68) 0

0

0

for 0 < t < B +

whenever f is a solution and f ∈ C  Q . In general, for f ∈  Q+  satisfying the hypotheses of Theorem VI.4.1, let fm x t be a sequence of C  solutions on Q+ satisfying: (i) for each 0 ≤ t ≤ B, fm  t → f t in  −a a; (ii) fm x 0 → fx 0 and fm x B → fx B in L1 −a a. We now apply inequality (VI.68) to fm − fn , let m and n tend to , and use (i) and (ii) above to conclude that (VI.68) also holds for f . Cases 2 and 3 are also treated in a similar fashion. Finally we consider the case where p = . Suppose we are in the situation of Case 1 where M0 = m0 and

304

Some boundary properties of solutions

Ma = ma for some a > 0. Assume first that fx t ∈ C  Q and for k fixed as before, let Uk = x + iy k < x < k  x 0 < y < x B and fx y = Fk Zx y on k  k × 0 B , Fk holomorphic on Uk and continuous on the closure. We apply the maximum modulus principle to Fk and use the constancy of f on the vertical segments x = k and x = k to conclude that

fx y ≤

f 0

L 0a +

f B

L 0a If S is the set as before with 0 a =

" 

# k  k 

∀x y ∈ k  k × 0 B  

S

k

then fx y = fx B

∀x y ∈ S × 0 B, and so we conclude that

fx y ≤

f 0

L 0a +

f B

L 0a

(VI.69)

∀x y ∈ 0 a × 0 B For a solution f ∈  Q+  satisfying f 0 and f B ∈ L −A A, we use the refinement of the approximation theorem in Chapter II according to which fx y = lim E fx y →

a.e. in

0 a × 0 B

(VI.70)

provided that A and B are small enough. Moreover,  1  2

G fx B ≤ c1  2 e−c2  x−x fx  B

hx  dx ≤ c3

f B

L

∀ > 0

(VI.71)

and likewise,

G fx 0 ≤ c

f 0

L 

(VI.72)

Letting  → , and recalling that R f → 0 uniformly, we get lim E fx 0 ≤ C

f 0

L

→

lim E fx B ≤ C

f B

L

→

and (VI.73)

for some C > 0. From (VI.69) (applied to E f ), (VI.70) and (VI.73), we conclude that for every x y ∈ 0 a × 0 B,  

fx y ≤ C

f 0

L 0a +

f B

L 0a  (VI.74)

VI.4 The H p property for vector fields

305

Next we consider Case 2 where M0 = m0 and Mx > mx for every 0 < x ≤ A. As before, let a b > 0 such that E fx t → fx t

a.e. in −a a × 0 b 

(VI.75)

Let Ua = Z0 a × 0 b and consider the holomorphic function F such that F Zx t = E fx t. The maximum principle applied to F on Ua leads to

E fx y ≤

E f 0

L 0a +

E f b

L 0a +

E fa 

L 0b

∀x y ∈ 0 a × 0 b 

(VI.76)

As observed already, the terms

E f 0

L 0a and

E f b

L 0a are dominated by a constant multiple of

f 0

L 0a +

f b

L 0a  We therefore only need to estimate the term

E fa 

L 0b for which it suffices to estimate

G fa 

L 0b . Let 0 < a1 < a < a2 < A be as before, F holomorphic such that fx y = Fx + ix y

on

a1  a2 × 0 b 

Since bF = bf ∈ L a1  a2 , by the generalized maximum principle applied to F there exists M > 0 such that

Fx + ix y = fx y ≤ M

on

a1  a2 × 0 b 

for some a1 < a1 < a < a2 < a2 . We write G f = G1 f +G2 f as before, except that this time * is supported in a1  a2 . Recall that G2 f → 0 uniformly while

G1 fx t ≤ C sup *x fx  t ≤ CM Hence for some C > 0,

E fa 

L 0b ≤ C

∀ > 0

We have shown that f ∈ L 0 a × 0 b in this case. Case 3 is treated likewise. We conclude that f is bounded. Theorem VI.4.1 has now been proved. Corollary VI.4.9. Suppose f is a distribution solution of Lf = g in the rectangle Q = −A A × 0 B. Suppose f has a weak boundary value bf = fx 0 at y = 0 and that g is a Lipschitz function. Then there exist A0 > 0 and T0 > 0 such that for any 0 < T ≤ T0 and 0 < a < A0 , if f 0 and f T ∈ Lp −A0  A0 , f t ∈ Lp −a a for any 0 < t < T .

306

Some boundary properties of solutions

Proof. Using Proposition VI.3.6 we may find a function f0 , uniformly continuous on Q, such that Lf0 = g. Then, f1 = f − f0 satisfies the hypothesis of Theorem VI.4.1. It follows that (i) holds for f1 if 1 ≤ p <  or (ii) if p =  and the same conclusion applies to f = f0 + f1 because f0 is continuous up to the boundary. Corollary VI.4.10. Let L be as above, f ∈  Q+ , Lf = g in Q+ where g ∈ C  Q+ . Let A0 and T0 be as in Theorem VI.4.1. If f has a weak trace fx 0 ∈ C  −A0  A0  and f T0  is in C  −A0  A0 , then for all 0 < a < A0 and 0 < T < T0 , f ∈ C   −a a × 0 T . In particular, f is smooth up to the boundary t = 0. Proof. By Proposition VI.3.6, we can get u ∈ C 0 −A A × 0 B that solves Lu = g in Q+ . Hence Lu − f = 0 in Q+ and so by Theorem VI.4.1 and the continuity of u up to the boundary, for any 0 < a < A0 and 0 < t ≤ T0 there is a constant C > 0 such that  a

fx t 2 dx ≤ C ∀t ∈ 0 T0  (VI.77) −a

 1 Define the vector field M = Z xt . Since the bracket L M = 0 and Lf = g, x x the distribution Mf is also a solution of LMf = Mg in Q+ . Moreover, since the traces Mf T0  and Mf 0 are smooth, by repeating the same arguments, for any 0 < a < A0 and 0 < T < T0 there is a constant C > 0 such that  a

Mfx t 2 dx ≤ C ∀t ∈ 0 T  (VI.78) −a

Since

f t

= −ax t f + gx t, (VI.78) implies that for some constant C  , x &2  a && f & & x t& dx ≤ C  ∀t ∈ 0 T  & & −a t

By iterating this argument, we derive that for every m n = 1 2    , there exists C = Cm n > 0 such that  a

Dxm Dtn fx t 2 dx ≤ C ∀t ∈ 0 T  (VI.79) −a

From (VI.79) we conclude that f ∈ C   −a a × 0 T . Smoothness up to the boundary now follows from the case p =  in Theorem VI.4.1. Remark VI.4.11. Conversely, if a locally integrable vector field L shares the H p property as in Theorem VI.4.1, then L has to satisfy condition  +  at the origin in 0 = −A A × 0 . See [BH6] for the proof.

Notes

307

Corollary VI.4.12. Let L satisfy  +  at the origin as above. Suppose Lf = g in Q+ , g ∈ C  Q+ , and f ∈ C  Q+ . If the trace bf = fx 0 exists and fx 0 ∈ C  −A A, then f is C  up to the boundary t = 0. Example 4.3 in [BH6] provides a real-analytic vector field L for which Corollary VI.4.12 is not valid even for a solution of the homogeneous equation Lf = 0. Example 4.4 in the same paper shows that in Theorem VI.4.1, one needs to assume the integrability of two traces. That is, if we only assume that bf = fx 0 ∈ L1 , the traces f t may not be in L1 .

Notes The results of this chapter in the holomorphic case are classical. For a discussion of the conditions that guarantee the existence of a boundary value we refer to the books [BER] and [H2]. The basic theory of Hardy spaces for bounded, simply connected domains in the complex plane is exposed in [Du] (see also [Po]). The paper [L] and the references in it contain more recent developments on the subject. The planar case of Theorem VI.1.3 as well as the necessity in the real-analytic, planar situation was proved in [BH5]. Lemma VI.4.8 is taken from [L]. Theorem VI.4.1 and its corollaries appeared in [BH6]. The work [HH] extends Theorem VI.4.1 to the case 0 < p < 1 for vector fields with real-analytic coefficients.

VII The differential complex associated with a formally integrable structure

In this chapter we shall introduce the differential complex associated with a formally integrable structure and discuss several aspects of its exactness.

VII.1 The exterior derivative Let  be a differentiable manifold of dimension N . As in Chapter I, we shall denote by X the space of all complex vector fields over . We then set N0  = C   and if q ≥ 1 is an integer we shall denote by Nq  the space of all C  -multilinear, alternating forms  X ×    × X −→ C   > ?@ A q

Notice that, according to Section I.4, we have N1  = N; notice also that Nq  has, for each q, the structure of a C  -module. We then generalize the concept of one-forms introduced in Section I.4 and call the elements of the direct sum ⊕ q=0 Nq  differential forms over . If  ∈ Nq  we shall say that  is a differential form of degree q (or q-form for short). The exterior product between  ∈ Nq  and  ∈ Nr  is the q + r-form  ∧  ∈ Nq+r  defined by the formula  ∧X1      Xq+r  = sg  X1      Xq Xq+1      Xq+r  AB

(VII.1) where Xj ∈ X and the summation is over all partitions A B of 1     q + r with A = q, B = r and  ∈ S q+r is such that 1     q = A, q + 1     q + r = B. It is easy to see that (VII.1) defines indeed a q + rform, that the map 308

VII.2 The local representation of the exterior derivative

309

  →  ∧  is C  -bilinear, and that the operation so defined is associative. It follows  that ⊕ q=0 Nq  has a structure of a graded C -algebra. We also remark that  ∧  = −1qr  ∧ 

 ∈ Nq   ∈ Nr 

(VII.2)

The exterior differentiation operator is a C-linear map  d ⊕ q=0 Nq  → ⊕q=0 Nq 

whose restriction to N0  = C   coincides with the operator introduced in Definition I.1.6 [that is, dfX = Xf if f ∈ C   and X ∈ X] and is characterized by the following additional properties: d1  dNq  ⊂ Nq+1  for every q ≥ 0; d2  d d = 0; d3  if  ∈ Nq  and  ∈ Nr  then d ∧  = d ∧  + −1q  ∧ d

(VII.3)

The only operator d which satisfies these properties can be defined by the expression: 

q+1

dX1     Xq+1  = +



, ˆ j      Xq+1  −1j+1 Xj X1      X

j=1

ˆ j     X ˆ k      Xq+1  −1j+k  Xj  Xk  X1      X

j r

(VIII.8)

We then apply the Cauchy–Kowalevsky theorem in order to solve, in a neighborhood of the origin, the problems  Lr uJ = −1q−1 fJ ∪r (VIII.9) uJ tr =0 = 0 Since the vector fields Lj are pairwise commuting, (VIII.8) implies Lr Ls uJ = 0

s > r

Since moreover Ls uJ = 0 when tr = 0 it follows from the uniqueness part in the Cauchy–Kowalevsky theorem that Ls uJ = 0 for all s > r and all J .  Consequently, if we set u = J uJ dtJ then d u =



r−1 

Lj uJ dtj ∧ dtJ +

J =q−1J ⊂1r−1 j=1



Lr uJ dtr ∧ dtJ

J =q−1J ⊂1r−1

and hence (VIII.9) implies that the d -closed form d u − f only involves dt1      dtr−1 . The proof can then be concluded by an elementary inductive argument, whose details are left to the reader.

VIII.3 Elliptic structures When the structure  is elliptic the discussion presented at the end of Section I.12 shows that the differential complex associated with  can be locally  realized as the standard elliptic complex in Cm × Rn , n = n − m, which we now describe and study in some detail.

