tesi di dottorato - polito.itcalvino.polito.it/~nicola/research/revised-tesi2.pdf · tesi di...

Universita degli Studi di TorinoDottorato di Ricerca in Matematica

XVI ciclo

Tesi di Dottorato

Fabio Nicola

LOWER BOUNDS FOR

PSEUDODIFFERENTIAL OPERATORS

Relatori: Prof. Alberto Parmeggiani

Prof. Luigi Rodino

Coordinatore del Dottorato: Prof. Ferdinando Arzarello

Anno Accademico 2003-2004

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to Prof. Alberto Parmeggiani, whointroduced me to the fascinating topic of lower bounds for pseudodifferential opera-tors. In particular, I am very grateful to him for suggesting several topical argumentsof research and for communicating his enthusiasm.

I thank also very much Prof. Luigi Rodino for his encouragement and his constantsupport during the achievement of this Ph.D. His many advices, that did not concernmathematics alone, have been really invaluable.

Finally I am very indebted to Dr. Marco Mughetti for many helpful discussionson the subject of this thesis, and also for accepting to collaborate with me when Istill was studying the basic facts of the theory.

CONTENTS

Sunto (Italian) iii

I Introduction and discussion of the results 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 A generalization of Hormander’s inequality . . . . . . . . . . . . . . . 81.3 On the positive parts of second order operators . . . . . . . . . . . . 151.4 The Weak-Hormander inequality for systems . . . . . . . . . . . . . . 19

II Proof of the generalization of Hormander’s inequality 252.1 A preliminary lower bound . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Proof of Theorem 1.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3 Proof of Proposition 2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3.1 Estimate of (PΠΨu,ΠΨu) . . . . . . . . . . . . . . . . . . . . 452.3.2 Estimate of (P (Id− Π)Ψu, (Id− Π)Ψu) . . . . . . . . . . . . 512.3.3 Estimate of 2Re (PΠΨu, (Id− Π)Ψu) . . . . . . . . . . . . . . 52

2.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.4.1 Some further remarks on localized operators . . . . . . . . . . 542.4.2 Two counterexamples . . . . . . . . . . . . . . . . . . . . . . . 552.4.3 Lower bounds in absence of Levi-type conditions . . . . . . . . 58

Appendices 632.A A temperate calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 632.B A technical result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.C On the composition of pseudodifferential operators . . . . . . . . . . 71

IIIAn approximate construction of positive parts for second orderoperators 753.1 Preliminaries on the theory of Hermite-operators . . . . . . . . . . . . 753.2 Construction of the positive and negative parts . . . . . . . . . . . . 80

3.2.1 Case I: Aρ0 = ∅ . . . . . . . . . . . . . . . . . . . . . . . . . . 813.2.2 Case II: Aρ0 6= ∅ . . . . . . . . . . . . . . . . . . . . . . . . . 85

i

ii CONTENTS

IV Proof of the Weak-Hormander inequality 91

Appendices 974.A Hermite operators with matrix valued symbols . . . . . . . . . . . . . 97

Bibliography 101

iii

SUNTO

Questa tesi riguarda stime dal basso per operatori e sistemi di operatori (pseudo)dif-ferenziali a caratteristiche multiple.

Precisamente, sia X un aperto di Rn e P = P ∗ ∈ OPSm(X) un operatorepseudodifferenziale classico, a supporto proprio e formalmente autoaggiunto in X.Assumiamo che P abbia caratteristiche di molteplicita pari k ≥ 2, i.e. il terminepositivamente omogeneo di ordine m − j nello sviluppo asintotico del simbolo siannulla sulla varieta caratteristica Σ all’ordine k − 2j, per ogni j ≤ k/2. Diamoquindi condizioni sufficienti per la validita della seguente stima dal basso:

Per ogni compatto K ⊂ X esiste una costante CK > 0 tale che

(Pu, u) ≥ −CK‖u‖2m/2−(k+2)/4, ∀u ∈ C∞

0 (K).

Il nostro risultato e una continuazione di quelli ottenuti da Parenti e Parmeggianiin [53]. Inoltre, per operatori a caratteristiche doppie (k = 2), esso si riduce al nototeorema di Hormander [29], Thm. 3.3.1.

Passiamo poi a considerare il seguente problema:Fino a che punto si possono dedurre le proprieta spettrali di un operatore (pseudo)-

differenziale formalmente autoaggiunto P = p(x,D), a partire dal suo simbolo p?In particolare, si vorrebbe costruire esplicitamente, a partire da p, proiettori spettraliapprossimati positivi e negativi, vale a dire operatori Π+, Π− lineari e continui in L2

tali che Π+ +Π− = Id+ “termini trascurabili” e soddisfacenti, per ogni u ∈ C∞0 (K),

K ⊂⊂ X,

Re(PΠ+u, u) ≥ −CK‖u‖20, −Re(PΠ−u, u) ≥ −CK‖u‖2

0.

Si tratta di un problema inizialmente studiato da D. Fujiwara [17], che ha costruitotali proiettori nel caso in cui P ha ordine 1, basandosi su una decomposizione dellospazio delle fasi simile a quella introdotta da Beals e Fefferman in [1]. Noi quiotteniamo un risultato microlocale per una classe di operatori del secondo ordine,combinando le tecniche di [1, 17] con quelle di Boutet de Monvel [4] e Parenti eParmeggiani [53, 54].

Infine per una classe di sistemi N × N di operatori pseudodifferenziali a carat-teristiche doppie studiamo la seguente stima dal basso con guadagno di 3/2 derivate(Weak-Hormander):

Per ogni compatto K ⊂ X esiste una costante CK > 0 tale che

(Pu, u) ≥ −CK‖u‖2(m−3/2)/2, ∀u ∈ C∞

0 (K; CN).

Si tratta di una disuguaglianza intermedia tra la Sharp Garding e quella di Hormander.Il nostro risultato migliora una stima provata recentemente da Brummelhuis [8] eParenti e Parmeggiani [54].

Chapter I

INTRODUCTION AND DISCUSSION OF THE

RESULTS

The present thesis deals with lower bounds for pseudodifferential operators andsystems with multiple characteristics.

We start off by a short survey of the problem. Then in Sections 1.2, 1.3 and 1.4we present the main results, namely Theorems 1.2.2, 1.3.1 and 1.4.1. Finally we willprove these theorems in Chapters 2, 3 and 4 respectively.

1.1 Introduction

We deal with pseudodifferential operators P = p(x,D) in an open subset X ⊂ Rn,namely linear operators of the form

Pu(x) = (2π)−n∫eixξp(x, ξ)u(ξ)dξ,

with u ∈ C∞0 (X). The smooth function p(x, ξ), (x, ξ) ∈ X × Rn, is referred to as

the symbol of P and belongs to Hormander’s classes Sm(X × Rn), m ∈ R, i.e. itsatisfies the following growth estimates

|∂αx∂βξ p(x, ξ)| ≤ Cα,β,K(1 + |ξ|)m−|β|,

uniformly for x in compact subsets K ⊂ X and ξ ∈ Rn. As usual we denote byOPSm(X) the space of the corresponding (properly supported) pseudodifferentialoperators. Indeed, although our results concern operators (or systems of operators)of this type, we will make use of the more general calculus developed by Beals andFefferman [1] and Hormander [30]. We will consider classical operators, namelyoperators P = p(x,D) whose total symbol p(x, ξ) admits an asymptotic expansionin homogeneous terms,

p(x, ξ) ∼∑j≥0

pm−j(x, ξ),

1

2 INTRODUCTION AND DISCUSSION OF THE RESULTS

withpm−j(x, tξ) = tm−jpm−j(x, ξ), t > 0.

We suppose that the operator P is formally self-adjoint, namely

(Pu, v)L2(Rn) = (u, Pv)L2(Rn), ∀u, v ∈ C∞0 (X).

Loosely speaking, lower bounds for P are estimates from below for the quadraticform (Pu, u) associated with P , in terms of Sobolev norms of u ∈ C∞

0 (X).One of the most celebrated lower bounds is the Garding inequality, that concerns

elliptic operators.

Theorem 1.1.1 ([20]). The following two properties are equivalent:

pm(x, ξ) > 0, ∀(x, ξ) ∈ T ∗X \ 0;

For any µ < m/2 and any compact K ⊂ X there exist cµ,K , Cµ,K > 0 such that

(Pu, u) ≥ cµ,K‖u‖2m2− Cµ,K‖u‖2

µ, ∀u ∈ C∞0 (K).

Indeed, from the ellipticity of P and by means of the symbolic calculus, one canconstruct an elliptic approximate square root Q, so as P = Q∗Q + R where R is aregularizing operator (namely having kernel in C∞(X×X)). Then elliptic estimatesfor Q give the result.

