EMOIRS

of the

American Mathematical Society

Volume 221

•

Number 1039 (third of 5 numbers)

•

January 2013

Wave Front Set of Solutions

to Sums of Squares

of Vector Fields

Paolo Albano

Antonio Bove

ISSN 0065-9266 (print)

ISSN 1947-6221 (online)

American Mathematical Society

Number 1039

Wave Front Set of Solutions

to Sums of Squares

of Vector Fields

Paolo Albano

Antonio Bove

January 2013

• Volume 221 • Number 1039 (third of 5 numbers)

• ISSN 0065-9266

Library of Congress Cataloging-in-Publication Data

Albano, Paolo, 1969-

Wave front set of solutions to sums of squares of vector ﬁelds / Paolo Albano, Antonio Bove.

p. cm. – (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; number 1039)

“January 2013, volume 221, number 1039 (third of 5 numbers).”
Includes bibliographical references and index.
ISBN 978-0-8218-7570-4 (alk. paper)
1. Wavelets (Mathematics)

2. Summability theory.

3. Vector ﬁelds.

I. Bove, Antonio.

II. Title.
QA403.3.A426

2013

515

.2433–dc23

2012035212

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18 17 16 15 14 13

Contents

Chapter 1.

Introduction

vii

Chapter 2.

The Poisson–Treves Stratiﬁcation

2.1.

Analytic Stratiﬁcation of an Analytic Set

2.2.

Symplectic Stratiﬁcation of an Analytic Submanifold

2.3.

Poisson Stratiﬁcation

2.4.

Poisson Stratiﬁcation Associated to Vector Fields

Chapter 3.

Standard Forms for a System of Vector Fields

3.1.

The Symplectic Case of Depth > 1

3.2.

The Symplectic Case of Depth 1

3.3.

The Nonsymplectic Case of Depth > 1

3.4.

The Nonsymplectic Case of Depth 1

Chapter 4.

Nested Strata

Chapter 5.

Bargman Pseudodiﬀerential Operators

5.1.

The FBI Transform

5.2.

Pseudodiﬀerential Operators

5.3.

Some Pseudodiﬀerential Calculus

Chapter 6.

The “A Priori” Estimate on the FBI Side

6.1.

Proof of Theorem 6.1

6.2.

First Part of the Estimate: Estimate from Below

6.3.

Second Part of the Estimate: Estimate from Above

Chapter 7.

A Single Symplectic Stratum

7.1.

dim Σ = 2 and X

, . . . , X

Quasi-homogeneous

7.2.

codim Σ > 2

7.3.

One Symplectic Stratum of Depth 1

Chapter 8.

A Single Nonsymplectic Stratum

8.1.

The Case rank σ

Char(P )

= 2 and X

Quasi-homogeneous

8.2.

The Transversally Elliptic Case

8.3.

A Class of Nontransversally Elliptic Operators

Chapter 9.

Microlocal Regularity in Nested Strata

9.1.

Symplectic Stratiﬁcations

9.2.

A Case of Nonsymplectic Stratiﬁcation

9.3.

A Case of Two Strata

iii

CONTENTS

Chapter 10.

Known Cases and Examples

10.1.

The Case of codim Σ = 2

10.2.

Okaji’s Theorem

Appendix A.

A Bracket Lemma

Appendix B.

Nonsymplectic Strata Do Not Have the Reproducing Bracket

Property

Bibliography

Index

Abstract

We study the (micro)hypoanalyticity and the Gevrey hypoellipticity of sums

of squares of vector ﬁelds in terms of the Poisson–Treves stratiﬁcation. The FBI
transform is used. We prove hypoanalyticity for several classes of sums of squares
and show that our method, though not general, includes almost every known hy-
poanalyticity result. Examples are discussed.

Received by the editor November 2, 2010.
Article electronically published on May 21, 2012; S 0065-9266(2012)00663-0.
2010 Mathematics Subject Classiﬁcation. Primary 35A18; Secondary 35H10, 35H20.
Key words and phrases. Analytic Hypoellipticity, FBI Transform, Wave Front Set, Canonical

Forms.

2012 American Mathematical Society

CHAPTER 1

Introduction

This paper is concerned with the problem of the analytic regularity of the

solutions to sums of squares operators. More precisely let X

(x, D), . . . , X

(x, D)

be N vector ﬁelds in

with coeﬃcients belonging to C

(U ), the space of all real

analytic functions deﬁned in the open subset U

⊂ R

We are concerned with the analytic regularity of the solutions for the operator

(1.1)

P (x, D) =

j=1

(x, D)a

(x, D)X

(x, D)

j=1

(x, D)X

(x, D) + c(x, D),

where a

, b

, c are real analytic pseudodiﬀerential operators of order zero and

moreover

(1.2)

(x, ξ) + a

(x, ξ) > 0,

where a

(x, ξ) denotes the principal symbol of a

(x, D).

In some cases it is not diﬃcult to see that the vector ﬁelds may be replaced by

ﬁrst order pseudodiﬀerential operators, but we will in general stick to the vector
ﬁelds case, which is the most popular and rich in structure.

Condition (1.2) just implies that the characteristic set for the operator P ,

Char(P ), is the characteristic set associated to the vector ﬁelds

Char(X) =

{(x, ξ) | X

(x, ξ) = 0,

j = 1, . . . , N

(x, ξ) denoting the symbol of the vector ﬁeld X

We are always assuming that the vector ﬁelds X

, . . . , X

satisfy the H¨

ormander

condition

(H) The Lie algebra generated by the vector ﬁelds and their commutators has

dimension n, equal to the dimension of the ambient space.

The importance of Condition (H) is due to the paper [15], by L. H¨

ormander, where

it is proved that (H) is suﬃcient for C

∞

hypoellipticity. This means that if P u =

∈ C

∞

(U ), for a certain distribution u on U , then u is actually smooth on U , i.e.

∈ C

∞

(U ).

Using Nagano’s Theorem it has been shown by Derridj ([9]) that Condition

(H) is also necessary for C

∞

hypoellipticity, provided the coeﬃcients of the vector

ﬁelds have real analytic regularity.

As a further step in studying the hypoellipticity properties of P one may ask if

it is analytic hypoelliptic, i.e. if P u = f

∈ C

for a certain distribution u

∈ D

(U )

implies that actually u

∈ C

(U ).

vii

viii

1. INTRODUCTION

In this case the answer is diﬃcult and in fact not much is known. We brieﬂy

mention the most important results on this problem, but we stress the fact that
the list is by no means exhaustive or complete.

First of all in the early seventies M. S. Baouendi and C. Goulaouic, [2], showed

that there are operators with real analytic coeﬃcients, satisfying (H), and thus C

∞

hypoelliptic, that are not analytic hypoelliptic, thus wreaking havoc in this part of
the theory.

Here is a short list of the major positive results proved since then.
In 1978 D. S. Tartakoﬀ [25] and F. Treves [28] independently proved that if P

vanishes exactly of the second order on Char(P ) and if Char(P ) is a symplectic C

submanifold of the cotangent bundle, then P is analytic hypoelliptic. A diﬀerent
proof has been given in 1983 by J. Sj¨

ostrand in [22], using FBI methods.

We recall in passing that a submanifold is symplectic if the restriction of the

symplectic form, σ = dξ

∧ dx, to the submanifold in question has rank equal to its

dimension. Another way of stating the latter fact is to say that the 2-form induced
by σ on Char(P ) is non degenerate. Moreover we say that P vanishes exactly of
the second order on Char(P ) if dim ker F

(ρ) = dim Char(P ), for every ρ

∈ W , W

conic open set of Char(P ). Here F

(ρ) = dH

(ρ) is the Hamilton matrix of P at ρ

and H

(ρ) = ∂

p(ρ)∂

− ∂

p(ρ)∂

denotes the usual Hamilton ﬁeld of the principal

symbol p of P .

We note that the operator studied by Baouendi and Goulaouic has a charac-

teristic set which is a nonsymplectic manifold.

In 1980 M´

etivier, following the ideas of F. Treves, showed that if P is a

(pseudo)diﬀerential operator of symbol p(x, ξ) = p

(x, ξ) + p

−1

(x, ξ) +

· · · , where

vanishes exactly to the order m on a symplectic real analytic manifold and p

−j

vanishes at least to the order (m

− 2j)

on the same manifold, then P is analytic

hypoelliptic. The vanishing conditions on the lower order terms are also called the
Levi conditions.

A few years later Okaji extended M´

etivier’s results to the case when p

∼

(

| ξ | + | x |

)

for a certain positive integer μ. He also needed Levi conditions,

which are slightly more involved; we do not state them here, since we are interested
in the second order case with no given lower order terms.

On the other hand Oleinik and Radkeviˇ

c, in [19] and [20] and later Christ,

[7], Bove and Tartakoﬀ, [3], showed that there are operators, whose characteristic
variety is a symplectic manifold, which are not analytic hypoelliptic.

Moreover in 1991 Hanges and Himonas, [12], and Christ, [6], considered the

operator

∂

− x

−1

∂

If m = 2 the above operator is known to be analytic hypoelliptic, due to the
Tartakoﬀ–Treves theorem. In [12] it was shown that for m = 3, 5, 7, . . . the above
operator fails to be analytic hypoelliptic and in [6] it was proved that the operator
fails to be analytic hypoelliptic for every integer m

≥ 3. Note that the above

operator corresponds to the Kohn’s Laplacian for a degenerate three dimensional
CR manifold.

In an attempt to elucidate and unify the theory F. Treves in 1999, [29], intro-

duced the concept of stratiﬁcation associated to a number of vector ﬁelds with real

here denotes k if k

≥ 0 and 0 if k < 0.

1. INTRODUCTION

analytic coeﬃcients. The stratiﬁcation, henceforth called Poisson–Treves stratiﬁ-
cation, is a disjoint union of real analytic manifolds, called the strata.

Below (see Chapter 2) we give more details on the Poisson–Treves stratiﬁcation

following the approach of [5].

In [29] the author formulated the

Conjecture

1.1. The operator P =

N
j=1

(x, D), where the X

are vector

ﬁelds with real analytic coeﬃcients satisfying H¨

ormander’s condition, is analytic

hypoelliptic if and only if each stratum in the stratiﬁcation associated to P is a
symplectic manifold.

Treves formulated also some variants of the above Conjecture in [30], but clearly

the above conjecture is microlocal rather than local.

Conjecture 1.1, to our knowledge, is neither proved nor disproved. We must

however say that it agrees with all the known theorems.

In this paper we consider mainly operators with a single stratum, although the

technique may apply to multi-strata operators as well. Let us consider single strata
ﬁrst.

Instead of proving local analytic hypoellipticity we focus on a microlocal state-

ment of the form:

Assume that P as given in ( 1.1) has a characteristic set whose stratiﬁcation

is the (disjoint) union of just single strata. This means that in a microlocal neigh-
borhood of a given point (x

, ξ

) the characteristic set is actually a real analytic

manifold, Σ, where the symplectic form has constant rank. Denote by F

,ξ

)

the

Hamilton leaf going through the point (x

, ξ

). Denote by U

a (conic) neighborhood

of (x

, ξ

) in Σ. Then if (x

, ξ

) /

∈ W F

(P u) and ∂U

∩ F

,ξ

)

∩ W F

(u) =

∅,

we can conclude that (x

, ξ

) /

∈ W F

(u). Here ∂U

denotes the boundary of U

a set.

In particular if Σ is a symplectic manifold, i.e. if the rank of the restriction of

the symplectic form σ to Σ is maximal, then the Hamilton leaf F

,ξ

)

{(x

, ξ

)

so that ∂U

∩ F

,ξ

)

∩ W F

(u) is always empty and thus (x

, ξ

) /

∈ W F

(u).

Actually, for technical reasons, we have to make an assumption on the foliation,

in the case of a non symplectic stratum: we assume that, roughly speaking, the
foliation has an injective projection from Σ onto the base of T

∗

. This rules out

sums of squares like the M´

etivier operator:

+ (x

+ y

Basically the method of proof relies on deducing an a priori estimate for the op-
erator once we take an FBI transform of it. Thus the a priori estimate has to
be in spaces suitable for the FBI transform, such as those given by weighted L

norms of holomorphic functions deﬁned on open subsets of

. A general estimate

of this kind, being the counterpart of the subelliptic estimate of H¨

ormander and

Rothschild–Stein, has been proved in [1]. To make the paper more readable we
shall give below an idea of the proof of such an estimate.

Unfortunately this type of estimate is not enough since any estimate leading

to a (propagation of) regularity theorem has to be a uniqueness estimate, i.e. an
estimate ensuring that, when the data are zero in some sense, the solution is zero
in the same sense.

This is the reason why we use a deformation argument resembling a microlocal

version of Holmgren theorem. Such an argument has been used ﬁrst in the FBI

1. INTRODUCTION

framework by Sj¨

ostrand in [22] and, since then, in diﬀerent ﬂavors as well as context,

by many authors.

Actually the following estimate is proved in Chapter 6:

2
r

Φ,Ω

≤ C

Φ,Ω

+ λ

Φ,Ω

\Ω

where r denotes the minimum length of an elliptic bracket of the vector ﬁelds at
the point we consider, Ω

⊂⊂ Ω are neighborhoods of that point in C

and the

function Φ is a weight function. Note that the above estimate is an estimate on
the FBI side and that it resembles the well known subelliptic estimate proved by
Rothschild and Stein, [21].

The regularity theorem follows once one can get rid of the last term in the

above estimate. It is not diﬃcult to see that the weight function corresponding to
the FBI transform is not enough for that. Here the Holmgren type argument plays
a role.

The idea is that we must do a deformation of the complex path in such a

way that the right hand side of the estimate is exponentially decreasing if our
assumptions are satisﬁed. But doing this involves changing the reality properties
of the symbols of the vector ﬁelds. Although the deformation is accomplished in a
canonical way as we shall see below, this forces us to introduce some assumptions.
Next we discuss these assumptions.

First of all we may argue in the real domain, i.e. in T

∗

or rather T

∗

Ω, since

everything will be then easily carried on to

via the diﬀeomorphism induced by

the FBI transform. Thus denote by r(x, ξ) the Hamiltonian that we use to generate
the deformation. Basically we want the characteristic variety of the vector ﬁelds
to be contained in its complexiﬁcation after we apply the complex ﬂow exp(itH

This is obtained if, denoting by X(x, ξ) = (X

(x, ξ), . . . , X

(x, ξ)) the vector of

the symbols of the vector ﬁelds, we have

{X, r} = αX,

where α denotes a suitable N

× N matrix of symbols and r is the Hamiltonian

function used in the deformation. It is not diﬃcult to see that such an r can always
be found, but to get rid of the error terms in the above estimate we need r to vanish
just at the point under consideration and to be strictly positive away from it.

Evidently the above condition is a symplectic invariant, but we do not know

if it has a solution verifying the supplementary positivity conditions for any set of
vector ﬁelds admitting e.g. a single symplectic stratum. At present a study of the
above diﬀerential equation, which should have an intrinsic geometric nature, seems
diﬃcult and too long to be included here.

To overcome this diﬃculty we deal with classes of vector ﬁelds satisfying addi-

tional conditions insuring that a Hamiltonian r with the required properties exists.
The additional assumptions may vary from case to case and they will be the subject
of Chapter 7.

We do however know that all the known single stratum cases can be studied

using this method. This is certainly true for the transversally non degenerate case of
Tartakoﬀ and Treves (see also J. Sj¨

ostrand paper [22].) In Chapter 10 we (re)prove

using this method the most famous known cases of sums of squares of vector ﬁelds
admitting a single symplectic stratum.

The paper is organized as follows: in Chapter 2 we deﬁne the Treves–Poisson

stratiﬁcation. In Chapter 3 some standard forms for the vector ﬁelds are proved.

1. INTRODUCTION

The standard forms give some information on the vanishing of the symbols of the
vector ﬁelds in special coordinates where the equations of the characteristic manifold
are extremely simple. Chapter 4 is devoted to the standard forms in the case of
nested strata. In Chapter 5 a sketch of the proof of the a priori estimate on the FBI
side is given. Chapter 7, 8 and 9 are devoted to our results in the single symplectic,
single nonsymplectic and nested stratum, respectively. Chapter 10 contains the
proofs of known results, in the case of a single stratum, using the methods of the
present paper. The Appendices contain some material which is not used in the
proofs of the paper but that we thought might useful and non-trivial.

Finally the authors would like to acknowledge a number of very stimulating

discussions with Fran¸cois Treves, as well as his constant encouragement. We would
like also to acknowledge a number of discussions with Paulo Cordaro, Nick Hanges
and David Tartakoﬀ.

CHAPTER 2

The Poisson–Treves Stratiﬁcation

In this chapter we recall the deﬁnition of stratiﬁcation associated to a ﬁnite

number of vector ﬁelds satisfying H¨

ormander’s condition. Basically we follow the

paper [5].

2.1. Analytic Stratiﬁcation of an Analytic Set

Throughout this section V denotes the set of common zeroes of a ﬁnite family

, ..., F

of real-valued C

functions in an open subset U of

We shall use the analytic stratiﬁcation of V provided by the ranks of the analytic

map

F = (F

, ..., F

) : U

−→ R

and of its diﬀerentials, deﬁned as follows.

Let m

≤ min (n, r) be the maximum value of rank F in V . We denote by

(V )

⊂ V

the set in which rank F = m; R

(V ) is a C

submanifold of U of codimension m.

Note that R

(V ) is a relatively open subset of the regular part of V , i.e. the subset

of V consisting of the points in a neighborhood of which V is a C

submanifold.

The set

= V

\ R

(V )

is deﬁned by the vanishing of F and of all the m

× m minors D

,...,i

,...,j

≤ i

· · · < i

≤ r, 1 ≤ j

· · · < j

≤ n ) of the n × r matrix

∂(F

,...,F

)

∂(x

,...,x

)

. This means

that V

is an analytic subset of U of the same type as V . We introduce the map

, ..., F

,...,i

,...,j

≤i

···<i

≤r,

≤j

···<j

≤n

: U

−→ R

with r

= r + #

minors D

,...,i

,...,j

We denote by m

the maximum value of rank F

in V

and by

)

⊂ V

the subset in which rank F

= m

Remark

2.1. It may happen that m

< m as shown in the classical example

of Milnor: in

take F

= x (1

− x) , F

= y (1

− x). Here m = 2 and R

(V) =

{(0, 0)} whereas m

= 1 and V

is the vertical line x = 1.

We can repeat the above argument indeﬁnitely. For each k = 0, 1, . . . we obtain

an analytic map

(k)

: U

−→ R

(k)

≤ r

(k)

2. THE POISSON–TREVES STRATIFICATION

With the understanding that V

(0)

= V we call R

(k)

the subset of V

(k)

which F

(k)

has maximum rank m

(k)

and we deﬁne

(k+1)

= V

(k)

\ R

(k)

In particular R

(k)

is a C

submanifold of U of codimension m

(k)

; we denote

by Λ

(k)
α

( α = 1, 2, ...) its connected components. The end result of this procedure

is a decomposition

(2.1)

V =

∞

k=0

∞

α=1

(k)
α

The connected submanifolds Λ

(k)

are pairwise disjoint and locally ﬁnite. Let us

emphasize the following property of the Λ

(k)
α

, as it will play a simplifying role

in the sequel. Call F

(k)

∈ C

(U ), j = 1, ..., r

(k)

, the components of the map

(k)

: U

−→ R

(k)

. Then

• There is an open set U

(k)

⊂ U such that

(2.2)

(k)
α

∈ U

(k)

| F

(k)

(x) = 0, j = 1, ..., r

(k)

and the rank of the map F

(k)

, ..., F

(k)

is equal to m

(k)

at every

point of Λ

(k)
α

Definition

2.2. The decomposition (2.1) will be called the analytic stratiﬁ-

cation of V and each submanifold Λ

(k)

will be referred to as an analytic stratum

of V .

Implicit in this deﬁnition is the role of the map F : U

−→ R

. But if G is a

diﬀeomorphism of an open neighborhood of the origin in

onto another such

neighborhood, the analytic stratiﬁcation of V viewed as the null set of G

◦ F is the

same as its stratiﬁcation when V is viewed as the null set of F.

2.2. Symplectic Stratiﬁcation of an Analytic Submanifold

In this section we take the dimension n to be even and actually, slightly abusing

our notation, we write it 2n. Let σ =

n
j=1

n+j

∧ dx

be the symplectic form in

. If x

∈ R

we denote by σ

the nondegenerate skew-symmetric bilinear form

induced by σ on the tangent space T

To each germ of real-valued C

function f at x there is a unique germ of C

vector ﬁeld at x, which we denote by H

, deﬁned by the property that, for any

tangent vector v to U at x, σ

, v) =

df (x) , v. H

is usually referred to

as the Hamiltonian vector ﬁeld of f . If g is another germ of C

function at x we

denote the Poisson bracket of f and g by

{f, g} = σ (H

, H

). Let Σ be a connected

submanifold of U of class C

endowed with the property analogous to (2.2):

• there are functions G

∈ C

(U ) , j = 1, . . . , s, and an open set U

⊂ U

such that

(2.3)

Σ =

{x ∈ U

| G

(x) = 0, j = 1, ..., s

}

and the rank of the map G = (G

, ..., G

) is constant at every point of Σ.

2.2. SYMPLECTIC STRATIFICATION OF AN ANALYTIC SUBMANIFOLD

Henceforth we shall assume that (2.3) holds. Then, if d = codim Σ is the rank of
G (x), x

∈ Σ, each point x

∈ Σ has an open neighborhood N

⊂ U in which

there are indices 1

≤ i

· · · < i

≤ s such that the following is true:

(1) dG

∧ · · · ∧ dG

= 0 at x

;

(2) Σ

∩ N

∈ N

; G

(x) =

· · · = G

(x) = 0

We denote by σ

the restriction of σ to Σ: for each x

∈ U the restriction to

the tangent space T

Σ of the nondegenerate skew-symmetric bilinear form σ

is a

skew-symmetric bilinear form σ

, possibly degenerate.

The rank of the bilinear form σ

, i.e. the rank of the linear map T

−→

∗

Σ deﬁned by σ

, is related to that of the matrix (

, G

} (x))

≤j,k≤s

by the

formula

rank σ

+ codim Σ = rank (

, G

} (x))

≤j,k≤s

+ dim Σ.

We refer to rank σ

as the symplectic rank of the submanifold Σ at the point x.

Denote by Σ

the open and dense subset of Σ consisting of the points x at which the

symplectic rank of Σ is maximum, say equal to μ

≥ 0. Each connected component

of Σ

is a submanifold of U of class C

whose symplectic rank is everywhere equal

to μ.

The subset Σ

\ Σ

is an analytic subset of Σ: it can be deﬁned in U

the set of common zeros of G

, ..., G

and of all the ν

× ν minors of the matrix

(

, G

})

≤j,k≤s

where ν = μ + codim Σ

− dim Σ. It is an analytic subset of U

precisely of the type considered in Section 2.1 and as such it admits an analytic
stratiﬁcation of type (2.1) in U

. The dimension of each analytic stratum of Σ

\ Σ

is strictly less than dim Σ. Furthermore these strata also have Property (2.3). This
means that we can repeat with each one of them the construction started with
Σ; and that it will suﬃce to repeat this same construction a number of time not
exceeding dim Σ to obtain a decomposition

(2.4)

Σ =

∞

β=1

where each Σ

is a submanifold of an open subset U

of U

and satisﬁes the analogue

of Condition (2.3) in U

. The submanifold Σ

are pairwise disjoint.