334

Local solvability 

Let m ∈ Z+ and write the variables in Cm × Rn as z t = z1      zm  t1      tn  We shall also write zj = xj + iyj , j = 1     m, and n = m + n .   The elliptic complex on Cm ×Rn is defined as follows: given  ⊂ Cm ×Rn q  open and 0 ≤ q ≤ n we set C  #  as being the space of all smooth differential forms of the kind  fJK dzJ ∧ dtK  fJK ∈ C   (VIII.10) f=

J + K =q

We define the differential operator Dq C   #q  −→ C   #q+1 

(VIII.11)

by the formula 

D0 u =

m n   u u dzj + dtk t z k j j=1 k=1

if u ∈ C   = C   #0 , and by  D0 fJK ∧ dzJ ∧ dtK Dq f =

(VIII.12)

(VIII.13)

J + K =q

if f is as in (VIII.10). In particular, when m = 0, we have Dq = dq , the exterior derivative acting on q-forms. It is clear that Dq+1 Dq = 0 and consequently (VIII.11) defines a complex D of differential operators, whose cohomology will be denoted by HDq  q = 0     n . In particular, HD0  is the space of all solutions u ∈ C   of the system D0 u = 0, that is, the space of all smooth functions on  that are holomorphic in z and locally constant in t. Furthermore, when m = 0, there are isomorphisms between HDq  and H q  C, the cohomology groups of  with complex coefficients (de Rham’s theorem). 

Theorem VIII.3.1. Let U ⊂ Cm be open and pseudo-convex and let " ⊂ Rn be open and convex. Then D is solvable in U × " in degree q, for every q ≥ 1. Proof. For the proof it is convenient to introduce the natural decomposition D  C U × " #rs  C  U × " #q  = r+s=q 

where C U × " #  is the space of forms of the kind  fJK dzJ ∧ dtK  f= rs

J =r K =s

VIII.3 Elliptic structures

335

Notice that C  U × " #rs  = 0 if either r > m or s > n . We also observe that we have homomorphisms z C  U × " #rs  −→ C  U × " #r+1s   dt C  U × " #rs  −→ C  U × " #rs+1  such that D = z + dt .  Let f ∈ C  U × " #q  satisfy Dq f = 0 and decompose f = rs frs , where frs ∈ C  U × " #rs  and the sum runs over the pairs r s such that r + s = q, r ≤ m, s ≤ n . Consider, in this decomposition, the term frs whose value of s is maximum. From the fact that Df = 0 it follows that dt frs = 0 and consequently we can apply the Poincaré Lemma (Section VII.3) in order to find h ∈ C  U × " #rs−1  such that dt h = frs . If we set f • = f − Dq−1 h  • the maximum it follows that in the analogous decomposition f • = rs frs • value of s that occurs has dropped by one and Dq f = 0. If we iterate the argument we will, after a finite number of steps, either solve the equation Dq−1 u = f or at least find v ∈ C  U × " #q−1  such that  g = f − Dq−1 v does not involve dt1      dtn . If this is the case we can write  gJ z tdzJ g=

J =q

and the fact that Dq g = 0 gives in particular that dt gJ = 0 for all J , that is, gJ are independent of t. Hence g defines a Dolbeault class in U and by the  standard complex analysis theory we can determine w = L =q−1 wL zdzL  solving z w = g. If we set u = v + w we obtain Dq−1 u = f , which completes the proof. Likewise we can introduce the spaces   #q , which are the spaces of all currents of the form (VIII.11) where now the coefficients are allowed to be elements of  . By the same expressions (VIII.12) and (VIII.13) we obtain new differential complexes Dq   #q  −→   #q+1 

(VIII.14)

HDq    q 

q = 0     n . whose cohomology will be denoted by The natural injections C   #q    #  commute with the operator D and then induce homomorphisms HDq  −→ HDq   

(VIII.15)

Finally we shall also consider the spaces Cc  #q  = f ∈ C   #q  supp f ⊂⊂  

(VIII.16)

336

Local solvability

   #q  = f ∈   #q  supp f ⊂⊂  

(VIII.17)

The natural pairing C   #q  × Cc  #n−q  −→ C defined by f * −→



f ∧ dz ∧ *

where dz = dz1 ∧ · · · ∧ dzm , extends to a bilinear form

  #q  × Cc  #n−q  −→ C which allows us to identify   #q  with the topological dual of Cc  #n−q , when the latter carries its natural structure of an inductive limit of Fréchet spaces. We shall use the standard notation of the de Rham theory: if T ∈   #q  and * ∈ Cc  #n−q  we shall set  T * = T ∧ * 1 = T ∧ dz ∧ * Likewise we have a natural identification between    #q  and the topological dual of C   #n−q , where now the latter carries its natural topology of a Fréchet space. We shall always consider the weak topology in the spaces   #q  and    #q . Lemma VIII.3.2. If T ∈   #q , * ∈ C   #n−q−1  and one of them has compact support then   Dq T ∧ dz ∧ * = −1q+m−1 T ∧ dz ∧ Dn−q−1 * Proof. Using the fact that C   #q  ⊂   #q  as well as Cc  #q  ⊂    #q  are dense inclusions we can assume that T = f is smooth. We have d2m+n −1 f ∧ dz ∧ * = dq f ∧ dz ∧ * + −1q f ∧ dm+n−q−1 dz ∧ * = Dq f ∧ dz ∧ * + −1q+m f ∧ dz ∧ dn−q−1 * = Dq f ∧ dz ∧ * + −1q+m f ∧ dz ∧ Dn−q−1 * Since



d2m+n −1 f ∧ dz ∧ * = 0

we obtain the desired conclusion.

VIII.4 The Box operator associated with D

337

VIII.4 The Box operator associated with D 

If f g ∈ C  Cm × Rn  #q  and one of them has compact support we set    f gq = fJK gJK dxdydt (VIII.18)  m n

J + K =q C ×R

The formal adjoint of the operator (VIII.13) is the differential operator 



D∗q C  Cm × Rn  #q+1  −→ C  Cm × Rn  #q  defined by the expression



Dq f u

 q+1

  = f D∗q u q 

(VIII.19)

(VIII.20)





where u ∈ C  Cm × Rn  #q+1  and f ∈ C  Cm × Rn  #q , the latter with compact support. We then set D−1 = Dn+1 = 0 and define q = Dq−1 D∗q−1 + D∗q Dq 

(VIII.21)

Notice that q is a second-order differential operator acting on the space  C  Cm × Rn  #q . Actually an elementary but long computation shows that  PfJK  dzJ ∧ dtK  (VIII.22) q f =

J + K =q

where f is as in (VIII.10) and 

n  2 2 −  P=− 2 j=1 zj zj k=1 tk m 

(VIII.23)

The following crucial properties of the operators q , q = 0 1     n will be used in the sequel: Dq q = q+1 Dq = Dq D∗q Dq 

(VIII.24)

D∗q q+1 = q D∗q = D∗q Dq D∗q 

(VIII.25)



0 is hypoelliptic in Cm × Rn 

(VIII.26)

n

Moreover, since any open subset of Cm ×R is P-convex for singular supports ([H3]), we also have 

Given any open set  ⊂ Cm × Rn the maps q   #q  −→   #q  are surjective

(VIII.27)

338

Local solvability 

Proposition VIII.4.1. For any open set  ⊂ Cm × Rn the maps (VIII.15) are isomorphims. More precisely: (i) If u ∈   satisfies D0 u = 0 then u ∈ C  . (ii) If q ≥ 1 and if u ∈   #q−1  is such that Dq−1 u ∈ C   #q  then there is v ∈ C   #q−1  such that Dq−1 u = Dq−1 v. (iii) If q ≥ 1 and if f ∈   #q  satisfies Dq f = 0 then there are g ∈ C   #q  and u ∈   #q−1  such that f − g = Dq−1 u. Proof. (i) is a consequence of (VIII.26), since 0 = D∗0 D0 . Next take u as in (ii) and apply (VIII.27). We can solve q−1 w = u for some w ∈   #q−1 . Then, by (VIII.24), Dq−1 u = Dq−1 q−1 w = q Dq−1 w If we apply (VIII.26) we conclude that Dq−1 w ∈ C   #q  and consequently  v = D∗q−1 Dq−1 w ∈ C   #q−1 . Since using (VIII.24) we also have Dq−1 u = Dq−1 q−1 w = Dq−1 D∗q−1 Dq−1 w = Dq−1 v (ii) is proved. Finally let f be as in (iii) and solve q U = Dq−1 D∗q−1 U + D∗q Dq U = f   for some U ∈   #q−1 . We set u = D∗q−1 U and g = D∗q Dq U . In order to conclude the proof it remains to show that g is smooth. But (VIII.24) and (VIII.25) imply q g = q D∗q Dq U = D∗q q+1 Dq U = D∗q Dq q U = D∗q Dq f = 0 By (VIII.26) g is smooth and we are done. Remark VIII.4.2. The preceding argument gives indeed the proof of a stronger statement than (iii): every cohomology class in HDq    contains a representative which is in the kernel of q (and consequently it is realanalytic). By a similar argument we have: 

Proposition VIII.4.3. If  is any open set on Cm × Rn then HDn  = 0

VIII.4 The Box operator associated with D

339

Proof. Given f ∈ C   #n  we apply (VIII.26) and (VIII.27) in order to find v ∈ C   #n  solving n v = f

(VIII.28)

in . Since moreover n = Dn−1 D∗n−1 we then have Dn−1 u = f , where u = D∗n−1 v ∈ C   #n−1 , thanks to (VIII.28). 

Consider the function E ∈ L1loc Cm × Rn  defined by ⎧ 

2 2 −m−n /2+1 ⎪ if m ≥ 1, −1 ⎪ mn z + t/2 ⎪ ⎨ −t/2 t  if m = 0, n = 1, Ez t = ⎪ −log t /2) if m = 0, n = 2, ⎪ ⎪ 

⎩ −1 2 2 −m−n /2+1 if m = 0 and n ≥ 3, mn z + t/2 



where mn = 2n −2 2m + n − 2 S 2m+n −1 . It is a well-known fact that E is a  fundamental solution of P. If we then set, for U ∈   Cm × Rn  #q ,   E  UJK  dzJ ∧ dtK  (VIII.29) EU =

J + K =q

we obtain q E  U = q E  U = U

(VIII.30)

    Dq−1 D∗q−1 E  U + D∗q E  Dq U = U

(VIII.31)

We push the argument further. Let  be a regular, bounded open subset of   Cm × Rn . If f ∈ C   #q  we consider f  ∈   Cm × Rn  #q , where  denotes the characteristic function of . We obtain, from (VIII.31),     f  = Dq−1 D∗q−1 E  f   + D∗q E  Dq f + D∗q E  D0  ∧ f  (VIII.32) If we now introduce the operators Gq C   #q  −→ C   #q−1 

Hq C   #q  −→ C   #q 

defined by the expressions Gq f = D∗q−1 E  f    

Hq f = D∗q E  D0  ∧ f  

(VIII.33)

formula (VIII.32) then gives a natural extension of the so-called Bochner– Martinelli formula: 

Theorem VIII.4.4. If  is a regular, bounded open subset of Cm × Rn with a smooth boundary and if f ∈ C   #q  then Dq−1 Gq f + Gq+1 Dq f + Hq f = f

(VIII.34)

340

Local solvability

Observe that E is real-analytic in the complement of the origin and that supp D0  ⊂  and so there exists a neighborhood • of  in the complex ification of Cm × Rn such that the following is true: for every f ∈    #q  the coefficients of Hq f extend as holomorphic functions to • . This fact, in conjunction with Proposition VIII.2.1, provides another proof for the local solvability of the complex D.