The problems arise when the principal symbol is allowed to vanish. The followingresult, known as Sharp Garding inequality, was proved by Hormander in [28].


pm(x, ξ) ≥ 0, ∀(x, ξ) ∈ T ∗X \ 0;

For any compact K ⊂ X there exists a constant CK > 0 such that

(Pu, u) ≥ −CK‖u‖2m−1

2, ∀u ∈ C∞

0 (K).

Up to now there exist many proofs of this result, see for example [30], Theorems18.1.14 and 18.6.7. A short proof was given by Cordoba and Fefferman in [11] bymeans of wave-packets methods, see also Tataru [64].

As shown by Fefferman and Phong [15], a gain of two derivatives is reached ifthe total symbol is nonnegative. More precisely, for an operator P (not necessarilyformally self-adjoint), we have the following result.

Theorem 1.1.3 ([15]). Let us suppose that the total symbol is nonnegative. Thenfor any compact K ⊂ X there exists a constant CK > 0 such that

Re(Pu, u) ≥ −CK‖u‖2m−2

2, ∀u ∈ C∞

0 (K).

1.1. INTRODUCTION 3

A sharper result was then established by Fefferman and Phong in [16] (“MainTheorem”, page 278); see also Fefferman [14]. The proof is very hard and in di-mension n = 1, 2 it was then simplified by Herau [24], who used his proof of the1-dimensional SAK principle given in [25]. Some efforts have also been done totreat symbols which can take large negative values. In this connection a conjectureof Fefferman and Phong [16] was proved by Lerner and Nourrigat [37] yet whenn = 1.

The Fefferman-Phong inequality is certainly a very strong result; however thecondition on the symbol is not necessary in order for the corresponding lower boundto hold. This is easily seen by taking the second order differential operator in R2

P = P ∗ = D21 + x2

1D22 −D2,

where, as usual, Dj = −i∂xj, j = 1, 2. Its total symbol p(x, ξ) = ξ2

1 + x21ξ

22 − ξ2,

when x1 = ξ1 = 0 and ξ2 → +∞, goes to −∞ as fast as −|ξ|. On the other hand,P = MM∗ ≥ 0 where

M = D1 + ix1D2

is the Mizohata operator. Basically the reason for this is that the harmonic oscillatorD2 + x2 − 1 is still nonnegative, although its symbol is negative in a neighborhoodof the origin. This fact is in turn an expression of the uncertainty principle, see [14]and Lerner [36].

In any case, assuming that the principal symbol pm is nonnegative, one looks fornecessary conditions in order for the lower bound

(Is) (Pu, u) ≥ −CK‖u‖2(m−s)/2 ∀u ∈ C∞

0 (K), K ⊂⊂ X,

to hold, with a gain of s > 1 derivatives. It is just in this setting that someinvariants naturally associated with a classical symbol appeared in the literature(see [43]). Precisely, denote by

Σ = (x, ξ) ∈ T ∗X \ 0 : pm(x, ξ) = 0

the so-called characteristic set of P . One defines the subprincipal symbol psm−1 as

psm−1(x, ξ) = pm−1(x, ξ) +i

2〈∂x, ∂ξ〉pm(x, ξ).

As it is well known, it has an invariant meaning at points of Σ (actually, if we thinkof P as acting on 1/2-density on X, psm−1 is actually invariant on the whole T ∗X \0).We observe that the subprincipal symbol of a formally self-adjoint operator is real-valued. Another invariant object at any point (x, ξ) ∈ Σ is the fundamental matrixFx,ξ, defined by

σ(v, Fx,ξw) =1

2〈Hess pm(x, ξ) v, w〉, ∀v, w ∈ T(x,ξ)T

∗X. (1.1)


As proved by Melin [43], if the principal symbol pm is nonnegative, the spectrum ofthe corresponding fundamental matrix consists of the eigenvalue 0 and of eigenvalues±iµj, with µj > 0. Then one sets

Tr+Fx,ξ =∑

µj.

We have the following result (Melin’s inequality).


For every ε > 0, µ < (m− 1)/2 and any compact K ⊂ X there exists Cε,µ,K > 0such that

(Pu, u) ≥ −ε‖u‖2m−1

2− Cε,µ,K‖u‖2

µ, ∀u ∈ C∞0 (K); (1.2)

pm(x, ξ) ≥ 0, ∀(x, ξ) ∈ T ∗X \ 0;

psm−1(x, ξ) + Tr+Fx,ξ ≥ 0, ∀(x, ξ) ∈ Σ.(1.3)

Therefore we see that conditions (1.3) are necessary to have a gain s > 1, for(Is) with s > 1 trivially implies (1.2).

In general, under Melin’s assumptions (1.3) one cannot expect any better esti-mate (we refer to the counterexamples by Hormander [29], Section 3.4). On theother hand, estimate (1.2) is too weak if one wants to study the Cauchy problemfor as simple operators as D2

t − P , where P is a second order non-elliptic differen-tial operators satisfying (1.3). Indeed, one needs to know then that P is boundedfrom below in the L2 norm. Motivated by this application to the Cauchy problem,Hormander [29] proved the following lower bound with a gain of two derivativesunder suitable assumptions on the symplectic geometry of the characteristic set Σ.

Theorem 1.1.5 ([29]). Let P = p(x,D) = P ∗ ∈ OPSm(X) be a classical pseudod-ifferential operator, with p(x, ξ) ∼

∑j≥0 pm−j(x, ξ). Suppose that

(a) the characteristic set Σ is a smooth submanifold of T ∗X \ 0;

(b) the rank of the restriction to Σ of the canonical symplectic 2-form σ =∑n

j=1 dξj∧dxj is locally constant, i.e. the map Σ 3 ρ 7→ dim(TρΣ ∩ TρΣσ) is locally con-stant (TρΣ

σ being the symplectic orthogonal space of TρΣ);

(c) pm(x, ξ) vanishes exactly to second order on Σ, namely (1)

|pm(x, ξ)| ≈ |ξ|mdistΣ(x, ξ)2,

where distΣ(x, ξ) denotes the distance of (x, ξ/|ξ|) to Σ.

1For two nonnegative functions f, g defined on some open conic subset Γ ⊆ T ?Rn \ 0, f .g (or g & f) means that, for every subcone Γ′ ⊆ Γ with compact base and for any ε > 0, thereexists a constant C > 0 such that f(x, ξ) ≤ Cg(x, ξ), for any (x, ξ) ∈ Γ′, |ξ| ≥ ε (f ≈ g stands forf . g and g . f).

1.1. INTRODUCTION 5

Then the following inequality

(Pu, u) ≥ −CK‖u‖2m/2−1, ∀u ∈ C∞

0 (K),

holds for any compact set K ⊂ X, if and only if P verifies (1.3).

Recently Parenti and Parmeggiani [53] generalized this result to a class of pseu-dodifferential operators with higher characteristics. More precisely, they consideredoperators in the class OPNm,k(X,Σ) introduced by Sjostrand in [63], that we nowbriefly recall for the sake of completeness.For a smooth conic and closed submanifold Σ of T ∗X \ 0, m ∈ R, k ∈ Z+, wedenote by Nm,k(X × Rn,Σ) the set of classical symbols p(x, ξ) of order m, withp(x, ξ) ∼

∑j≥0 pm−j(x, ξ), such that

|pm−j(x, ξ)| . |ξ|m−jdistΣ(x, ξ)k−2j, 0 ≤ j ≤ k/2. (1.4)

Accordingly, one denotes by OPNm,k(X,Σ) the corresponding class of (properly sup-ported) ψdo’s. For operators with multiple characteristics satisfying these Levi-typevanishing conditions, a detailed study of hypoellipticity with loss of k/2 derivativeshas been performed by Boutet de Monvel, Grigis and Helffer [5]. Applications tothe analysis of the Kohn Laplacian were also given by Boutet de Monvel [4] and,more recently, by Parmeggiani [55]; see also Helffer and Nourrigat [23].

It turns out that the spaces OPNm,k(X,Σ) are invariant under canonical changesof variables (see [4]); to be definite:

Let X, Y ⊂ Rn be open sets and let χ : T ∗X \ 0 → T ∗Y \ 0 be a smoothhomogeneous (of degree one in the fibres) canonical transformation. Let Λχ ⊂ (T ∗Y \0)×(T ∗X\0) (resp. Λχ−1 ⊂ (T ∗X\0)×(T ∗Y \0)) be the canonical relation associatedwith χ (resp. χ−1) and finally denote by

F ∈ I0(Y ×X,Λχ) (resp. F−1 ∈ I0(X × Y,Λχ−1))

an elliptic Fourier integral operator of order 0 associated with Λχ (resp. Λχ−1), withFF−1 ≡ I, F−1F ≡ I. Then

P ∈ OPNm,k(X,Σ) =⇒ P := FPF−1 ∈ OPNm,k(Y, χ(Σ)).