We can carry out the decomposition (2.4) taking Σ to be any of the analytic

strata Λ

(k)

of the analytic set V in Section 2.1, thus obtaining a new decomposition

into pairwise disjoint C

submanifolds of U ,

(2.5)

V =

∞

k=0

∞

α=1

∞

β=1

(k)
α,β

Definition

2.1. The decomposition (2.5) will be referred to as the symplectic

stratiﬁcation of the analytic set V and each stratum Λ

(k)
α,β

will be referred to as a

stratum in the sense of the symplectic stratiﬁcation of V .

In (2.5) the family of C

submanifolds

(k)
α,β

≤k≤m

is locally ﬁnite in U . Each

submanifold Λ

(k)
α,β

satisﬁes Property (2.3) for an appropriate choice of the functions

and of the open set U

⊂ U. The symplectic rank of each Λ

(k)
α,β

is constant.

2. THE POISSON–TREVES STRATIFICATION

2.3. Poisson Stratiﬁcation

We continue to deal with the analytic set V and with the functions F

∈ C

(U ),

i = 1, ..., r. For each multi-index I = (i

, ..., i

) with 1

≤ i

, ..., i

≤ r , ν ≥ 2, we

write

, ..., F

} =

, ...

−1

, F

...

We refer to ν as the length of the multi-index I; we also write

|I| = ν. When

|I| = 1, when I = {i} for some i, 1 ≤ i ≤ r, we equate F

to F

Definition

2.1. We say that the functions F

, ..., F

∈ C

(U ) satisfy the

H¨

ormander condition if for every x

∈ U there is a multi-index I, |I| ≥ 1, such

that F

(x)

= 0.

We can deﬁne the monotone decreasing sequence of analytic subsets of U : for

each ν

≥ 2,

(ν)

= V

∩ {x ∈ U | ∀I, |I| ≤ ν, F

(x) = 0

} .

In particular V =

(1)

. The H¨

ormander condition states that

∞
ν=1

(ν)

∅.

Note that there is a subsequence of integers 1 = ν

< ν

· · · such that

(1)

(ν

p+1

)

(ν

)

;

(2) if ν

< ν

p+1

then

V (

) =

(ν

)

for every ν

, ν

≤ ν

< ν

p+1

Now consider, for any given integer p

≥ 1, the symplectic stratiﬁcation of the

analytic set

(ν

)

(Deﬁnition 2.1):

(ν

)

∞

k=0

∞

α=1

∞

β=1

(k,ν

)

α,β

In each stratum Λ

(k,ν

)

α,β

the set

(k,ν

)

α,β

of points x

∈

(ν

)

(ν

p+1

)

is either empty

or else, it is an open and dense subset of Λ

(k,ν

)

α,β

(as the latter is a connected C

submanifold). If

(k,ν

)

α,β

= ∅ we denote by Λ

(k,ν

)

α,β,γ

its connected components. We

obtain thus the decomposition

(ν

)

(ν

p+1

)

∪

∞

k=0

∞

α,β,γ=1

(k,ν

)

α,β,γ

Letting p range over the set of positive integers yields a decomposition

(2.6)

V =

∞

ν=1

∞

j=0

(ν

)

in which, whatever p and j,

(1) the C

submanifolds Σ

(ν

)

are connected and pairwise disjoint;

(2) at every point of Σ

(ν

)

the rank of

T Σ

(ν

)

∩

T Σ

(ν

)

⊥

is equal to one

and the same (even) nonnegative integer;

(3) at every point of Σ

(ν

)

all Poisson brackets F

of length ν < ν

p+1

vanish

but at least one of length ν

p+1

does not.

Definition

2.2. The decomposition (2.6) will be called the Poisson strat-

iﬁcation of V deﬁned by the functions F

, ..., F

and each submanifold Σ

(ν

)

will

2.4. POISSON STRATIFICATION ASSOCIATED TO VECTOR FIELDS

be called a Poisson stratum of V deﬁned by these functions. We shall refer to ν

as the depth of the Poisson stratum Σ

(ν

)

It follows immediately from the elementary properties of the Poisson bracket that
the Poisson stratiﬁcation of V deﬁned by the functions F

, ..., F

is invariant under

nonsingular C

substitutions, i.e. substitutions of the form

(2.7)

k=1

k
j

j = 1, ..., r,

with a

∈ C

(U ) and det

≤j,k≤r

= 0 at every point of U.

If a Poisson stratum Σ of V is not symplectic (i.e. if the restriction to Σ of

the symplectic form in T

∗

is degenerate) then the intersection T Σ

∩ (T Σ)

⊥

a nonvanishing vector bundle over Σ which satisﬁes the Frobenius condition: the
commutation bracket of two of its smooth sections is a section of T Σ

∩ (T Σ)

⊥

. As

a consequence T Σ

∩(T Σ)

⊥

deﬁnes a foliation on Σ in which all the leaves have the

same dimension. We shall refer to the leaves of this foliation as the Hamiltonian
leaves.

2.4. Poisson Stratiﬁcation Associated to Vector Fields

We consider N real vector ﬁelds X

(x, D), ..., X

(x, D) of class C

in a con-

nected and open subset Ω of R

and the “sum of squares” operator P = X

· · · +

. Let T

∗

\ 0 denote the cotangent bundle of R

from which the zero section

has been deleted and π the base projection T

∗

\ 0 → R

The symplectic manifold U of the preceding sections will be the open subset

−1

π (Ω) of T

∗

\ 0.

The variety V will be the set of common zeros of the symbols X

(x, ξ) of the

vector ﬁelds X

; in other words, V = Char(P ).

We apply the concepts of the previous sections with the choice of

(x, ξ) = X

(x, ξ)

j = 1, . . . , N.

This choice will deﬁne once for all the meaning of the Poisson strata of Char(P ).
We recall that a subset of phase-space T

∗

is said to be conic if it is invariant

under the dilations (x, ξ)

→ (x, λξ), λ > 0. Of course Char(P ) is conic. We

can repeat the constructions in Sections 1.1, 1.2, 1.3, making use only of functions
F (x, ξ) that are homogeneous with respect to ξ, i.e. F (x, λξ) = λ

F (x, ξ) for

some integer m and all λ

∈ R. We obtain

Proposition

2.1. Every Poisson stratum of Char(P ) is conic.

We list here a few examples that are going to guide us in the sequel. In the

ﬁrst example the characteristic variety is smooth but nonsymplectic:

Example

2.2. ([2]) The characteristic set of the Baouendi-Goulaouic operator

P = D

+ D

+ x

2
1

consists of the two open half-spaces x

= ξ

= 0, ξ

> 0 (resp. ξ

< 0).

Given a point (0, x

, x

, 0, 0, ξ

)

∈ Char(P ) the Hamiltonian leaf through such

a point is the line (0, x

, x

, 0, 0, ξ

), x

∈ R.

The next example generalizes the Baouendi-Goulaouic operator in that the

characteristic variety is smooth, but now it is symplectic.

2. THE POISSON–TREVES STRATIFICATION

Example

2.3. ([19]) The characteristic set of the Oleinik operator

P = D

+ x

2(p

−1)

+ x

2(q

−1)

(1 < p < q)

can be identiﬁed to the phase-space (x

, x

, ξ

) with the null section excised.

It admits the following stratiﬁcation:

(1) two symplectic strata x

= ξ

= 0, ξ

> 0 (resp., ξ

< 0) at depth 1;

(2) two nonsymplectic strata x

= ξ

= 0, ξ

> 0 (resp., ξ

< 0) at

depth p.

The latter remains true down to depth q

− 1. At depth q we encounter the zero

section (which, by our convention, is not part of Char(P )).

The Hamiltonian leaf passing through the point (0, x

, x

, 0, 0, ξ

) is the line

(0, x

, x

, 0, 0, ξ

), x

∈ R.

In the next example at depth 1 the Poisson strata are symplectic; nonsymplectic

strata occur at depth p > 1; symplectic strata re-appear at depth r > p.

Example

2.4. At depth one the characteristic set of the operator

P = D

+ x

2(p

−1)

+ x

2(r

−1)

2(q

−r)

+ x

2
2

(p < r < q

≤ p + r, > 1) admits the symplectic strata x

= ξ

= 0, ξ

> 0

(resp., ξ

< 0 ). It admits the strata x

= ξ

= 0, ξ

> 0 (resp., ξ

< 0) at

depth p and the strata x

= x

= ξ

= 0, ξ

> 0 (resp., ξ

< 0 ) at depth r.

We encounter the zero section at depth q. The only nonsymplectic strata occur at
depth p.

CHAPTER 3

Standard Forms for a System of Vector Fields

The purpose of this Chapter is to deduce some standard forms for a system

of vector ﬁelds whose stratiﬁcation has a single stratum. We deal with both the
symplectic and non symplectic case.

The argument for the symplectic case of depth > 1 is inspired by Fran¸cois

Treves, whom we thank for making available to us his unpublished manuscript
[31]. The argument for the nonsymplectic case is somewhat similar and is certainly
inspired by the former.

Let X

, . . . , X

denote a system of vector ﬁelds deﬁned in an open set Ω

⊂

which we may assume wlog to contain the origin.

Let us denote by Σ =

Char(X

, . . . , X

) the characteristic variety of the vector ﬁelds under exam and

deﬁne

m = codim Σ.

We take ρ

∈ Σ.

Using possibly a translation and a rotation with constant coeﬃcients we may

assume that ρ

= (0, e

We recall that we may perform any change of variables as well as any nonsin-

gular linear substitution of the vector ﬁelds, i.e. any operation of the form

(3.1)

(x, D

) =

j=1

(x)X

(x, D

where the functions a

∈ C

(Ω) and the matrix [a

(x)]

i,j=1,...,N

is nonsingular for

every x

∈ Ω.

We shall argue in two diﬀerent cases, depending on the depth of the single

stratum i.e. when its depth is equal to one or it is greater than one. Here by depth
we mean the minimum length of an elliptic Poisson bracket of the ﬁelds. Denote
by

κ the number of linearly independent vector ﬁelds at 0. We may then choose

the coordinates in such a way that the vector ﬁelds have the form

(3.2)

∂

∂x

κ+1

(x)

∂

∂x

i = 1, . . . ,

κ;

(3.3)

κ+1

(x)

∂

∂x

j =

κ + 1, . . . , N;

here a

∈ C

(Ω) and a

(0) = 0 for all j = 1, . . . , N , k =

κ + 1, . . . , n.

3.1. The Symplectic Case of Depth > 1

We make the assumption

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

(Sympl) The characteristic set Σ is a symplectic real analytic submanifold of

∗

\ {0}, where all Poisson brackets {X

, X

} vanish identically, for

i, j = 1, . . . , N .

Denote by π the projection from T

∗

\ {0} onto the base. Then we have

Proposition

3.1.1. Under the assumption (Sympl), the restriction to Σ of

the base projection π has constant rank. Moreover its rank is equal to n

− κ and

κ = codim Σ.

Proof.

Denote by m the codimension of Σ. Since the symbols of the ﬁelds in

(3.2) are independent equations of Σ, the vanishing of the symbols in (3.3) yields
m

− κ independent real analytic equations of the form

(3.1.1)

(x, ξ

) = 0, . . . , ϕ

−κ

(x, ξ

) = 0,

where

= (ξ

κ+1

, . . . , ξ

From (3.2), (3.3) it follows that

(x = 0, ξ

= 0, . . . , ξ

= 0, ξ

) = (0, ξ

)

∈ Σ.

Hence, we have

(3.1.2)

(0, ξ

) = 0,

for

j = 1, . . . , m

− κ.

Because of (Sympl), the matrix of the Poisson brackets of the functions X

(x, ξ),

in (3.2) and ϕ

(x, ξ

) in (3.1.1) has maximal rank m on Σ.

We have, for i = 1, . . . ,

κ, j = 1, . . . , m − κ,

, ϕ

}(0, ξ) =

∂ϕ

∂x

(0, ξ

because of the properties of the vector ﬁelds in (3.3) and of (3.1.2). Moreover for
the same reason we have that

{ϕ, ϕ}(0, ξ

) = 0. We then deduce that the matrix

{X, ϕ}(0, ξ) must have maximal rank, min{κ, m − κ} = m/2, and thus it has to be
a square matrix, i.e. m = 2

κ. We point out that the last identity implies that the

number of linearly independent vector ﬁelds is constant in π(Σ). In order to show
that

rank π

dim Σ = n

− κ

we remark that a tangent vector to Σ at (0, ξ

), (δx, δξ), is a solution of the following

linear system

δξ

κ+1

s=1

∂a

∂x

(0)ξ

δx

= 0,

j = 1, . . . ,

κ,

s=1

∂ϕ

∂x

(0, ξ

)δx

= 0,

i = 1, . . . , m

− κ.

Since rank[∂

]

i=1,...,m

−κ,s=1...,n

(0, ξ

) =

κ we ﬁnd that

rank dπ(0, ξ

) = n

− κ

This proves the proposition.

Corollary

3.1.2. We have the following

(i) π(Σ) is a real analytic submanifold of Ω of dimension n

− κ.

3.1. THE SYMPLECTIC CASE OF DEPTH > 1

(ii) The vector ﬁelds X

, with j =

κ +1, . . . , N, vanish at every point of π(Σ).

(iii) The rank of the system of vector ﬁelds X

, . . . , X

is equal to

κ at any

point of π(Σ).

Proof.

The ﬁrst assertion is a direct consequence of Proposition 3.1.1. As for

the second we remark that the vector ﬁelds X

κ+1

, . . . , X

are a linear combinations

of X

, . . . , X

at every point of π(Σ). In fact, if this wouldn’t be true, there would

exist a vector ﬁeld X

, j

∈ {κ + 1, . . . , N}, linearly independent with the ﬁrst κ

vector ﬁelds. This would alter the number

κ and we would have a rank for π

diﬀerent from n

− κ; this is impossible because of (Sympl). Due to the form of

the vector ﬁelds we deduce then that the X

, j =

κ + 1, . . . , N, vanish identically

on π(Σ).

Finally the last assertion is a trivial consequence of the second.

Next we perform a number of real analytic changes of variables. The equations

deﬁning π(Σ) are given by

κ real analytic functions ψ

(x) = 0, ψ

(0) = 0, j =

1, . . . ,

κ. We point out that X

(x, ξ) = 0, ψ

(x) = 0, j = 1, . . . ,

κ, are the equations

deﬁning Σ. In fact we have the inclusion of the two real analytic manifolds

= 0, j = 1, . . . ,

κ} ⊂ Σ, because of (ii) of Corollary 3.1.2. Then, since the

codimension is the same we get equality.

We thus proved the lemma

Lemma

3.1.3. We have

Σ =

{(x, ξ) | X

(x, ξ) = ψ

(x) = 0, j = 1, . . . ,

κ}.

Since the Poisson bracket matrix

, X

}

, ψ

}

−{X

, ψ

}

where i, j

∈ {1, . . . , κ}, has maximal rank 2κ, we may always assume that ∂

0, i = 1, . . . ,

κ. Hence we may write the equations ψ

(x) = 0 as x

+ ˜

) = 0,

for i = 1, . . . ,

κ. Changing variables according to

= x

+ ˜

)

i = 1, . . . ,

κ,

= x

j =

κ + 1, . . . , n,

the form of the vector ﬁelds X

, . . . , X

is preserved.

We need a further change of variables. First rectify X

. A second transforma-

tion rectiﬁes the vector ﬁeld

= ∂

κ+1

(0, x

, . . . , x

)∂

This is a change of variables in the coordinates x

, . . . , x

, leaving x

unchanged.

As a consequence

= ∂

+ x

κ+1

(1)
2k

(x)∂

This argument can be iterated and we obtain for X

the form

= ∂

κ+1

−1

()
ik

(x)∂

for i = 1, . . . ,

κ. We have thus the lemma

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

Lemma

3.1.4. There is a real analytic change of variables such that the vector

ﬁelds ( 3.2) and ( 3.3) have the form

(3.1.3)

= ∂

κ+1

−1

()
ik

(x)∂

i = 2, . . . ,

κ.

(3.1.4)

κ+1

(x)∂

j =

κ + 1, . . . , N.

Here the a

∈ C

(Ω), for j = 1, . . . , N , k =

κ + 1, . . . , n, and

≤

κ+1

(x, ξ)

≤ C|x

where c, C are suitable positive constants, M is a suitable positive integer and
x

= (x

, . . . , x

Because of Corollary 3.1.2, we obtain the

Theorem

3.1.5. If (Sympl) holds, then the characteristic set of the vector

ﬁelds X

, . . . , X

, has the form

(3.1.5)

Σ =

{(x, ξ) ∈ T

∗

\ {0} | x

= ξ

= 0, i = 1, . . . ,

κ}

Proof.

It is a consequence of Lemma (3.1.4) and the above changes of vari-

ables.

We complete this section with an example of vector ﬁelds satisfying the as-

sumptions:

Example

3.1.6. Let

(x, D) = D

+ x

f (x

, x

(x, D) = x

n
1

(x, D) = x

m
2

where m, n denote positive integers

≥ 2 and f is a real analytic function deﬁned in

a neighborhood of the origin in

and vanishing at the origin. We note explicitly

that the H¨

ormander bracket condition is satisﬁed.

3.2. The Symplectic Case of Depth 1

We make the assumption

(Sympl-1) The characteristic set Σ is a symplectic real analytic submanifold of

∗

\{0} and there exists a Poisson bracket of two vector ﬁelds whose

symbol does not vanish.

We may assume that the vector ﬁelds still have the form (3.2) and (3.3) and retain
the notation of the preceding section. Furthermore let

r = rank [

, X

}]

i,j=1,...,

(0, e

Set

(3.2.1)

ρ = m

− 2κ + r.

We recall that m = codim Σ.

3.2. THE SYMPLECTIC CASE OF DEPTH 1

Proposition

3.2.1. Under the assumption (Sympl-1), the restriction to Σ of

the base projection π has constant rank equal to n

− κ + r − ρ. Moreover ρ is a

non-negative even integer and 0

≤ ρ ≤ r.

Proof.

Since the symbols of the ﬁelds in (3.2) are independent equations of

Σ, the vanishing of the symbols in (3.3) yields m

− κ independent real analytic

equations of the form

(3.2.2)

(x, ξ

) = 0, . . . , ϕ

−κ

(x, ξ

) = 0.

From (3.3) it follows that (0, ξ

)

∈ Σ for every ξ

= 0, so that

(3.2.3)

(0, ξ

) = 0,

for

j = 1, . . . , m

− κ.

Because of (Sympl-1), the matrix of the Poisson brackets of the symbols X

(x, ξ),

in (3.2) and ϕ

(x, ξ

) in (3.1.1) has maximal rank m on Σ.

We have, for i = 1, . . . ,

κ, j = 1, . . . , m − κ,

, ϕ

}(0, ξ) =

∂ϕ

∂x

(0, ξ

because of the properties of the vector ﬁelds in (3.3) and of (3.2.3). Moreover for
the same reason we have that

{ϕ, ϕ}(0, ξ

) = 0.

Now the corank of the symplectic form on Σ is the rank of the matrix

− κ

, X

}(0, ξ

)

, ϕ

}(0, ξ

)

{ϕ

, X

}(0, ξ

)

which coincides with the codimension of Σ, m, because of the assumptions. Thus
the rank of the matrix [

, ϕ

}]

i=1,...,

κ,k=1,...,m−κ

(0, e

) is m

−κ. Hence m−κ ≤ κ

and

κ − r ≤ m − κ, so that 0 ≤ ρ ≤ r.

This proves the second assertion in the statement.

Remark

3.2.2. As a consequence of the above argument we point out that the

matrix [

, ϕ

}]

i=1,...,

κ,k=1,...,m−κ

, ξ

= 0, has maximal rank m − κ.

As for the ﬁrst assertion we use a direct argument. For reference purposes it is

useful to state the

Lemma

3.2.3. We have that dim π(Σ) = n

− m + κ.

Proof of Lemma

3.2.3. The equations deﬁning the characteristic manifold

are X

(x, ξ) = 0, i = 1, . . . ,

κ, ϕ

(x, ξ

) = 0, k = 1, . . . , m

− κ, we can compute a

generic form of a vector tangent to Σ on the point (0, ξ). Diﬀerentiating the ϕ

obtain

j=1

∂

(0, ξ

)δx

= 0,

since ∂

(0, ξ

) = 0. By the above remark

rank[∂

(0, ξ

)]

j=1,...,n; k=1,...,m

−κ

= m

− κ.

This proves the lemma.

Since m

−κ = κ−r+ρ, the dimension of π(Σ) can also be written as n−m+κ =

− κ + r − ρ; if r = 0 and hence ρ = 0, one reobtains the result of the previous

section.

This proves the proposition.

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

Arguing as in the preceding section we may ﬁnd

κ − r + ρ equations deﬁning

π(Σ), ϕ

(x) = 0, k = 1, . . . ,

κ − r + ρ. The equations deﬁning Σ will then be

(x, ξ) = 0, i = 1, . . . ,

κ and ϕ

(x) = 0, k = 1, . . . ,

κ − r + ρ. Possibly performing

a nonsingular substitution in the ﬁrst

κ vector ﬁelds (and possibly changing the

coordinates labels), we may assume that

(3.2.4)

[

, X

}]

i,j=1,...,

[

, X

}]

i,j=1,...,r

at (0, e

)

∈ Σ. Indeed, by deﬁnition,

rank[

, X

}]

i,j=1,...,

(0, e

) = r.

Hence, possibly changing the coordinates labels, we may assume that

rank[

, X

}]

i,j=1,...,r

(0, e

) = r.

We deduce that for every i = 1, . . . , r and j = r + 1, . . . ,

κ there exist β

∈ R, with

s = 1, . . . , r, such that

, X

}(0, e

) =

s=1

, X

}(0, e

Then, we may perform the substitution in the vector ﬁelds

(3.2.5)

(x, D) = X

(x, D)

i = 1, . . . , r

(x, D) = X

(x, D)

−

r
s=1

(x, D)

j = r + 1, . . . ,

κ.

Since the above substitution is associated to the lower triangular matrix

−β I

κ−r

we deduce that the substitution in the vector ﬁelds (3.2.5) is non singular. Here I

and I

κ−r

denote the r

× r and (κ − r) × (κ − r) identity matrix respectively. A

change of the variables x

, . . . , x

puts the ﬁelds X

, . . . , X

back into the original

form (3.2) Then (3.2.4) follows since the matrix [

, X

}]

i,j=1,...,

is antisymmetric

and

rank[

, X

}]

i,j=1,...,

(0, e

) = r.

Thus we have that the matrix

(3.2.6)

⎡
⎢

⎣

[

, X

}]

i,j=1,...,r

−

⎤
⎥

⎦ ,

has rank 2

κ − r + ρ at (0, e

). Here

A = [∂

]

i=1,...,r
k=1,...,

κ−r+ρ

(0, e

B = [∂

]

i=r+1,...,

k=1,...,

κ−r+ρ

(0, e

Recalling that the rank of the matrix in Formula (3.2.6) is equal to m and that
rank[

−

B] = m

− κ we obtain that

rank

[

, X

}]

i,j=1,...,r

κ .