VIII.5 The intersection number 

We fix a pair    of open subsets of Cm × Rn , with  ⊂ , and an integer q ≥ 1. The intersection number for the pair    in degree q is the C-bilinear form defined on the product f ∈ C   #q  Dq f = 0 × " ∈ Cc   #n−q  Dn−q " = 0 defined by Iq  f " =



f ∧ dz ∧ "

The intersection number for the pair    in degree 0 is the C-bilinear form defined on the product  f ∈ C   D0 f = 0 ×" ∈ Cc   #n  F dz ∧ " = 0 

∀F ∈ C  Cm × Rn  D0 F = 0 defined by I0  f " =



f dz ∧ "

We have the following result: Proposition VIII.5.1. Let q ≥ 1. The intersection number Iq  vanishes identically if and only if for every f ∈ C   #q  satisfying Dq f = 0 its restriction to  belongs to the closure of the image of the map Dq−1 C    #q−1  −→ C    #q  Proof. Let f ∈ C   #q  satisfy Dq f = 0. If f  = lim Dq−1 u →

in C    #q 

(VIII.35)

VIII.5 The intersection number

341

for some sequence u  in C    #q−1 , and if " ∈ Cc   #n−q  satisfies Dn−q " = 0, we have   Iq  f " = lim Dq−1 u ∧ dz ∧ " = lim −1q+m u ∧ dz ∧ Dn−q " = 0 →

→

thanks to Lemma VIII.3.2. For the converse we reason by contradiction and apply the Hahn–Banach theorem. Thus we assume that there are f0 ∈ C   #q  satisfying Dq f0 = 0 and T ∈     #n−q  such that T f0 = 1

T Dq−1 u = 0

∀u ∈ C    #q−1 

In particular we have Dn−q T = 0.   Let now & ∈ Cc Cm × Rn  be such that & = 1 and set, for % > 0, z t  1  &% z t = 2m+n &  % % % If we introduce the regularizations 

"% = &%  T ∈ Cc Cm × Rn #n−q  then there is %0 > 0 such that "% ∈ Cc   #n−q  if 0 < % ≤ %0 . Moreover, "% → T in     #n−q  and Dn−q "% = &%  Dn−q T = 0 for every % > 0. Now  f0 ∧ dz ∧ "% = "% f0 −→ T f0 = 1 and consequently there is 0 < %1 ≤ %0 such that    Iq  f0  "%1 = f ∧ dz ∧ "%1 = 0 Next we turn to the case q = 0: Proposition VIII.5.2. The intersection number I0  vanishes identically if and only if the following holds: For every f ∈ C   satisfying D0 f = 0 there is

(VIII.36)



F ⊂ C  Cm × Rn  satisfying D0 F = 0 such that F  −→ f  in C    Proof. The proof that (VIII.36) implies the vanishing of I0  is immediate. We then prove the converse and for this we argue by contradiction as in the proof of Proposition VIII.5.1. Thus we assume that there is f0 ∈ C    satisfying D0 f = 0 for which no sequence F ⊂ C  Cm × Rn  as stated

342

Local solvability

exists and apply once more the Hahn–Banach theorem: there is T ∈     #n  such that T f0 = 1 T F = 0

(VIII.37)



∀F ∈ C  Cm × Rn  D0 F = 0

(VIII.38)



We next observe that the vanishing of HD1 Cm × Rn  (Theorem VIII.3.1) implies, in particular, that the homomorphism of Fréchet spaces 



D0 C  Cm × Rn  −→ C  Cm × Rn  #1  has closed image. Consequently its transpose, which is the map 



Dn   Cm × Rn  #n−1  −→   Cm × Rn  #n  has a weakly closed image, that is, its image is precisely the orthogonal of    the kernel of D0 C  Cm × Rn  → C  Cm × Rn  #1  in   Cm × Rn  #n . Returning to our argument we conclude from (VIII.38) that there exists  S ∈   Cm × Rn  #n−1  such that Dn−1 S = T . As in the proof of Proposition VIII.5.1 we introduce once more the regularizations 

"% = &%  T ∈ Cc Cm × Rn  #n  There is %0 > 0 such that "% ∈ Cc   #n  if 0 < % ≤ %0 . Furthermore, if  F ∈ C  Cm × Rn  satisfies D0 F = 0 then    F dz ∧ "% = F dz ∧ &%  Dn−1 S = F dz ∧ Dn−1 &%  S  = −1m−1 D0 F ∧ dz ∧ &%  S = 0 for 0 < % ≤ %0 and also I0  f0  "% =



%→0

f0 ∧ dz ∧ "% −→ 1

thanks to (VIII.37), which leads to the desired contradiction. Remark VIII.5.3. It follows from the argument in the proof of Proposition  VIII.5.2 that the space Cm × Rn can be replaced in (VIII.36) by any open set containing  and of the form U × ", where U and " are as in Theorem VIII.3.1. Corollary VIII.5.4. Assume that m = 0 and let  ⊂  be open subsets  of Rn . Then, if q ≥ 1, the vanishing of Iq  is equivalent to the vanishing of the natural map induced by restriction H q  C → H q   C. Also, the

VIII.6 The intersection number

343

vanishing of I0  is equivalent to the property that  is contained in a single connected component of . Proof. Thanks to de Rham’s theorem we can assert: (a) The exterior derivative defines a map with closed image when defined on an arbitary smooth manifold. (b) The d-cohomology is isomorphic to the standard singular cohomology with complex coefficients for any smooth manifold. These two properties in conjunction with Proposition VIII.5.1 prove the assertion for q ≥ 1. Furthermore, since a scalar function is d-closed if and only if it is locally constant, the assertion for q = 0 is an immediate consequence of Proposition VIII.5.2. We shall now draw an important corollary of Propositions VIII.5.1 and  VIII.5.2. Let  be a regular, bounded open subset of Cm × Rn . Since we are  dealing with an elliptic structure on Cm × Rn it follows that  is noncharacteristic and consequently we can apply the one-sided approximate Poincaré Lemma (Theorem VII.8.4) and obtain: 

Corollary VIII.5.5. Let p ∈ . Given any neighborhood W of p in Cm ×Rn there is another such neighborhood W  ⊂⊂ W such that IqW ∩W  ∩ = 0 for all q = 0     m + n .

VIII.6 The intersection number under certain geometrical assumptions In this section we shall give a special meaning to one of the complex variables. Thus we shall assume m ≥ 1 and write the complex variables as  z1      z  w, where now m =  + 1. If  is an open subset of Cm × Rn we shall denote by  the space of all u ∈ C   which satisfy u/w = 0.  If  is an open subset of Cm × Rn and if w0 ∈ C we shall write w0  = z w t ∈  w = w0  In the sequel, when dealing with functions defined on w0 , we shall identify   the latter with the open subset of C × Rn given by z t ∈ C × Rn z w0  t ∈ w0  . We start with an important result: 

Proposition VIII.6.1. Let  ⊂ Cm × Rn be open and /w-convex, that is: The homomorphism /w C   → C   is surjective.

(VIII.39)

344

Local solvability

Then given w0 ∈ C the restriction map  → C  w0  is surjective. Proof. There is a continuous function  w0  → 0  such that the open set  = z w t z w0  t ∈ w0  w − w0 < z t is contained in . Let f = fz t ∈ C  w0  and select f • ∈ C   such that f • z w t = fz t if z w t ∈ /2 . In particular, f • w0  = f and f • =0 w

in /2 

(VIII.40)

We must find g ∈ C   such that Fz w t = f • z w t + w − w0 gz w t belongs to . For this we must have g • f • + w − w0  = 0 w w which is possible to achieve, since by hypothesis and by (VIII.40) we can solve the equation f • g • = −w − w0 −1 w w in order to determine the desired g. Denote by C   #q , q = 0     n−1, the space of all forms in C   #q  which do not involve dw, that is, the space of all forms of the kind  fJK dzJ ∧ dtK  (VIII.41) f=

J + K =q

with fJK ∈ C  . It is important to observe that if f is as in (VIII.41) and satisfies Dq f = 0 then a fortiori we have fJK ∈  for every J and K. Notice also that the pullback of an element in C   #q  to any slice w0  is simply obtained by setting w = w0 in its coefficients. 

Proposition VIII.6.2. Let  ⊂  be open subsets of Cm ×Rn , both satisfying (VIII.39). If for some q ≥ 1 the homomorphism HDq  → HDq   is trivial then for every w0 ∈ C and every f ∈ C  w0  #q−1  satisfying Dq−1 f = 0 there is F ∈ C    #q−1  satisfying Dq−1 F = 0 and Fz w0  t = fz t on  w0 .

VIII.6 The intersection number

345

Proof. Let f = fz t ∈ C  w0  #q−1  satisfy Dq−1 f = 0. We apply Proposition VIII.6.1 in order to get f • z w t ∈ C   #q−1 , with coefficients in , such that f • z w0  t = fz t. We have Dq−1 f • z w0  t = Dq−1 fz t = 0 and consequently we can write Dq−1 f • = w −w0 G for some G ∈ C   #q , also with coefficients in . It is clear that Dq G = 0 and thus by hypothesis there is u ∈ C    #q−1  satisfying Dq−1 u = G in  . Write u = u0 + u1 ∧ dw with uj ∈ C    #q−j−1 , j = 0 1. We now use the fact that  also satisfies (VIII.39) in order to solve v = −1q u1  w with v ∈ C    #q−2  (we set v = u1 = 0 if q = 1). A simple computation shows that u − Dq−2 v ∈ C    #q  and consequently if we set    F = f • − w − w0  u − Dq−2 v we obtain F ∈ C    #q , Fz w0  t = fz t, and Dq−1 F = Dq−1 f • − w − w0 Dq−1 u = w − w0 G − Dq−1 u = 0 We can now prove: 

Theorem VIII.6.3. Let  ⊂  ⊂  be open subsets of Cm × Rn , all of them satisfying (VIII.39). Let q ≥ 1 and assume that Iq−1    = 0 and that q q q−1  HD  → HD   is the trivial map. Then Iw0  w0  = 0 for every w0 ∈ C. Proof. First we observe that, after taking regularizations, the vanishing of Iq−1    = 0 allows one to assert that  F ∧ dz ∧ dw ∧ T = 0 (VIII.42) for every F ∈ C    #q−1  satisfying Dq−1 F = 0 and every T∈   #n−q+1  satisfying Dn−q+1 T = 0 (when q = 1 this condition must be replaced by  T G = 0 for every G ∈ C  Cm × Rn  satisfying D0 G = 0). Fix w0 ∈ C and let f ∈ C  w0  #q−1 , " ∈ Cc  w0  #n−q  be both  D-closed (we assume " ∈ g ∈ C  C ×Rn  D0 g = 0 ⊥ when q = 1). Thanks to our hypotheses we can apply Proposition VIII.6.2 in order to obtain F ∈ C    #q−1  satisfying Dq−1 F = 0 and F w=w0 = f .

346

Local solvability

On the other hand, if we write  "=

"JK z t dzJ ∧ dtK

J + K =n−q

and define T" ∈     #n−q+1  by the formula   "JK z t ⊗ w − w0  dzJ ∧ dw ∧ dtK T" =

J + K =n−q 

we have Dn−q+1 T" = 0 and also T" ∈ G ∈ C  Cm × Rn  D0 G = 0 ⊥ when q = 1. Then (VIII.42) gives     Iq−1 f " = F

∧ dz ∧ " = ± F ∧ dz ∧ dw ∧ T" = 0  w=w0 w0  w0  which concludes the proof.