Moreover P ∈ OPNm,k(X,Σ) implies2 P t ∈ OPNm,k(X,−Σ), with −Σ :=(x, ξ) ∈ T ∗X : (x,−ξ) ∈ Σ , P ∗ ∈ OPNm,k(X,Σ) and

P ∈ OPNm,k(X,Σ), Q ∈ OPNm′,k′(X,Σ) =⇒ PQ ∈ OPNm+m′,k+k′(X,Σ).

2The transposed operator P t is defined by (P tu, v) = (u, Pv), for u, v ∈ C∞0 (X).


We also recall that a symbol p ∈ Nm,k(X ×Rn,Σ) (or the related operator p(x,D))is called transversally elliptic (with respect to Σ) if its principal symbol pm(x, ξ)vanishes exactly to k−th order on Σ, namely

|pm(x, ξ)| ≈ |ξ|mdistΣ(x, ξ)k.

Note that Hormander’s inequality applies to transversally elliptic operators P =P ∗ ∈ OPNm,2(X,Σ).In [53] Parenti and Parmeggiani established sufficient conditions for an operatorP = P ∗ ∈ OPNm,k(X,Σ) to satisfy a lower bound with a gain of k/2+1 derivatives,namely

(Pu, u) ≥ −CK‖u‖2m/2−(k+2)/4, ∀u ∈ C∞

0 (K), K ⊂⊂ X. (1.5)

The reason for this Sobolev exponent is that this inequality turns out to be triviallyinvariant for perturbations by operators of order m − k/2 − 1, which is the orderof the first term, in the asymptotic expansion of the symbol, that does not have aninvariant meaning at Σ.

They considered two cases depending on the symplectic nature of Σ. Moreprecisely they supposed that Σ is

either regular involutive, namely for any ρ ∈ Σ, TρΣσ ⊂ TρΣ, and the canon-

ical 1-form∑n

j=1 ξjdxj does not vanish on TρΣ,

or symplectic, namely TρΣ ∩ TρΣσ = 0 for every ρ ∈ Σ.

In order to state their results, we now briefly recall the definition of the maininvariant associated with an operator P = p(x,D) ∈ OPNm,k(X,Σ), with p ∼∑

j≥0 pm−j (see [4, 52]).

From now on we suppose that P is formally self-adjoint and transversally elliptic,with nonnegative principal symbol. Moreover k will be an even integer.

Consider the Weyl symbol pw(x, ξ) of P defined by

pw(x, ξ) = e〈Dx,Dξ〉/2ip(x, ξ) ∼∑j≥0

qm−j(x, ξ),

where qm−j(x, ξ) =∑

l+r=j

(〈Dx, Dξ〉/2i

)rpm−l(x, ξ)/r! and observe that pw(x, ξ) ∈

Nm,k(X × Rn,Σ) if and only if p(x, ξ) ∈ Nm,k(X × Rn,Σ). For any ρ ∈ Σ and anyv ∈ TρT

∗X, let V be a smooth vector field of TT ∗X near ρ, with V (ρ) = v. Wethen define the smooth map p(k) : TT ∗X|Σ −→ R by

p(k)(ρ, v) =∑

0≤j≤k/2

1

(k − 2j)!

(V k−2jqm−j

)(ρ), ∀(ρ, v) ∈ TT ∗X|Σ. (1.6)

1.1. INTRODUCTION 7

First of all we observe that this map is well defined, i.e. it does not depend on thechoice of the extension V . Moreover, it has an invariant meaning in the followingsense:

With χ : T ∗X \ 0 → T ∗Y \ 0, F , P as above, the map p(k) associated with Pverifies

p(k) = p(k) dχ. (1.7)

Notice that, in view of the vanishing conditions in the definition of the classNm,k(X×Rn,Σ), the map pk descends to the quotient bundle NΣ := TT ∗X|Σ/TΣ.

Theorem 1.1.6 ([53]). Let P = P ∗ ∈ OPNm,k(X,Σ) be transversally elliptic andsuppose Σ is regular involutive. Moreover, suppose that

p(k)(ρ, v) ≥ 0, ∀ρ ∈ Σ, ∀v ∈ NρΣ, (1.8)

p(k)(ρ, v) = 0 =⇒ v = 0, (1.9)

p(k)(ρ, 0) = 0 =⇒(d2vp

(k))(ρ, 0) is invertible. (1.10)

Then (Is) holds with s = k/2 + 1.

For operators in the class OPNm,k(X,Σ) with Σ involutive, condition (1.8) isalso necessary to have a gain greater than k/2 derivatives (see [52]), whereas oncek > 2, (1.9) and (1.10) are not necessary in general. On the other hand one easilysees that, under (1.8) alone, when k > 2 estimate (Is) with s = k/2+1 is, in general,false.

Suppose now that Σ is a symplectic sub-manifold of T ∗X \ 0. Let ρ ∈ Σ and let

ζ : T ∗Rν → TρΣσ (2ν = codim Σ)

be any linear symplectomorphism. Setting pζ(y, η) = p(k)(ρ, ζ(y, η)) for (y, η) ∈T ∗Rν , we then consider the Weyl quantization

Pρ,ζ = Opw(pζ)(y,Dy) : S(Rν) → S(Rν).

As shown in [53], the spectrum of Pρ,ζ , as an unbounded operator on L2(Rν), isindependent of the parameterization ζ, and it turns out to be discrete and boundedfrom below. Thus the lowest eigenvalue

λ(ρ) := min Spec(Opw(pζ))

is a continuous function on Σ, independent of ζ. Similarly, the dimension of thecorresponding (finite dimensional) eigenspace Vρ,ζ ⊂ S(Rν) is invariantly defined, aswell.


Theorem 1.1.7 ([53]). Let P = P ∗ ∈ OPNm,k(X,Σ) be transversally elliptic, andsuppose Σ symplectic. Moreover, assume that:

(i) for any ζ : T ∗Rν → TρΣσ as above, one has

(Opw(pζ)(y,Dy)f, f) ≥ 0, ∀f ∈ S(Rν), ∀ρ ∈ Σ;

(ii) if λ(ρ0) = 0, there exists a conic neighborhood Γ ⊂ Σ of ρ0 such that

dimVρ,ζ = const, ∀ρ ∈ Γ.

Then estimate (Is) holds with s = k/2 + 1/2.If we suppose, in addition, that

(iii) the eigenspace Vρ,ζ consists of functions which are either all even for everyρ ∈ Γ or all odd for every ρ ∈ Γ,

then estimate (Is) holds with s = k/2 + 1.

By standard localization techniques it is easy to see that Hypothesis (i) is neces-sary to reach a gain s > k/2 (see [52]), whereas Hypothesis (ii) was then relaxed bythe authors in the subsequent paper [54]. However, an open question was whetherHypothesis (iii) is really necessary, or if it is caused by the approach of [53]. It turnsout, as it is shown in [46], that Hypothesis (iii) is strictly linked to the nature ofthe problem, and cannot be omitted in order to obtain (1.2). In fact, when k > 2there are operators which satisfy (i) and (ii) but do not verify none of the estimates(Is) with s > k/2 + 1/2. In this sense Theorem 1.1.7 is sharp.

We observe that in the double characteristic case treated by Hormander [30], theeigenspace Vρ,ζ has dimension 1, hence conditions (ii) and (iii) are automaticallysatisfied, (see [53]).

1.2 A generalization of Hormander’s inequality

The results presented here are contained in [47]; they can be regarded as a con-tinuation of [53] and deal with Inequality (1.5) under the following geometricalhypotheses:

(H1) Σ is a smooth conic and connected submanifold of T ∗X \ 0;

(H2) the canonical 2−form σ has constant rank when restricted to Σ;

(H3) the canonical 1−form∑n

j=1 ξjdxj does not vanish on TρΣ, for any ρ ∈ Σ.

1.2. A GENERALIZATION OF HORMANDER’S INEQUALITY 9

As a consequence of these assumptions, the dimension of the linear spaces TρΣ∩TρΣσ

and TρΣσ/(TρΣ ∩ TρΣσ) is independent of ρ ∈ Σ, hence we can put

dim (TρΣ ∩ TρΣσ) = l, dimTρΣσ/(TρΣ ∩ TρΣσ) = 2ν, ∀ρ ∈ Σ.

In fact,2ν + l = codim Σ, 2(n− (ν + l)) = rkσ|Σ.