3.2. THE SYMPLECTIC CASE OF DEPTH 1

Hence we deduce that

rank B =

κ − r

and

rank A = m

− κ − (κ − r) = ρ.

We may add or subtract columns and rows in the matrix (3.2.6) (possibly changing
the labels of the ﬁrst r coordinates) obtaining

⎡
⎢

⎣

∗

−

⎤
⎥

⎦ ,

with

C =

[

, X

}]

i,j=1,...,r

−ρ

Using the implicit function theorem we see that the equations ϕ

(x) = 0, k =

1, . . . ,

κ − r + ρ, become y

+ ˜

, . . . , y

−ρ

, x

) = 0, s = r

− ρ + 1, . . . , κ, where y

denotes the new variables. We change variables by ﬂattening the latter equations;
the vector ﬁelds in the new variables still have the form (3.2) and (3.3), after a
non-singular substitution.

Applying the iterated rectiﬁcation procedure of the previous section to the

vector ﬁelds X

−ρ+1

, . . . , X

so far we have proved the

Theorem

3.2.4. There exists a real analytic change of variables such that the

vector ﬁelds X

, . . . , X

assume the form

= ∂

n
k=

κ+1

(x)∂

i = 1, . . . , r

− ρ,

−ρ+1

= ∂

−ρ+1

= ∂

n
k=

κ+1

−1

−ρ+1

()
ik

(x)∂

i = r

− ρ + 2, . . . , κ,

n
k=

κ+1

(x)∂

j =

κ + 1, . . . , N.

Here the a

∈ C

(Ω) and a

(0) = 0, for j = 1, . . . , r

− ρ, κ + 1, . . . , N, k =

κ + 1, . . . , n. Furthermore

N
j=

κ+1

(x, ξ)

= 0 if and only if x

−ρ+1=···=x

;

we have that

(3.2.7)

rank[

, X

}]

i,j=1,...,r

−ρ

= r

− ρ

and

π(Σ) =

−ρ+1

· · · = x

= 0

Finally, the equations deﬁning the characteristic manifold are

(3.2.8)

(x, ξ) = ξ

κ+1

, . . . , x

−ρ

, 0, . . . , 0, x

)ξ

= 0,

for i = 1, . . . , r

− ρ, and

(3.2.9)

−ρ+1

· · · = x

= 0,

−ρ+1

· · · = ξ

= 0.

If r = ρ we are done and the result is completely analogous to that of the

preceding section (Lemma 3.1.4 and Theorem 3.1.5.)

Obviously, if r > ρ, there are pairs of symplectically conjugated coordinates

among those in (3.2.8). We shall need to identify the tangent variables to the

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

stratum under exam, for this purpose we need also to perform a canonical transfor-
mation. As a result our vector ﬁelds given as in Theorem 3.2.4 will turn into ﬁrst
order pseudodiﬀerential operators. Let ψ

as in (3.2.8). We have that the matrix

[

{ψ

(x, ξ), ψ

(x, ξ)

}]

i,j=1,...,r

−ρ

has rank r

− ρ on Σ. We may always assume that there exists an index j

∈

{2, . . . , r − ρ} such that

{ψ

, ψ

, . . . , x

−ρ

, x

, ξ

)

} =

∂ψ

∂x

, . . . , x

−ρ

, x

, ξ

)

= 0.

Hence

, . . . , x

−ρ

, x

, ξ

)

= e

, . . . , x

−ρ

, x

, ξ

)

+ ˜

, . . . , x

−ρ

, x

, ξ

)

Here e

and ˜

are homogeneous of degree 1 and 0 respectively. Moreover e

= 0.

We may now use Darboux theorem to perform a homogeneous canonical transforma-
tion in the variables (x

, . . . , x

−ρ

, x

) and their duals, leaving the other coordinates

unchanged, such that

(3.2.10)

= ψ

= x

+ ˜

, . . . , x

−ρ

, x

, ξ

The vector ﬁelds X

and X

become

+ Y

(x, D

, D

, . . . , x

−ρ

, x

, D

+ Y

(x, D

, D

where Y

and Y

are pseudodiﬀerential operators homogeneous of degree 1 whose

symbol vanishes on the surface x

−ρ+1

· · · = x

= 0.

Now, recalling that

we are working microlocally near the point ρ

= (0, e

), we can perform a linear

substitution in the operators X

and X

and we may reduce the operator to the

following form

(x, D) = D

+ Y

(x, D

, D

(x, D) = x

+ Y

(x, D

, D

where Y

and Y

are pseudodiﬀerential operators homogeneous of degree 1 whose

symbol vanishes on the surface x

−ρ+1

· · · = x

= 0.

We denote by ψ

, . . . , ψ

the functions in (3.2.8), computed in the new coor-

dinates, where ψ

and ψ

have been left out. Moreover we may suppose that the

matrix [

{ψ

, ψ

}]

i,j=3,...,r

−ρ

has maximal rank r

−ρ−2. The latter are real analytic

functions of (x

, . . . , x

−ρ

, x

, ξ

, . . . , ξ

−ρ

, ξ

First of all we restrict ψ

, . . . , ψ

−ρ

to the plane x

= ξ

= 0. Since ψ

linearly independent with the radial vector ﬁeld, we may apply Darboux theorem
and perform a homogeneous canonical transformation such that

= η

Arguing as before we may ﬁnd ψ

, 4

≤ j

≤ r − ρ such that

{ψ

, ψ

, . . . , x

−ρ

, x

, ξ

, . . . , ξ

−ρ

, ξ

)

}

∂ψ

∂x

, . . . , x

−ρ

, x

, ξ

, . . . , ξ

−ρ

, ξ

)

= 0.

3.2. THE SYMPLECTIC CASE OF DEPTH 1

Hence

, . . . , x

−ρ

, x

, ξ

, . . . , ξ

−ρ

, ξ

)

= e

, . . . , x

−ρ

, x

, ξ

, . . . , ξ

−ρ

, ξ

)

+ ˜

, . . . , x

−ρ

, x

, ξ

, . . . , ξ

−ρ

, ξ

)

Here e

and ˜

are homogeneous of degree 1 and 0 respectively. Moreover e

= 0.

We may now use Darboux theorem to perform a homogeneous canonical trans-

formation in the variables (x

, . . . , x

−ρ

, x

) and their duals, leaving the other co-

ordinates unchanged, such that

(3.2.11)

= ξ

= x

+ ˜

, . . . , x

−ρ

, x

, ξ

, . . . , ξ

−ρ

, ξ

The vector ﬁelds X

and X

become

+ Y

(x, D

, . . . , D

−ρ

, D

) + A

, . . . , x

−ρ

, x

, D

, . . . , D

−ρ

, D

, . . . , x

−ρ

, x

, D

, . . . , D

−ρ

, D

+ Y

(x, D

, . . . , D

−ρ

, D

)

+ B

, . . . , x

−ρ

, x

, D

, . . . , D

−ρ

, D

where Y

, Y

are pseudodiﬀerential operators homogeneous of degree 1 vanishing

on the surface x

−ρ+1

· · · = x

= 0, A

and B

are pseudodiﬀerential operators,

homogeneous of degree 1, whose symbol vanishes on the surface x

= ξ

= 0. Now,

performing a non singular substitution in the pseudodiﬀerential operators X

, X

and X

we can rewrite the operators in the following form

(x, D) = D

+ Y

(x, D

, D

(x, D) = x

+ Y

(x, D

, D

(x, D) = D

+ Y

(x, D

, . . . , D

−ρ

, D

(x, D) = x

+ Y

(x, D

, . . . , D

−ρ

, D

where Y

, Y

and Y

are pseudodiﬀerential operators homogeneous of degree

1 whose symbol vanishes on the surface x

−ρ+1

· · · = x

= 0.

The argument can be iterated to get a canonical form for X

, . . . , X

−ρ

Let us now consider X

−ρ+1

, . . . , X

. Since the canonical transformations per-

formed above involve the variables x

, . . . , x

−ρ

, x

and their duals, the symbols of

the vector ﬁelds X

−ρ+1

, . . . , X

take the form

−ρ+1

+ a

(x, ξ

, . . . , ξ

−ρ

, ξ

where a

denotes a homogeneous symbol of degree 1 and j = r

− ρ + 2, . . . , κ. Let

x = (x

−ρ)/2+1

, . . . , x

−ρ

, x

)

and denote by

(˜

x, ˜

ξ),

j = r

− ρ + 2, . . . , κ

the restriction of a

to the set

{(x

= . . . = x

−ρ)/2

= 0, x

−ρ+1

= . . . = x

= 0, ξ

= . . . = ξ

−ρ)/2

= 0

Set χ

−ρ+1

= ξ

−ρ+1

and

(ξ

−ρ+1

, . . . , ξ

, ˜

x, ˜

ξ) = ξ

+ ˜

(˜

x, ˜

ξ),

j = r

− ρ + 2, . . . , κ.

It is evident that the functions x

−ρ+1

, . . . , x

and χ

−ρ+1

, . . . , χ

satisfy the as-

sumptions of the Darboux theorem, so that there is a canonical transformation

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

involving the variables (x

−ρ+1

, . . . , x

, ˜

x) and their duals that turns the χ

into

, j = r

− ρ + 1, . . . , κ.

Recalling that we are also allowed to perform a linear nonsingular substitution

in the vector ﬁelds, we may state the

Theorem

3.2.5. There exists a real analytic canonical tranformation deﬁned

in U , conic neighborhood of (0, e

)

∈ Σ, and a linear nonsingular substitution in

the vector ﬁelds, such that the vector ﬁelds in Theorem 3.2.4 take the form

(x, D)

+ Y

(x, D)

j = 1, . . . ,

− ρ

, r

− ρ + 1, . . . , κ,

−ρ)/2+j

(x, D)

+ Y

−ρ)/2+j

(x, D)

j = 1, . . . ,

− ρ

(3.2.12)

(x, D) = Y

(x, D),

j =

κ + 1, . . . , N.

Here Y

−ρ+1

(x, ξ)

≡ 0 and the symbols Y

(x, ξ), = 1, . . . , r

− ρ, r − ρ + 2, . . . , N,

vanish on the surface x

−ρ+1

· · · = x

= 0, are homogeneous of degree 1 w. r. t.

ξ.

Furthermore we have that

N
j=

κ+1

(x, ξ)

= 0 if and only if x

−ρ+1

· · · =

= 0 and the characteristic manifold is given by the equations

(3.2.13)

Σ =

(x, ξ)

| x

= ξ

= 0, i = 1, . . . ,

− ρ

, r

− ρ + 1, . . . , κ

We complete this section with an example of vector ﬁelds satisfying the as-

sumptions:

Example

3.2.6. Let

(x, D) = D

+ x

(x, D) = D

(x, D) = x

where h denotes a positive integer. We note explicitly that the H¨

ormander bracket

condition is satisﬁed. Furthermore, we have that n = 4,

κ = 3, m = 4, r = 2 and

ρ = m

− 2κ + r = 0.

Example

3.2.7. Let

(x, D)

+ x

(x, D)

+ x

(x, D)

where h denotes a positive integer. Also in this case the H¨

ormander bracket condi-

tion is satisﬁed and we have that n = 5,

κ = 4, m = 6, r = 4 and ρ = 2.

3.3. The Nonsymplectic Case of Depth > 1

We make the assumption

(NonSympl) The characteristic set Σ is a real analytic submanifold of T

∗

{0}. The rank of the symplectic form restricted to Σ is constant
and all Poisson brackets

, X

} vanish identically. Moreover the

projection π onto the base maps the foliation of Σ injectively onto
a foliation of the same dimension of π(Σ).

3.3. THE NONSYMPLECTIC CASE OF DEPTH > 1

The latter assumption is technical; it implies that the leaves are transverse to the
ﬁber and prevents a possible propagation of the regularity along the ﬁbers.

Let as before

κ denote the number of linearly independent vector ﬁelds at the

origin. We may then choose the coordinates in such a way that the vector ﬁelds
have the form

(3.3.1)

∂

∂x

κ+1

(x)

∂

∂x

i = 1, . . . ,

κ;

(3.3.2)

κ+1

(x)

∂

∂x

j =

κ + 1, . . . , N;

here a

∈ C

(Ω) and a

(0) = 0 for all j = 1, . . . , N , k =

κ + 1, . . . , n.

Denote by σ the symplectic form in T

∗

Ω and by F the foliation in Σ. Let be

the dimension of the leaves in F . Then rank σ

|Σ

= 2n

− 2h − 2. We have the

Proposition

3.3.1. Under the assumption (NonSympl), the restriction to Σ

of the base projection π has constant rank. Moreover its rank is equal to n

− h and

2h + = codim Σ.

Proof.

Because of (NonSympl) and the above deﬁnitions, at any point γ

∈ Σ

we have that = dim T

∩ T

Σ and 2h = dim T

Σ/(T

∩ T

Σ), where T

denotes the orthogonal of T

Σ with respect to the symplectic form. Arguing as in

the proof of Proposition 3.2.1 we can show that the rank of π

|Σ

is equal to n

−h.

Remark

3.3.2. We have that dim Σ = 2(n

− h) − , and a direct computation

shows that dim Σ = 2n

− κ − h; then we obtain that κ = h + .

Corollary

3.3.3. We have the following

(i) π(Σ) is a real analytic submanifold of Ω of dimension n

− h.

(ii) The vector ﬁelds X

, with j =

κ +1, . . . , N, vanish at every point of π(Σ).

(iii) The rank of the system of vector ﬁelds X

, . . . , X

is equal to

κ at any

point of π(Σ).

Proof.

The only assertion to prove is the second, the other being trivial. The

proof of the second statement is done exactly as the proof of the second statement
in Corollary 3.1.2.

From assertion (i) in the above Corollary we deduce that π(Σ) is deﬁned by a

set of equations of the form ϕ

(x) = 0, for j = 1, . . . , h. Σ is then deﬁned by the

equations ξ

n
k=

κ+1

(x)ξ

= 0, i = 1, . . . ,

κ, and ϕ

(x) = 0, j = 1, . . . , h.

Possibly renaming the variables we may assume that

det

∂(ϕ

, . . . , ϕ

)

∂(x

, . . . , x

)

(0)

= 0.

We may thus rewrite ϕ

(x) = x

+ ˜

h+1

, . . . , x

for j = 1, . . . , h.

Let us now perform the change of variables:

(3.3.3)

+ ϕ

h+1

, . . . , x

≤ j ≤ h

h + 1

≤ i ≤ n

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

We have that ∂

= ∂

, for j = 1, . . . , h, while ∂

= ∂

h
s=1

∂

, for

j = h + 1, . . . , n.

As a consequence,

= ∂

s=1

κ+1

∂

κ+1

∂

for 1

≤ i ≤ h;

= ∂

s=1

∂

κ+1

∂

κ+1

∂

for h + 1

≤ i ≤ h + = κ; ﬁnally

s=1

κ+1

∂

κ+1

∂

for

κ+1 ≤ j ≤ N. We point out explicitly that in the expressions above a

(0) = 0,

for 1

≤ j ≤ N, κ + 1 ≤ k ≤ n.

Let us look at the ﬁrst

κ vector ﬁelds. Forgetting about the fact that the

derivatives are not in the self-adjoint form, we observe that, denoting by η

(η

, . . . , η

), η

= (η

h+1

, . . . , η

) the covariables of the ﬁrst

κ derivatives, modulo

terms vanishing at the origin the ﬁrst

κ vector ﬁelds are given by

∂ϕ

∂x

We conclude thus that there is a linear nonsingular substitution allowing us to write
the vector ﬁelds in the form

(3.3.4)

∂

n
k=

κ+1

(y)∂

i = 1, . . . ,

κ;

n
k=

κ+1

(y)∂

j =

κ + 1, . . . , N,

where a

(0) = 0, for j = 1, . . . , N , k =

κ + 1, . . . , n. Moreover the equations of Σ

are

(3.3.5)

n
k=

κ+1

(y)η

= 0

i = 1, . . . , h +

= y

= . . . = y

= 0.

We now apply to the vector ﬁelds X

, . . . , X

the argument of Lemma 3.1.4, starting

by rectifying X

. We emphasize that in this process the coordinates x

, . . . , x

are

unchanged (here we reverted to the x notation.) We may thus write

(3.3.6)

∂

n
k=

κ+1

−1

m=1

(m)
ik

(x)

∂

i = 1, . . . , h;

∂

n
k=

κ+1

(x)∂

i = h + 1, . . . , h + ;

n
k=

κ+1

(x)∂

j =

κ + 1, . . . , N.

Let us denote by X

(x, ξ), X

(x, ξ) the vector-valued functions whose compo-

nents are the symbols of X

, i = 1, . . . , h, and of X

, j = h + 1, . . . , h + , re-

spectively. Let us furthermore write x = (x

, x

), where x

= (x

, . . . , x

= (x

h+1

, . . . , x

), x

= (x

h++1

, . . . , x

3.4. THE NONSYMPLECTIC CASE OF DEPTH 1

The equations of Σ then may be written as x

= 0, X

(0, x

, x

, ξ) = 0, and

(0, x

, x

, ξ) = 0. Note that X

(0, x

, x

, ξ) = ξ

, where ξ

= (ξ

, . . . , ξ

It is well known then that the block matrix

⎡
⎢

⎢

⎣

, X

} {X

, X

} I

, X

} {X

, X

} 0

−I

⎤
⎥

⎥

⎦

has a rank equal to the corank of σ

|Σ

, which is 2h. Here e.g.

, X

} denotes the

matrix of all the Poisson brackets between the components of X

and those of X

By I

we denoted the h

× h identity matrix. This implies easily that

, X

}(0, x

, x

, ξ)

≡ 0.

We may apply Frobenius theorem in the (x

, x

) variables, leaving unchanged the

origin as well as the x

variables, and conclude that

(0, x

, ξ) = (ξ

h+1

, . . . , ξ

We obtain thus for the vector ﬁelds the following form
(3.3.7)

∂

n
k=

κ+1

−1

m=1

(m)
ik

(x)

∂

i = 1, . . . , h;

∂

n
k=

κ+1

h
m=1

(m)
ik

(x)

∂

i = h + 1, . . . , h + ;

n
k=

κ+1

(x)∂

j =

κ + 1, . . . , N,

where a

(0) = 0, for j =

κ+1, . . . , N, k = κ+1, . . . , n and

N
j=

κ+1

(x, ξ)

= 0

if and only if x

· · · = x

= 0.

From this point on we may argue as in Section 4.1 and thus obtain the proof

of the following

Theorem

3.3.4. If (NonSympl) holds, then the characteristic set of the vector

ﬁelds X

, . . . , X

, has the form

(3.3.8) Σ =

{(x, ξ) ∈ T

∗

\ {0} | x

= 0, i = 1, . . . , h, ξ

= 0, i = 1, . . . , h +

where denotes the dimension of the leaves and rank σ

|Σ

= 2n

− 2h − 2.

Moreover the vector ﬁelds can be written as in Equation ( 3.3.7), where a

= 0

when x

= 0, for j =

κ + 1, . . . , N, k = κ + 1, . . . , n and κ = h + .

We complete this section with an example of vector ﬁelds satisfying the as-

sumptions:

Example

3.3.5. Let

(x, D) = D

(x, D) = x

m
1

where m denotes an integer > 1. We note explicitly that the H¨

ormander bracket

condition is satisﬁed.

3.4. The Nonsymplectic Case of Depth 1

We make the assumption

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

(NonSympl-1) The characteristic set Σ is a real analytic submanifold of T

∗

{0}. The rank of the symplectic form restricted to Σ is constant
and there exists a Poisson bracket

, X

} diﬀerent from zero.

Moreover the projection π onto the base maps the foliation of Σ
injectively onto a foliation of the same dimension of π(Σ).

We may assume that the vector ﬁelds have the form (3.2), (3.3) and retain the

notation of the preceding section. Let

r = rank [

, X

}]

i,j=1,...,

(0, e

Denote by σ the symplectic form in T

∗

Ω and by F the foliation in Σ. Let be the

dimension of the leaves in F . Deﬁne m = codim Σ and set

(3.4.1)

h =

−

and

ρ = m

− 2κ + r + .

Then rank σ

|Σ

= 2n

− 2h − 2. We have the

Proposition

3.4.1. Under the assumption (NonSympl-1), the restriction to

Σ of the base projection π has constant rank equal to n

− κ + r − ρ + . Moreover

ρ is an integer and ρ

∈ [0, r].

Proof.

By Lemma 3.2.3, the rank of π

is n

− m + κ. Furthermore, using

(3.4.1), we deduce that

− m + κ = n − (2κ + ρ − r − ) + κ = n − κ + r − ρ + .

In order to show that ρ

≤ r it suﬃces to verify that

(3.4.2)

− 2κ + ≤ 0.

We need the following general lemma:

Lemma

3.4.2. Let M be a submanifold of T

∗

\ {0}, deﬁned by the inde-

pendent equations ψ

(x, ξ) =

· · · = ψ

(x, ξ) = 0. Let ρ

∈ M and denote by

A = [

{ψ

, ψ

}(ρ

)]

i,j=1,...,m

. Assume that the symplectic form, σ, has constant

rank on M and denote by F

the Hamilton leaf of M through ρ

. Then

⎧

⎨
⎩

j=1

(ρ

)

| v = (v

, . . . , v

)

∈ ker A

⎫

⎬
⎭

Proof of Lemma

3.4.2. If v

∈ ker A then

m
j=1

σ(H

, H

= 0, for every

i. Hence

m
j=1

∈ [H

, . . . , H

]

(the symplectic orthogonal of the space

generated by the H

.) This allows us to conclude.

Suppose that

Σ =

(x, ξ)

∈ T

∗

\ {0} | X

(x, ξ) = 0, i = 1, . . . ,

κ;

(x, ξ

) = 0, j = 1, . . . , m

− κ

Let

A be the matrix

A =

{X, X} {X, ϕ}

{ϕ, X} {ϕ, ϕ}

(0, e

Then

(3.4.3)

rank

A = m − .

3.4. THE NONSYMPLECTIC CASE OF DEPTH 1

Lemma

3.4.3.

{ϕ, ϕ}(0, e

) = 0.

Proof of Lemma

3.4.3. We have that, for every ξ

= 0, (0; 0, ξ

)

∈ Σ. Thus

(3.4.4)

(0, ξ

) = 0

for every ξ

= 0 and i = 1, . . . , m − κ,

and the conclusion follows.

Using the above lemma we deduce that

A =

{X, X} {X, ϕ}

{ϕ, X}

(0, e

Lemma

3.4.4. The columns of the

κ × (m − κ) matrix {X, ϕ} are linearly

independent. In particular, we have that rank

{X, ϕ} = m − κ ≤ κ.

Proof of Lemma

3.4.4. Lemma 3.4.2 yields that T

(0,e

)

is the kernel of the

matrix A. Suppose that the columns of the block

{X, ϕ} are linearly dependent.

It follows that there exist v

, . . . , v

−κ

∈ R, with

−κ

i=+1

= 0, such that

−κ

i=1

(0, e

)

∈ T

(0,e

)

Recalling (3.4.4), we deduce that

−κ

i=1

(0, e

) is a vector tangent to the ﬁber

and this fact contradicts the fact that, according to Assumption (NonSympl-1),
the leaves of F project onto the base space. The last assertion in the statement
of the lemma is a trivial consequence of the above arguments. This proves the
lemma.