VIII.7 A necessary condition for one-sided solvability We keep the notation established in Section VIII.6 and consider now a regular  open subset  of Cm × Rn . We fix a defining function & for : thus & is a smooth, real-valued function such that  is defined by the equation & = 0, with d& = 0 on . Theorem VIII.7.1. Let p ∈  be such that & p = 0  w

(VIII.43)

Then if for some q ≥ 1 D is solvable near p in degree q with respect to  it follows that the following property holds: given any open neighborhood U  of p in Cm × Rn there is another such neighborhood V ⊂ U such that, for every w0 ∈ C, the intersection number Iq−1 w0 ∩Uw0 ∩V  vanishes identically. This result is a direct consequence of Theorem VIII.6.3 in conjunction with Corollary VIII.5.5 and the following proposition: Proposition VIII.7.2. Suppose that (VIII.43) is satisfied. Then there is an  open neighborhood W of p in Cm × Rn such that given any open convex set D ⊂ W the set D ∩  is /w-convex. Proof. It suffices to prove the analogous statement for the operator $w = 4

2  ww

VIII.7 A necessary condition for one-sided solvability

347

since every open set which is $w -convex is a fortiori /w-convex. We write w = s + ir, p = z0  s0 + ir0  t0 , and assume, say, that &/r = 0 at p. By the implicit function theorem, there are an open neighborhood W of  p and a smooth function * C × R × Rn → R such that *z0  s0  t0  = r0 and W ∩  = z w t ∈ W r < *z s t  

Now the set  = z w t ∈ Cm × Rn r < *z s t is $w -convex since $w is real and any normal to  is not a characteristic vector for $w ([H1],  theorem 3.7.4). Consequently, given any open convex set D ⊂ Cm × Rn , the set D ∩ , being the intersection of $w -convex open sets, is also $w -convex. If we finally observe that if D ⊂ W then D ∩  = D ∩ , the result follows at once. Remark VIII.7.3. As in Section VII.12, we introduce the spaces of germs: C p #q  = lim C  U ∩  #q  U →p

C p #q  = lim C  U ∩  #q  U →p

It can be proved, via methods of hyperfunction theory, that if solvability for D near p in degree q with respect to  does not occur then there is f ∈ C p #q  for which no u ∈ C p #q−1  satisfies Du = f . In particular, Corollary VIII.5.5 also gives a necessary condition for solvability for D near p in degree q with respect to . In the particular case when m = 1, Corollary VIII.5.4 allows us to state the necessary condition in terms of the de Rham cohomology. We give first a definition. Definition VIII.7.4. Assume that m = 1 and let p ∈ . We shall say condition q (q ≥ 1) holds at p with respect to  if given any open neighborhood U  of p in C × Rn there is another such neighborhood V ⊂ U such that, for all w0 ∈ C, the natural homomorphism H q w0  ∩ U C → H q w0  ∩ V C is trivial. We further say that condition 0 holds at p with respect to   if given any open neighborhood U of p in C × Rn there is another such neighborhood V ⊂ U such that, for all w0 ∈ C, w0  ∩ V is contained in one of the connected components of w0  ∩ U . Corollary VIII.7.5. Suppose that m = 1 and that (VIII.43) is satisfied. Then if for some q ≥ 1, D is solvable near p ∈  in degree q with respect to  it follows that condition q−1 holds at p with respect to .

348

Local solvability

VIII.8 The sufficiency of condition 0 We shall now show the sufficiency, in a weak form, of condition 0 for solvability near p ∈  in degree 0 with respect to  under the stronger assumption that The boundary of  is real-analytic.

(VIII.44)

In other words, we shall assume that  is defined by & > 0, where & is realvalued, real-analytic and such that  is defined by & = 0, with &/z = 0 near p. The next result is the key tool for the proof of the result. In all the arguments  that follow we shall denote by ) C × Rn → C the projection )z t = z. We assume that the central point in the analysis is p = z0  t0  ∈  in  C × Rn . By applying the implicit function theorem we can assume that &z t = y − !x t

z = x + iy

(VIII.45)

where ! is real-analytic and !x0  t0  = y0 . We shall also denote by Vp the set of all open sets D of the form R × ", where R (resp. ") is an open square in C with sides parallel to the coordinate   axes (resp. open ball in Rn ) centered at z0 ∈ C (resp. t0 ∈ Rn ). Proposition VIII.8.1. Assume that both (VIII.44) and condition 0 hold. Then given any D ∈ Vp there is D• ⊂⊂ D also belonging to Vp and a constant M > 0 such that, for any z ∈ C, any two points in z ∩ D• can be joined by a piecewise real-analytic curve contained in z ∩ D and with length ≤ M. Proof. Given D as in the statement we take D1 ⊂⊂ D also in Vp and apply condition 0 : there is D• ⊂ D1 , also in Vp such that, for any z ∈ C, z ∩ D• is contained in a single component of z ∩ D1 .  Next we observe that the set K = D1 ∩  is compact and sub-analytic. We then apply a standard result on the theory of subanalytic sets which can be found in ([Har], section 8): there is M > 0 such that any two points in a component of ) −1 z ∩ K may be joined by a piecewise analytic arc in ) −1 z ∩ K of length ≤ M. Hence if t s belong to z ∩ D• they belong to a component of z ∩ D1 and consequently to a component of ) −1 z ∩K. Since ) −1 z ∩K ⊂ z∩D the result follows. The key point in the argument is the following result:

VIII.8 The sufficiency of condition 0

349

Proposition VIII.8.2. Under the same hypotheses as in Proposition VIII.8.1, given D ∈ Vp there are D ∈ Vp, D ⊂⊂ D and a constant C > 0 such that the following is true: given u ∈   ∩D ∩  ∩ D there is v ∈ ∩ D  such that dt v = dt u and

vz ty − !x t ≤ Cdt uL ∩D 

sup

(VIII.46)

zt∈∩D

Before we embark on the (rather long) proof of this result, we will show how it can be used to derive our one-sided solvability result. Corollary VIII.8.3. Assume (VIII.44) and that condition 0 holds. Then given any D0 ∈ Vp there is D ⊂⊂ D0 also belonging to Vp such that for every f ∈ C   ∩ D0  #1  satisfying D1 f = 0 there is v ∈ C   ∩ D  satisfying D0 v = f in  ∩ D and

vz ty − !x t < 

sup zt∈∩D

Notice that, in particular, (VIII.44) and condition 0 imply solvability for D near p in degree 1 with respect to  (cf. Remark VIII.7.3). Proof. Write 

f = f0 dz +

n 

fj dtj 

j=1

If we extend f0 to a smooth function on D0 and then solve v/z = f0 in D0 we obtain a new form f −D0 v ∈ C  ∩D0  #1  which has no dz-component. In other words, we can start with f ∈ C   ∩ D0  #1  of the form f=

n 

fj dtj 

j=1

Notice that D1 f = 0 means that dt f = 0 and that each coefficient fj is holomorphic in z, that is, fj ∈  ∩ D0 . We apply the Approximate Poincaré Lemma: there is D ∈ Vp, D ⊂ D0 (which is independent of f ) and a sequence u ∈ C  ∩D such that D0 u → f in C   ∩ D #1 . Notice that this means dt u → f

in

C   ∩ D #1  

u →0 z

in

Consider now a linear, continuous extension operator E C   ∩ D −→ C  D

C   ∩ D

350

Local solvability

and if D = R × " let

  u 1 1  E dx dy  z  t A z t = z z − z ) R

It is easily seen that A → 0 as  →  in C  D and, clearly, A u =  z z

in

 ∩ D

If we substitute u −A for u we then obtain a new sequence u ∈ C  ∩D such that dt u → f in C   ∩ D #1 , u is holomorphic in z.

(VIII.47)

Finally we take D• ⊂⊂ D as in Proposition VIII.8.2 and apply its conclusion to u = u : we can find v ∈  ∩ D  such that dt v = dt u and, for some constant C > 0, sup

v z ty − !x t ≤ C

∀

zt∈∩D

But then some subsequence vk converges weakly to a function v which satisfies the required properties. This concludes the proof of Corollary VIII.8.3.

VIII.9 Proof of Proposition VIII.8.2 We take D• = R• ×"• ⊂⊂ D as in Proposition VIII.8.1 and start by constructing   a suitable covering of  ∩ D• . Set  = ) ∩ D•  and for each a ∈ Rn we set  Wa = z ∈ C z a ∈  ∩ D•  Notice that Wa is an open covering of . We also set  Ua = ) −1 Wa  ∩  ∩ D• 

(VIII.48)

Then Ua is an open covering of  ∩ D• and z t ∈ Ua implies z a ∈  ∩ D• . Using the curves given in Proposition VIII.8.1 and the corresponding bound for their lengths we obtain

uz t − uz a ≤ Mdt uL ∩D

z t ∈ Ua 

(VIII.49)

The family u· a defines a holomorphic one-cochain with respect to the open covering Wa of  which satisfies

uz a − uz b ≤ Mdt uL ∩D

z ∈ Wa ∩ Wb 

(VIII.50)

VIII.9 Proof of Proposition VIII.8.2

351

We shall now construct a new one-cochain wa ∈ Wa  such that wa − wb = u· a − u· b on Wa ∩ Wb and for which each term wa can be estimated, on Wa , in terms of the right-hand side of (VIII.50). Such a one-chain will be constructed through the following standard argument: start with a partition of unity *j , 0 ≤ *j ≤ 1, subordinate to the covering Wa , that is for each j there corresponds aj such that *j ∈ Cc Waj  and set  ga z = *k z uz a − uz ak   k 

Then ga ∈ C Wa  and ga − gb = u· a − u· b in Wa ∩ Wb . Notice that this last equality implies ga /z = gb /z in Wa ∩ Wb and consequently there is G ∈ C  , G = ga /z in Wa for every a. Finally we solve F =G z

(VIII.51)

in , with F ∈ C  , and set wa = ga − F . Observe that such a solution F always exists (every open subset of C is a domain of holomorphy!) but in order to obtain (VIII.46) we will be forced to make a further contraction in the domain. We have 

ga z ≤ *k z uz a − uz ak  ≤ Mdt uL ∩D z ∈ Wa k

for every a and thus the proof will be completed if we can show that, for some suitable choice of the partition of unity *j , we can obtain a solution F to (VIII.51) on R ∩ , with R ⊂ R• another square centered at z0 , satisfying

Fzy − !z t ≤ M1 dt uL ∩D  z t ∈  ∩ D   where D = R∗ × "• ∈ Vp. Indeed v ∈  ∩ D , defined on Ua ∩ D as

(VIII.52)

u − u· a − wa = u − u· a − ga + F satisfies dt v = dt u and (VIII.46). In order to achieve (VIII.52) we start by observing that & & & & & ga &  & *k & & & & & & z z& ≤ Mdt uL ∩D & z z&  z ∈ Wa  k and take a closer look on the coverings Ua and Wa . We have  = z ∈ R• ∃t ∈ "•  &z t > 0 

(VIII.53)

352

Local solvability Wa = z ∈ R• &z a > 0 

We set z = sup &z t = max &z t  t∈"•

t∈"•

In particular,  is Lipschitz continuous. Also  = z ∈ R• z > 0  Set z = minz dist z  and observe that  is also Lipschitz continuous. We then recall Lemma IV.3.11: Lemma VIII.9.1. Let % > 0 be arbitrary. There is an open covering of  by squares Qj , with sides parallel to the coordinate axes, having the following properties: diam Qj ≤ % inf z

(VIII.54)

Qj

There are *j ∈ Cc Qj , 0 ≤ *j ≤ 1, such that & &  & *j & −1 & & & z z& ≤ Cz  j



*j = 1 and

(VIII.55)

Next we claim that if we take % < 1/2K, where K is a Lipschitz constant for &, then for each j there is aj such that Qj ⊂ Waj . Indeed let z ∈ Qj and take t ∈ "• such that z  = &z  t . If z ∈ Qj we have z − z ≤ %z  and consequently &z t  = &z t  − &z  t  + &z  t  ≥ z  − K z − z ≥ z  − %Kz  1 ≥ z  > 0 2 whence our assertion. For this choice of partition of unity (VIII.53) gives

Gz ≤ 2MCdt uL ∩D z−1 

z ∈ 

(VIII.56)

Since &z0  t0  = 0 we have z0  ≥ 0. The case z0  > 0 is almost elementary, for we can take R ⊂  in such a way that z ≥ c > 0 in R and consequently the solution to (VIII.51), given by F defined via the formula 1   Gz    dx dy Fz = )  z − z

VIII.9 Proof of Proposition VIII.8.2

353

satisfies

Fz ≤ M1 dt uL ∩D 

z ∈ R 

Let us then assume that z0  = 0. We now take R as stated and such that z ∈ R

&⇒

z = z

(VIII.57)

Notice that, thanks to (VIII.45), we have z = y − .x

.x = inf !x t t∈"•

and then  ∩ R = z ∈ R∗ y > .x  We now apply the standard identity 1 1 1 = + z − z z −  z − z



z −  z−



in order to obtain  Gz      Gz  dx dy dx dy =    ∩R z − z ∩R z − x − i.x   Gz y − .x  −i dx dy     2 ∩R z − z z − x − i.x  Since the first term on the right-hand side is holomorphic in  ∩ R we can solve (VIII.51) by taking Fz =

Gz y − .x  1  dx dy  )i ∩R z − z z − x − i.x 

z ∈  ∩ R  (VIII.58)

It remains to verify (VIII.52). From (VIII.56) and (VIII.57) we obtain

×

y − .x Fz ≤M2 dt uL ∩D

 ∩R

y − .x 1 · dx dy 

z − z x − x + y − .x 

z ∈  ∩ R 

To conclude we just observe that, since . is Lipschitz, y − .x

x − x + y − .x  and hence (VIII.59) implies

≤

y − .x  + M3 x − x ≤ M3 + 1

x − x + y − .x 

(VIII.59)

354

Local solvability y − .x Fz ≤ M4 dt uL ∩D 

We have thus proved (VIII.52) since z t ∈  ∩ D &⇒ z ∈ R  t ∈ "•  y > !x t &⇒ y − !x t ≤ y − .x The proof of Proposition VIII.8.2 is now complete.