In the following we will use a particular class of symplectic coordinates near Σ.Indeed, in view of Theorem 21.2.4 of [30], for any ρ0 ∈ Σ there exist

· an open conic neighborhood Γ ⊂ T ∗X \ 0 of ρ0,

· an open conic set Γ ⊂ R2νy′,η′ × R2l

y′′,η′′ ×(R2n−2ν−2ly′′′,η′′′ \ 0

),

· a smooth symplectomorphism (homogeneous of degree 1 in the fibers) χ : Γ −→ Γ,such that

χ(Γ ∩ Σ) = (y, η) ∈ Γ | y′ = η′ = η′′ = 0, η′′′ 6= 0.In the sequel, following [52], such a map will be called a canonical flattening (of Σ)near ρ0.Denote by NΣ = TT ∗X|Σ/TΣ the normal bundle to Σ. Any canonical flattening χinduces a local trivialization of NΣ defined as follows

ζχ : χ(Γ ∩ Σ)× Rl+2νζ′′;z′,ζ′ → NΣ|Γ∩Σ

ζχ(χ(ρ), (ζ ′′; z′, ζ ′)

)=(ρ,[dχ−1(χ(ρ))(z′, ζ ′, 0, ζ ′′, 0, 0)

]),

ρ ∈ Γ ∩ Σ, (z′, ζ ′) ∈ R2ν , ζ ′′ ∈ Rl, where [v] denotes the residue class of v ∈TρT

∗X in NρΣ. It will be important to know the structure-group associated withthe trivializations ζχ.

Lemma 1.2.1 ([52]). Let χ : Γ → Γ, χ′ : Γ′ → Γ′, be two canonical flattening of Σwith Γ ∩ Γ′ ∩ Σ 6= ∅. Then the map

ζ−1χ′ ζχ : χ(Γ ∩ Γ′ ∩ Σ)× R2ν+l −→ χ′(Γ ∩ Γ′ ∩ Σ)× R2ν+l

takes the form

ζ−1χ′ ζχ(χ(ρ), z′, ζ ′; ζ ′′) =

(χ′(ρ);

(α(ρ) β(ρ)

0 γ(ρ)

)[z′ζ ′]

ζ ′′

), (1.11)

where α ∈ C∞(Γ ∩ Γ′ ∩ Σ, Sp(2ν,R))

γ ∈ C∞(Γ ∩ Γ′ ∩ Σ,GL(l,R))

β ∈ C∞(Γ ∩ Γ′ ∩ Σ,Mat(2ν × l,R)).


Any canonical flattening χ induces trivializations ηχ : χ(Γ ∩ Σ)× Rlζ′′ → (TΣ ∩

TΣσ)′|Γ∩Σ for the dual bundle (TΣ ∩ TΣσ)′

ηχ(χ(ρ), ζ ′′

)=(ρ, σ(dχ−1(χ(ρ))(0, 0, 0, ζ ′′, 0, 0), ·

)).

In what follows, we hence consider NΣ and (TΣ ∩ TΣσ)′ as vector-bundles over Σwith respect to the structure given by the trivializations above.

Now, by using a canonical flattening χ : Γ → Γ near ρ0 ∈ Σ, we can define thelocalized polynomial p

(k)χ := p(k) ζχ of P at any ρ ∈ Γ ∩ Σ, which is given by

p(k)χ (χ(ρ), ζ ′′; z′, ζ ′) =

∑|α|+|β|+|γ|+2j=k

1

α!β!γ!

(∂αz′∂

βζ′∂

γζ′′(qm−j χ

−1))(χ(ρ))ζ ′′

γz′αζ ′β,

where, recall, pw ∼∑

j≥0 qm−j is the Weyl symbol of P .Since pm is assumed to be nonnegative and transversally elliptic, its Weyl-

quantization with respect to the (z′, ζ ′)-variables defines an unbounded operator

Pχ,χ(ρ),ζ′′ = Opw(p(k)χ )(χ(ρ), ζ ′′; z′, Dz′) in L2(Rν

z′), smoothly depending on the pa-rameters (ρ, ζ ′′) and globally elliptic (see [22]). It turns out that Pχ,χ(ρ),ζ′′ has abounded-from-below discrete spectrum, made of real eigenvalues, with finite mul-tiplicities, diverging to +∞. In particular, its lowest eigenvalue λχ(χ(ρ), ζ ′′) is acontinuous function of the parameters (χ(ρ), ζ ′′). If we choose another canonical

flattening χ′ : Γ′ → Γ′ near ρ0, it follows from Lemma 1.2.1 and the Segal Theorem(Thm. 18.5.9 of [30]) that the operators Pχ′,χ′(ρ),γ(ρ)ζ′′ and Pχ,χ(ρ),ζ′′ are unitarilyequivalent (via a metaplectic transformation associated with an affine symplecticmap), whence one immediately gets

λχ′(η−1χ′ ηχ(χ(ρ), ζ ′′)

)= λχ′

(χ′(ρ), γ(ρ)ζ ′′

)= λχ

(χ(ρ), ζ ′′

).

As a consequence, the local functions λχ(ρ, ζ′′) can be glued together into a contin-

uous function λ(ρ, v) : (TΣ ∩ TΣσ)′ −→ R, that we call the “ground energy” of P .Furthermore, if we consider the corresponding eigenspace

Vχ(χ(ρ), ζ ′′) = Ker(Pχ,χ(ρ),ζ′′ − λχ(χ(ρ), ζ ′′)IdL2(Rν)

)⊂ S(Rν).

the same arguments above show that dim Vχ′(η−1χ′ ηχ(χ(ρ), ζ ′′)

)= dim Vχ

(χ(ρ), ζ ′′

).

Therefore, we can define a global function (ρ, v) 7→ dim V (ρ, v) defined on (TΣ ∩TΣσ)′ by setting

dim V (ρ, v) = dim Vχ(η−1χ (ρ, v)),

where χ is any canonical flattening near ρ ∈ Σ.If λ(ρ, v) > 0 for every (ρ, v) ∈ (TΣ∩TΣσ)′, Theorem 3.1 of [5] and Proposition

(5.6) of [44] apply and yield the following strong lower bound (Mohamed [44]):


For every compact K ⊂ X and any µ < m/2 − k/4 there exist c > 0, C > 0 suchthat

(Pu, u) ≥ c‖u‖2m/2−k/4 − C‖u‖2

µ, ∀u ∈ C∞0 (K). (1.12)

Our main result, which completes [53] with the non-symplectic and non-involutivecase, instead deals with the situation in which the ground energy λ(ρ, v) is allowedto vanish.

Theorem 1.2.2. Let P = P ∗ ∈ OPNm,k(X,Σ) be transversally elliptic, with non-negative principal symbol. Assume that Σ satisfies conditions (H1)-(H2)-(H3), andthat

λ(ρ, v) ≥ 0 ∀ρ ∈ Σ, ∀v ∈ (TρΣ ∩ TρΣσ)′. (1.13)

Moreover, if λ(ρ0, v0) = 0, suppose that:

(i) v0 = 0;

(ii) dimV (ρ, v) is constant in a neighborhood of (ρ0, 0) in (TΣ ∩ TΣσ)′;

(iii) given any canonical flattening χ near ρ0, the eigenfunctions in Vχ(χ(ρ0), 0) areeither all even or all odd;

(iv) the Hessian map d2vλ(ρ0, 0) is non-degenerate.

Then, for any compact subset K ⊂ X, there exists CK > 0 such that

(Pu, u) ≥ −CK‖u‖2m/2−(k+2)/4, ∀u ∈ C∞

0 (K). (1.14)

Theorem 1.2.2 will be proved in Chapter 2. We observe that for operatorswith double characteristics (k = 2) it reduces to Hormander’s theorem (see be-low), whereas for general (even) k ≥ 2 we recapture the results of [53] (Thm. 1.2and Thm. 1.5) when Σ is regular involutive or symplectic.

Let us now discuss Hypotheses (i)–(iv) in more detail.The ground energy λ is a-priori a continuous function on (TΣ ∩ TΣσ)′, that turnsout to be smooth in a neighborhood of (ρ0, 0) because of (ii). Moreover, in view of(1.13) and (i), λ(ρ0, ·) vanishes to second order at (ρ0, 0), hence Hypothesis (iv) onthe Hessian map of λ makes sense. Finally Hypothesis (iii) applies to points of thezero section of (TΣ ∩ TΣσ)′ and from Lemma 1.2.1 it follows that the eigenspaceVχ′(χ

′(ρ0), 0) corresponding to another choice of canonical flattening is the imageof Vχ(χ(ρ0), 0) via a metaplectic transformation in L2(Rν) associated with a linearsymplectic map (and not merely affine symplectic), that hence is parity preserving.Therefore Hypothesis (iii) is actually independent of the choice of χ.