Using the form of the matrix

A it is easy to see that

2 rank

{X, ϕ} ≤ rank A ≤ rank{X, X} + 2 rank{X, ϕ}

The conclusion of Lemma 3.4.4 and (3.4.3) imply that the inequalities above can
be rewritten as

2(m

− κ) ≤ m − ≤ r + 2m − 2κ.

Since ρ = m

− 2κ + r + , we deduce that ρ ≥ 0 and that 2(m − κ) ≤ m − (i.e.

Formula (3.4.2) holds).

This proves Proposition 3.4.1.

Remark

3.4.5. We point out that, as a consequence of Proposition 3.4.1, r

− ρ

is constant on Σ, since

κ and are constant on Σ.

Notice that, arguing as in the proof of Lemma 3.1.3, we may conclude that

(3.4.5)

Σ =

(x, ξ)

| X

(x, ξ) = 0, i = 1, . . . ,

κ, ψ

(x) = 0,

j = 1, . . . ,

κ − r + ρ −

denoting the equations of π(Σ). Arguing as in Section 4.2, we may suppose that

(3.4.6)

A =

⎡
⎢

⎣

[

, X

}]

i,j=1,...,r

−

⎤
⎥

⎦ .

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

Here

A = [∂

]

i=1,...,r
k=1,...,

κ−r+ρ−

(0, e

B = [∂

]

i=r+1,...,

k=1,...,

κ−r+ρ−

(0, e

Recalling that rank

A = m − and that, by Lemma 3.4.4,

rank[

−

B] = m

− κ

we obtain that

rank

[

, X

}]

i,j=1,...,r

κ − .

Hence we deduce that

rank B =

κ − − r

and

rank A = m

− κ − (κ − − r) = ρ.

We may add or subtract columns and rows in the matrix (3.2.6) (possibly changing
the labels of the ﬁrst r coordinates) obtaining

⎡
⎢

⎣

∗

−

⎤
⎥

⎦ ,

with

C =

[

, X

}]

i,j=1,...,r

−ρ

Consider now the matrix

[

, ψ

}]

i=r

−ρ+1,...,κ

k=1,...,

κ−r+ρ−

(0, e

);

this matrix has (maximal) rank

κ − r + ρ − . On the other hand

, ψ

}(0, e

) =

∂ψ

∂x

(0).

Possibly changing the labels to x

−ρ+1

, . . . , x

and thus also to the ﬁelds X

−ρ+1

. . . , X

, which amounts to performing a nonsingular substitution, we may assume

that

det

∂(ψ

, . . . , ψ

κ−r+ρ−

)

∂(x

−ρ+1

, . . . , x

κ−

)

(0)

= 0.

Using the implicit function theorem we ﬁnd that, modulo elliptic factors,

= x

i+r

−ρ

+ ˜

, . . . , x

−ρ

, x

κ−+1

, . . . x

i = 1, . . . ,

κ − r + ρ − . We change variables by ﬂattening the latter equations:

i+r

−ρ

= ψ

i = 1, . . . ,

κ − r + ρ −

= x

j = 1, . . . , r

− ρ, κ − + 1, . . . , n.

We point out that the vector ﬁelds in the new variables still have the form (3.2)
and (3.3), after a nonsingular substitution.

3.4. THE NONSYMPLECTIC CASE OF DEPTH 1

Applying the iterated rectiﬁcation procedure of the previous section to the

vector ﬁelds X

−ρ+1

, . . . , X

κ−

we obtain that the symbols of the vector ﬁelds have

the form

(x, ξ)

n
i=

κ+1

(x)ξ

i = 1, . . . , r

− ρ,

−ρ+1

(x, ξ)

−ρ+1

(x, ξ)

n
k=

κ+1

−1

−ρ+1

(x)ξ

j = r

− ρ + 2, . . . , κ −

(x, ξ)

n
i=

κ+1

(x)ξ

i =

κ − + 1, . . . κ,

(x, ξ)

n
k=

κ+1

(x)ξ

i =

κ + 1, . . . , N,

with a

(0) = 0 and

π(Σ) =

{x | x

−ρ+1

= . . . = x

κ−

= 0

Let us denote by X

(x, ξ), X

(x, ξ) and X

(x, ξ) the vector-valued functions

whose components are the symbols of X

, i = 1, . . . , r

− ρ, X

, j = r

− ρ +

1, . . . ,

κ − , and of X

, k =

κ − + 1, . . . , n, respectively. Let us further-

more write x = (x

, x

), where x

= (x

, . . . , x

−ρ

), x

= (x

−ρ+1

, . . . , x

κ−

= (x

κ−+1

, . . . , x

The equations of Σ then may be written as x

= 0,

, 0, x

, ξ) = 0, X

, 0, x

, ξ) = ξ

= 0 and X

, 0, x

, ξ) = 0. Set

B =

⎡
⎢

⎢

⎣

, X

} {X

, X

} {X

, X

} {X

, x

}

, X

} {X

, X

} {X

, X

} {X

, x

}

, X

} {X

, X

} {X

, X

} {X

, x

}

, X

}

, X

}

, X

}

, x

}

⎤
⎥

⎥

⎦

We claim that

, X

}(x

, 0, x

, ξ) = 0.

Indeed, by construction, rank

, X

} = r − ρ and {X

, x

} = I, the (κ − −

r + ρ)

× (κ − − r + ρ) identity matrix. Hence

rank

⎡
⎢

⎢

⎣

, X

}

−I 0 0

⎤
⎥

⎥

⎦

= r

− ρ + 2(κ − − r + ρ) = m − .

Recalling that rank

B = m − we deduce that {X

, X

} = 0.

As a consequence of our claim, we may apply the Frobenius theorem in the

, x

) variables (leaving unchanged the origin as well as the x

variables) and

conclude that

, 0, x

, ξ) = (ξ

κ−+1

, . . . , ξ

We proved the

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

Theorem

3.4.6. There exists a real analytic change of variables such that the

vector ﬁelds X

, . . . , X

assume the form

= ∂

n
k=

κ+1

(x)∂

i = 1, . . . , r

− ρ

−ρ+1

= ∂

−ρ+1

= ∂

n
k=

κ+1

−1

−ρ+1

(x)∂

j = r

− ρ + 2, . . . , κ −

= ∂

n
k=

κ+1

κ−
=r

−ρ+1

(x)∂

j =

κ − + 1, . . . , κ

n
k=

κ+1

(x)∂

j =

κ + 1, . . . , N.

Here the a

∈ C

(Ω) and a

(0) = 0, for j = 1, . . . , N , k =

κ + 1, . . . , n. Further-

more, we have that

rank[

, X

}]

i,j=1,...,r

−ρ

= r

− ρ

and

π(Σ) =

−ρ+1

= . . . = x

κ−

= 0

Finally, we have that

N
j=

κ+1

(x, ξ)

= 0 if and only if x

−ρ+1

· · · = x

κ−

0 and the characteristic manifold is given by the equations

(x, ξ) = ξ

κ+1

, . . . , x

−ρ

, 0, . . . , 0, x

)ξ

= 0,

for i = 1, . . . , r

− ρ, and

−ρ+1

= . . . = x

κ−

= 0,

−ρ+1

= . . . = ξ

= 0.

In order to identify the variables tangent to the stratum we argue as in the

symplectic case. For this purpose, let us now consider the symbols of the vector
ﬁelds X

, . . . , X

−ρ

. Arguing as in section 4.2, we ﬁnd a standard pseudodiﬀerential

form for X

, . . . , X

−ρ

as well as a simple form for the corresponding equations of

Σ. In fact we have the following form for X

, . . . , X

−ρ

(x, D)

+ Y

(x, D),

(3.4.7)

−ρ)/2+j

(x, D)

+ Y

−ρ)/2+j

(x, D),

j = 1, . . . ,

−ρ

, where

• The symbols Y

(x, ξ), = 2, . . . , r, vanish on the surface x

−ρ+1

· · · =

κ−

= 0, are homogeneous of degree 1 w. r. t. ξ.

Furthermore the characteristic manifold is given by the equations

(3.4.8)

Σ =

(x, ξ)

| x

= ξ

= 0, i = 1, . . . ,

− ρ

= 0, j = r

− ρ + 1, . . . , κ − , X

−ρ+1

(x, ξ) =

· · · = X

(x, ξ) = 0

Let us now consider the symbols X

−ρ+1

(x, ξ), . . . , X

κ−

(x, ξ). We point out ex-

plicitly that the performed canonical tranformation does not involve the variables
x

−ρ+1

, . . . , x

κ−

and their duals.

3.4. THE NONSYMPLECTIC CASE OF DEPTH 1

The argument for X

, j = r

− ρ + 1, . . . , κ − , is similar to the one in Section

4.2; however, because of the presence of a Hamiltonian leaf in the present case, we
need a careful control on the variables involved in the canonical tranformations.

We argue recursively on the X

Let us start with X

−ρ+1

We restrict

−ρ+1

(x, ξ) to x

= 0, for i

∈ {1, . . . , (r −ρ)/2, r −ρ+1, . . . , κ −}, thus obtaining

the symbol

−ρ+1

(x, ξ) = ξ

−ρ+1

+ a

−ρ+1

κ−+1

, . . . , x

, ˜

x, ξ

κ−+1

, . . . , ξ

, ˜

ξ),

where

x = (x

−ρ)/2+1

, . . . , x

−ρ

, x

Since x

, . . . , x

−ρ)/2

, ξ

, . . . , ξ

−ρ)/2

, x

−ρ+1

, ψ

−ρ+1

(x, ξ) are a set of canonically

conjugate coordinates, noncollinear to the radial ﬁeld, we may apply Darboux
theorem and obtain a set of canonical coordinates. In particular the coordinates
x

, . . . , x

−ρ)/2

and their duals as well as x

−ρ+1

are not changed by the canonical

tranformation.

In these coordinates the ﬁrst r

− ρ symbols retain the same form as in (3.4.7),

while

−ρ+1

(x, D) = D

−ρ+1

+ Y

−ρ+1

(x, D),

where Y

−ρ+1

(x, ξ) is a symbol homogeneous of degree 1 w.r.t. ξ, vanishing where

· · · = x

−ρ)/2

= 0 = x

−ρ+1

· · · = x

κ−

and ξ

· · · = ξ

−ρ)/2

= 0.

Furthermore the characteristic manifold is given by the equations

(3.4.9)

Σ =

(x, ξ)

| x

= ξ

= 0, i = 1, . . . ,

− ρ

, r

− ρ + 1,

= 0, j = r

− ρ + 2, . . . , κ − , X

−ρ+2

(x, ξ) =

· · · = X

(x, ξ) = 0

The argument can be iterated to give the form (3.4.7) for the symbols X

, . . . , X

−ρ

and the following form for

(3.4.10)

(x, D) = D

+ Y

(x, D),

j = r

− ρ + 1, . . . , κ − .

Here Y

(x, ξ) is a symbol homogeneous of degree 1 w.r.t. ξ, vanishing where x

· · · = x

−ρ)/2

= 0 = x

−ρ+1

· · · = x

κ−

and ξ

· · · = ξ

−ρ)/2

= 0.

Furthermore the characteristic manifold is given by the equations

(3.4.11)

Σ =

(x, ξ)

| x

= ξ

= 0, i = 1, . . . ,

− ρ

, r

− ρ + 1, . . . , κ − ,

κ−+1

(x, ξ) = 0, . . . , X

(x, ξ) = 0

Let us now consider the symbols X

κ−+1

(x, ξ), . . . , X

(x, ξ). We restrict X

(x, ξ)

to x

= ξ

= 0, for i

∈ {1, . . . , (r − ρ)/2, r − ρ + 1, . . . , κ − }, thus obtaining the

symbol

(x, ξ) = ξ

+ a

κ−+1

, . . . , x

, ˜

x, ˜

ξ),

j =

κ − + 1, . . . , κ.

Here we remark that the symbol a

does not depend on the covariables ξ

, j

∈

{κ − + 1, . . . , κ}.

3. STANDARD FORMS FOR A SYSTEM OF VECTOR FIELDS

Now, because of Proposition 3.4.1, or rather its proof, the Poisson brackets

{ψ

, ψ

} are identically zero in a neighborhood of the characteristic manifold. This

involutivity property implies that there is a homogeneous canonical transformation,
which is the identity in the variables x

, ξ

, i

∈ {1, . . . , (r−ρ)/2, r−ρ+1, . . . , κ−},

such that ψ

becomes the coordinate ξ

Recalling that we are also allowed to perform a linear non singular substitution

in the vector ﬁelds, we obtain a proof of the

Theorem

3.4.7. There exists a real analytic, homogeneous canonical trans-

formation deﬁned in U , conic neighborhood of (0, e

)

∈ Σ, and a linear non singular

substitution in the vector ﬁelds, such that the vector ﬁelds in Theorem 3.4.6 take
the form

(x, D)

+ Y

(x, D)

j = 1, . . . ,

− ρ

(3.4.12)

−ρ

(x, D)

+ Y

−ρ

(x, D)

j = 1, . . . ,

− ρ

(x, D)

+ Y

(x, D)

j = r

− ρ + 1, . . . , κ

(3.4.13)

(x, D) = Y

(x, D),

j =

κ + 1, . . . , N.

Here

• The symbols Y

(x, ξ), = 2, . . . , N , vanish on the surface x

−ρ+1

· · · =

κ−

= 0, are homogeneous of degree 1 w. r. t. ξ.

Furthermore we have that

N
j=

κ+1

(x, ξ)

= 0 if and only if x

−ρ+1

· · · =

κ−

= 0 and the characteristic manifold is given by the equations

(3.4.14)

Σ =

(x, ξ)

| x

= ξ

= 0, i = 1, . . . ,

− ρ

, r

− ρ + 1, . . . , κ − ,

= 0, j =

κ − + 1, . . . , κ

We complete this section with an example of vector ﬁelds satisfying the as-

sumptions:

Example

3.4.8. Let

(x, D) = D

+ x

(x, D) = D

We note explicitly that the H¨

ormander bracket condition is satisﬁed. Furthermore,

we have that n = 4,

κ = m = 3, r = 2, = 1, h = 1 and ρ = 0.

Example

3.4.9. Let

(x, D) = D

+ x

(x, D) = D

(x, D) = x

(x, D) = D

+ x

(x, D) = x

s
4

(x, D) = D

where s is a positive integer. We have that n = 6,

κ = 5, r = 4, m = 7, = 1,

h = 3 and ρ = 2.

CHAPTER 4

Nested Strata

In this chapter we consider the case when the given vector ﬁelds deﬁne a non-

trivial Poisson-Treves stratiﬁcation, i.e. the stratiﬁcation has more than one stra-
tum. For a deﬁnition of Poisson-Treves stratiﬁcation we refer to Chapter 2 (see
also [29] and [5].)

In this chapter we consider nested strata of the stratiﬁcation, when each stra-

tum has depth bigger than 1. The canonical forms of Sections 3.1 and 3.3 then
apply. We point out that our method can be applied also to particular classes
exhibiting strata of depth 1; we mention some such cases in Chapter 9.

We assume that

(NS) Each stratum of the stratiﬁcation associated to the vector ﬁelds is either a

symplectic manifold or has a Hamilton leaf that projects injectively onto
a submanifold of the same dimension in the base. Moreover the restriction
to it of the symplectic form has constant rank.

Since the Poisson brackets of symbols of vector ﬁelds are symbols of vector

ﬁelds, we may suppose that Σ

, i.e. a p-th stratum in the stratiﬁcation, is deﬁned

by the vanishing of N

symbols of vector ﬁelds, deﬁned in Ω, of the form X

(x, ξ),

|I| ≤ ν

, for a suitable index ν

. Moreover there exists a multiindex, J ,

|J| =

+ 1, such that X

is elliptic in Ω. Evidently Σ

is a real analytic submanifold of

∗

\ {0}, σ

has constant rank and Σ

is connected.

Obviously N

≥ N and N

> N , if p > 1. Moreover we may assume that

the ﬁrst N vector ﬁelds among the N

vector ﬁelds are the original vector ﬁelds

, . . . , X

Denote by

the number of the linearly independent vector ﬁelds at the origin.

Clearly

≥ κ.

The stratum Σ

p+1

is a real analytic submanifold of Char(X

, . . . , X

) where

all the brackets X

with

|I| ≤ ν

p+1

vanish and such that there is a bracket X

|J| = ν

p+1

+ 1, which is elliptic. We point out that, because of our assumptions,

p+1

> ν

+ 1.

We may thus state the following theorem, whose proof is an application of the

arguments of Sections 3.1, 3.3.

Theorem

4.10. Let X

, . . . , X

be vector ﬁelds as in Section 3 and assume

that the associated stratiﬁcation has more than one stratum with diﬀerent depths as
well as (NS). Denote by Σ

, Σ

p+1

⊂ ∂Σ

, two strata at depth ν

, ν

p+1

respectively,

p+1

> ν

+ 1.

4. NESTED STRATA

Assume that

{(x, ξ) ∈ T

∗

\ {0} | ξ

· · · = ξ

= 0,

· · · = x

= 0,

≤ k

where

, k

are suitable positive integers.

Then there exists a real analytic (C

) change of variables such that

p+1

{(x, ξ) ∈ T

∗

\ {0} | ξ

· · · = ξ

p+1

= 0,

· · · = x

p+1

= 0,

p+1

≤ k

p+1

, k

p+1

≥ k

p+1

≥

p+1

> k

Furthermore the vector ﬁelds in these coordinates retain the form of Lemma 3.1.4
in the symplectic case or of ( 3.3.7) in the nonsymplectic case.

CHAPTER 5

Bargman Pseudodiﬀerential Operators

We are going to use a pseudodiﬀerential and FIO calculus introduced by Grigis

and Sj¨

ostrand in the paper [10]. We recall below the main deﬁnitions and properties

to make this paper self-consistent and readable. For further details we refer to the
paper [10] and to the lecture notes [24].

5.1. The FBI Transform

Deﬁne the FBI transform of a temperate distribution u as

(5.1.1)

T u(x, λ) =

iλϕ(x,y)

u(y)dy,

where λ

≥ 1 is a large parameter, ϕ is a holomorphic function such that det ∂

∂

0, Im ∂

ϕ > 0. Here ∂

denotes the complex derivative with respect to the complex

variable x.

Example

5.1.1. A typical phase function may be ϕ(x, y) =

− y)

To the phase function ϕ there corresponds a weight function Φ(x), deﬁned as

Φ(x) = sup

∈R

− Im ϕ(x, y),

∈ C

We may take a slightly diﬀerent perspective. Let us consider a point (x

, ξ

)

∈ C

and a real-valued real analytic function Φ(x) deﬁned near x

, such that Φ is

strictly plurisubharmonic and

∂

Φ(x

) = ξ

Denote by ψ(x, y) the holomorphic function deﬁned near (x

, ¯

) by

(5.1.2)

ψ(x, ¯

x) = Φ(x).

Because of the plurisubharmonicity of Φ, we have

(5.1.3)

det ∂

∂

= 0

and

(5.1.4)

Re ψ(x, ¯

−

[Φ(x) + Φ(y)]

∼ −|x − y|

To end this chapter we recall the deﬁnition of s–Gevrey wave front set of a distri-
bution. In particular, for s = 1, we obtain the deﬁnition of an analytic wave front
set.

5. BARGMAN PSEUDODIFFERENTIAL OPERATORS

Definition

5.1.2. Let (x

, ξ

)

∈ U ⊂ T

∗

\0. We say that (x

, ξ

) /

∈ W F

(u)

if there exist a neighborhood Ω of x

− iξ

∈ C

and positive constants C

, C

such

that

−λΦ

(x)

T u(x, λ)

| ≤ C

−λ

1/s

for every x

∈ Ω. Here T denotes the classical FBI transform, i.e. that using the

phase function of Example 5.1.1.

Next we need to deﬁne pseudodiﬀerential operators on the FBI side.

5.2. Pseudodiﬀerential Operators

Let λ

≥ 1 be a large positive parameter. We write

D =

∂.

Denote by q(x, ξ, λ) an analytic classical symbol and by Q(x, ˜

D, λ) the formal clas-

sical pseudodiﬀerential operator associated to q.

Using “Kuranishi’s trick” one may represent Q(x, ˜

D, λ) as

(5.2.1)

Qu(x, λ) =

2iπ

(

2λ(ψ(x,θ)

−ψ(y,θ))

q(x, θ, λ)u(y)dydθ.

Here ˜

q denotes a formal classic analytic symbol deﬁned in a neighborhood of

, ¯

), which we may write as Ω

× Ω.

To realize the above operator we need a prescription for the integration path.
This is accomplished by transforming the classical integration path via the

Kuranishi change of variables and eventually applying Stokes theorem:

(5.2.2)

u(x, λ) =

(

2λψ(x,¯

q(x, ¯

y, λ)u(y)e

−2λΦ(y)

L(dy),

where L(dy) = (2i)

−n

∧ d¯y, the integration path is θ = ¯y and Ω is a small

neighborhood of x

. We remark that Q

u(x) is a holomorphic function of x.

The advantages of such a deﬁnition are:

1- if the principal symbol is real, Q

is formally self-adjoint in L

(Ω, e

−2λΦ

2- If ˜

q is a classical symbol of order zero, Q

is uniformly bounded as λ

→

∞, from H

(Ω) into itself.

Here H

(Ω) denotes the space of all holomorphic functions u(x, λ) such that for

every ε > 0 and for every compact K

⊂⊂ Ω there exists a constant C > 0 such that

|u(x, λ)| ≤ Ce

λ(Φ(x)+ε)

for x

∈ K and λ ≥ 1.

For future reference we also recall that the identity operator can be realized as

(5.2.3)

u(x, λ) =

(

2λψ(x,¯

i(x, ¯

y, λ)e

−2λΦ(y)

u(y, λ)L(dy),

for a suitable analytic classical symbol i(x, ξ, λ). Moreover we have the following
estimate (see [10] and [23])

(5.2.4)

− u

−d

≤ C

Φ+d

for suitable positive constants C and C

. Here we denoted by

(5.2.5)

d(x) = dist(x,

Ω),

5.3. SOME PSEUDODIFFERENTIAL CALCULUS

the distance of x to the boundary of Ω, and by

(5.2.6)

2
Φ

−2λΦ(x)

|u(x)|

L(dx).

5.3. Some Pseudodiﬀerential Calculus

We start with a proposition on the composition of two pseudodiﬀerential oper-

ators.

Proposition

5.3.1 ([10]). Let Q

and Q

be of order zero. Then they can be

composed and

Ω
1

◦ Q

Ω
2

= (Q

◦ Q

)

+ R

where R

is an error term whose norm is

O(1) as an operator from H

Φ+(1/C)d

−(1/C)d

We shall need also a lower bound for an elliptic operator of order zero.

Proposition

5.3.2. Let Q a zero order pseudodiﬀerential operator deﬁned on

Ω as above. Assume further that its principal symbol q

(x, ξ, λ) satisﬁes

ΛΦ∩π−

1 (Ω)

| ≥ c

> 0.

Here π denotes the projection onto the ﬁrst factor in

× C

n
ξ

. Then

(5.3.1)

≥ Cu

where

(5.3.2)

Φ(x) = Φ(x) +

(x),

and d has been deﬁned in ( 5.2.5).