VIII.10 Solvability for corank one analytic structures Since the solution v obtained in Corollary VIII.8.3 is holomorphic with respect to z and has tempered growth when z t →  ∩ D the results in Chapter VI show that its boundary value is a well-defined distribution on  ∩ D of order ≤ 2. If in addition we also assume the validity of condition 0 at p  with respect to C × Rn \, and if we denote by • the bundle spanned by the  vector fields /z, /tj , j = 1     n and by L = D the complex induced by the elliptic complex D on , an almost immediate extension of (a) in Theorem VII.12.1 gives: Given an open neighborhood U of p in  there is another (VIII.60) • such neighborhood V ⊂ U such that, given f ∈ U U  V solving Lu = f in V . satisfying Lf = 0 there is u ∈ 2 Consider the complex vector fields  −1 &   & L•j = −  tj z tj z

j = 1     n 

Near p the vector fields L•j are tangent to  and their restriction to  span • . As before we describe  by the equation y − !x t = 0, with ! real-analytic and take x y t as local coordinates in . In these local  coordinates the vector fields Lj = L•j  are written as Lj =

i!tj    − tj 1 + i!x x

j = 1     n 

(VIII.61)

Hence •  is exactly the locally integrable structure defined on a neigh borhood of the point p in Rn +1 which is orthogonal to the sub-bundle of  CT ∗ Rn +1  spanned by dZ, where Zx t = x + i!x t. The reverse argument is also true, that is, any smooth locally integrable  structure  of corank one, say in a neighborhood of the origin in Rn +1 , arises

VIII.10 Solvability for corank one analytic structures

355



from the restriction of the elliptic structure • on C × Rn to a hypersurface 0  in C × Rn . Indeed if we choose local coordinates x t = x t1      tn  in a  neighborhood of the origin in Rn +1 in such a way that the orthogonal of  is generated by the differential of Zx t = x +i!x t, with ! smooth and realvalued, and if we denote by 0 the image of the imbedding x t → Zx t t, it follows easily that  = • 0. Keeping this notation, and recalling Corollary I.10.2, we can (and will) even assume that !0 0 = !x 0 0 = 0. We emphasize that  is spanned by the pairwise commuting vector fields (VIII.61). We further take a small  open neighborhood V of the origin in C × Rn and set V+ = z t ∈ V z = x + iy y > !x t  V− = z t ∈ V z = x + iy y < !x t  Definition VIII.10.1. We shall say that condition (P)0 holds at the origin for the locally integrable structure  if condition 0 holds at the origin in  C × Rn with respect to both V+ and V− . We shall then prove: Theorem VIII.10.2. Let  be a corank one, real-analytic, locally integrable  structure defined in an open neighborhood of the origin in Rn +1 and let d be the associated differential complex. Then d is solvable near the origin in degree one if and only if condition (P)0 holds at the origin. Proof. The necessity of condition (P)0 follows from Theorem VII.12.1, Corollary VIII.8.3 and Remark VIII.7.3. We now embark on the proof of the sufficiency. Let us denote by W0 the  family of all open neighborhoods of the origin in Rn +1 of the form U = I × ", where I (resp. ") is an open interval (resp. ball) centered at the origin in  U R (resp. Rn ). If p q ∈ R2 and if U ∈ W0 we shall denote by L2rs loc the local Sobolev space of order r with respect to x and of order s with respect to t. We recall that if we set M = Zx−1 /x then the vector fields L1      Ln  M, (cf. (VIII.61)), are pairwise commuting, linearly independent (see (I.38)). We now make use of (VIII.60). Then there is r0  s0  ∈ R2 such that the following is true: given U ∈ W0 there is U  = I  × " ∈ W0, U  ⊂⊂ U , such that given 

fx t =

n  j=1

fj x t dtj ∈ U U Lf = 0

(VIII.62)

356

Local solvability

there is v ∈ Lloc 0 0 U   satisfying Lv = f in U  . We then fix f as in (VIII.62). Noticing that, for each k ∈ N, M 2k f is also L-closed (here M 2k acts componentwise on the one-form f ), we can find 2r s vk ∈ Lloc 0 0 U   solving Lvk = M 2k f in U  . Next we solve, in U  , M 2k wk = vk . Thus 2r s

M 2k Lwk − f = 0 and consequently we can write Lwk − f =

2k−1 

gjk t Zx tj 

j=0

where gjk are dt -closed one-forms with distributional coefficients. We can find distributions Gjk ∈  "  such that dt Gjk = gjk and hence we have C B 2k−1  j (VIII.63) Gjk t Zx t = f L wk + j=0 2r +2ks0

Since wk ∈ Lloc 0

U   it follows that 2r +2k−1s0 −1

Lwk − f ∈ Lloc 0

U  

2s −1

and hence gjk ∈ Lloc 0 " . Consequently Gjk ∈ Lloc 0 "  and then, if we set 2s

2k−1   Gjk t Zx tj uk = wk + j=0

we have Luk = f

2r +2ks0

uk ∈ Lloc 0

U  

(VIII.64)

Explicitly (VIII.64) means i!tj uk uk = + fj  tj 1 + i!x x

j = 1     n 

This expression implies that it is possible to trade differentiability with respect to x for differentiability with respect to tj , j = 1     n , that is, we also have 2r +ks +k uk ∈ Lloc 0 0 U  . Let U• ∈ W0, U• ⊂⊂ U  . By the Sobolev imbedding theorem it then follows that for each  ∈ N we can find a solution u• ∈ C  U•  to the equation Lu• = f in U• . We finally apply, for each  ∈ N, the C  -version of the Baouendi–Treves approximation formula (cf. Theorem II.1.1). There are U1 ∈ W0, U1 ⊂⊂ U• ,

Notes

357

depending only on U• and on  , and a sequence of holomorphic polynomials p ⊂ C z such that u•+1 − u• − p ZC  U1  ≤

1  2

(VIII.65)

If we set u1 = u•1 

u = u• − p1 Z −    − p−1 Z  ≥ 2

then (VIII.65) gives u+1 − u C  U1  ≤

1  2

This shows that, for each p ∈ N, the sequence u ≥p converges to an element u ∈ C p U1 , of course independent of p, and belonging to   U1 . Since moreover Lu = f in U1 for every  we also have Lu = f in U1 . The proof of Theorem VIII.10.2 is complete.

Notes Until now, complete answers for local solvability in locally integrable structures, besides the cases n = 1 (a situation which has already been discussed in Chapter IV),  defines an essentially real structure (Section VIII.1) and when  defines an elliptic structure (Theorem VIII.3.1) are known in the following cases: (i)  defines a nondegenerate locally integrable CR structure of codimension one ([AH2]); (ii)  defines a nondegenerate real-analytic structure ([T9]); (iii) m = 1 ([CorH3]). We also mention a necessary condition for nondegenerate CR structures of arbitrary codimension proved in [AFN], which was extended to general locally integrable structures with additional nondegeneracy conditions in [T5]. The notion of intersection number and the necessary condition given in Theorem VIII.11.4 is due to [CorT1]. As far as sufficiency is concerned, we point out the works by Kashiwara– Schapira ([KaS]) and Michel ([Mi]), which deal with locally integrable CR structures of codimension one and whose Levi form satisfies weaker nondegeneracy conditions. Locally integrable structures with m = 1: for this class of locally integrable structures we have seen in Sections VIII.7 and VIII.8 that condition (P)0 is necessary and (in the real-analytic category) sufficient for local solvability near the origin (cf. Corollary VIII.7.5 and Theorem VIII.10.2). This result

358

Local solvability

can be generalized much more. Let us introduce, for each q = 0 1     n − 1, the following property: (P)q Given any open neighborhood V of the origin there is another such neighborhood V  ⊂ V such that, for every regular value z0 ∈ C of the map Z, either Z−1 z0 ∩ V  = ∅ or else the homomorphism ˜ q Z−1 z0 ∩ V   −→ H ˜ q Z−1 z0 ∩ V H induced by the inclusion map Z−1 z0 ∩ V  ⊂ Z−1 z0 ∩ V ˜ ∗ denotes the reduced homology with complex vanishes identically. Here H coefficients. In 1981 F. Treves proposed the following conjecture: local solvability near the origin holds for  if and only if property Pq−1 is verified. Several articles were published towards its verification; see [MenT], [CorH1], [CorH2], [CorT3], [ChT]. The complete proof of the conjecture was finally achieved in [CorH3]. The main point in the proof of Theorem VIII.10.2 that we presented is the use of the special covering (VIII.48), an idea inspired by the work [H10]. Solvability in top degree: one of the main questions in the theory is how to generalize condition (P)q in order to state a plausible conjecture for local solvability for general locally integrable systems. Observe that when m ≥ 2 the sets Z−1 z0 no longer carry enough information: for instance, in the CR case they are reduced to points. There is one particular situation where a conjecture can be stated and at least verified in some particular but important cases: this is when q = n (local solvability in maximum degree). Returning to the notation established at the beginning of this chapter, in particular to the vector fields (VIII.4), the equation under study is now n 

Lj uj = f

(VIII.66)

j=1

where no compatibility condition occurs. This makes this case, in some sense, the closest to the single equation situation. Before we introduce the solvability condition for (VIII.66) we introduce the following definition: a real-valued function F defined on a topological space X is said to assume a local minimum over a compact set K ⊂ X if there exist a ∈ R and K ⊂ V ⊂ X open such that F = a on K and F > a on V \K.

Notes

359

Definition VIII.10.3. We shall say that  satisfies condition Pn−1 near the origin if there is an open neighborhood U ⊂  of the origin such that given any open set V ⊂ U and given any h ∈ C  V satisfying Lj h = 0, j = 1     n, then h does not assume a local minimum over any nonempty compact subset of V . By using a classical device due to Lars Hörmander [H6], it was proved in [CorH1] that local solvability near the origin for (VIII.66) implies condition Pn−1 . This result would be of limited importance if no evidence that Pn−1  is also a sufficient condition could be presented. This however is not the case, as the discussion that follows will show, and we can even conjecture at this point that local solvability of (VIII.66) near the origin is equivalent to Pn−1 . When n = 1 condition P0  is equivalent to the Nirenberg–Treves condition : this result was proved in [CorH1] in the analytic category and in the general case in [T3]. When m = 1 condition Pn−1  is equivalent to condition (P)n−1 ([CorH1], [T3]). Thus, in these extreme cases, (Pn−1 ) unifies both known solvability conditions. Let us pause here to discuss again the case when m = 1. The proof of the Treves conjecture in top degree as presented in [CorH2] is obtained by proving that (P)n−1 implies, when n ≥ 2, the following property: there are an open neighborhood U of the origin and constants C > 0 and k ∈ Z+ such that n   D Lj    ∈ Cc U (VIII.67)  ≤ C j=1  ≤k

Indeed k can be taken any integer ≥ n/2 + 1 and equal to zero when the structure is real-analytic ([CorH1]). The completion of the argument is quite standard, and holds whatever the value of m: by applying the Hahn–Banach theorem it is easily seen that (VIII.67) implies the existence of weak solutions to (VIII.66), and a general result due to [T5] proves the existence of smooth solutions. For the tube structures it is not difficult to prove that property Pn−1  implies (VIII.67) and consequently the preceding discussion shows that our conjecture is also satisfied for this particular class. When  defines a CR structure of codimension one then condition Pn−1  is equivalent to the existence of an open neighborhood U of the origin such that at every characteristic point over U the Levi form is not definite. In this case a partial answer was given in [Mi], where the existence of hyperfunction solutions is proved.