It is important to point out that the hypotheses of Theorem 1.2.2 are also invari-ant under canonical changes of variables. Precisely, let ψ : T ∗X → T ∗Y be a smooth


homogeneous symplectic transformation, and F (resp. F−1) an elliptic Fourier in-tegral operator of order zero associated with ψ (resp. ψ−1), satisfying FF−1 ≡ I,F−1F ≡ I. Setting P := FPF−1 ∈ OPNm,k(Y, χ(Σ)) we saw that p(k) = p(k) dψ.Furthermore, given any canonical flattening χ of Σ near ρ0 ∈ Σ, it turns out thatχ := χ ψ−1 is a canonical flattening of χ(Σ) near ρ0 := ψ(ρ0). Therefore, fromdψ ζχ = ζχψ−1 it follows that p(k) ζχ = p(k) ζχ and hence Pχ,χ(ρ),ζ′′ = Pχ,χ(ρ),ζ′′ .

As regards necessity of the conditions in Theorem 1.2.2 some further commentsare in order.As a consequence of Theorem (3.1) of [5] and Theorem 3.1 of [52], if Inequality (1.14)holds, the lowest eigenvalue λ(ρ, v) must be nonnegative as required in (1.13). Theother conditions in Theorem 1.2.2 are, loosely speaking, far from being necessary for(1.14) to hold; in fact, given any pseudodifferential operator P ∈ OPSm(X), one mayconsider the operator P ∗P which is always nonnegative, but in general satisfies noneof Hypotheses (i), (ii), (iii), (iv). However, without assuming (iii) or (iv), Estimate(1.14) (and actually every estimate with a gain greater than k/2+1/2 derivatives) isin general false; we refer to Section 2.4.2 for the discussion of two counterexamplesin this direction.At any rate an inspection of the proof of Theorem 1.2.2 shows that if we removeHypotheses (iii), (iv) in Theorem 1.2.2, then the following weaker inequality (witha gain of k/2 + 1/2 derivatives) still holds:

(Pu, u) ≥ −CK‖u‖2m/2−(k+1)/4, ∀u ∈ C∞

0 (K).

This lower bound was first proved by Parenti and Parmeggiani in [53] when Σ iseither regular involutive or symplectic. Similarly one sees that this inequality alsoholds if we replace the remaining hypotheses (i), (ii) by the following strong versionof (ii)

dimV (ρ, v) = const, ∀ρ near ρ0, ∀v ∈ (TρΣ ∩ TρΣσ)′.

We mention that actually, in verifying the hypotheses of Theorem 1.2.2, we do notneed to know explicitly a canonical flattening of Σ near every ρ0 ∈ Σ; we refer toSection 2.4.1 for a discussion of this point.

Finally we address the reader to Section 2.4.3 (see [51, 12]) for the analysis of acase in which the vanishing conditions (1.4) are not satisfied. We also point out thepaper [49] where a special situation in absence of transversal ellipticity is studied.

The case of operators with double characteristicsWe see that Theorem 1.2.2 reduces to Hormander’s theorem (Theorem 1.1.5

above) when k = 2. Indeed we now verify that (1.13) reduces to the second conditionin (1.3), whereas Hypotheses (i), (ii), (iii) and (iv) are also satisfied in that case.

First of all we observe that here

p(2)(ρ, v) = σ(v, Fρv) + psm−1(ρ), ρ ∈ Σ, v ∈ NρΣ.


Fix now ρ0 ∈ Σ. Let us show that there exists a canonical flattening χ for Σ nearρ0 such that, at ρ0,

p(2) ζχ(χ(ρ0), ζ′′; z′, ζ ′) =

ν∑j=1

µj(ρ0)(z′j2+ ζ ′j

2) +

l∑j=1

ζ ′′j2+ psm−1(ρ0). (1.15)

Indeed, consider any canonical flattening χ′ : Γ −→ Γ′ for Σ near ρ0. Then

p(2) ζχ′(χ′(ρ0), ζ′′; z′, ζ ′) = Qρ0(z, ζ) + psm−1(ρ0),

where Qρ0(z, ζ) is a nonnegative quadratic form on T ∗Rn. By Hormander’s theoremon the classification of semidefinite quadratic form (Thm. 21.5.3 of [30]) there existsa linear symplectic map χ : T ∗Rn −→ T ∗Rn such that

Qρ0 χ(y, η) =ν∑j=1

µj(ρ0)(y′j2+ η′j

2) +

l∑j=1

η′′j2.

Since the operator P is assumed to be transversally elliptic, i.e. KerFρ = TρΣ∀ρ ∈ Σ, χ−1 will be a canonical flattening for χ(Γ∩Σ) near χ(ρ0). Hence χ := χ−1χ′is the canonical flattening we were looking for.

From (1.15) it follows that

Pχ,χ(ρ0),ζ′′ =ν∑j=1

µj(ρ0)(z′j2+D2

z′j) +

l∑j=1

ζ ′′j2+ psm−1(ρ0).

Let

hk(t) := π−1/4(2kk!)−1/2

(d

dt− t

)ke−t

2/2 k = 0, 1, . . .

be the k−th Hermite function, and define

φβ(z′) = Πν

j=1hβj(z′j), β = (β1, . . . , βν) ∈ Zν

+.

It is well known that the spectrum of the harmonic oscillatory D2t + t2 consists of

the simple eigenvalues 2k+1, k ∈ Z+, and the corresponding eigenfunctions are thehk(t). As a consequence, the spectrum of Pχ,χ(ρ0),ζ′′ is given by

SpecPχ,χ(ρ0),ζ′′ = ν∑

j=1

µj(ρ0)(2βj + 1) +l∑

j=1

ζ ′′j2+ psm−1(ρ0), β ∈ Zν

+

.

Whence we have

λχ(χ(ρ0), ζ′′) =

∑νj=1 µj(ρ0) +

∑lj=1 ζ

′′j

2 + psm−1(ρ0)

= Tr+Fρ0 + psm−1(ρ0) +∑l

j=1 ζ′′j

2. (1.16)


Therefore Hypothesis (1.13), (i) and (iv) in Theorem 1.2.2 are satisfied if the secondcondition in (1.3) holds. Moreover, as regards the eigenspace belonging to the lowesteigenvalue,

Vχ(χ(ρ0), ζ′′) = span e−|z′|2/2.

is one dimensional and consists of even functions. In particular Hypothesis (ii) and(iii) in Theorem 1.2.2 are fulfilled.

Examples.Consider the following operator in Rn

P = P ∗ = (|Dx′|2 + |x′|2|Dx′′′ |2 + |Dx′′|2)2 + µ|Dx′′′ |2, µ ∈ R,with x = (x′, x′′, x′′′) ∈ Rν × Rl × Rn−ν−l, for which k = 4 and Σ = x′ = ξ′ = ξ′′ =0, ξ′′′ 6= 0.Its localized operator associated with the map χ = Id at ρ0 = (0, x′′0, x

′′′0 , 0, 0, ξ

′′′0 ) is

given byPχ,χ(ρ0),ζ′′ = (|Dz′|2 + |z′|2|ξ′′′0 |2 + |ζ ′′|2)2 + µ|ξ′′′0 |2.

We therefore have

λχ(χ(ρ0), ζ′′) = (|ξ′′′0 |+ |ζ ′′|2)2 + µ|ξ′′′0 |2.

Hence P satisfies (1.13) if and only if µ ≥ −1. If µ > −1 we have that Pχ,χ(ρ0),ζ′′

is actually invertible, therefore P verifies (1.12). For µ = −1, (i) and (iv) are ofcourse satisfied, as well as (ii) and (iii) because we have

Pχ,χ(ρ0),ζ′′ = (|Dz′|2 + |z′|2|ξ′′′0 |2 + |ζ ′′|2 + |ξ′′′0 |)(|Dz′|2 + |z′|2|ξ′′′0 |2 + |ζ ′′|2 − |ξ′′′0 |),(notice that the two factors commute) whence

Vχ(χ(ρ0), ζ′′) = span e−|ξ′′′0 ||z′|2/2.