Proof.

We have

u(x, λ)

− q

ΛΦ

(x, λ)I

u(x, λ)

(

2λψ(x,¯

)

q(x, ¯

y, λ)

− q

ΛΦ

(x, λ)i(x, ¯

y, λ)

× e

−2λΦ(y)

u(y)L(dy).

The absolute value of the term in square brackets may be estimated by C(

|x − y| +

−1

). Then

− q

ΛΦ

2
Φ

≤ Cλ

−2

2
Φ

+ C

(

−λΦ(x)+2λψ(x,¯

−λΦ(y)

|x − y|e

−λΦ(y)

u(y)L(dy)

L(dx)

≤ C

(

−λ/C|x−y|

|x − y|L(dy)

(

−λ/C|x−y|

|x − y|e

−2λΦ(y)

|u(y)|

L(dy)

(

L(dx)

+ Cλ

−2

2
Φ

≤ Cλ

−1

2
Φ

5. BARGMAN PSEUDODIFFERENTIAL OPERATORS

Using (5.2.4) we may conclude that

≥ q

ΛΦ

− Cλ

−1/2

≥ q

ΛΦ

− q

ΛΦ

− 1)u

− Cλ

−1/2

≥ c

− Cu

− Cλ

−1/2

This proves the assertion.

CHAPTER 6

The “A Priori” Estimate on the FBI Side

In this chapter we state an a priori estimate for an operator of the type “sum

of squares” on the FBI side. The estimate is optimal as far as the subellipticity
index (or the Gevrey regularity) is concerned. We give a sketch of the proof and
refer to [1] for the details.

Let X

(x, ξ), . . . , X

(x, ξ) be classical analytic symbols of the ﬁrst order deﬁned

in an open neighborhood Ω of (x

, ξ

)

∈ Λ

. We assume also that the X

j |

ΛΦ

are

real valued, so that we may think of the corresponding pseudodiﬀerential operators
as formally self-adjoint in H

. Let

(6.1)

L(x, ˜

D) =

j=1

(x, ˜

D).

Arguing as in [10] we see that the Ω-realization of L can be written as

(6.2)

j=1

)

O(λ

where

O(λ

) is continuous from H

to H

−(1/C)d

with norm bounded by C

We assume also that there is a commutator of length ν(x

, ξ

) which is elliptic

at (x

, ξ

)

∈ Λ

and that it involves the minimal number of operators.

In this chapter we sketch the proof of the following result

Theorem

6.1. Let L

be as in ( 6.2). We write r = ν(x

, ξ

). Then

(6.3)

≤ C

−

2
r

+ λ

S (λ)u

where ˜

Φ = Φ + (1/C)d

, for a suitable positive constant C. Here d = d(x) denotes

the distance of x from the boundary of a ﬁxed tubular neighborhood of ∂Ω and

S (λ),

deﬁned by ( 6.3.5) below, is an “error” term that is continuous from H

to H

A precise estimate of the norm containing

S (λ) in (6.3) above will be given at

the end of Section 6.3 (see Corollary 6.3.4 below.)

We recall that the number ν(x

, ξ

) in the statement above is the depth of the

stratiﬁcation at the point (x

, ξ

Remark

6.2. (i) In inequality (6.3) the term

is clearly an exponentially

decreasing term—and hence a good error—away from ∂Ω in

(ii) Estimate (6.3) is the analog of the subelliptic estimate in H¨

ormander’s theorem.

Moreover (6.3) is sharp. We stress that this sharpness is necessary to get optimal
Gevrey regularity.

6. THE “A PRIORI” ESTIMATE ON THE FBI SIDE

6.1. Proof of Theorem 6.1

We recall the Baker-Campbell-Hausdorﬀ formula (BCH) for vector ﬁelds in the

real domain.

Theorem

6.1.1. Let X

, . . . , X

be vector ﬁelds with real analytic coeﬃcients.

Fix a positive integer r. Then there exists a positive integer N and ε

∈ {±1},

j = 1, . . . , N , such that

itε

· · · e

itε

= e

Z(t)

where Z(t) is a real analytic vector ﬁeld such that Z(t) =

O (|t|

We point out that the BCH formula holds also on the FBI side, i.e. denoting

by X

the realization in Ω of the FBI transformed vector ﬁeld X

we have that

itε

· · · e

itε

= e

Z(t)

here Z(t)

is a suitable ﬁrst order pseudodiﬀerential operator and Z(t)

O (|t|

6.2. First Part of the Estimate: Estimate from Below

We start the proof of Theorem 6.1 with a lower bound for an elliptic commutator

of the vector ﬁelds. For the sake of brevity we write

(6.2.1)

r = ν(x

, ξ

where ν(x

, ξ

) is the depth of the Poisson-Treves stratiﬁcation at the point (x

, ξ

i.e. the minimum length of an elliptic commutator of the vector ﬁelds.

We recall the following

Proposition

6.2.1 ([10]). Let Q be a formally self-adjoint pseudodiﬀerential

operator of order 1. Let Q

be the realization of Q on Ω as deﬁned in ( 5.2.2).

Then there exist a suitable constant C > 0 and an operator, E

t,Q

, such that

t,Q

− E

t,Q

O(λ),

where

O(λ) denotes an operator from H

→ H

whose L

norm is bounded by Cλ.

The operator E

t,Q

is sought of the form

(6.2.2)

t,Q

u(x, λ) =

(

2λψ(t,x,¯

a(t, x, ¯

y, λ)e

−2λΦ(y)

u(y, λ)L(dy),

for t

∈] − ε, ε[. Here a denotes a classical symbol of order zero. The phase ψ is

determined, via geometrical optics, by

(6.2.3)

∂

ψ(t, x, y)

− q

∂

ψ(t, x, y)

= 0

t=0

= ψ(x, y)

The amplitude a is determined by the transport equations and is assumed to satisfy
the initial condition a

t=0

= i(x, ¯

y, λ) (the symbol of I

, see (5.2.3)).

Proposition

6.2.2. Denote by Q the elliptic bracket of length r

. . . ad

ir−1

Then for t

∼ δλ

−

1
r

, δ a small positive constant,

≤ C

− I

6.3. SECOND PART OF THE ESTIMATE: ESTIMATE FROM ABOVE

Proof.

Using the equation deﬁning E

t,Q

− I

s,Q

u +

O(λ)u

u + i[E

s,Q

− I

u +

O(λ)u

For the second term we remark that

s,Q

− I

]vds = i

dσ

σ,Q

vdσds

−

σ,Q

O(λ)]vdσds

and that, as a consequence,

s,Q

− I

]vds

≤ C

λt

+ t

≤ C

Then

u + i[E

s,Q

− I

u +

O(λ)u

≥ Ct

− C

+ λ

In the last inequality we used Proposition 5.3.2. Then we complete the proof taking
in the inequality above t

∼ δλ

−

1
r

, with δ small enough.

6.3. Second Part of the Estimate: Estimate from Above

We want to estimate the term

− I

of the previous section from

above, where Q has been deﬁned in Proposition 6.2.2.

We start with the

Proposition

6.3.1. Let Q be a classical analytic symbol of the ﬁrst order,

independent of t and let E

t,Q

be deﬁned as in Proposition 6.2.1. Then E

t,Q

−e

itQ

O(λ), where O(λ) denotes an operator from H

→ H

whose norm is

≤ Cλ.

Proof.

The proof is a straightforward consequence of the deﬁnitions of both

t,Q

and e

itQ

Using BCH we have

(6.3.1)

−itε

· · · e

−itε

= e

Z(t)

with ε

∈ {±1} and

Z(t) =

O(t

r+1

)

We deﬁne

(6.3.2)

= E

−tε

· · · E

−tε

One can show, see [1], that H

is a FIO modulo the sum of two error terms,

one of which is an analytically regularizing operator while the second is bounded
from H

Φ+(1/C)d

to H

−(1/C)d

. Here d

= d

(x) denotes the distance of x to

the boundary of Ω

, Ω

being a tubular neighborhood of ∂Ω of diameter Kt, t

suﬃciently small.

6. THE “A PRIORI” ESTIMATE ON THE FBI SIDE

Let us now compute E

from the expression (6.3.2):

tε

= E

tε

−tε

· · · E

−tε

= (I

− 1 + R

,λ

−tε

· · · E

−tε

+ E

−tε

· · · E

−tε

From the above we have

(6.3.3)

= E

tε

· · · E

tε

−

tε

· · · E

tε

i+1

− 1 + R

,λ

)

−tε

i+1

· · · E

−tε

We deduce that

(6.3.4)

− I

= E

t,X

. . . E

t,X

− I

S (λ),

where

(6.3.5)

S (λ) = −

tε

· · · E

tε

i+1

− 1 + R

,λ

)

−tε

i+1

· · · E

−tε

The above quantity can be rewritten

(6.3.6)

t,X

. . . E

t,X

− I

) + E

t,X

. . . (E

t,X

− I

)

+ E

t,X

. . . E

t,X

− I

) +

· · · + E

t,X

− I

S (λ).

Our task is to prove an estimate from above of the quantity in the l.h.s.

(6.3.4), i.e. of the various terms in (6.3.6). These terms are all but one dealt
with in the same way using the fact that a FIO of the form E

t,X

is bounded in

(Ω, e

−2λΦ

L(dx)), i.e.

t,X

≤ Cu

. We have the

Proposition

6.3.2 (See [10]). We have the estimate

t,X

− I

≤ C|t|X

O(λ),

where

O(λ) is bounded from H

to H

by Cλ.

The other term that we have to estimate is H

− I

. We have

Lemma

6.3.3. We have the estimate

(6.3.7)

− I

≤ Cλ|t|

r+1

O(λ),

Proof.

First we point out that we may compose the FIO’s involved in the

deﬁnition of H

obtaining

u(x) =

(

2λψ(t,x,¯

a(t, x, ¯

y, λ)e

−2λΦ(y)

u(y)L(dy) + R

u(x),

6.3. SECOND PART OF THE ESTIMATE: ESTIMATE FROM ABOVE

where a is a classical analytic symbol of order zero depending analytically on t, ψ
is holomorphic with respect to x, ¯

y and real analytic in t. Now

(6.3.8)

− I

)u(x)

(

2λψ(t,x,¯

− e

2λψ(x,¯

a(t, x, ¯

y, λ)e

−2λΦ(y)

u(y)L(dy)

(

2λψ(x,¯

(a(t, x, ¯

y, λ)

− i(x, ¯y, λ))e

−2λΦ(y)

u(y)L(dy)

+ R

u(x).

Let us consider the ﬁrst term in the r.h.s. of the above relation. We want to show
that

(6.3.9)

ψ(t, x, ¯

− ψ(x, ¯y) = O(t

r+1

First we remark that ψ(t, x, ¯

y) is the phase function obtained by composing N + 1

FIO’s and therefore it is the function, holomorphic w.r.t. x, ¯

y, such that

(6.3.10)

ψ(t,

(y), ¯

y) =

(Φ(

(y)) + Φ(y)) ,

where

(6.3.11)

(y) =

1,t

◦ κ

2,t

◦ · · · ◦ κ

N +1,t

(y),

where

j,t

denotes the (space component of the) Hamilton ﬂow of X

if j

≤ N and

of Q if j = N + 1.

On the other hand the solution of the eikonal equation corresponding to Z

(t),

which is

O(t

), also satisﬁes relation (6.3.10) since the Hamilton ﬂow of the principal

part of Z

is the composition in the r.h.s. of (6.3.11). Moreover it is trivially equal

to ψ(x, ¯

y) +

O(t

r+1

). Since, for small t, there exists a unique function, holomorphic

w.r.t. x, ¯

y, satisfying (6.3.10) we obtain (6.3.9).

Denote by A

u the ﬁrst term in the r.h.s. of (6.3.8). Then A

u =

O(t

r+1

) and

(6.3.12)

≤ C|t|

r+1

Now

− I

= e

Z(t)

− I + R

where

= I

− I

N +1

j=1

−iε

· · · e

−iε

−1

ij−1

O(λ)E

−ε

j+1

t,X

ij+1

· · · E

−ε

t,X

O(λ) having the same meaning as in Proposition 6.3.1.

Now (e

Z(t)

− I)u = O(t

r+1

) so that, calling B

the second term in the r.h.s. of

(6.3.8), we have that

+ R

− R(λ))u = O(t

r+1

This implies that B

u =

O(t

r+1

) +

− R

u. Since both B

and the error terms

are continuous in H

we get the desired conclusion.

6. THE “A PRIORI” ESTIMATE ON THE FBI SIDE

Let us now start from

− I

. By (6.3.4) and (6.3.6) we have that

− I

≤ C

− I

t,X

− I

S (λ)u

By Lemma 6.3.3 and Proposition 6.3.2 we obtain

− I

≤ C

λt

r+1

|t|X

+ λ

S (λ)u

Using Proposition 6.2.2 and choosing, as we did in its proof, t

∼ δλ

−

1
r

, we have

that

≤ C

−

1
r

+ λ

S (λ)u

By (6.2) we get

≤ C

−

1
r

(

u, u

)

1/2

+ λ

S (λ)u

from which we deduce

≤ C

−

2
r

+ λ

S (λ)u

and this ends the proof. A variation of the ﬁnal part of the above proof yields the

Corollary

6.3.4. With the same notation of Theorem 6.1, we also have

(6.3.13)

2
r

2
Φ

j=1

2
Φ

≤ C

u, u

+ λ

2(2

−

1
r

)

S (λ)u

2
Φ

Inequality (6.3.13) can be written in a more friendly way. Let us consider the

third term on the right of (6.3.13). We recall that, by (6.3.5),

S (λ) = −

tε

· · · E

tε

i+1

− 1 + R

,λ

)

−tε

i+1

· · · E

−tε

Hence

S (λ)u

≤ C

− 1 + R

,λ

−tε

i+1

· · · E

−tε

≤ λC

−tε

i+1

· · · E

−tε

Recall that ˜

Φ(x) = Φ(x) +

(x). Let us estimate a term of the form

Here E denotes a FIO of the form appearing in the estimate above. We denote by
κ

the associated Hamilton ﬂow. Iterating the argument will then suﬃce to rewrite

(6.3.13) in a slightly diﬀerent form.

6.3. SECOND PART OF THE ESTIMATE: ESTIMATE FROM ABOVE

Let Ω

⊂⊂ Ω. We have

−2λΦ(x)−

(x)

|Ev(x)|

L(dx)

\Ω

−2λΦ(x)−

(x)

|Ev(x)|

L(dx)

≤ e

−2λ/C

2
Φ

\Ω

−2λΦ(x)−

(x)

|Ev(x)|

L(dx)

since d(x) is bounded away from zero in Ω

. We have to estimate the second integral

above. Let Ω

⊂⊂ Ω

. Then the integral on the r.h.s. of the above inequality can

be bounded by

(

\Ω

−Cλ|x−κ

(y)

−λΦ(y)

|v(y)|L(dy)

L(dx)

+ C

(

\Ω

−Cλ|x−κ

(y)

−λΦ(y)

|v(y)|L(dy)

(

L(dx).

The second integral is easily estimated by e

−2λ/C

2
Φ

provided (Ω

\Ω

)

∩κ

(Ω

) =

∅, which is always possible if both t and Ω

are suitably small. We are left with

the ﬁrst integral. We have

(

\Ω

−Cλ|x−κ

(y)

−λΦ(y)

|v(y)|L(dy)

L(dx)

≤

(

\Ω

−λC|x−κ

(y)

L(dy)

(

\Ω

−λC|x−κ

(y)

−2λΦ(y)

|v(y)|

L(dy)

L(dx)

≤ Cv

2
Φ,Ω

\Ω

We have thus proved the following

Theorem

6.3.5. Let L

be as in ( 6.2). Let Ω

⊂⊂ Ω. Then

(6.3.14)

2
r

2
Φ

j=1

2
Φ

≤ C

u, u

+ λ

2
Φ,Ω

\Ω

where α is a positive integer.

From (6.3.14) we easily deduce

Corollary

6.3.6. With the same notation of Theorem 6.3.5 we have

(6.3.15)

2
r

≤ C

+ λ

Φ,Ω

\Ω

CHAPTER 7

A Single Symplectic Stratum

We start by considering an operator of the form

(7.1)

P (x, D) =

i,j=1

(x, D)a

(x, D)X

(x, D)

j=1

(x, D)X

(x, D) + c(x, D),

where D

= D

= i

−1

∂

and the a

(x, ξ), b

(x, ξ), c(x, ξ) are analytic symbols of

order zero such that

(7.2)

]

i,j=1,...,N

+ [¯

]

i,j=1,...,N

≥ c,

where c > 0 is a positive constant.

It is clear that such an operator is invariant if a non singular linear substitution

is applied to the vector ﬁelds.

Let (x

, ξ

)

∈ Char(X

, . . . , X

) = Char(P ). We assume that Char(P ) is

locally given by a single symplectic stratum in the Poisson-Treves stratiﬁcation,
i.e. Char(P ) is a real analytic manifold on which the symplectic form has constant
rank and is nondegenerate (Char(P ) is symplectic.) Moreover all brackets, X

|I| ≤ ν, ν ≥ 1, vanish and there exists J, |J| = ν + 1, such that X

, ξ

)

= 0.

Denote by U a neighborhood of the point (x

, ξ

) in

. Let r : U

→ [0, +∞[

a real analytic function such that

(1) r(x

, ξ

) = 0 and r(x, ξ) > 0 in U

\ {(x

, ξ

)

(2) There exist real analytic functions, α

j,k

(x, ξ), deﬁned in U , such that

(7.3)

{r(x, ξ), X

(x, ξ)

} =

(x, ξ)X

(x, ξ),

where j = 1, . . . , N .

We point out that

i) Condition (7.3) is homogeneous both with respect to the vector ﬁelds X

and to r, α.

ii) Condition (7.3) is invariant with respect to canonical transformations.

The following is a statement in a general framework; we discuss it in particular,

more concrete, classes further on. We have

Theorem

7.1. Under the above assumptions, we have that

if (x

, ξ

) /

∈ W F

(P u) then (x

, ξ

) /

∈ W F

(u).

This immediately implies the

7. A SINGLE SYMPLECTIC STRATUM

Corollary

7.2. Assume that there is a homogeneous canonical change of vari-

ables such that in the new variables Condition ( 7.3) holds. Then (x

, ξ

) /

∈ W F

(u)

if (x

, ξ

) /

∈ W F

(P u).

Proof of Theorem

7.1. Let us write ˜

D = λ

−1

D, where λ denotes a large

positive parameter. The operator P then becomes

(7.4)

P (x, ˜

D; λ) =

i,j=1

(x, ˜

D)a

(x, ˜

D; λ)X

(x, ˜

+ λ

−1

j=1

(x, ˜

D; λ)X

(x, ˜

D) + λ

−2

c(x, ˜

D; λ),

with condition (7.2) still holding.

We now perform an FBI tranformation on P and we still denote by P the

reasulting pseudodiﬀerential operator. The cotangent bundle T

∗

is thus locally

transformed into Λ

, where Φ

denotes the weight function corresponding to the

FBI transform phase function ϕ

. Note that Λ

is contained in

and has real

dimension 2n.

The next step consists in moving away from Λ

and, following Sj¨

ostrand, [22],

we use a canonical deformation of Φ

for this purpose.

Let r(x, ξ) be the real anlytic function whose existence is guaranteed by our

assumptions, or rather its FBI transform. Deﬁne the deformed weight function Φ

where t denotes a small non negative parameter, as the solution to the following
Hamilton–Jacobi equation:

(7.5)

⎧

⎪

⎨
⎪

⎩

∂Φ

(x)

∂t

− r

∂Φ

(x)

∂x

(

= 0,

(x)

t=0

= Φ

(x).

We have that Λ

= exp itH

(Λ

Next we want to deduce a priori estimates for P with the weight function Φ

replaced by Φ

First we write (7.3) as

(7.6)

{r(x, ξ), X(x, ξ)} = α(x, ξ)X(x, ξ),

where X denotes a vector whose components are the symbols of the vector ﬁelds
X

(x, ξ), . . . , X

(x, ξ) and α denotes a N

×N matrix with entries being real analytic

symbols.

Denote by Y

the symbol X

◦ exp(itH

), or the restriction to Λ

of the

holomorphic extension of X

, j = 1, . . . , N . We have

∂

(x, ξ) = i

{r, X} ◦ exp(itH

)(x, ξ),

for t small enough. We deduce then that

⎧

⎨
⎩

∂

(x, ξ) = iα

◦ exp(itH

(x, ξ),

(x, ξ)

t=0

= X(x, ξ).

From the above equation we obtain that there is a N

× N matrix whose entries

are real analytic symbols with a real analytic dependence on the real parameter t,

7. A SINGLE SYMPLECTIC STRATUM

(x, ξ), such that

(7.7)

(x, ξ) = b

(x, ξ)X(x, ξ),

and that b

t=0

(x, ξ) = Id

. In particular b

is nonsingular, provided t is small.

Denote by X

the holomorphic extension of Re Y

; since X is real on Λ

, using

(7.7), we have that

(7.8)

(x, ξ) = β

(x, ξ)X(x, ξ),

where β

t=0

(x, ξ) = Id

. In particular β

is nonsingular, provided t is small.

Then we may write

(7.9)

P (x, ˜

D) =

i,j=1

(x, ˜

D)a

t
ij

(x, ˜

D; λ)X

(x, ˜

+ λ

−1

j=1

t
j

(x, ˜

D; λ)X

(x, ˜

D) + λ

−2

(x, ˜

D; λ),

where a

, b

, c

are symbols of order zero with real analytic dependence on t.

It is also obvious from what has been said before that the ﬁelds X

, j =

1, . . . , N , also satisfy H¨

ormander condition with the same bracket length, r, as the

We may thus apply Theorem 6.1 and conclude that the following a priori esti-

mate holds ([1]):

(7.10)

2
r

,Ω

≤ C

P u

,Ω

+ λ

,Ω

\Ω

where Ω

⊂⊂ Ω, α is a ﬁxed positive integer and P denotes the realization on Ω of

the given operator P .

Let us now assume that (x

, ξ

) /

∈ W F

(P u). We may choose Ω in such a way

that

(7.11)

P u

,Ω

≤ Ce

−λ/C

for a suitable positive constant C. From

(7.12)

(x) = Φ

(x) +

∂

(x)

ds,

using the fact that r

Φ0

≥ 0, and recalling that Λ

= exp(itH

)Λ

, we deduce

that r

Φt

≥ 0 so that

(7.13)

(x)

≥ Φ

(x),

∈ Ω.

Hence, by (7.13) and (7.11),

(7.14)

P u

,Ω

≤ Ce

−λ/C

for a suitable positive constant C.

Let us now estimate the second term in the right hand side of (7.10). We point

out that

Φ0

∩Ω\Ω

≥ a > 0.

It follows, because of (7.12), that

(7.15)

(x)

≥ Φ

(x) + c

∈ Ω \ Ω

7. A SINGLE SYMPLECTIC STRATUM

Then

2
Φ

,Ω

\Ω

−2λΦ

(x)

|u(x)|

L(dx)

≤

\Ω

−2λΦ

(x)

−2λc

|u(x)|

L(dx)

≤ Ce

−2λc

≤ Ce

−λc

t > 0.