360

Local solvability

Finally we mention another general class of locally integrable structures that satisfy condition Pn−1 : these are the hypocomplex structures (cf. Definition VIII.5.4). For hypocomplex structures it is not still known if Pn−1  implies the local solvability of (VIII.66). Neverthless, again in this case we can find hyperfunction solutions, as a consequence of more general results proved in [CorTr].

Epilogue

In this epilogue we describe briefly some results that are closely connected with the theory and tools developed in previous chapters and have been obtained in recent years but, in spite of their importance, could not be fully treated without increasing too much the size of this book.

1 The similarity principle and applications In this section we will briefly discuss the first-order equation Lu = Au + Bu

(1)

where L is a complex vector field in the plane and A and B are bounded, measurable functions. We will also present two applications of equation (1). The first application concerns uniqueness in the Cauchy problem for a class of semilinear equations. The second application will be to the theory of bending of surfaces. Equation (1) generalizes the classical elliptic equation u = Au + Bu z

(2)

which was investigated by numerous researchers (see for example [Be], [CoHi], [Re], and [V]). In the literature, solutions of (2) are called pseudo-analytic functions or generalized analytic functions. Such functions share many properties with holomorphic functions of one variable. These properties follow easily from the similarity principle according to which every continuous solution of (2) has the local form u = expg h

(3)

where h is a holomorphic function and g is Hölder continuous. Thus, for example, the zero set of u is the same as that of h. The similarity principle holds for any elliptic vector field L (where the holomorphy of h is replaced by the condition Lh = 0) since any such vector field is a multiple of z in appropriate coordinates. In [Me2] Meziani explored the validity of the similarity principle for the following three classes of vector

361

362

Epilogue

fields: Lk =

  − iy2k  y x

Kn =

  − ixn  x y

M=

  − iy y x

where k and n are non-negative integers. It was proved in [Me2] that the similarity principle is valid for the Lk and Kn (under some vanishing assumption on Bx y on the characteristic sets of the vector fields) in the following sense: if w is a continuous solution of Lw = Aw + Bw where L ∈ Lk  Kn , then w has the local form w = expg h where Lh = 0 and g is Hölder continuous. It was also shown in [Me2] that this principle does not hold for M. The vector fields Lk and Kn are locally solvable while M is not. With this observation as a point of departure, it was shown in [BHS] and [HdaS] that a weaker version of the similarity principle is valid for all locally solvable vector fields L. In this weaker version, the functions g and h in the representation w = expg h may no longer be continuous. However, this representation was still good enough to yield the uniqueness result mentioned below.

1.1 Application to uniqueness in the Cauchy problem Let the vector field L=

n    +i bk x t t x k k=1

satisfy condition  in some neighborhood  = 1 × −T T of the origin in Rn+1 . Here each bk is real-valued, of class C 1+r  0 < r < 1. Let fx t   be a bounded measurable complex-valued function defined for x t ∈ ,  ∈ C satisfying the Lipschitz condition in 

fx t   − fx t       ≤ K  −    If L and f are as above, the following result on uniqueness in the Cauchy problem was proved in [HdaS] (see also [BHS]): Theorem 1.1. Suppose ux t wx t ∈ Lp , p ≥ 2, satisfy Lu = fx u u, Lw = fx w w, and ux 0 = wx 0. Then u ≡ w in a neighborhood of the origin. If the coefficients of L are smooth, Theorem 1.1 was proved in [BHS] under the weaker assumption that u and w belong to Lp  p > 1. These results were proved by applying the similarity principle to the difference v = u − w which in view of the assumptions satisfies an equation of the form Lv = Av + Bv with A and B bounded. The fact that L satisfies condition  is then used to reduce matters to a planar situation.

1 The similarity principle and applications

363

1.2 Application to infinitesimal bendings of surfaces In a series of papers (see [Me3], [Me4], and the references therein) Meziani has demonstrated an intimate link between the study of the equation Lu = Au + Bu (L a planar vector field) and the study of infinitesimal deformations of surfaces with non-negative curvature. Here we will summarize some of the results in [Me4] to indicate this link. Let S be a surface of class C l , l > 2, embedded in R3 and given by parametric equations as S = Rs t = xs t ys t zs t ∈ R3 

s t ∈ D ⊂ R2

(4)

with D an open subset of R . An infinitesimal bending of S is a deformation 2

S% = R% s t = Rs t + %Us t s t ∈ D 

− < % < 

(5)

for some  > 0 and Us t = s t s t s t

(6)

satisfying dRs t · dUs t = 0

∀s t ∈ D

(7)

This means that the first fundamental forms of S and S% satisfy dR2% = dR2 + O%2  Note that equation (7) is equivalent to the system of three equations Rs · Us = 0

Rt · Ut = 0

Rs · Ut + Rt · Us = 0

(8)

Recall that the coefficients of the first fundamental form of S are E = Rs · Rs 

F = Rs · Rt 

G = Rt · Rt

(9)

g = Rtt · N

(10)

and those of the second fundamental form are e = Rss · N

f = Rst · N

where N=

Rs × Rt

Rs × Rt

is the unit normal to S. The Gaussian curvature of S is K=

eg − f 2  EG − F 2

We will assume that the curvature K ≥ 0 everywhere on S. The (complex) asymptotic directions of S are given by the quadratic equation 2 + 2f + eg = 0

364 That is,

Epilogue +  = −f + i eg − f 2 

Let L be a vector field of asymptotic direction:      L = as t gs t + s t s t

(11)

where a is any function defined in D. Note that since K ≥ 0, if a = 0, then L is an elliptic vector field that degenerates along the set where the curvature K = 0. Let w be the C-valued function defined by w = LR · U

(12)

where U is as given in (6). In [Me4], the following theorem was proved. Theorem 1.2. With w as in (12) and L as in (11), if U(s,t) is a field of infinitesimal bending for the surface S, then the function w satisfies the equation CLw = Aw + Bw where A, B, and C are invariants of the surface S.

1.3 Application to uniqueness in the Cauchy problem in elliptic structures Let    define an elliptic structure. If u ∈ L1loc  we shall say that u is an approximate solution for the structure  if for any smooth section L of  , Lu has coefficients belonging to L1loc  and given any point p ∈ , there is an open neighborhood U of p and a constant M > 0 such that

Lu ≤ M u a.e. in U . In [Cor2] the author established a similarity principle for approximate solutions p in the following sense: every approximate solution which belongs to Lloc  with p > N = dim  can locally be written as u = expS h, where S is Hölder continuous and h is a solution. This similarity principle was then used to show that every approximate solution that vanishes on a maximally real submanifold  vanishes identically in a neighborhood of  .

2 Mizohata structures The vector field in R2 , where the coordinates are denoted x t, given by M=

  − it t x

(13)

2 Mizohata structures

365

is called the (standard) Mizohata vector field (or operator) after the work of S. Mizohata ([M]) who studied the analytic hypoellipticity of a class of related operators of which M is the simplest example. A globally defined first integral of M is the function Zx t = x + it2 /2. Notice that t → t 2 fails to be monotone in any neighborhood of a point x0  0, i.e., condition  in not satisfied at any point of the x-axis and, as discussed in Chapter IV, fails to be locally solvable at those points. Thus, it is the simplest example of a nonlocally solvable operator and, in fact, its lack of local solvability at points of the x-axis can be proved by ad hoc elementary arguments, as shown by L. Nirenberg ([N1]). Off the x-axis, M is elliptic. In his Lectures Notes, Nirenberg constructed a perturbation of the Mizohata operator L=

  − it1 + &x t t x

(14)

with &x t real-valued and vanishing to infinite order at t = 0, which is not locally integrable in any neighborhood of the origin. As a matter of fact, any smooth function u that satisfies the homogeneous equation Lu = 0 in a connected open set U that contains the origin must be constant. In spite of the fact that the perturbed vector fields L and M behave differently with respect to local integrability, they have important geometric features in common. We have (1) M and its conjugate M are linearly dependent precisely on the x-axis; (2) M and M M are linearly independent whenever M and M are linearly dependent. These properties are shared by L in a neighborhood of the origin. Definition 2.1. A vector field L defined on a connected 2-manifold  is called a Mizohata vector field if for a nonempty subset 0 ⊂  the following holds: (1) L and L are linearly dependent precisely on 0; (2) L and L L are linearly independent on 0. We also say that a Mizohata vector field L is of standard type at p ∈ 0 if there exist local coordinates x t in a neighborhood of p in terms of which 0 is given by t = 0 and  has the form (13). A Mizohata structure  on  is a structure which is locally generated in the neighborhood of every point by a Mizohata vector field. Notice that (1) means that 0 is the image of the characteristic set p  ∈ T ∗  p  = 0 ,  being the symbol of L, under the canonical projection 1 T ∗  −→ . With this terminology, the vector field (13) is a Mizohata vector field of standard type and (14) is also a Mizohata vector field but not of standard type. Indeed, (14) cannot be of standard type because it is not locally integrable. Notice that a Mizohata vector field is elliptic on \0, which is a relatively small set, since an application of the implicit function theorem shows that 0 is an embedded curve. The following question was considered by Treves [T7]: when is a Mizohata vector field L of standard type at a given point? Of course, since this is a local question, it is enough to study the case when L is defined in a neighborhood of the origin in R2 . He showed that local coordinates can be found so that L becomes of the form (14) with &x t real-valued and vanishing to infinite order at t = 0, in other words, every Mizohata vector field has this form locally and it will be of standard type

366

Epilogue

if we are able to take & ≡ 0. Furthermore, L is of standard type at the origin if and only if it is locally integrable. Then Sjöstrand ([Sj2]) took a closer look into the nonlocally integrable case. To describe his results, let us consider the problem of finding a smooth function Z+ x t satisfying dZ0 0 = 0 and LZ+ = 0 on U + = U ∩ t ≥ 0 , where U is a small disk centered at the origin. By the proof of Lemma I.13.4, to find Z+ it is enough to find a smooth function u that satisfies Lu = t&x on U + . This is, in fact, possible because L satisfies condition  for t > 0 ([BH6]). Similarly, shrinking U if necessary, we can also find a smooth function Z− x t satisfying dZ− 0 0 = 0 and LZ− = 0 on U − = U ∩ t ≤ 0 . We can always choose Z+ and Z− satisfying Z± 0 0 = 0, JZx± 0 0 = 0, and Zx± 0 0 > 0 and we will do so. If we are so lucky that Z+ x 0 = Z− x 0, x 0 ∈ U , we may patch Z+ and Z− to get a single continuous solution Z of LZ = 0 on U and it is easy to see using the equation that Z is actually smooth. So the obstruction to the local integrability of L is related to the difficulty of finding a pair Z+  Z−  such that LZ± = 0 on U ± and Z+ = Z− on U + ∩ U − . Given such a pair, it can be shown that the range of Z± lies on one side of the smooth curve Z± x 0 (in fact, above the curve because Zx± 0 0 > 0), so let H ± z be a smooth function defined on the range of Z± and holomorphic in its interior with H ± 0 = 0, H ±  0 = H ±  0 > 0. Then, Z˜ ± = H ± Z± satisfies dZ˜ ± 0 0 = 0 and LZ˜ ± = 0 on U ± . By the Riemann mapping theorem we may find H + and H − so that the range of Z˜ + and Z˜ − is the upper half-plane. In other words, we may restrict ourselves to consider pairs Z+  Z−  such that Z± U ±  = Jz ≥ 0 and Z± U + ∩ U −  = R. Given such a pair and a smooth function H defined on Jz ≥ 0, holomorphic on Jz > 0, real for z real and satisfying H0 = 0, H  0 > 0, a new pair Z+  Z˜ −  = Z+  H Z−  may be considered and L will be locally integrable if Z+ x 0 = Z˜ − x 0. It turns out that L is locally integrable if and only if there exists a pair Z+  Z−  such that Hz = Z+ Z− −1 z is holomorphic for Jz > 0 and smooth up to Jz = 0. Since Hz is real for z real, H has, by the reflection principle, an extension to a holomorphic function. By uniqueness, Hx + iy is determined by its trace bHx = Hx + i0 so it is enough to look at the restrictions bZ± x = Z± x 0  and check whether 2 = bZ+ bZ− −1 R −→ R has a holomorphic extension to a neighborhood. Summing up, to each Mizohata vector field L we have associated an increasing diffeomorphism 2 R −→ R such that L is locally integrable if an only if 2 = bH for some H ∈  C, i.e., 2 has a holomorphic extension. More generally, we may consider the following question: given two Mizohata vector fields L1 , L2 , when are they equivalent in the sense that one can be locally transformed into a multiple of the other by a change of variables? The answer, due to Sjöstrand, can be stated as follows. Consider the associated diffeomorphisms 21 = bZ1+ bZ1− −1 and 22 = bZ2+ bZ2− −1 , then L1 and L2 are equivalent if and only if there are holomorphic functions, H1 z, H2 z, real and increasing for z real, such that 21 H1 x = 22 H2 x, x ∈ R. The local questions of standardness and equivalence for Mizohata vector fields have their global counterpart. For instance, it was established in [BCH] that a locally standard Mizohata planar vector field has a first integral globally defined in a tubular neighborhood of the characteristic set 0. The standardness of a particular class of Mizohata structures on the sphere S 2 was proved in [Ho4] and Jacobowitz ([J2]) studied Mizohata structures on compact surfaces , in particular, he proved that the existence of a first integral is equivalent to the fact that the genus is even. In the case of the sphere, he gave a classification of Mizohata structures in the spirit of Sjöstrand’s

2 Mizohata structures

367

result, proving in particular the existence of nonstandard Mizohata structures. These topics were developed further by Meziani in [Me5] and [Me6].