Finally we briefly discuss a source of examples after [52]. Let Pj ∈ OPNm,2(X,Σ),j = 1, . . . , N (with Σ satisfying conditions (H1),(H2),(H3)) be transversally ellipticoperators with positive principal symbol and assume that their fundamental matricesFj(ρ), ρ ∈ Σ, commute. Consider then the operator

R =∑|α|≤µ

AαPα, Pα = Pα1

1 . . . PαNN ,

where Aα = A∗α are classical pseudodifferential operators of order 0, with principal

symbol aα. If, say,∑

|α|=µ aαζα > 0 for ζ 6= 0, then P := ReR ∈ OPNmµ,2µ(X,Σ)

is (formally self-adjoint and) transversally elliptic. As detailed in [52], for suchan operator one can write down explicitly both the function λ(ρ, v) in terms ofthe eigenvalues of the matrices Fj, and a basis of the corresponding eigenspace,which consists of certain Hermite functions. Hence, in this case, the assumptions ofTheorem 1.2.2 reduce to explicit algebraic conditions (see [52]).

1.3. ON THE POSITIVE PARTS OF SECOND ORDER OPERATORS 15

1.3 On the positive parts of second order opera-

tors

In the previous section we were concerned with the lower bound (1.14), which re-duces to Hormander’s inequality for operators with double characteristics. On theother hand, in many cases, one is interested in extracting more detailed spectralinformation from the symbol of the operator.More generally, one deals with the following question: To what extent can we deducethe spectral properties of a formally self-adjoint operator P = p(x,D) ∈ OPS2(Rn)by analyzing its symbol p(x, ξ)? Precisely, we would like to construct explicitly fromthe symbol p(x, ξ), by microlocalization, an approximate positive part Π+ and anapproximate negative part Π− for the operator P ; namely, Π+ and Π− are linearbounded operators in L2(Rn) such that Π+ + Π− = Id + “negligible terms” and, forevery u ∈ C∞

0 (K), K ⊂⊂ Rn,

Re(PΠ+u, u) ≥ −CK‖u‖20, −Re(PΠ−u, u) ≥ −CK‖u‖2

0.

This program was carried out for a first order pseudodifferential operator Q =q(x,D) by Fujiwara [17]. The main tool used in [17] is a kind of cutting and stoppingargument introduced by Beals and Fefferman [1], that allows a reduction of theoperator Q to suitable normal forms. The symbol of these normal forms turnsout to be either nonnegative or non positive, or just bounded, or finally, (aftera symplectomorphism) of the type q(x, ξ) = ξ1. For each of these operators thecorresponding positive and negative parts Π+ and Π− are easily written; in fact,for a nonnegative symbol one just takes Π+ = Id and Π− = 0 as a consequence ofthe Sharp Garding inequality, whereas for the symbol q(x, ξ) = ξ1 one sets Π+ =Op(H(ξ1)), Π− = Op((1 − H(ξ1)), (as bounded operators L2

comp → L2loc), where

H(ξ1) is the Heaviside function on the real line. Finally these operators are patchedtogether by means of a microlocal partition of unity.

A similar problem for general second order pseudodifferential operators seemsto be quite difficult. As an attempt to attack this problem, in [48] we consideredthe particular case of classical operators, with principal symbol transversally ellipticand satisfying some additional conditions ((h1), (h2), (h3) below).

Precisely, let X ⊂ Rn be an open subset and let p ∼∑

j≥0 p2−j be the symbol of

the classical operator P = P ∗ ∈ OPS2(X). As above we denote by

Σ = (x, ξ) ∈ T ∗X \ 0 : p2(x, ξ) = 0

its characteristic set. We suppose that

(h1) Σ is a symplectic sub-manifold of T ∗X \0, namely TρΣ∩TρΣσ = 0 for everyρ ∈ Σ.


Furthermore we assume that the operator P is transversally elliptic, namely itsprincipal symbol p2 vanishes exactly to second order on Σ. Consider now the fun-damental matrix F associated with p2 at ρ ∈ Σ, defined in (1.1).Since the operator P is formally self-adjoint, its principal symbol p2 is real-valuedand has locally constant sign because of the transversal ellipticity assumption.Hence, without loss of generality, we may assume that p2 is nonnegative everywhere;as a consequence, its Hessian map is positive semi-definite. Furthermore the spec-trum of the fundamental matrix Fρ consists of the eigenvalue 0 and of eigenvalues±iµj(ρ), j = 1, . . . , ν, where

0 < µ1(ρ) ≤ µ2(ρ) ≤ . . . ≤ µν(ρ), 2ν = codim Σ,

are real continuous functions of ρ ∈ Σ, positively homogeneous of degree 1 in thefibers.Also, we recall that the positive trace of the fundamental matrix Fρ is given byTr+Fρ :=

∑νj=1 µj(ρ), and the sub-principal symbol is defined by ps1(ρ) := p1(ρ) +

i2

∑nj=1

∂2p2∂xj∂ξj

(ρ). We know that ps1 is invariantly defined at points of Σ and is real

valued (for p2 vanishes to second order there and P = P ∗).Let us now consider the functions

mα(ρ) =ν∑j=1

µj(ρ)(2αj + 1) + ps1(ρ), ρ ∈ Σ, α ∈ Zν+. (1.17)

It is well known that the operator P is hypoelliptic with loss of 1 derivative if andonly if mα(ρ) 6= 0 for every ρ ∈ Σ and every α ∈ Zν

+ (see e.g. [4]). However, thesefunctions can, in general, vanish and in this case we require the following additionalconditions.

Suppose ρ0 ∈ Σ and mα(ρ0) = 0 for some α ∈ Zν+; set Aρ0 = α ∈ Zν

+ : mα(ρ0) =0. Then we assume that

(h2) µj(ρ0) 6= µk(ρ0), for j, k = 1, . . . , ν, j 6= k;

(h3) |α| is either even ∀α ∈ Aρ0 or odd ∀α ∈ Aρ0 .

Notice that, under Hypothesis (h2), the functions µj, and therefore the mα’s, α ∈Aρ0 , are actually smooth in a conic neighborhood of ρ0 in Σ. It is also clear thatconditions (h2) and (h3) are automatically satisfied if, for example, codim Σ = 2.

Furthermore, we observe that, just under these assumptions (h1)–(h3), Helffer([21], Thm. 4.3.14) characterized the hypoellipticity with loss of 3/2 derivatives oftransversally elliptic pseudodifferential operators.

We can now state our main result.

1.3. ON THE POSITIVE PARTS OF SECOND ORDER OPERATORS 17

Theorem 1.3.1. Let P = P ∗ ∈ OPS2(X) be a classical (properly supported) pseu-dodifferential operator, transversally elliptic. Suppose that Hypothesis (h1) holds.Furthermore, if Aρ0 6= ∅ for any ρ0 ∈ Σ, assume the conditions (h2) and (h3).

Then, for every fixed ρ0 ∈ Σ, there exists a conic neighborhood V of ρ0 forwhich we can construct, starting directly from the symbol of P , two linear continuoussymmetric operators Π+, Π− : L2

comp(X) −→ L2comp(X) and R ∈ OPS

−1/21/2,1/2(X) such

that

Π+ + Π− = Id +R,

PR, RP ∈ OPS1/21/2,1/2(X),

and satisfying, for every Ψ = ψ(x,D) ∈ OPS0(Rn) with suppψ ⊂ V , the estimates

Re(PΠ+Ψu,Ψu) ≥ −CK‖u‖20, ∀u ∈ C∞

0 (K), (1.18)

−Re(PΠ−Ψu,Ψu) ≥ −CK‖u‖20, ∀u ∈ C∞

0 (K). (1.19)

If Aρ0 = ∅ then the construction yields R = 0 and there exist constants cK > 0,CK > 0 such that

Re(PΠ+Ψu,Ψu)≥ cK‖Π+Ψu‖21/2 − CK‖u‖2

0, ∀u ∈ C∞0 (K), (1.20)

−Re(PΠ−Ψu,Ψu) ≥ cK‖Π−Ψu‖21/2 − CK‖u‖2

0, ∀u ∈ C∞0 (K). (1.21)

We observe that if ρ0 6∈ Σ and, say, p2(ρ0) > 0, P is elliptic near ρ0 and, uponchoosing Π+ = Id and Π− = 0, estimate (1.18) is a trivial consequence of Garding’sinequality

(PΨu,Ψu) ≥ cK‖Ψu‖21 − CK‖u‖2

0, ∀u ∈ C∞0 (K).

Moreover, from the proof of Theorem 1.3.1, one will see that if P satisfies the Melintrace+ condition

m0(ρ) = Tr+Fρ + ps1(ρ) ≥ 0, ∀ρ ∈ Σ, (1.22)

then one gets Hormander’s inequality

(Pu, u) ≥ −CK‖u‖20, ∀u ∈ C∞

0 (K). (1.23)

Unfortunately, we are not able in general to patch together the microlocal operatorsΠ+,Π− so as to obtain a local version of Theorem 1.3.1, and the problem does notseem to us to be merely a technical obstruction.