By (7.10) we deduce that

,Ω

≤ C exp(−λt/C), for a suitable positive constant

C. Let now Ω

⊂⊂ Ω

be a neighborhood of x

such that Φ

≤ Φ

+ t/(2C) in Ω

We conclude that

2
Φ

,Ω

≤ Ce

−λt/C

t > 0.

This proves the theorem.

Next we turn to more particular cases in which the existence of the function r

can be shown.

7.1. dim Σ = 2 and X

, . . . , X

Quasi-homogeneous

Let X

, . . . , X

satisfy Condition (Sympl) of Section 4.1. Without loss of

generality, we may suppose also that (x

, ξ

) = (0, e

We assume that

(1) dim Σ = 2 (up to a change of coordinates as in Section 4.1. we have

that, denoting x = (x

, x

), the characteristic manifold is given by Σ =

{(0, x

, 0, ξ

)

| x

∈ R, ξ

> 0

}.)

(2) The symbols of the vector ﬁelds X

, . . . , X

are of the form

(x, ξ)

+ a

)ξ

j = 2, . . . , n

− 1,

(7.1.1)

(x, ξ)

)ξ

j = n, . . . , N,

and satisfy the following quasi-homogeneity conditions: there exist θ

, q

∈ N, k = 1, . . . , N, such that

(7.1.2)

−1

k=1

∂

) = q

j = 2, . . . , N,

with θ

+ q

= α,

k = 1, . . . , n

− 1.

We point out that, for k = n, . . . , N , there are no conditions on the numbers θ

, q

Under the above assumption, consider an operator P of the form (7.1). Then we
have the following

Theorem

7.1.1. If (0, e

) /

∈ W F

(P u) then (0, e

) /

∈ W F

(u).

Proof.

In order to prove the result it suﬃces to show that the operator P

satisﬁes the assumptions of the Theorem 7.1. For this purpose deﬁne

r(x, ξ) =

⎛
⎝x

−1

j=1

⎞
⎠

+ (ξ

− 1)

j=1

(x, ξ)

7.1. dim Σ = 2 AND X

, . . . , X

QUASI-HOMOGENEOUS

We may think of r as a real analytic symbol of order 0. Moreover, we have that
r(0, e

) = 0 and r(x, ξ) > 0 if (x, ξ)

= (0, e

). Let us verify Condition 7.3. We have

that

{r(x, ξ), ξ

} =

⎛
⎝x

−1

j=1

⎞
⎠ θ

j=1

{r(x, ξ), X

(x, ξ)

(x, ξ).

Furthermore, let us compute

{r(x, ξ), X

(x, ξ)

}, for k = 2, . . . , n − 1. There are

three types of terms:

(1) the Poisson bracket

{

N
j=1

(x, ξ)

, X

(x, ξ)

} can be written as a linear

combination of the symbols of the vector ﬁelds,

(2)

{(ξ

−1)

, X

(x, ξ)

} = 0 (since the symbol of X

is independent of x

, ξ

the variables tangent to Σ.)

(3)

−1

j=1

, X

(x, ξ)

In order to verify Condition (7.3) for a term of type (3) it suﬃces to check that

−1

j=1

, X

is a multiple of the symbol X

. We have

⎧

⎨
⎩

−1

j=1

, X

(x, ξ)

⎫

⎬
⎭

⎧

⎨
⎩

−1

j=1

, ξ

+ a

)ξ

⎫

⎬
⎭

−a

)ξ

−

−1

j=1

∂

)ξ

Using the quasi-homogeneity condition (7.1.2) we ﬁnd that the above Poisson
bracket is equal to

−

(ξ

+ a

)ξ

A similar argument can be used to show that

{r(x, ξ), X

}, k = n, . . . , N, satisfy

(7.3). Hence, Theorem 7.1 applies to the present situation and this completes our
proof.

Example

7.1.2. Let X

(x, ξ) = ξ

−1

)ξ

j = 2, . . . , n

− 1,

where a

) =

with c

real valued and the multiindex α running over

| β = (β

, . . . , β

−1

i=1

(β

+ δ

) = q

Furthermore let

(x, ξ) = a

)ξ

j = n, . . . , N,

7. A SINGLE SYMPLECTIC STRATUM

where the functions a

are homogeneous as above and

≤

j=n

2
j

)

≤ C|x

In particular X

= D

, X

= D

+ (x

+ x

, X

= x

p
1

, X

= x

q
2

, p, q

positive integers, belong to the above class.

7.2. codim Σ > 2

Assume that codim Σ = 2k, k < n. Let X

, . . . , X

be C

vector ﬁelds such

that Char(X) = Σ is a single symplectic stratum. We also suppose that dX

j = 1, . . . , 2k

− 1 are linearly independent at a point ρ

∈ Char(X).

We point out that if dX

is linearly independent with the diﬀerentials of the

other vector ﬁelds, the problem was solved in the papers [25] and [28]. Hence we
consider the case when dX

(ρ

) = 0.

Let P be as in (7.1) with N = 2k, then we have

Theorem

7.2.1. If ρ

∈ W F

(P u) then ρ

∈ W F

(u).

Proof.

Due to our assumptions we know that there exists a real analytic

function ϕ deﬁned in Ω, where Ω is a neighborhood of ρ

in T

∗

\ {0}, such that

(x, ξ), . . . , X

−1

(x, ξ), ϕ(x, ξ) are the deﬁning functions of Char(X). Consider

the restriction, ˜

, of X

(x, ξ) to the manifold X

(x, ξ) = 0, j = 1, . . . , 2k

− 1.

Then there exists a positive integer h such that

= ϕ

for a suitable real analytic function g, such that g(ρ

)

= 0. We may suppose that

ϕ is homogeneous of degree 0 and g is homogeneous of degree 1 with respect to ξ.
Thus

(x, ξ) =

−1

j=1

(x, ξ)X

(x, ξ) + ϕ

(x, ξ)g(x, ξ),

for suitable real analytic symbols α

homogeneous of degree 0 w.r.t. ξ.

There is a homogeneous canonical change of coordinates such that X

, . . . ,

−1

, ϕ are transformed into y

, . . . , y

and their duals η

, . . . , η

, possibly modulo

an elliptic factor, and such that ϕ goes into y

Thus

(y, η) =

j=1

(y, η)η

−1

j=1

j+k

(y, η)y

+ y

g(y, η),

for suitable real analytic symbols ˜

and ˜

g having the natural homogeneity, ˜

g(ρ

)

Let

r(y, η) =

j=k+1

− y

)

+ (η

− η

)

+ C

j=1

(y, η).

7.3. ONE SYMPLECTIC STRATUM OF DEPTH 1

It is enough to check that (7.3) is satisﬁed. The brackets of r with X

, . . . , X

−1

being obvious, we have to check only the bracket of r with X

. Since

, X

} = −

j=1

∂

(y, η)η

−

−1

j=1

∂

j+k

(y, η)y

− y

g(y, η)

∂

g(y, η)

(7.3) easily follows in the variables (y, η). Applying the inverse canonical tranfor-
mation we obtain (7.3) in the variables (x, ξ).

This ends the proof of the theorem.

Example

7.2.2. Let P (x, D) = D

+ (D

+ g(x)D

)

+ x

, in

where g denotes an analytic function and h > 1 is a positive integer.

7.3. One Symplectic Stratum of Depth 1

Let us suppose that the vector ﬁelds X

, . . . , X

, are such that Char(X) is

a single symplectic stratum of depth 1. By Theorem 3.2.4, we may, via a (real
analytic) change of variables, write them in the form

(7.3.1)

= ∂

n
k=

κ+1

(x)∂

i = 1, . . . , r

− ρ,

−ρ+1

= ∂

−ρ+1

= ∂

n
k=

κ+1

−1

−ρ+1

()
ik

(x)∂

i = r

− ρ + 2, . . . , κ,

n
k=

κ+1

(x)∂

j =

κ + 1, . . . , N.

Here the a

∈ C

(Ω) and a

(0) = 0, for j = 1, . . . , N , k =

κ + 1, . . . , n and the

integers r, ρ and

κ have been deﬁned in Section 4.2.

We need supplementary assumptions:

(H1) –

κ = n − 1, i.e. dim Σ = 2 + r − ρ. Moreover the ﬁelds X

, i = 1, . . . , r

− ρ,

do not depend on the variables x

−ρ+1

, . . . , x

(H2) – The ﬁelds X

−ρ+1

, . . . , X

depend only on the variables x

−ρ+1

, . . . , x

—

actually this is not an assumption for X

−ρ+1

This amounts to say that, denoting by x

= (x

, . . . , x

−ρ

, x

), and x

−ρ+1

, . . . , x

−1

), the ﬁelds in (7.3.1) can be written as

(7.3.2)

= ∂

+ a

)∂

i = 1, . . . , r

− ρ,

−ρ+1

= ∂

−ρ+1

= ∂

+ a

)∂

i = r

− ρ + 2, . . . , n − 1,

= a

)∂

j = n, . . . , N.

Here the a

denote functions in C

(Ω) vanishing at the origin.

(H3) – The ﬁelds X

, . . . , X

−ρ

commute with the ﬁelds X

−ρ+1

, . . . , X

This implies that the functions a

), i = 1, . . . , r

− ρ, do not depend on

the variable x

(H4) – The symbols of the ﬁelds X

−ρ+1

, . . . , X

satisfy the following quasi-

homogeneity conditions: there exist θ

, q

, α

∈ N, k = r − ρ + 1, . . . , N,

7. A SINGLE SYMPLECTIC STRATUM

such that,

(7.3.3)

−1

k=r

−ρ+1

∂

) = q

j = r

− ρ + 2, . . . , N,

with θ

+ q

= α,

k = r

− ρ + 1, . . . , n − 1.

Then we have the theorem

Theorem

7.3.1. Let the vector ﬁelds X

, . . . , X

satisfy the above assumptions

(H1)–(H4). Let P be as in ( 7.1). Then P is microlocally analytic hypoelliptic at
(0, e

)

∈ Char(X).

Proof.

Since, by (H3), the ﬁrst r

− ρ vector ﬁelds in (7.3.2) do not depend on

, it is convenient to redeﬁne x

as the vector x

= (x

, . . . , x

−ρ

). Contrary to the

case of Section 6.1, in the present case the characteristic manifold cannot be easily
described due to the presence of the Heisenberg–type vector ﬁelds X

, . . . , X

−ρ

We thus need the microlocal reduction at the end of Section 4.2 or rather an easy
change of its proof. In particular we can ﬁnd a canonical transformation such that

(a) it goes from a conical neighborhood of (0, e

) into itself, say from coordi-

nates (x, ξ) to coordinates (y, η);

(b) it is the identity in the variables x

−ρ+1

, . . . , x

−1

;

= ξ

, because of (H3);

(d) in the new coordinates (y, η) the vector ﬁelds have the form

(y, D)

j = 1, . . . ,

− ρ

;

(7.3.4)

−ρ)/2+j

(y, D)

j = 1, . . . ,

− ρ

;

(y, D)

+ a

j = r

− ρ + 1, . . . , n − 1;

(y, D)

j = n, . . . , N.

(e) The characteristic manifold is given, in the new coordinates, by the equa-

tions

(7.3.5)

Σ =

(y, η)

| y

= η

= 0, i = 1, . . . ,

− ρ

, r

− ρ + 1, . . . , n − 1

Thus a candidate to be the function r in the coordinates (y, η) will have a “tangential
part” of the form

(y, η) =

−ρ

i=(r

−ρ)/2+1

+ η

⎛
⎝y

−1

j=r

−ρ+1

⎞
⎠

+ (η

− 1)

so that the function r will be given by

(7.3.6)

r(y, η) = r

(y, η) + C

j=1

(y, η)

Let us verify that r

satisﬁes the reproduction condition (7.3).

First of all, due to the form (7.3.4), it is evident that both y

and η

, for

i = (r

− ρ)/2 + 1, . . . , r − ρ, have zero Poisson bracket with all the vector ﬁelds in

the (y, η) coordinates. Again η

− 1 also has zero Poisson bracket with the vector

ﬁelds. Thus we are left with the function y

−1

j=r

−ρ+1

7.3. ONE SYMPLECTIC STRATUM OF DEPTH 1

Evidently we have no problem with the vector ﬁelds η

, since the Poisson

bracket is zero and with the vector ﬁelds y

, because they get just reproduced,

when j = 1, . . . , (r

− ρ)/2.

The discussion of the bracket

−1

j=r

−ρ+1

, X

}, j = r − ρ +

1, . . . , N is literally the same as that in Section 6.1.

This proves the theorem.

Finally we give an example of the type of operators studied in the present

section.

Example

7.3.2. We need for this kind of example a higher space dimension.

Let x

∈ R

and let X

= D

, X

= D

+ x

f (x

, x

, X

= D

, X

= D

g(x

, x

, X

= x

, X

= x

. Here h, k are positive integers larger than

1. Moreover (0, e

) belongs to the characteristic manifold which is also a single

symplectic stratum. We assume that f (0, 0)

= 0 and that g is quasi–homogeneous

with respect to x

and x

CHAPTER 8

A Single Nonsymplectic Stratum

We consider an operator of the form

(8.1)

P (x, D) =

i,j=1

(x, D)a

(x, D)X

(x, D)

j=1

(x, D)X

(x, D) + c(x, D),

where D

= D

= i

−1

∂

and the a

(x, ξ), b

(x, ξ), c(x, ξ) are analytic symbols of

order zero such that

(8.2)

]

i,j=1,...,N

+ [¯

]

i,j=1,...,N

≥ c,

where c > 0 is a positive constant.

It is clear that such an operator is invariant if a nonsingular linear substitution

is applied to the vector ﬁelds.

Let (x

, ξ

)

∈ Char(X

, . . . , X

) = Char(P ). We assume that Char(P ) is

locally given by a single nonsymplectic stratum in the Poisson-Treves stratiﬁcation,
i.e. Char(P ) is a real analytic manifold on which the symplectic form has constant
rank. Moreover all brackets, X

|I| ≤ ν, ν ≥ 1, vanish and there exists J, |J| =

ν + 1, such that X

, ξ

)

= 0.

Let F

be the Hamilton leaf through (x

, ξ

) in Char(P ). We assume that the

base projection of F

has the same dimension as F

Denote by U a neighborhood of the point (x

, ξ

) in

. Let r : U

→ [0, +∞[

a real analytic function such that

(1) r

∩F

= 0 and r(x, ξ) > 0 in U

\ F

(2) There exist real analytic functions, α

j,k

(x, ξ), deﬁned in U , such that

(8.3)

{r(x, ξ), X

(x, ξ)

} =

(x, ξ)X

(x, ξ),

where j = 1, . . . , N .

We point out that, as in the preceding chapter,

i) Condition (8.3) is homogeneous both with respect to the vector ﬁelds X

and to r, α.

ii) Condition (8.3) is invariant with respect to canonical transformations.

The following is a statement in a general framework; we discuss it in particular

more concrete classes further on. Then we may state the

Theorem

8.3. Let 1

≤ s < ν + 1. Under the above assumptions, we have that

8. A SINGLE NONSYMPLECTIC STRATUM

i) Assume that (x

, ξ

) /

∈ W F

(P u) and that W F

(u)

∩ F

∩ ∂U = ∅, then

, ξ

) /

∈ W F

(u).

ii) P is s Gevrey hypoelliptic for any s

≥ ν + 1.

As before this immediately implies the

Corollary

8.4. Assume that there is a homogeneous canonical change of vari-

ables such that in the new variables Condition ( 8.3) holds. Then (x

, ξ

) /

∈ W F

(u)

if (x

, ξ

) /

∈ W F

(P u) and W F

(u)

∩ F

∩ ∂U = ∅, for 1 ≤ s < ν + 1.

Furthermore P is s Gevrey hypoelliptic for any s

≥ ν + 1.

Proof of Theorem

8.3. Assertion ii) is proved in [1]. The proof of the ﬁrst

assertion follows the same lines of the proof of Theorem 7.1, with minor modi-
ﬁcations in the ﬁnal argument. More precisely we prove the estimate (7.10) ex-
actly as before. The fact that (x

, ξ

) /

∈ W F

(P u) implies that we may choose

Ω in such a way that

P u

,Ω

≤ C exp(−λ

1/s

/C).

From (7.13) we deduce

that

P u

,Ω

≤ C exp(−λ

1/s

/C), for a suitable positive constant C. Let now

\Ω

= K

∩K

, with K

⊂⊂ Ω and such that K

∩F

∅ and K

⊃ F

∩(Ω\Ω

By assumption we have that r

≥ a > 0 so that

≤ C exp(−λ/C) and u

≤ C exp(−λ

1/s

/C).

This completes the proof of the theorem.

8.1. The Case rank σ

Char(P )

= 2 and X

Quasi-homogeneous

Let X

, . . . , X

satisfy Condition (NonSympl) of Section 4.3. Without loss

of generality, we may suppose that (x

, ξ

) = (0, e

). We assume that

(1) rank σ

Char(P )

= 2.

Let 2k be the rank of σ

(0,en)

Char(P )

, where T

Char(P ) denotes the orthogonal of

T Char(P ) with respect to the symplectic form.

Then there is a change of coordinates as in Section 4.3. such that, the charac-

teristic manifold is given by

Σ =

{(0, x

, x

; 0, 0, ξ

)

| (x

, x

)

∈ R

−k

, ξ

> 0

where x = (x

, x

) with x

= (x

, . . . , x

) and x

= (x

k+1

, . . . , x

−1

). We note

that in this case n

− 1 − k is the dimension of the Hamiltonian leaf in Char(P ).

The other assumption we make is

(2) The symbols of the vector ﬁelds X

, . . . , X

are of the form

(x, ξ)

+ a

)ξ

j = 2, . . . , n

− 1,

(8.1.1)

(x, ξ)

)ξ

j = n, . . . , N,

with

j=n

)

= 0

if and only if

= 0 ,

8.2. THE TRANSVERSALLY ELLIPTIC CASE

and satisfy the following quasi-homogeneity conditions: there exist θ

, q

∈ N, = 1, . . . , N, such that

(8.1.2)

−1

∂

) = q

j = 2, . . . , N,

with θ

+ q

= α,

= 1, . . . , n

− 1.

We point out that, for = n, . . . , N , there are no conditions on the numbers θ

, q

Under the above assumption, consider an operator P of the form (7.1). Then

we have the following

Theorem

8.1.1. Let U be a neighborhood of (0, e

) in

and let ν + 1 be

the depth of (0, e

). Let 1

≤ s < ν + 1. If (0, e

) /

∈ W F

(P u) and if W F

(u)

∩

∂U

∩ {(0, x

, 0; e

)

| x

∈ R

−k−1

} = ∅ then (0, e

) /

∈ W F

(u). Furthermore P is

microlocally Gevrey s hypoelliptic for every s

≥ ν + 1.

Proof.

The second assertion has been proved in [1]. In order to prove the

ﬁrst assertion it suﬃces to show that there exists a function r satisfying Conditions
(1) and (2) above. Set

r(x, ξ) =

⎛
⎝x

j=1

⎞
⎠

+ (ξ

− 1)

j=1

(x, ξ)

Arguing as in the proof of Theorem 6.2. it is easy to see that r satisﬁes (8.3). This
completes the proof.

8.2. The Transversally Elliptic Case

We assume Condition (NonSympl-1) and that the principal symbol of the

operator P in (8.1) vanishes exactly to the second order on Σ. This means that
dim ker F

= dim T Σ, where F

= dH

is the fundamental matrix of the principal

symbol P

of P . In this case, the standard forms given in Theorem 3.4.7 can be

reﬁned according to the following

Lemma

8.2.1. There exists a real analytic, homogeneous canonical transfor-

mation deﬁned in U , conic neighborhood of (0, e

)

∈ Σ, and a linear nonsingular

substitution in the vector ﬁelds, such that the vector ﬁelds in Theorem 3.4.7 take
the form

(x, D)

j = 1, . . . ,

− ρ

, r

− ρ + 1, . . . , κ

−ρ)/2+i

(x, D)

j = 1, . . . ,

− ρ

, r

− ρ + 1, . . . , κ − .

Let P be an operator of the form (8.1). Then we have the

Theorem

8.2.2. Let U be a neighborhood of (0, e

) in

, denote by F

the

Hamiltonian leaf through (0, e

) and let 1

≤ s < 2. If (0, e

) /

∈ W F

(P u) and

if W F

(u)

∩ F

∩ ∂U = ∅ then (0, e

) /

∈ W F

(u). Furthermore P is s Gevrey

hypoelliptic for every s

≥ 2.

Proof.

In this case too the second assertion can be proved arguing as in [1].

In order to verify the ﬁrst assertion, we set

(x, ξ) =

∈{1,...,

−ρ

−ρ+1,...,κ−}

2
j

+ ξ

κ−+1

8. A SINGLE NONSYMPLECTIC STRATUM

(x, ξ) =

∈{(r−ρ)/2+1,...,r−ρ,κ+1,...,n}

2
j

+ (ξ

− δ

)

and ﬁnally

r(x, ξ) = r

(x, ξ) + r

(x, ξ),

it is easy to see that r satisﬁes Condition (8.3). This completes our proof.

8.3. A Class of Nontransversally Elliptic Operators

Let μ be a positive integer and consider an operator of the form

P (x, D) =

2k+

i,j=1

(x, D)a

(x, D)X

(s, D) +

2k+

j=1

(x, D)X

(s, D) + c(x, D)

where a

(x, ξ), b

(x, ξ), c(x, ξ) analytic symbols of order 0,

]

i,j=1,...,2k+

+ [a

]

i,j=1,...,2k+

≥ c

where c is a positive constant and

(x, D) = D

j = 1, . . . , k + ,

k+j

(x, D) = x

μ
j

j = 1, . . . , k.

Here k, denote positive integers such that k + < n.
Let us write x = (x

, x

), where

= (x

, . . . , x

= (x

k+1

, . . . , x

= (x

k++1

, . . . , x

Then we have the

Theorem

8.3.1. Let U be a neighborhood of (0, e

) in

and let 1

≤ s <

μ + 1.

If (0, e

) /

∈ W F

(P u) and if W F

(u)

∩ ∂U ∩ {(0, x

, 0; e

)

} = ∅ then

(0, e

) /

∈ W F

(u).

Furthermore P is microlocally Gevrey s hypoelliptic for every s

≥ μ + 1.

Proof.

The second assertion has been proved in [1].

The proof of the ﬁrst assertion is a consequence of Theorem 8.3. Indeed it

suﬃces to construct a nonnegative function r satisfying (8.3). It is easy to see that
the function

r(x, ξ) =

−1

j=1

+ (ξ

− 1)

j=1

2μ
j

j=k++1

2
j

has all the required properties. This completes our proof.

CHAPTER 9

Microlocal Regularity in Nested Strata

In the present chapter we study operators of the form (8.1) with a fully ﬂedged

stratiﬁcation. We start with a symplectic stratiﬁcation, i.e. a stratiﬁcation in which
each stratum is a symplectic real analytic manifold.

9.1. Symplectic Stratiﬁcations

Let us consider an operator of the form (8.1). We assume that the strata in

the stratiﬁcation associated to P , Σ

, j = 1, . . . , q, are symplectic.

First we prove the

Lemma

9.1.1. Let 1

≤ p ≤ q. Let ρ

∈ Σ

and let U be a conic neighborhood

of ρ

in T

∗

. Assume that

(9.1.1)

W F

(u)

∩ Σ

∩ U = ∅,

j = 1, . . . , p

− 1,

(9.1.2)

∈ W F

(P u).