2.1 Mizohata structures in higher-dimensional manifolds The questions discussed in the previous section admit natural generalization to higher dimension. A formally integrable structure  defined on a manifold  of dimension N is said to be a Mizohata structure if the following holds: (1)  has rank n = N − 1; (2) the characteristic set T 0 = T  ∩ T ∗  is not empty; (3) the Levi form is nondegenerate at every point of T 0 \0 . Example 2.2. Denote by t = t1      tn  the variables in Rn , n ≥ 1, and write t = t   t , t = t1      t , t = t+1      tn , for some 1 ≤  ≤ n. Consider the function Zx t = x + i t 2 − t 2 /2 defined on Rx × Rt and the locally integrable structure  determined by imposing that T  is spanned by dZx t. Then,  is spanned by the vector fields   Mj = (15) − i-j tj  j = 1     n tj x with -j = 1 for 1 ≤ j ≤  and -j = −1 for  +1 ≤ j ≤ n. Then  is a Mizohata structure such that at every characteristic point its Levi form has  eigenvalues with one sign and n −  eigenvalues with the opposite sign and when this happens we say that  has type  n −  . Thus, we have examples of Mizohata structures with all possible types. Notice that the projection of the characteristic set is the curve 0 = t = 0 , i.e., the x-axis. A Mizohata structure with type  n −  is standard if for any point lying in the projection of the characteristic set we can choose local coordinates x t so that the vector fields (15) span  in a neighborhood of that point. Let  be a Mizohata structure with type  n −  . By analogy with the case n = 1, it turns out that for any n ∈ N and 1 ≤  ≤ n ([T5]) it is possible to find local coordinates in a neighborhood U of a generic point p in the projection of 0 such that xp = tp = 0 and  is generated over U by the vector fields Lj =

  − i-j tj 1 + &j x t  tj x

j = 1     n

(16)

where the functions &j x t, j = 1     n, vanish to infinite order at t = 0. In other words, every Mizohata structure has at a given point a contact of infinite order with a standard Mizohata structure of the same type. In particular, if we can take all the functions &j identically zero  will have a first integral in U and will be standard in U . Conversely, if  has a first integral it is possible to choose the coordinates so that  is generated by the vector fields (15). For the case  = 1, i.e., if the type is 1 n − 1 , Treves showed the existence of functions &j x t vanishing to infinite order at t = 0 such that the structure  spanned by (16) is formally integrable (i.e.,    ⊂  ) and not locally integrable. On the other hand, Meziani proved in [Me7] that Mizohata structures of all other types  n −  = 1 n − 1 are always locally integrable. His proof is delicate and beyond

368

Epilogue

the scope of this book: he first constructs first integrals on the connected components of x t  t  ∈ Rx × Rt t 2 = t 2 which can be 2 (if n > 2 and  < n − 2), 3 (if n > 2 and  = n − 2), or 4 (if n = 2 and  = 0). When the components are 2 or 4, these first integrals can be patched together to yield a globally defined first integral of class C 1 which, by the hypoellipticity of the structure, is in fact smooth. The possibility of patching together these partially defined first integrals depends on a careful analysis of the holonomy of a certain foliation with leaves of dimension n − 1 defined by the structure. For the case of type 1 n − 1 he gives a classification of Mizohata structures analogous to Sjöstrand’s result for a single vector field. The local integrability for Mizohata structures of type 0 n , n ≥ 3, was first proved in [HMa2], by techniques akin to those used in the proof of Kuranishi’s embedding theorem for CR structures ([Ku1], [Ku2], [Ak], [W2], [W3]), which also fall beyond the scope of this book. The restriction n ≥ 3 comes from a technical fact: Kuranishi’s approach depends on the existence of certain so-called homotopy formulas that do not exist when n = 2 ([HMa3]). However, the local integrability of Mizohata structures of type 0 n in Rn+1 , n ≥ 2, can be proved by elementary methods. Consider a system of n commuting vector fields Lj =

  − itj 1 + &j x t  tj x

j = 1     n

(17)

Here a generic point is described by coordinates x t1      tn  and the smooth functions &j x t vanish to infinite order at + = t = 0 = Rx × 0 . We regard the Lj ’s as perturbations of the Mizohata vector fields Mj =

  − itj  tj x

j = 1     n

A simple computation using polar coordinates, t = r, r > 0,  ∈ S n−1 shows that the standard Mizohata structure spanned by the Mj ’s is also spanned on Rn+1 \+ by ⎧   ⎪ M= − ir ⎪ ⎪ ⎨ r x ⎪ ⎪  ⎪ ⎩k = k

k=1,…,n −1

with 1      n−1  angular variables in S n−1 . Then, the change of variables s = r 2 /2 (x and  are kept unchanged) takes M into a multiple of the Cauchy–Riemann operator z¯ =

1 2



   +i  x s

z = x + is s > 0

and does not change k . If we perform the same operations on the perturbed system (17) n−1 we may find a set of generators of  in the variables x s  ∈ Rx × R+ of the s × form

2 Mizohata structures

369

⎧   ⎪ ˜ = + 1 ⎪L ⎪ ⎨ 1 ¯z z ⎪ ⎪ ⎪ ⎩L ˜k =

(18)   + k k−1 z

k=2,…,n

with smooth coefficients j x s , j = 1     n, that converge to zero when s ( 0 together with their derivatives of any order. Thus, we may smoothly extend the coefficients j as zero for Jz = s ≤ 0 and obtain an elliptic system defined on C × S n−1  Rx × Rs × S n−1 which for Jz < 0 has the first integral z = x + is. The process that produced an elliptic system starting from a nonelliptic one was obtained by a combination of singular changes of variables (polar coordinates that are singular at the origin of Rnt and s = r 2 /2 which is singular at r = 0) and blows up the line Rx × t = 0 to the n-manifold Rx × S n−1 . Although we know from Theorem I.12.1 that elliptic structures are locally integrable, applying that result to (18) would only give us a first integral defined in a neighborhood of a point s = 0, x = 0,  = 0 ∈ S n−1 while only a first integral defined for all  ∈ S n−1 can give us a first integral defined in a neighborhood of the origin of the original variables x t. Let’s consider first the case n = 2, that is the system of two vector fields ⎧   ⎪ ˜ = + 1  z = x + is ∈ C ⎪L ⎪ ⎨ 1 ¯z z (∗) ⎪ ⎪   ⎪ ⎩L ˜2 = + 2  0 ≤  ≤ 2)  z defined in C × S 1 , where the j x s , j = 1 2, are C  functions, 2)-periodic in , and vanish for s = Jz ≤ 0. Choose a smooth function  = x s  such that ˜ 2 + L ˜ 1 is a real vector. It is easy to check that this is possible if 1 < 1 (in X=L particular for x s close to the origin). Thus, X is a real generator of the structure ˜2 ˜ 1 and L ˜ 2 for x < 1, s < - and 0 ≤  ≤ 2). It is clear that X = / spanned by L for s ≤ 0, and that the orbits of X stemming from points x0  s0  0, s0 ≤ 0, are the closed circles  → x0  s0  , 0 ≤  ≤ 2). Notice also that the component of X along / is 1, i.e.,    + &1 + &2 X=  x s for some smooth functions &1 and &2 which are 2)-periodic in  and vanish for s ≤ 0. ˜ 1 ∈ ˜2 it must be a linear combination of L ˜ 1 and L ˜ 2; Since the commutator X L on the other hand, it does not contain derivations with respect to  so it has to be ˜ 1 . This shows that there exists a smooth function  = x s  such proportional to L that ˜ 1 ˜ 1 = L (19)

X L Now pick once and for all a local solution Wx s of W  W ˜ 10 W = + 1 x s 0 = 0 L ¯z z Wx 00 = 0

(20)

370

Epilogue

We may assume that in a neighborhood of the origin any other solution W x s of ˜ 10 W = 0 is a holomorphic function of W , in fact, W is a local diffeomorphism that L ˜ 10 into a multiple of the Cauchy–Riemann operator. Let  denote the closed takes L orbit of X stemming from 0 0 0, given by  → 0 0 , 0 ≤  ≤ 2). We now solve the Cauchy problem XV = 0

(21)

Vx s 0 = Wx s ˜ 1 V and in a tubular neighborhood of  made up of orbits of X. Let us set U = L observe that it follows from (19), (20) and (21) that U satisfies the Cauchy problem XU − U = 0 Ux s 0 = 0 so it must vanish identically in a tubular neighborhood of . This proves that dV is ˜ 1 and X form a basis of ˜2 . Differentiating (21) with respect orthogonal to ˜2 because L to x and setting s = x = 0 it is easy to conclude that Vx 0 0  = 0, 0 ≤  ≤ 2), so Vx 0 0  = Wx 0 0 is constant, in particular it does not vanish in a neighborhood of . This already implies that dV is a generator of the orthogonal of ˜2 , but we ˜ 1 are 2)do not know yet that V is 2)-periodic in . Since the coefficients of L ˜ 10 Vx s 2) = 0 and therefore, there periodic we have that Vx s 2) satisfies L exists a holomorphic function G such that Vx s 2) = G Vx s 0 = G Wx s hold for x s in a neighborhood of the origin. But X = / for s ≤ 0, which implies that Vx s 0 = Vx s 2) for s ≥ 0, and it turns out that Gz = z. Thus, Vx s 0 = Vx s 2) in a neighborhood of x = s = 0. This proves that V is welldefined in C × S 1 and is a first integral globally defined in  ∈ S 1 of the system (∗). Furthermore, using  = , x = V and s = JV as local coordinates in a neighborhood ˜ 1, L ˜ 2. of the origin we see that x + is and  generate the same structure as L In the case of the system (18) with n > 2 the arguments above can be applied to the first two equations keeping the variables 2      n−1 as parameters. Thus, after a change of variables x s → x  s , we may now assume that 1 ≡ 0 in (18). But ˜ 1 commutes with L ˜ k , it then we have k ≡ 0 for all values of k. Indeed, since L follows that k , k ≥ 2, depends holomorphically on z and then has to be identically zero because it vanishes for Jz ≤ 0. Thus, all the k are identically zero in the new variables and z = x + is is a first integral of the system. Returning to the original variables x s  this shows the existence of a solution Vx s  of system (18) for

x and s small and  ∈ S n−1 that satisfies Vx 0 0 0 = Vx 0 0  = 0. Finally, the function x t → Vx t 2 /2 t is smooth in a neighborhood of the origin and its differential spans  .