Remark 1.3.2. However, the following weaker local construction can be performed.For any given compact set K ⊂ X, let ψj ∈ S0(R2n), j = 1, . . . , N , be real valuedsymbols supported in conic neighborhoods Vj where Theorem 1.3.1 applies and such


that∑N

j=1 ψj(x, ξ)2 = 1 for x near K. Let thus Π+

j and Π−j be the corresponding op-

erators given in Theorem 1.3.1. Upon defining the operators P+ =∑N

j=1 Ψ∗jPΠ+

j Ψj,

P− =∑N

j=1 Ψ∗jPΠ−

j Ψj, with Ψj = ψj(x,D), we have P+ + P− = P + R′, with

R′ ∈ OPS1/21/2,1/2(X), and, for every u ∈ C∞

0 (K),

Re(P+u, u) ≥ −CK‖u‖20, −Re(P−u, u) ≥ −CK‖u‖2

0.

We will prove Theorem 1.3.1 in Chapter 3 by considering the two cases Aρ0 = ∅and Aρ0 6= ∅. The construction of the operators Π+ and Π− in Theorem 1.3.1is based on both the above-mentioned result by Fujiwara [17] and on the theoryof Hermite-operators due to Boutet de Monvel [4] and then developed by Helffer[21]. Basically, one uses particular wave-packets strictly tailored to the symbol ofthe operator: the eigenfunctions of the so-called localized operator associated withP . This suggestive interpretation was first put in evidence by Parmeggiani in [57].Indeed, we also use some new ideas from Parenti and Parmeggiani [53, 54], where thistechnique is applied to prove strong lower bounds for pseudodifferential operatorswith multiple characteristics and for systems.

We now want to illustrate Theorem 1.3.1 by a very simple example, that howeverrepresents a model case.To this purpose, assume that Σ = (x1, x2, ξ1, ξ2) ∈ R4 : x1 = ξ1 = 0, ξ2 6= 0, andfix ρ0 = (0, x2, 0, ξ2) ∈ Σ. Let P be any pseudodifferential operator in R2, takingthe following form, microlocally near ρ0,

P = D2x1

+ x21D

2x2

+ p1(x2, Dx2),

where p1(x2, ξ2) is real and positively homogeneous of degree 1. We know that Pis microlocally hypoelliptic at ρ0 with loss of 1 derivative if and only if its localizedoperator at ρ = (0, x2, 0, ξ2) (see Definition 3.1.4 in Section 3.1)

Pρ(y,Dy) = D2y + y2|ξ2|2 + p1(x2, ξ2) : S(Ry) → S(Ry)

is invertible when ρ = ρ0, that is, all its eigenvalues

mk(x2, ξ2) = (2k + 1)|ξ2|+ p1(x2, ξ2), k ∈ Z+,

do not vanish at ρ = ρ0. Consider the k−th Hermite function

hk(t) = π−1/4(2kk!)−1/2

(d

dt− t

)ke−t

2/2, t ∈ R, (1.24)

and notice that hk(|ξ2|1/2y) spans the eigenspace of the localized operator Pρ =Pρ(y,Dy) corresponding to the eigenvalues mk(x2, ξ2). Furthermore, we observe

1.4. THE WEAK-HORMANDER INEQUALITY FOR SYSTEMS 19

that m0(x2, ξ2) = min Spec(Pρ).If the spectrum of Pρ is nonnegative in a neighborhood U of ρ0, then (1.22) isfulfilled in U and hence P satisfies the Hormander inequality (1.23), microlocallynear ρ0. This fact suggests to construct the operators Π+ and Π− of Theorem1.3.1 by suitably “following” the spectral projectors onto the positive and negativeeigenspaces of Pρ, as ρ varies near ρ0.The most interesting case occurs when 0 belongs to the spectrum of Pρ, i.e. thereexists k0 ∈ Z+ such that mk0(x2, ξ2) = 0. In accordance with our approach, weconsider Fujiwara’s approximate projectors π+, π− : L2(R) → L2(R) for the firstorder operator mk0(x2, Dx2), which is symmetric modulo OPS0(R), and set T =π+ + π− − Id. For any k ∈ Z+, we define the operator Hk : L2(R) → L2(R2) by

(Hkf)(x1, x2) = (2π)−1

∫|ξ2|1/4eix2ξ2hk(|ξ2|1/2x1)f(ξ2) dξ2,

in such a way that the localized operator of HkH∗k is the spectral projector onto the

eigenspace Ker (Pρ −mk(x2, ξ2)Id).Finally we define the operators Π+ and Π− as

Π+ = Hk0(π+ − T )H∗

k0+∑k>k0

HkH∗k ,

Π− = Hk0π−H∗

k0+∑

0≤k<k0

HkH∗k ,

with convergence in OPS01/2,1/2(R2).

1.4 The Weak-Hormander inequality for systems

with double characteristics

Let X be an open subset of Rn and P a N × N formally self-adjoint system ofpseudodifferential operators of order m in X. Let p ∼

∑j≥0 pm−j be its symbol,

where the matrices pm−j(x, ξ) are positively homogeneous of degree m − j withrespect to ξ. We are interested in the following lower bound:

For any given compact K ⊂ X there exists CK > 0 such that

(WH) (Pu, u)L2(Rn;CN ) ≥ −CK‖u‖2(m−3/2)/2, ∀u ∈ C∞

0 (K; CN),

where ‖u‖2s =

∫(1 + |ξ|2)s|u(ξ)|2CN dξ denotes the Sobolev norm.

Following Parmeggiani [58], this estimate will be referred to as the Weak-Hormanderinequality. More generally, lower bounds of the type

(Is) (Pu, u)L2(Rn;CN ) ≥ −CK‖u‖2(m−s)/2, ∀u ∈ C∞

0 (K; CN),


with a gain of s ≥ 1 derivatives, have been studied by several authors, see for examplethe papers [35, 29, 6, 7, 8, 9, 54, 58]. Estimate (I1) is indeed the Sharp Gardinginequality, which is equivalent to requiring pm ≥ 0 (as Hermitian matrix), as provedby Lax and Nirenberg [35]. Instead, once s > 1 the estimate (Is) imposes conditionson pm−1 as well. Contrary to what happens in the scalar case (N = 1), the non-negativity of the total symbol is no longer sufficient for the validity of (I2), namelythe Fefferman-Phong inequality, see Brummelhuis’ counterexample [6]. Basically thereason for this is that the Weyl quantization of positive semidefinite matrix-valuedquadratic forms may be not nonnegative; we address the reader to Parmeggiani [58]for a deep discussion in this connection, and for several geometrically characterizedclasses of examples of operators for which the Sharp Garding inequality cannot beimproved, namely none of inequalities (Is), with s > 1, may hold (we observe thatthe classes considered in [58] contain Brummelhuis’ counterexample as a particularcase).

An intermediate estimate, which is stronger than the Sharp Garding inequalitybut weaker than each (Is), with s > 1, is Melin’s inequality:

For any given ε > 0, any µ < (m − 1)/2 and any compact K ⊂ X there existsCε,µ,K > 0 such that

(M) (Pu, u)L2(Rn;CN ) ≥ −ε‖u‖2(m−1)/2 − Cε,µ,K‖u‖2

µ, ∀u ∈ C∞0 (K; CN).

This estimate was studied in the scalar case by Melin [43] and for systems by Brum-melhuis [7, 8], Brummelhuis and Nourrigat [9], Parenti and Parmeggiani [54], fromslightly different points of view.

We now recall in detail the setting of [54], which will also be assumed here.Let

Σ = (x, ξ) ∈ T ∗X \ 0 : det pm(x, ξ) = 0be the characteristic set of P . Let us suppose that

pm(x, ξ) ≥ 0, ∀(x, ξ) ∈ T ∗X \ 0, (1.25)

and assume the following conditions:Σ is a conic, connected symplectic manifold of codimension 2ν, 1 ≤ ν < n,

det pm(x, ξ) ≈ |ξ|mNdistΣ(x, ξ)2l, 1 ≤ l ≤ N,

dim (Ker pm(x, ξ)) = l, ∀(x, ξ) ∈ Σ,

(1.26)where distΣ(x, ξ) is the distance of (x, ξ/|ξ|) to Σ. We are therefore considering aclass of systems with a symplectic characteristic manifold and with double charac-teristics, namely Ker pm(x, ξ) has constant dimension l and det pm vanishes exactlyto order 2l at Σ.


A notion of localized operator is then introduced: roughly speaking it is a param-eter dependent, second order differential system Pρ : S(Rν ; Cl) → S(Rν ; Cl), ρ ∈ Σ,which reads as a matrix oscillator (see [7, 8, 54]).More precisely, consider a system P = P ∗ ∈ OPSm(X; CN) satisfying the hypothe-ses (1.25) and (1.26).