Suppose that there exists a real analytic function r such that r : U

→ [0, +∞[ such

that

1– r > 0 on Σ

\ {ρ

};

2– r(ρ

) = 0;

3– There exist real analytic functions α

, deﬁned in U , such that

(9.1.3)

{r(x, ξ), X

(x, ξ)

} =

(x, ξ)X

(x, ξ),

where j = 1, . . . , N .

Then ρ

∈ W F

(u).

Proof.

We proceed along the same lines of the proof of Theorem 7.1. The

ﬁrst step is the deduction of an a priori estimate for P on the FBI side. To this end
we construct a weight function Φ

as in (7.5). A literal repetition of the argument

in the proof of Theorem 7.1 yields the a priori estimate

(9.1.4)

2
r

,Ω

≤ C

P u

,Ω

+ λ

,Ω

\Ω

where Ω

⊂⊂ Ω, α is a ﬁxed positive integer and P denotes the realization on Ω of

the given operator P .

Since ρ

∈ W F

(P u) we may choose Ω in such a way that

(9.1.5)

P u

,Ω

≤ Ce

−λ/C

for a suitable positive constant C. Because of the fact that r

Φ0

≥ 0, and recalling

that Λ

= exp(itH

)Λ

, we deduce that

(9.1.6)

(x)

≥ Φ

(x),

∈ Ω.

9. MICROLOCAL REGULARITY IN NESTED STRATA

Hence, by (9.1.6) and (9.1.5),

(9.1.7)

P u

,Ω

≤ Ce

−λ/C

for a suitable positive constant C.

Let us now estimate the second term in the right hand side of (9.1.4).
Let W

, i = 1, 2, be a relatively compact open cover of Ω

\ Ω

, such that

(9.1.8)

\ Ω

∩ Σ

⊂ W

\ Ω

∩ Σ

∩ W

∅.

Then

,Ω

\Ω

≤ u

. Because of our assumptions the function

r is strictly positive away from ρ

on Σ

, therefore

(9.1.9)

(x)

≥ Φ

(x) + c

∈ W

Then

2
Φ

−2λΦ

(x)

|u(x)|

L(dx)

≤

−2λΦ

(x)

−2λc

|u(x)|

L(dx)

≤ Ce

−2λc

≤ Ce

−λc

t > 0.

On the other hand in W

we have that

is exponentially decreasing w.r.t.

λ because of (9.1.1). Thus, due to (9.1.6), we obtain that

is also expo-

nentially decreasing.

By (9.1.4) we deduce that

,Ω

≤ C exp(−λt/C), for a suitable positive

constant C. Let now Ω

⊂⊂ Ω

be a neighborhood of x

such that Φ

≤ Φ

+t/(2C)

in Ω

. We conclude that

2
Φ

,Ω

≤ Ce

−λt/C

t > 0.

This proves the theorem.

Next we state

Theorem

9.1.2. Let ρ

∈ Char(P ). Then there exists an index p ∈ {1, . . . ,

}, such that

∈ Σ

Let U be a conic neighborhood of ρ

. We assume that for every j = 1, . . . , p

− 1, for

every ρ

∈ Σ

∩ U there exists a real analytic function r

: U

→ [0, +∞[, such that

∩U\{ρ}

> 0,

(ρ) = 0.

Moreover we assume that r

satisﬁes ( 9.1.3). Assume also that there exists a real

analytic function r

: U

→ [0, +∞[ such that

∩U\{ρ

}

> 0,

(ρ

) = 0,

and satisﬁes ( 9.1.3).

If ρ

∈ W F

(P u), then ρ

∈ W F

(u).

A straightforward consequence of the above theorem is the

Corollary

9.1.3. If the assumptions of Theorem 9.1.2 are satisﬁed for every

ρ in a conic subset of Char(P ), then P is micro-hypoanalytic.

9.2. A CASE OF NONSYMPLECTIC STRATIFICATION

Proof of Theorem

9.1.2. All we have to do is to verify that the hypotheses

of the theorem imply (9.1.1) and then apply Lemma 9.1.1.

This can be done arguing by induction.
First observe that since ρ

∈ W F

(P u), implies that, possibly shrinking the

open set U ,

∩ W F

(P u) =

∅.

We start with the ﬁrst stratum Σ

. Apply Theorem 7.1 to Σ

using the function

. We get that

W F

(u)

∩ Σ

∩ U = ∅.

Apply now Lemma 9.1.1 to a generic point of U

∩ Σ

, using the function r

, we ﬁnd

that

W F

(u)

∩ Σ

∩ U = ∅.

Iterating we conclude the proof.

Example

9.1.4. Let

P (x, D) = D

−2

j=1

2(p

−1)

j+1

+ x

j+1

) + x

2(q

−1)

where n > 2, p

< p

j+1

, p

< q, p

, q

∈ N. To ﬁx the ideas we also assume that

j+1

> q.

The stratiﬁcation is given by the strata Σ

, . . . , Σ

−1

, at depth 1, p

, . . . , p

−2

respectively, where

(x, ξ)

| x

· · · = x

= 0,

· · · = ξ

= 0,

j+1

+ x

2
j+1

> 0

Moreover if (ˆ

x, ˆ

ξ)

∈ Σ

then we may use the function

(x, ξ) =

k=j+1

)

− ˆx

)

+ (ξ

− ˆξ

)

+ CP (x, ξ),

for a suitable C > 0.

9.2. A Case of Nonsymplectic Stratiﬁcation

Let k be an integer, 3

≤ k ≤ n. Given a sequence of positive integers

≤ p

< p

· · · < p

−1

we deﬁne

(x, ξ) = ξ

(x, ξ) = x

−1

j = 2, . . . , k.

We consider an operator of the form

P (x, D) =

i,j=1

(x, D)a

(x, D)X

(s, D) +

j=1

(x, D)X

(s, D) + c(x, D),

where a

(x, ξ), b

(x, ξ), c(x, ξ) analytic symbols of order 0,

]

i,j=1,...,n

+ [a

]

i,j=1,...,k

≥ c,

for a suitable c > 0. Then we have the

9. MICROLOCAL REGULARITY IN NESTED STRATA

Theorem

9.2.1. Let j = 3, . . . , k. Assume that (0, e

) /

∈ W F

(P u). Then

(0, e

) /

∈ W F

pj−1

(u).

Proof.

In order to prove a hypoellipticity result for the operator P we need to

construct a function r vanishing only at the point (0, e

) and satisfying a duplication

condition similar to the one in formula (9.1.3). On the other hand, since the point
(0, e

), j = 3, . . . , k, belongs to a non symplectic stratum there is a geometric

obstruction to the construction of such a function r (see Appendix 2.) For this
reason we need to modify our construction so that the function r is an analytic
symbol of order zero. Let j = 3, . . . , k and deﬁne

r(x, ξ) = ξ

+ x

2(p

−1

−1)

−1

[ξ

+ λ

−1+

pj−1

] + (ξ

− 1)

+ x

2
j

=j+1

[ξ

+ x

We point out that the function r is non negative and that it vanishes only at the
point (0, e

Clearly, we have that there exists a real analytic function α

such that

{r(x, ξ), X

(x, ξ)

} = α

(x, ξ)X

(x, ξ).

Let X

(x, ξ) = x

−1

, = j + 1, . . . , k, then we have

{r(x, ξ), X

(x, ξ)

} =

)

2(p

−1

− 1)x

−1

−2

− 2x

−1

We recall that we are working in a microlocal neighbourhood of (0, e

), hence the

above identity can be rewritten as

{r(x, ξ), X

(x, ξ)

} = α

(x, ξ)X

(x, ξ) + α

(x, ξ)X

(x, ξ),

= j + 1, . . . , k,

where α

and α

are suitable real analytic functions.

Furthermore

{r(x, ξ), X

(x, ξ)

} =

)

2(p

−1

− 1)x

−1

−2

)

− 2x

−1

= α

(x, ξ)X

(x, ξ) + α

(x, ξ)X

(x, ξ)

for suitable real analytic functions α

and α

. Let = 2, . . . , j

− 1. Then we have

{r(x, ξ), X

(x, ξ)

} =

)

2(p

−1

− 1)x

−1

−2

)

− 2x

−1

−1+

pj−1

= α

(x, ξ)X

(x, ξ) + β

(x, ξ)x

−1

−1+

pj−1

where α

and β

are suitable real analytic functions.

We point out that the above formulas are of the form (9.1.3) modulo terms

having negative powers of λ and vanishing as x

−1

9.2. A CASE OF NONSYMPLECTIC STRATIFICATION

Arguing as in the proof of Theorem 7.1, see also the paper [1] for the treatment

of the lower order terms, we ﬁnd an estimate of the form

(9.2.1)

pj−1

≤ C

)

+ λ

,Ω

\Ω

+ t

pj−1

−1

where C is a positive constant independent of λ and t, α is a suitable positive
number and Y

, . . . , Y

−1

are pseudodiﬀerential operators of order 0 whose sym-

bols (restricted to Λ

) are x

2(p

−1)

, . . . , x

2(p

−2

−1)

respectively and Ω

⊂⊂ Ω is a

neighbourhood in

of the point

−ie

. We claim that, for = 2, . . . , j

− 1,

(9.2.2)

pj−1

≤ X

+ λ

pj−1

We point out that (9.2.1) and (9.2.2) imply that, taking t > 0 small enough,

(9.2.3)

pj−1

≤ C

)

+ λ

,Ω

\Ω

The basic idea for the proof of (9.2.2) is the elementary inequality

(9.2.4)

pj−1

2(p

−1

−1)

≤ λ

2(p

−1

−1)

+ λ

pj−1

Actually the proof goes along the following lines:

1- The symbol of Y

O((|x

| + |ξ

−1

2- The symbol of X

O((|x

| + |ξ

−1

3- The operator λ

−

pj−1

+ X

admits a parametrix which acts

continuously on L

(Ω, e

−2λΦ

As far as the wave front set of u is concerned, arguing as in the proof of Theorem
7.1, we deduce that

(9.2.5)

(x)

≥ Φ

(x)

∈ Ω.

Then, we obtain that

≤ Ce

−λ/C

for a suitable positive constant C. Moreover there is a positive constant c

such

that

(x)

≥ Φ

(x) + c

tλ

−1+

pj−1

for x

∈ Ω \ Ω

so that

,Ω

\Ω

≤ Ce

−ctλ

pj−1

for suitable positive constants c, C. Hence

,Ω

≤ Ce

−tλ

pj−1

for a suitable positive C, i.e. (0, e

) /

∈ W F

pj−1

(u). This completes the proof.

9. MICROLOCAL REGULARITY IN NESTED STRATA

9.3. A Case of Two Strata

we consider the two vector ﬁelds

(x, D) = D

− x

−1

m = 3, 4, . . . The set Char(X

, X

) has two Poisson strata: the ﬁrst one’s connected

components are symplectic manifolds of codimension 2 and the components are
labeled by x

≷ 0, ξ

≷ 0 while the second stratum connected components are

given by the equations x

= ξ

= 0, ξ

≷ 0 and are nonsymplectic. We recall

that, in accordance with the conjecture of Treves, the operator

(x, D)

+ X

(x, D)

is not analytic hypoelliptic (see [12] and [6]). We consider an operator of the form

P (x, D) =

i,j=1

(x, D)a

(x, D)X

(s, D) +

j=1

(x, D)X

(s, D) + c(x, D),

where a

(x, ξ), b

(x, ξ), c(x, ξ) analytic symbols of order 0,

]

i,j=1,...,n

+ [a

]

i,j=1,...,k

≥ c,

for a suitable c > 0.

If we take a point ρ

∈ Char(P ) which is in the ﬁrst

stratum then, by Tartakoﬀ–Treves Theorem, if ρ

∈ W F

(P u) we conclude that

∈ W F

(u). Hence, we assume that ρ

∈ Char(P ) is not in the ﬁrst stratum. To

be deﬁnite we may assume that ρ

= (0, e

). Then, let

{(0, x

, 0, e

)

| x

∈ R}

be the Hamiltonian leaf passing through the point ρ

. We have the

Theorem

9.3.1. Let 1

≤ s < m and let U be a neighbourhood of ρ

assume

that ρ

∈ W F

(P u) and that W F

(u)

∩ F

∩ ∂U = ∅. Then ρ

∈ W F

(u). Finally,

P is G

hypoelliptic for s

≥ m.

We omit the proof which is quite analogous to the proof of Theorem 8.1.1.

CHAPTER 10

Known Cases and Examples

10.1. The Case of codim Σ = 2

Let X

, . . . , X

satisfy Condition (Sympl) of Section 3.1., suppose that

codim Σ = 2

and let (x

, ξ

)

∈ Σ = Char(X

, . . . , X

Assume that the depth of Σ is > 1. We may suppose that the vector ﬁelds are

in the canonical form given in Lemma 3.1.4 and that (x

, ξ

) = (0, e

). Let P be

an operator of the form

P (x, D) =

i,j=1

(x, D)a

(x, D)X

(s, D) +

j=1

(x, D)X

(s, D) + c(x, D)

where a

(x, ξ), b

(x, ξ), c(x, ξ) analytic symbols of order 0,

]

i,j=1,...,2k

+ [a

]

i,j=1,...,2k

≥ c,

for a suitable c > 0. Then we have the

Theorem

10.1.1. Assume that (0, e

) /

∈ W F

(P u). Then (0, e

) /

∈ W F

(u).

Proof.

The standard form of the vector ﬁelds given in Lemma 3.1.4 and our

assumptions imply that we may suppose that the vector ﬁelds, X

, . . . , X

, are of

the form

(10.1.1)

(x, D) = ∂

(x, D) = x

j=2

(x)∂

j = 2, . . . , N,

where m

are suitable positive integer and for every j = 1, . . . , N there exists

k = 1, . . . , n such that

}

= 0.

Since we are assuming that there exists a single stratum, possibly changing the
label of the vector ﬁelds, we may assume that

(10.1.2)

(∂

)(0, e

)

= 0

and

(∂

)(0, e

) = 0

where i = 3, . . . , N . Now, we want to apply to the present situation Theorem 7.1.
Set

r(x, ξ) = X

(x, ξ)

+ X

(x, ξ)

−1

j=2

2
j

+ ξ

] + x

2
n

+ (ξ

− 1)

Clearly r is nonnegative and vanishes only at the point (0, e

). It remains to verify

that Condition (7.3) holds. We have that

{r(x, ξ), X

(x, ξ)

} = 2{X

(x, ξ), X

(x, ξ)

} X

(x, ξ).

10. KNOWN CASES AND EXAMPLES

Furthermore, let j = 2, . . . , N ,

{r(x, ξ), X

(x, ξ)

}

= 2

(x, ξ), X

(x, ξ)

} X

(x, ξ) + 2

(x, ξ), X

(x, ξ)

} X

(x, ξ)

k=2

, X

(x, ξ)

−1

k=2

{ξ

, X

(x, ξ)

}ξ

+ 2

{ξ

, X

(x, ξ)

}(ξ

− 1).

Now, using (10.1.1) and (10.1.2), it follows that the terms in the last line of the
above identity are multpile of X

. This completes our proof.

10.2. Okaji’s Theorem

Let μ be a positive integer and consider an operator of the form

P (x, D) =

i,j=1

(x, D)a

(x, D)X

(s, D) +

j=1

(x, D)X

(s, D) + c(x, D)

where a

(x, ξ), b

(x, ξ), c(x, ξ) analytic symbols of order 0,

]

i,j=1,...,2k

+ [a

]

i,j=1,...,2k

≥ c

where c is a positive constant and

(x, D) = D

k+j

(x, D) = x

μ
j

j = 1, . . . , k.

Then we have the

Theorem

10.2.1. Assume that (0, e

) /

∈ W F

(P u) then (0, e

) /

∈ W F

(u).

Proof.

The proof is a consequence of Theorem 7.1. Indeed it suﬃces to con-

struct a nonnegative function r satisfying (7.3) and such that r(x, ξ) = 0 if and
only if (x, ξ) = (0, e

). It is easy to see that the function

r(x, ξ) =

j=1

+ (ξ

− 1)

j=1

2μ
j

j=k+1

2
j

has all the required properties. This completes our proof.

APPENDIX A

A Bracket Lemma

Let us consider a system of real vector ﬁelds with real analytic coeﬃcients

deﬁned in an open set U

⊂ R

, X

, . . . , X

. We may always suppose that the

origin belongs to the open set U . On the vector ﬁelds we make the same assumptions
either as in Section 3.1 or as in Section 3.3.

Hence we may assume that the vector ﬁelds are written in the form (3.1.3)–

(3.1.4) if they correspond to a (single) symplectic stratum or in the form (3.3.7) if
they correspond to a nonsymplectic (single) stratum.

Let us denote by γ = (0, e

) a characteristic point for the vector ﬁelds and

by r the length of the ﬁrst Poisson bracket of the symbols of the vector ﬁelds
that does not vanish at γ. We use here the following standard notation. Denote by
X

(x, ξ) the symbol of the vector ﬁeld X

(x, D). If I = (i

, . . . , i

), with i

, . . . , i

∈

{1, 2, . . . , N}, is a multiindex, we write |I| = p and

(A.1)

(x, ξ) =

(x, ξ),

{· · · {X

−1

(x, ξ), X

(x, ξ)

} · · · }.

In this Appendix we want to prove the

Proposition

A.1. Let X

, . . . , X

be as in Section 3.1 (or 3.3) and let them

satisfy H¨

ormander condition with a bracket of length r at the point γ

∈ Char(X).

Then there exists a Poisson bracket of the symbols of the vector ﬁelds X

, . . . , X

, with

|I| = r, such that X

(γ)

= 0 and moreover if I = (i

, . . . , i

), we have

that i

∈ {1, . . . , κ} for α = 1, . . . , r − 1, and 1 ≤ i

≤ N.

We argue in the symplectic case. The other case is completely analogous. First

we recall that the integer r is just the depth of the single stratum associated to the
given vector ﬁelds.

Due to the standard forms of Section 3.1, a Poisson bracket of any length is a

vector ﬁeld with symbol

n
k=

κ+1

(x)ξ

, where the c

belong to C

(U ). Vanishing

on the characteristic manifold for a vector ﬁeld like that means vanishing when
x

· · · = x

= 0.

We shall have proved the above proposition if we prove the

Lemma

A.2. Same assumptions and notation as in Proposition A.1. Then the

following two assertions are equivalent:

(i) For any γ

∈ Char(X) there exists a multiindex I, |I| = r, I = (i

, . . . , i

with 1

≤ i

≤ N for α = 1, . . . , r, such that

(γ)

= 0.

(ii) For any γ

∈ Char(X) there exists a multiindex J, |J| = r, J = (j

, . . . , j

with 1

≤ j

≤ κ for α = 1, . . . , r − 1, 1 ≤ j

≤ N and such that

(γ)

= 0.

A. A BRACKET LEMMA

Proof.

The implication (ii)

⇒ (i) is trivial. We have thus to show only the

implication (i)

⇒ (ii). To this end we argue by contradiction: we assume that there

exists a point γ

∈ Char(X) such that for every J, |J| = r, J = (j

, . . . , j

), with

≤ j

≤ κ for α = 1, . . . , r − 1 and 1 ≤ j

≤ N, we have that X

(γ) = 0. On

the other hand from H¨

ormander’s condition it follows that there is a multiindex

I, I = (i

, . . . , i

), with 1

≤ i

≤ N for α = 1, . . . , r, such that X

(γ)

= 0. Thus

Char(X)

= 0.

To achieve the proof we are going to need the following lemma:

Lemma

A.3. Let the vector ﬁelds X

, . . . , X

be given as in Proposition A.1

and let p be a nonnegative integer. Then the following assertions are equivalent:

(a) For every multiindex I,

|I| = q ≤ p, I = (i

, . . . , i

) with 1

≤ i

≤ N for

α = 1, . . . , q, we have

(A.2)

vanishes for x

= 0,

where x

= (x

, . . . , x

(b) For every multiindex J ,

|J| = q ≤ p, J = (j

, . . . , j

) with 1

≤ j

≤ κ for

α = 1, . . . , q

− 1 and 1 ≤ j

≤ N, we have

(A.3)

vanishes for x

= 0.

It is then clear that assertion (a) of Lemma A.3, where q has been replaced

by r, is contradicted thus yielding a negation of assertion (b) of the same Lemma.
And this is absurd.

This ends the proof of Lemma A.2.

Let us prove now the second lemma.

Proof of Lemma

A.3. As before the implication (a)

⇒ (b) is trivial. Thus

we are left with the opposite implication.

Let at ﬁrst be p = 2. Assume (b). Consider the Poisson bracket

(x, ξ), X

(x, ξ)

There are two cases: (i) i

∈ {κ+1, . . . , N} and (ii) i

∈ {1, . . . , κ}, (no restrictions

on i

Case (ii) just restates the assumption. Let us look at case (i). Then X

has

the form

n
s=

κ+1

, x

)∂

, where a

∈ C

(U ) and a

(0, x

) = 0 for every

s. If i

∈ {1, . . . , κ}, then reversing the order of the arguments of the Poisson

bracket places us in a case coinciding with the assumption. Assume thus that
i

∈ {κ + 1, . . . , N} so that X

(x, ∂) =

n
=

κ+1

, x

)∂

, where b

∈ C

(U )

and b

(0, x

) = 0. Now

(x, ξ), X

(x, ξ)

} =

,s=

κ+1

∂

− b

∂

) ξ

which vanishes when x

= 0. We remark that the above expression vanishes when

= 0 if just one of the vector ﬁelds does. This concludes the case p = 2.

Let now p

≥ 3. We argue by induction. Assume that for every J, |J| = q ≤ p,

J = (j

, . . . , j

) with j

∈ {1, . . . , κ} for α = 1, . . . , q − 1 and 1 ≤ j

≤ N we have

that X

= 0 if x

= 0 and that the lemma holds if

|J| ≤ p − 1. We want to show

that then X

= 0 if x

= 0, when

|I| ≤ p and the components of I, i

, belong to

{1, . . . , N}.

A. A BRACKET LEMMA

To prove this we need the following

Lemma

A.4. Let us consider the same vector ﬁelds as in Lemma A.3. Assume

that condition (b) implies condition (a) for multiindices of length up to p

− 1 and

that (b) holds for multiindices of length up to p. Then if I is a multiindex of length
p and π(I) denotes a multiindex of length p whose components are a permutation
of those of I we have that

(A.4)

(x, ξ)

− X

π(I)

(x, ξ)

vanishes if x

= 0.

Let us now ﬁnish the proof of Lemma A.3. Consider a multiindex I which is

not of the type described in assertion (b). This implies that at least two of its
components belong to

{κ + 1, . . . , N}. Then by Lemma A.4 we know that X

diﬀers from X

π(I)

for a symbol vanishing when x

= 0. It is then enough to choose

π as the permutation that places two components in

{κ + 1, . . . , N} at the ﬁrst

and last position, so that X

π(I)

, X

}, with m

∈ {κ + 1, . . . , N} and X

|K| = p − 1 vanishing when x

= 0.

The lemma is then proved.

Proof of Lemma

A.4. It is enough to prove the assertion when π is a trans-

position of two adjacent elements.