3 Hypoanalytic structures Let  be a smooth manifold of dimension N . By a hypoanalytic structure on  (cf. [T5]) we mean a collection of pairs = U  Z  , with U an open subset of  and

4 The local model for a hypoanalytic manifold

371

Z = Z1      Zm  U → Cm a smooth map, where 1 ≤ m ≤ N is independent of , such that the following conditions are satisfied: (H)1 U is an open covering of ; (H)2 dZ1      dZm are C-linearly independent at each point of U ;   and if p ∈ U ∩ U there exists a biholomorphism F p of an open (H)3 if  = neighborhood of Z p in Cm onto one of Z p such that Z = F p Z in a neighborhood of p in U ∩ U . A complex-valued function f defined on an open subset U of  is called hypoanalytic if in a neighborhood of any point p of U we can write f = h Z , where  is such that p ∈ U and h is a holomorphic function in a neighborhood of Z p in Cm . By a hypoanalytic chart we shall mean a pair U Z where U ⊂ X is open, Z = Z1      Zm  U → Cm has hypoanalytic components and dZ1 ∧    ∧ dZm = 0 in U. If = U  Z  is a hypoanalytic structure on  and if • ⊂  is open then we can induce a hypoanalytic structure • by the rule • = U ∩ •  Z U ∩•   To each hypoanalytic structure = U  Z  on  we can canonically associate a locally integrable structure  on  in the following way: for each  its orthogonal on U is defined by T  U = span dZ1      dZm  By properties (H)1 , (H)2 , and (H)3 it follows that T  is indeed a subbundle of CT ∗  of rank m. Notice however that two different hypoanalytic structures can define the same locally integrable structure. Indeed, to give an example it suffices to take  = R and consider the hypoanalytic structure R Id , where Idx = x, and the hypoanalytic structure R f , where f R → R is smooth but not real-analytic and f  = 0 at each point. By a hypoanalytic manifold we shall mean a pair  , where  is a smooth manifold and is a hypoanalytic structure on . Notice that if   is a hypoanalytic manifold, endowed with the hypoanalytic structure = U  Z  , if  is another smooth manifold and if f  →  is a smooth submersion, then we can pull back the hypoanalytic structure to a hypoanalytic structure f ∗ on  by defining f ∗ = f −1 U  Z f  Finally we shall say that two hypoanalytic manifolds    and   are equivalent if there is a smooth diffeomorphism f  →  such that f ∗ =  .

4 The local model for a hypoanalytic manifold Let N ≥ 1 and write N = m + n. The variable in CN = Cm × Cn will be denoted by z z  with z = z1      zm , z = z1      zn . In this space we consider the hypoana-

372

Epilogue

lytic structure defined by • = CN  z1      zm  . The corresponding hypoanalytic functions are just the holomorphic functions of z that are locally independent of z . Let  and U  Z  be as in Section 3. An arbitrary point p of  has an open neighborhood Up in which there are defined hypoanalytic functions Z1      Zm and a complementary number of C  functions Z1      Zn , with m + n = N , such that dZ1 ∧ · · · ∧ dZm ∧ dZ1 ∧ · · · ∧ dZn = 0 at p Possibly after contracting Up about p we may assume that   Z Z  = Z1      Zm  Z1      Zn  is a smooth diffeomorphism of Up onto a smooth, maximally real submanifold 0p of Cm × Cn . We refer to the triplet Up  Z Z  as an extended hypoanalytic chart. The hypoanalytic • induces a hypoanalytic structure # on 0p , simply by setting # = 0p  z1 0p      zm 0p   and it is easily seen that Up = ∗ # 

(22)

This remark is crucial for what follows.

5 The sheaf of hyperfunction solutions on a hypoanalytic manifold The sheaf of hyperfunctions can be introduced on any real-analytic manifold. This is a fundamental result, due to M. Sato ([Sa]). It is also possible to extend such a concept to hypoanalytic manifolds where no real-analyticity is required, but in order to obtain an invariant meaning, we must restrict ourselves to the hyperfunctions that are solutions in some sense. We give now a brief description of this theory. It is a consequence of a result due to Harvey ([Ha]) that over any maximally real submanifold  of CN it is also possible to define the sheaf of hyperfunctions  . Moreover, the following description is valid: given q ∈  there is an open neighborhood V of q in  such that the following is true: if W ⊂⊂ V is open then  W =   W /  W

(23)

Here the boundary of W is taken in  and for a compact subset K of CN we are denoting by   K the space of analytic functionals of CN carried by K. We return to the discussion of Section 4. We fix p ∈  and 0p as described. Since the holomorphic derivatives act on   K by transposition we can consider the space of hyperfunctions u on 0p which satisfy the system u = 0 zj

j = 1     n

(24)

The main result presented in the monograph [CorT2] states that the sheaf of these hyperfunctions on 0p , when pulled back to Up , gives rise to a well-defined sheaf Sol

5 The sheaf of hyperfunction solutions

373

on , which is furthermore a hypoanalytic invariant. The proof of this fundamental result relies on (22). We call Sol the sheaf of germs of hyperfunction solutions on . This sheaf contains, as a subsheaf, the sheaf of germs of distribution solutions with respect to the associated locally integrable structure  . Moreover, if  and the maps Z are real-analytic then Sol equals the sheaf of hyperfunctions on  that are annihilated by the (real-analytic) sections of  . Many of the basic results that were proved in this book remain valid within this more general concept of solution, as for instance the propagation of the support of solutions by the orbits of the underlying structure and the uniqueness in the Cauchy problem ([CorT2]). Another important feature is that a certain class of infinite-order operators, which are local in the sense of Sato, act as endomorphisms of Sol ([Cor1]). It can then be proved that every hyperfunction solution can be obtained, locally, as the action of one such operator on a smooth solution and then, as a consequence, a version of the approximation formula for hyperfunction solutions can be derived (cf. [Cor1]).

Appendix A: Hardy space lemmas

A.1 Multipliers in h1 We recall that  is a modulus of continuity if  0  −→ R+ is continuous, increasing, 0 = 0 and 2t ≤ Ct, 0 < t < 1. A modulus of continuity determines the Banach space C R of bounded continuous functions f R −→ C such that

fy − fx  < 

f C = sup x=y  x − y  equipped with the norm f C = f L + f C . Note that C is only determined by the behavior of t for values of t close to 0. Consider a modulus of continuity t that satisfies   1  h 1 −1 n−1 tt dt ≤ K 1 + log  0 < h < 1 (A.1) h hn 0 and the corresponding space C Rn . Lemma A.1.1. Let b ∈ C Rn  and f ∈ h1 Rn . Then bf ∈ h1 Rn  and there exists C > 0 such that bf h1 ≤ CbC f h1 

b ∈ C Rn  f ∈ h1 Rn 

Proof. Let bx ∈ C . It is enough to check that bf  ≤ CbC for every h1 -atom a with C an absolute constant. This fact is obvious for atoms supported in balls B with radius & ≥ 1 without moment condition because b is bounded so ba/bL is again an atom without moment condition. If B = Bx0  &, & < 1, we may write axbx = bx0 ax + bx − bx0 ax = 1 x + 2 x. Then 1 x/bL is again an atom while 2 x is supported in B and satisfies 2 L ≤ 2bL aL ≤ 2 L1 ≤ CaL

 B

C  &n

 x − x0  dx ≤

374

C  1 + log & 

A.1 Multipliers in h1

375

We wish to conclude that m! 2 L1 < . Let B∗ = Bx0  2&. Since m! 2 x ≤ 2 L , we have  J1 = m! 2 x dx ≤ C B∗ &−n ≤ C   B∗

It remains to estimate J2 =

 R\B∗

m! 2 x dx =

 2&≤ x−x0 ≤2

m! 2 x dx

(A.2)

(observe that m! 2 is supported in Bx0  2 because supp ! ⊂ B0 1). If 0 < - < 1 and !- ∗ 2 x = 0 for some x − x0 ≥ 2& it is easy to conclude that - ≥ x − x0 /2, which implies & & C  1 C  x − x0 −n & & 2 L

!- ∗ 2 x ≤ & !- y2 x − y dy& ≤ ≤ n 1 + log &  so m! 2 x ≤

C

x − x0 n 1 + log & 

for

x − x0 ≥ 2&

(A.3)

It follows from (A.2) and (A.3) that  C dx ≤ C  J2 ≤ n 2&≤ x−x0 ≤2 x − x0 1 + log &  which leads to bah1 ≤ 1 h1 + 2 h1 ≤ C1 + J1 + J2 ≤ C2  Inspection of the proof shows that C2 may be estimated by CbC . Example A.1.2. Suppose that a modulus of continuity t satisfies: t/tn

is a decreasing function of t

and D=



1 0

t dt <  t

(A.4)

(A.5)

A short and elegant argument shows (cf. [Ta], page 25) that under these conditions h1 Rn  is stable under multiplication by elements of C Rn . On the other hand, (A.5) alone already implies that  1 t 1  1 h h log = dt ≤ dt ≤ D 0 < h < 1 h t t h h which keeping in mind the obvious estimate 1  h h ttn−1 dt ≤ hn 0 n shows that the modulus of continuity  satisfies (A.1) and Lemma A.1.1 can be applied, proving the mentioned stability of h1 Rn  under multiplication by elements of C Rn .

376

Hardy space lemmas

Consider now a modulus of continuity t such that t =

1 − n log t

 for 0 < t < 1/2 log2 t  1/2 Since t ≥ log t −1 it follows that 0 t/t dt =  and the Dini condition (A.5) is not satisfied. On the other hand,     1  h 1 −1 1 −1 n−1 tt dt = log ≈ 1 + log  as h → 0 hn 0 h h so criterion (A.1) is satisfied. This shows that (A.5) is strictly more stringent than (A.1).

A.2 Commutators We consider now a bounded smooth function *,  ∈ R, such that & k & &d & 1 & &  ∈ R k = 0 1 2    & dxk *& ≤ Ck 1 +  k  Then * is a symbol of order zero and defines the pseudo-differential operator 1  ix *Dux = e *$ u d u ∈ R 2) R In particular, *D is bounded in h1 R. The Schwartz kernel of *D is the tempered distribution kx − y defined by $ k = * which is smooth outside the diagonal x = y. Moreover, kx − y may be expressed as 1  ix−y−-  2 kx − y = lim e * d = lim k- x − y -→0 2) -→0 where the limit holds both in the sense of  and pointwise for x = y. Furthermore, the approximating kernels k- x − y satisfy uniformly in 0 < - < 1 the pointwise estimates Cj  j = 1 2    (A.6)

k- x − y ≤

x − y j which of course also hold for kx − y itself when x = y. We consider a function bx of class C 1+ , 0 <  < 1, and wish to prove that the commutator *D bx is bounded in h1 R. A simple standard computation that includes an integration by parts gives 

*D bx ux = k x − yby − bxuy dy − *Db u where the integral should be interpreted as the pairing #< k x − ·b· − bx u·$

A.2 Commutators

377

between a distribution depending on the parameter x and a test function u. Since multiplication by b is bounded in h1 R with norm controlled by b C  , we need only worry about the remaining integral term that can be rewritten as  bx − by uy dy Tux = y − xk x − y x−y  (A.7) = k1 x − y x yuy dy where x y =

 0

1

b x + 1 − y d

and

k1 x = −xk x

Observe that  ∈ C  R2 . Lemma A.2.1. Assume T is given by (A.7) with kernel Kx y = k1 x − y x y Then T is bounded in h1 R. Proof. It follows that $ k1  = k = * +  *  . In other words, $ k1  = *1  is a symbol of order zero and T has kernel k1 x − y x y. We may write x y = b x + x − y  rx y with rx y ∈ L R2  so  Tux = b x*1 Dux + k1 x − y x − y  rx yuy dy = T1 ux + T2 ux The first operator T1 is obviously bounded in h1 because it is the composite of *1 D with multiplication by a C  function. To analyze T2 we check—writing k1 = lim-→0 k1- and using (A.6) for k1- —that its Schwartz kernel is a locally integrable distribution given by the integrable function k2 x y = k1 x − y x − y  rx y. Hence, k2 x y ≤ C1 k1 x − y x − y  = k3 x − y. Observe that k3 x ≤ C min x −1  x −2  so k3 ∈ L1 R. We will now show that m! k3 x = sup !- ∗ k3 x ∈ L1 R 0