Under the hypotheses of the lemma, let us consider a multiindex I,

|I| = p,

I = (i

, . . . , i

), with 1

≤ i

≤ N for α ∈ {1, . . . , p}. Let us write I = (i

, i

, I

where of course

| = p − 2. We have

, X

}} = {X

, X

}} + {{X

, X

}, X

by Jacobi identity. Let us consider the second bracket; by our inductive assumption
the ﬁeld X

vanishes on x

= 0, since its length is p

− 2. On the other hand the

bracket

, X

} is a vector ﬁeld of the form

n
k=

κ+1

(x)∂

, so that the second

bracket above vanishes on x

= 0.

Thus

(A.5)

, X

}} = {X

, X

}}

mod a ﬁeld vanishing on x

= 0,

i.e. the assertion for the ﬁrst pair of indices in the multiindex I. Furthermore let
us consider the bracket in the r.h.s. of the above relation. Writing I

= (i

, I

with

| = p − 3, we have

, X

}}}

, X

}}} + {X

{{X

, X

}, X

}}

, X

}}}

{{X

, X

}, {X

, X

}} + {{X

, X

}}, X

}

Let us examine the brackets in the last line of the above formula. The ﬁrst bracket
has a symbol of the form

, X

} =

n
k=

κ+1

(x)ξ

bracketed with X

, where

I = (i

, I

), and the latter has the same form, i.e. a linear combination of the

, k =

κ + 1, . . . , n, and, by the inductive assumption, vanishes when x

= 0.

Thus the ﬁrst term vanishes when x

= 0. For the second term we apply the same

criteria: the triple bracket has a symbol of the form

n
k=

κ+1

(x)ξ

, because of

the standard form of our vector ﬁelds, while X

vanishes on x

= 0. We thus

A. A BRACKET LEMMA

conclude that

(A.6)

, X

}}} = {X

, X

}}}

mod a ﬁeld vanishing on x

= 0.

The argument can be iterated: let I

= (i

, I

(iv)

), with

(iv)

| = p − 4. We may

write

, X

(iv)

}}}}

, X

(iv)

}}}} + {X

{{X

, X

} , X

(iv)

}}}

, X

(iv)

}}}} + {X

{{X

, X

} , {X

(iv)

, X

}}}

{{X

, X

}} , X

(iv)

}}

, X

(iv)

}}}}

{{X

, X

} , {X

, X

(iv)

}}} + {{X

, X

}} , {X

, X

(iv)

}}

{{X

, X

}} , {X

, X

(iv)

}} + {{X

, X

}}} , X

(iv)

} .

Let us now examine the last four terms in the above formula. We have

, X

} is a linear combination of the x

-derivatives; the symbol

, X

(iv)

}} has the same form, but vanishes on x

= 0, by induc-

tion (its length is p

− 2.) Hence the ﬁrst of the last four terms vanishes

for x

= 0.

, X

}} is a linear combination of the derivatives w.r.t. the

variables x”; the symbol

, X

(iv)

} has the same form, but vanishes on

= 0, by induction (its length is p

− 3.) Hence the ﬁrst of the last four

terms vanishes for x

= 0.

, X

}} is a linear combination of the x

-derivatives; the symbol

, X

(iv)

} has the same form, but vanishes on x

= 0, by induction (its

length is p

− 3.) Hence the ﬁrst of the last four terms vanishes for x

= 0.

4- Finally

, X

}}} is a linear combination of the x

-deriva-

tives; the symbol X

(iv)

has the same form, but vanishes on x

= 0, by

induction (its length is p

− 4.) Hence the ﬁrst of the last four terms

vanishes for x

= 0.

We thus conclude that

(A.7)

, X

}}} = {X

, X

(iv)

}}}}

mod a ﬁeld vanishing on x

= 0.

This argument can be iterated and we conclude then that, denoting by

I = (i

, . . . , i

)

our starting multiindex and by I

, s = 2, . . . , p, the multiindex

= (i

, i

, . . . , i

, i

s+1

, . . . , i

we have

(A.8)

= X

mod a ﬁeld vanishing on x

= 0.

A. A BRACKET LEMMA

Let us now ﬁnish the proof of the lemma. Let s

∈ {1, . . . , p − 1}. We want to

exchange the adjacent indices i

and i

s+1

. First we use (A.8) for

, . . . , i

)

→ (i

, i

, . . . , i

, i

s+1

, i

, . . . , i

where the arrow means that there is an equation of the form (A.8) between the
corresponding brackets.

Next we do the same permutation so that i

goes to the (s + 1)

− th position:

, i

, . . . , i

, i

s+1

, i

, . . . , i

)

→ (i

, . . . , i

, i

s+1

, i

, . . . , i

Iterating this s

− 1 times we obtain the permutation

, i

s+1

, i

, . . . , i

−1

, i

s+2

, . . . , i

As it has been observed at the beginning of this proof we may interchange the ﬁrst
and the second index modulo a ﬁeld vanishing on x

= 0. We thus arrive at the

permutation

s+1

, i

, . . . , i

−1

, i

s+2

, . . . , i

Next we repeat what has been done before, i.e. we move the ﬁrst index to the
(s + 1)-th position twice, the ﬁrst time for i

s+1

, the second time for i

. We arrive

at the permutation

, i

, . . . , i

−1

, i

s+1

, i

s+2

, . . . , i

modulo a ﬁeld vanishing when x

= 0. This proves our claim and hence the lemma.

Using Proposition A.1 we want to prove the following theorem.

Theorem

A.5. Let the vector ﬁelds X

, . . . , X

be given either as Section 3.1.

or as in Section 3.3 and let

κ have the same meaning. Suppose that κ > 1. Assume

that there is a multiindex I, with

|I| = r, such that X

(γ)

= 0 and I is of the form

obtained in Proposition A.1, i.e. I = (i

, . . . , i

), and 1

≤ i

≤ κ for 1 ≤ α ≤ r − 1

whereas 1

≤ i

≤ N. Then there exists a nonsingular linear substitution of the

ﬁelds and a multiindex J , with

|J| = r, J = (j

, . . . , j

), for which 1

≤ j

≤ κ for

every α

∈ {1, . . . , r} such that X

(γ)

= 0.

Proof.

Consider the multiindex

I = (i

, . . . , i

)

and assume that i

∈ {1, . . . , κ} for 1 ≤ α ≤ r−1. If i

∈ {1, . . . , κ} there is nothing

to prove. Thus we may assume that i

∈ {κ + 1, . . . , N} and that X

(γ)

= 0. Let

us deﬁne the following linear nonsingular subtitution of the ﬁelds:

⎡
⎢

⎢

⎣

κ+1

⎤
⎥

⎥

⎦

⎡
⎢

⎢

⎣

κ−1

−κ−i

−1

−i

⎤
⎥

⎥

⎦

⎡
⎢

⎢

⎣

κ+1

⎤
⎥

⎥

⎦

where ε is a positive number and its entry is the (

κ, i

) entry. The size of ε will be

chosen in a short while. Here Id

denotes the h

× h identity matrix.

A. A BRACKET LEMMA

Clearly the above substitution is nonsingular. We only have to check that

the Poisson bracket Y

, with J = (i

, . . . , i

−1

κ) = (I

κ) is elliptic at γ. Now

= X

+ εX

and, by our assumption, X

κ)

(γ) = 0. Thus we can conclude

that the bracket Y

(γ) is a polynomial in ε of degree equal to the number of the

occurrences of the index

κ in I

plus one. Moreover the term of degree zero of this

polynomial vanishes, while the coeﬃcient of ε is nonzero. We may hence always
choose an ε

> 0 such that for 0 < ε

≤ ε

the bracket Y

(γ) is elliptic.

This ends the proof of the theorem.

APPENDIX B

Nonsymplectic Strata Do Not Have the

Reproducing Bracket Property

We consider an operator of the form (8.1) and we assume that the Poisson–

Treves stratiﬁcation associated to the vector ﬁelds X

, . . . , X

has a single stratum.

Let (x

, ξ

)

∈ Char(X

, . . . , X

) = Char(P ) In this section we show that, if the

single stratum is nonsymplectic, there is a geometric obstruction to the construction
of a function r = r(x, ξ) satisfying (7.3) and such that

(B.1)

r(x, ξ) = 0

if and only if

(x, ξ) = (x

, ξ

We assume that the depth of the stratum is bigger than 1. Let F

be the Hamil-

tonian leaf through (x

, ξ

) in Char(P ). We assume that the base projection of F

has the same dimension as F

(i.e. we make the same assumptions as in Section

3.3). Assume furthermore that there exist a neighbourhood of (x

, ξ

), U , a real

analytic function r : U

→ [0, +∞[ and real analytic functions, α

(x, ξ), deﬁned in

U , such that

(B.2)

{r(x, ξ), X

(x, ξ)

} =

(x, ξ)X

(x, ξ),

where j = 1, . . . , N . Then we have the following

Theorem

B.1. r

= 0.

In other words, if the stratum is nonsymplectic and a function r satisﬁes (B.2)

then Condition (B.1) cannot hold.

Proof.

We may assume that (x

, ξ

) = (0, e

) and we recall that, as shown in

Section 3.3, the vector ﬁelds X

, . . . , X

can be represented as follows:

∂

n
k=

κ+1

−1

m=1

(m)
ik

(x)

∂

i = 1, . . . , h;

∂

n
k=

κ+1

h
m=1

(m)
ik

(x)

∂

i = h + 1, . . . , h + ;

n
k=

κ+1

(x)∂

j =

κ + 1, . . . , N,

where a

(0) = 0, for j =

κ + 1, . . . , N, k = κ + 1, . . . , n. Moreover, the stratum is

given by the equation

Σ =

{(x, ξ) ∈ T

∗

\ {0} | x

= 0, i = 1, . . . , h, ξ

= 0, i = 1, . . . , h +

where denotes the dimension of the leaves and rank σ

|Σ

= 2n

− 2h − 2. In

particular, we have that

{(x, e

)

| x

= 0, i = 1, . . . , h, h + + 1, . . . , n

B. NONSYMPLECTIC STRATA AND THE REPRODUCING BRACKET PROPERTY

Since r(0, e

) = 0, if we show that

∂r

∂x

= 0

i = h + 1 . . . h + .

then we conclude that r

= 0. Let i

∈ {h + 1 . . . h + }, then we have

, r

}

∂r

∂x

−

κ+1

m=1

(m)
ik

(x)ξ

∂r

∂ξ

By (B.2), we have that

0 =

, r

}

, r

}

On the other hand, since the depth of the stratum is bigger than 1, we deduce that

0 =

, X

}

κ+1

(j)
ik

(x)ξ

j = 1, . . . , h,

hence we conclude that

∂r

∂x

= 0

i = h + 1, . . . h + .

This completes our proof.

Bibliography

[1] P. Albano, A. Bove and G. Chinni, Minimal Microlocal Gevrey Regularity for “Sums

of Squares”, International Mathematics Research Notices, Vol. 2009, No. 12, 2275-2302.
MR2511911 (2010e:35054)

[2] M. S. Baouendi and Ch. Goulaouic, Nonanalytic-hypoellipticity for some degenerate op-

erators, Bull. A. M. S. 78(1972), 483-486. MR0296507 (45:5567)

[3] A. Bove and D.S. Tartakoff, Optimal non-isotropic Gevrey exponents for sums of squares

of vector ﬁelds, Comm. Partial Diﬀerential Equations, 22(1997), 1263-1282. MR1466316
(98f:35026)

[4] A. Bove and D.S. Tartakoff, Propagation of Gevrey Regularity for a Class of Hypoelliptic

Equations, Trans. Amer. Math. Soc. 348(1996), 2533-2575. MR1340171 (96i:35017)

[5] A. Bove and F. Treves, On the Gevrey hypo-ellipticity of sums of squares of vector ﬁelds,

Ann. Inst. Fourier (Grenoble) 54(2004), 1443-1475. MR2127854 (2005k:35053)

[6] M. Christ, Certain Sums of Squares of Vector Fields Fail To Be Analytic Hypo-Elliptic,

Comm. Partial Diﬀerential Equations, 16(1991), 1695-1707. MR1133746 (92k:35056)

[7] M. Christ, Intermediate Optimal Gevrey Exponents Occur , Comm. Partial Diﬀerential

Equations, 22(1997), 359-379. MR1443042 (98c:35028)

[8] P. Cordaro and N. Hanges, A New Proof of Okaji’s Theorem for a Class of Sum of Squares

Operators, Ann. Inst. Fourier (Grenoble) 59(2009), 595-619. MR2521430 (2011e:35069)

[9] M. Derridj, Un probl`

eme aux limites pour une classe d’op´

erateurs du second ordre hypoel-

liptiques, Ann. Inst. Fourier, 21, no 4, (1971), 99-148. MR0601055 (58:29139)

[10] A. Grigis and J. Sj¨

ostrand

, Front d’onde analytique et sommes de carr´

es de champs de

vecteurs, Duke Math. J. 52 (1985), 35-51. MR791290 (86h:58136)

[11] V. V. Gruˇ

sin

, A certain class of elliptic pseudodiﬀerential operators that are degenerate on

a submanifold , Mat. Sb. 84 (1971), 163-195. MR0283630 (44:860)

[12] N. Hanges and A. Himonas, Singular solutions for sums of squares of vector ﬁelds, Comm.

Partial Diﬀerential Equations, 16(1991), 1503-1511. MR1132794 (92i:35031)

[13] B. Helffer, Hypoellipticit´

e analytique sur des groupes nilpotents de rang 2, S´

eminaire

Goulaouic-Schwartz (1979/80), ´

Ecole Polytechnique, Palaiseau, France, I.1-I.13. MR600685

(83f:35038)

[14] B. Helffer, Conditions n´

ecessaires d’hypoanalyticit´

e pour des op´

erateurs invariants `

gauche homog`

enes sur un groupe nilpotent gradu´

e, J. Diﬀ. Equations 44(1982), 460-481.

MR661164 (84c:35026)

[15] L. H¨

ormander

, Hypoelliptic second order diﬀerential equations, Acta Math. 119(1967), 147-

171. MR0222474 (36:5526)

[16] J.J. Kohn, Pseudo-diﬀerential operators and non-elliptic problems, Corso CIME 1968, 158-

165. MR0259334 (41:3972)

[17] G. M´

etivier

, Analytic hypoellipticity for operators with multiple characteristics, Comm. in

PDE 6(1980), 1-90. MR597752 (82g:35030)

[18] T. ¯

Okaji

, Analytic hypoellipticity for operators with symplectic characteristics, J. Math.

Kyoto Univ. 25 (1985), 489-514. MR807494 (87d:35036)

[19] O. A. Ole˘ınik, On the analyticity of solutions of partial diﬀerential equations and systems,

Colloque International CNRS sur les ´

Equations aux D´

eriv´

ees Partielles Lin´

eaires (Univ. Paris-

Sud, Orsay, 1972), 272-285. Ast´

erisque, 2 et 3. Societ´

e Math´

ematique de France, Paris, 1973.

MR0399640 (53:3483)

[20] O. A. Ole˘ınik and E. V. Radkeviˇ

, The analyticity of the solutions of linear partial diﬀer-

ential equations, (Russian) Mat. Sb. (N.S.) 90(132)(1973), 592-606. MR0433019 (55:5998)

BIBLIOGRAPHY

[21] L. P. Rothschild and E. M. Stein, Hypoelliptic diﬀerential operators and nilpotent groups,

Acta Math. 137(1977), 247-320. MR0436223 (55:9171)

[22] J. Sj¨

ostrand

, Analytic wavefront set and operators with multiple characteristics, Hokkaido

Math. J. 12(1983), 392-433. MR725588 (85e:35022)

[23] J. Sj¨

ostrand

, Singularit´

es analytiques microlocales, Ast´

erisque 95 (1982).

MR699623

(84m:58151)

[24] J. Sj¨

ostrand

, Lectures on resonances, unpublished lecture notes, 2002, http://www.

math.polytechnique.fr/~sjoestrand/CoursgbgWeb.pdf.

[25] D.S. Tartakoff, Local Analytic Hypoellipticity for

on Non-Degenerate Cauchy Riemann

Manifolds, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), 3027-3028. MR499657 (80g:58045)

[26] D.S. Tartakoff, On the Local Real Analyticity of Solutions to

and the ¯

∂-Neumann

Problem, Acta Math. 145 (1980), 117-204. MR590289 (81k:35033)

[27] D.S. Tartakoff, Analytic hypoellipticity for a sum of squares of vector ﬁelds in

whose

Poisson stratiﬁcation consists of a single symplectic stratum of codimension four , Advances
in Phase Space Analysis of Partial Diﬀerential Equations, 249–257, Progr. Nonlinear Diﬀeren-
tial Equations Appl., 78, Birkh¨

auser Boston, Boston, MA, 2009. MR2664615 (2011d:35107)

[28] F. Treves, Analytic Hypo-ellipticity of a Class of Pseudo-Diﬀerential Operators with Double

Characteristics and Application to the ¯

∂-Neumann Problem, Commun. Partial Diﬀ. Eq. 3

(6-7) (1978), 475-642. MR0492802 (58:11867)

[29] F. Treves, Symplectic geometry and analytic hypo-ellipticity, in Diﬀerential equations: La

Pietra 1996 (Florence), Proc. Sympos. Pure Math., 65, Amer. Math. Soc., Providence, RI,
1999, 201-219. MR1662756 (2000b:35031)

[30] F. Treves, On the analyticity of solutions of sums of squares of vector ﬁelds, Phase space

analysis of partial diﬀerential equations, 315-329, Progr. Nonlinear Diﬀerential Equations
Appl., 69, Birkh¨

auser Boston, Boston, MA, 2006. MR2263217 (2008b:35004)

[31] F. Treves, On a class of systems of vector ﬁelds, unpublished manuscript, 2009.

Index

∞

hypoellipticity, vii

(Ω) classes, 30

2
Φ

classes, 31

W F

(u), s-Gevrey wave front set, 29

Φ weight function, 29
Char(X), characteristic set , vii
{f, g} =

n
j=1

∂

f ∂

− ∂

f ∂

Poisson bracket, 2

H¨

ormander’s condition, vii

A priori estimate, 33
Admissible errors, 31
Analytic hypoellipticity, vii
Analytic stratiﬁcation, 1, 2

Baker-Campbell-Hausdorﬀ formula, 34
Baouendi-Goulaouic operator, 5

Canonical deformation, 42
Classical FBI phase, 29
Conjecture of Treves, ix

Depth of a Poisson stratum, 4

Eikonal equation, 34
Elliptic estimate, 31

FBI transform, 29
Fourier integral operators, 34

Gevrey wave front set, 29

H¨

ormander’s hypothesis, vii

Hamilton-Jacobi equation for a complex

deformation, 42

Hamiltonian leaf, 5

Kuranishi trick, 30

M´

etivier operator, ix

Nested strata, 27

Oleinik operator, 6

Path for realizing a pseudodiﬀerential

operator, 30

Poisson stratiﬁcation, 4, 5

Rank of the symplectic form, viii
Realization of a pseudodiﬀerential

operator, 30

Symplectic form, 2
Symplectic manifold, viii
Symplectic stratiﬁcation, 2, 3

Transversally elliptic degeneracy, viii

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Published Titles in This Series

1039 Paolo Albano and Antonio Bove, Wave Front Set of Solutions to Sums of Squares of

Vector Fields, 2013

1036 Matthias Lesch, Henri Moscovici, and Markus J. Pﬂaum, Connes-Chern

character for manifolds with boundary and eta cochains, 2012

1035 Igor Burban and Bernd Kreussler, Vector bundles on degenerations of elliptic curves

and Yang-Baxter equations, 2012

1034 Alexander Kleshchev and Vladimir Shchigolev, Modular branching rules for

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1033 Daniel Allcock, The reﬂective Lorentzian lattices of rank 3, 2012

1032 John C. Baez, Aristide Baratin, Laurent Freidel, and Derek K. Wise,

Inﬁnite-dimensional representations of 2-groups, 2012

1031 Idrisse Khemar, Elliptic integrable systems: A comprehensive geometric interpolation,

2012

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1029 Mikhail Khovanov, Aaron D. Lauda, Marco Mackaay, and Marko Stoˇ

si´

Extended graphical calculus for categoriﬁed quantum sl(2), 2012

1028 Yorck Sommerh¨

auser and Yongchang Zhu, Hopf algebras and congruence

subgroups, 2012

1027 Olivier Druet, Fr´

ed´

eric Robert, and Juncheng Wei, The Lin-Ni’s Problem for

Mean Convex Domains, 2012

1026 Mark Behrens, The Goodwillie Tower and the EHP Sequence, 2012

1025 Joel Smoller and Blake Temple, General Relativistic Self-Similar Waves that Induce

an Anomalous Acceleration into the Standard Model of Cosmology, 2012

1024 Mats Boij, Juan C. Migliore, Rosa M. Mir´

o-Roig, Uwe Nagel, and Fabrizio

Zanello, On the Shape of a Pure O-Sequence, 2012

1023 Tadeusz Iwaniec and Jani Onninen, n-Harmonic Mappings between Annuli, 2012

1022 Maurice Duits, Arno B.J. Kuijlaars, and Man Yue Mo, The Hermitian Two

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1021 Arnaud Deruelle, Katura Miyazaki, and Kimihiko Motegi, Networking Seifert

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1020 Dominic Joyce and Yinan Song, A Theory of Generalized Donaldson-Thomas

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1019 Abdelhamid Meziani, On First and Second Order Planar Elliptic Equations with

Degeneracies, 2012

1018 Nicola Gigli, Second Order Analysis on (

(M ), W

), 2012

1017 Zenon Jan Jablo´

nski, Il Bong Jung, and Jan Stochel, Weighted Shifts on Directed

Trees, 2012

1016 Christophe Breuil and Vytautas Paˇ

sk¯

unas, Towards a Modulo p Langlands

Correspondence for GL

, 2012

1015 Jun Kigami, Resistance Forms, Quasisymmetric Maps and Heat Kernel Estimates, 2012

1014 R. Fioresi and F. Gavarini, Chevalley Supergroups, 2011

1013 Kaoru Hiraga and Hiroshi Saito, On L-Packets for Inner Forms of SL

, 2011

1012 Guy David and Tatiana Toro, Reifenberg Parameterizations for Sets with Holes, 2011

1011 Nathan Broomhead, Dimer Models and Calabi-Yau Algebras, 2011

For a complete list of titles in this series, visit the AMS Bookstore at

www.ams.org/bookstore/series/.

ISBN 978-0-8218-7570-4

9 780821 875704

MEMO/221/1039

Document Outline

Chapter 1. Introduction
Chapter 2. The Poisson–Treves Stratification
Chapter 3. Standard Forms for a System of Vector Fields
Chapter 4. Nested Strata
Chapter 5. Bargman Pseudodifferential Operators
Chapter 6. The “A Priori” Estimate on the FBI Side
Chapter 7. A Single Symplectic Stratum
Chapter 8. A Single Nonsymplectic Stratum
Chapter 9. Microlocal Regularity in Nested Strata
Chapter 10. Known Cases and Examples
- 10.1. The Case of 𝑐𝑜𝑑𝑖𝑚Σ=2
- 10.2. Okaji’s Theorem
Appendix A. A Bracket Lemma
Appendix B. Nonsymplectic Strata Do Not Have the Reproducing Bracket Property
Bibliography
Index

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