In view of the third condition in (1.26) we have that

V =⊔ρ∈Σ

Vρ, Vρ = Ker pm(ρ),

defines a vector bundle on Σ, and one can write Σ × CN = V ⊕ V ⊥. Moreover,locally there exist extensions V and V ⊥ in a full neighborhood of points of Σ, whichare invariant with respect to pm, and any smooth section v of V (resp. V ⊥) admitsan extension v as section of V (resp. V ⊥).

We then consider, for v, v′ sections of V , ρ ∈ Σ, ζ ∈ TρΣσ, the map Apm(ρ, ζ)

defined by

〈Apm(ρ; ζ)v(ρ), v′(ρ)〉 =1

2〈Hess(αv′,v)(ρ)ζ, ζ〉TρT ∗X ,

where αv′,v(ρ) = 〈pm(ρ)v(ρ), v′(ρ)〉, v and v′ being any extensions of v and v′, re-spectively. Moreover one defines the subprincipal term psm−1,Σ(ρ) as

〈psm−1,Σ(ρ)v(ρ), v′(ρ)〉 = 〈

(psm−1(ρ) +

i

2

n∑j=1

bj(ρ)∗aj(ρ)− aj(ρ)

∗bj(ρ)]

)v, v′〉,

where

aj(ρ) =√q(ρ)∂xj

pm(ρ), bj(ρ) =√q(ρ)∂ξjpm(ρ), j = 1, . . . , n,

with q(ρ) :=(pm(ρ)|V ⊥ρ

)−1

, which is well defined, for

∂xjpm(ρ), ∂ξjpm(ρ) : Vρ → V ⊥

ρ , ∀ρ ∈ Σ, j = 1, . . . , n.

We can now define the localized symbol of P : it is the smooth section LP : TΣσ →HomC(π∗ΣV, π

∗ΣV ), given by

LP (ρ, ζ) = Apm(ρ, ζ) + psm−1,Σ(ρ),

(where πΣ : TΣσ → Σ is the projection and π∗Σ is the pull-back of vector bundles).At this point one fixes a smooth local orthonormal basis v = (v1, . . . , vl) of

sections of V and a so-called canonical flattening χ for Σ near a point ρ0 ∈ Σto obtain a “nice” local representative of LP . More precisely, Σ being symplectic,


there are a conic neighborhood Γ ⊂ T ∗X \ 0 of ρ0, an open conic set Γ ⊂ T ∗Rνy′,η′ ×(

T ∗Rn−νy′′,η′′ \ 0

), 2ν = codim Σ, and a smooth homogeneous symplectomorphism χ :

Γ → Γ such that

χ(Γ ∩ Σ) = (y′, y′′, η′, η′′) ∈ Γ : y′ = η′ = 0, η′′ 6= 0.

We therefore identify χ(Γ ∩ Σ) with an open subset Γ′ ⊂ T ∗Rn−ν \ 0.Any canonical flattening χ induces a trivialization χ : TΣσ|Γ∩Σ → Γ′×R2ν given

byχ(ρ, ζ) = (χ(ρ), dχ(ρ)ζ), ρ ∈ Γ ∩ Σ, ζ ∈ TρΣσ.

We can now define the local representatives

A(χ,v)pm

, ps,(χ,v)m−1,Σ : Γ′ × Rν

t × (Rν)∗τ → HomC(Cl,Cl),

by

⟨A(χ,v)pm

(y′′, η′′; t, τ)

c1...cl

,d1

...dl

⟩ =

⟨Apm

(χ−1(Y ′′), dχ−1(Y ′′)

t0τ0

) l∑j=1

cjvj(χ−1(Y ′′)),

l∑j=1

djvj(χ−1(Y ′′))

⟩,

and

⟨ps,(χ,v)m−1,Σ(y′′, η′′)

c1...cl

,d1

...dl

⟩ =

⟨psm−1

(χ−1(Y ′′)

) l∑j=1

cjvj(χ−1(Y ′′)),

l∑j=1

djvj(χ−1(Y ′′))

⟩,

with Y ′′ = (0, y′′, 0, η′′) ∈ χ(Γ∩Σ). Notice that A(χ,v)pm and p

s,(χ,v)m−1,Σ are l× l Hermitian

matrices; moreover the entries of A(χ,v)pm are quadratic forms in t, τ .

SetL

(χ,v)P (y′′, η′′; t, τ) = A(χ,v)

pm(y′′, η′′; t, τ) + p

s,(χ,v)m−1,Σ(y′′, η′′),

and consider its Weyl quantization

Opw(L

(χ,v)P

)(y′′, η′′; t,Dt) : S(Rν ; Cl) → S(Rν ; Cl),


as an operator depending on the parameter (y′′, η′′) ∈ Γ′. This is the so-calledlocalized operator; it has a unique realization as a self-adjoint operator in L2(Rν ; Cl)with domain

B2(Rν ; Cl) = f ∈ L2(Rν ; Cl) : tαDβt f ∈ L2(Rν ; Cl); ∀α, β, |α|+ |β| ≤ 2.

Furthermore, as a consequence of the second condition in (1.26), its spectrum isbounded from below and consists of a sequence of eigenvalues diverging to +∞.The key point is that changing canonical flattening χ or gauge v results in unitaryinterwinings of the various L

(χ,v)P (y′′, η′′; t, τ)’s, so that the spectrum of the localized

operator is a symplectic invariant of the system. In particular the lowest eigenvalue

λ0 : Σ → R

is well defined as a continuous function on Σ.We point out that Parmeggiani and Wakayama have recently studied in detail

the spectrum of some classes of matrix oscillators in [60, 61], see also [59]. Infact, a motivation of those works was the natural application to lower bounds forpseudodifferential systems.

We can now state one of the results of [54], Thm. 4.1 and [8], Thm. 5.6 that weare concerned with.

Assume (1.25) and (1.26). We have

λ0(ρ) ≥ 0, ∀ρ ∈ Σ ⇐⇒ (M). (1.27)

In the scalar case one can explicitly write down the lowest eigenvalue λ0, whichturns out to be nothing but Melin’s invariant λ0(ρ) = Tr+Fpm(ρ) + psm−1(ρ) (seeMelin [43], or Hormander [30], Section 21.6). However, when N = 1 this result doesnot recover Melin’s theorem [43] fully, which establishes the equivalence (1.27) forany operator P = P ∗ ∈ OPSm(X) with a nonnegative principal symbol, namelywithout any assumptions on the characteristic set. Instead, if N = 1, Σ is a smoothsymplectic manifold and the principal symbol pm vanishes exactly to second orderon Σ, then λ(ρ) ≥ 0, ∀ρ ∈ Σ ⇐⇒ (I2), that is Hormander’s inequality. This remarksuggests that also in the case of systems, under the additional assumptions (1.26),an inequality with a gain greater than 1 derivatives could hold ( see Remark 4.15 of[54]). Indeed, in [50] we proved the following improvement of (1.27).

Theorem 1.4.1. Let P = P ∗ ∈ OPSm(X; CN) be a classical pseudodifferentialsystem, whose principal symbol satisfies (1.25) and (1.26). Let λ0 : Σ → R bethe lowest eigenvalue of its localized operator, as defined above. Then the followingproperties are equivalent:


(1) λ0(ρ) ≥ 0, ∀ρ ∈ Σ;

(2) For every compact K ⊂ X there exists CK > 0 such that

(Pu, u)L2(Rn;CN ) ≥ −CK‖u‖2(m−3/2)/2, ∀u ∈ C∞

0 (K; CN). (1.28)

The proof is given in Chapter 4 and is based on the technique developed byParenti and Parmeggiani in [54] to study the stronger Hormander’s inequality (I2)for systems (Thm. 4.5 of [54]). We observe that in the proof of (I2) the authors of [54]assume a parity condition on the eigenfunctions belonging to the low-lying part ofthe spectrum of Pρ, and a condition of “regular vanishing” of the lowest eigenvalue.Then, as a byproduct of their proof, they already obtain Inequality (WH) whenthe parity assumption is dropped. The improvement contained here lies exactly inshowing that (WH) still holds without the regular vanishing assumption.

We will make use of the theory of Hermite operators (see [4] [21]) which is recalledin the scalar case in Section 3.1. For definiteness we collected the main definitionsand properties in the matrix valued case in Appendix 4.A. A key role will also beplayed by the Sharp Garding inequality for operator valued (no just matrix valued)symbols ([30], Thm. 18.6.14).

Chapter II

PROOF OF THE GENERALIZATION OF

HORMANDER’S INEQUALITY

This chapter is devoted to the proof of Theorem 1.2.2.

2.1 A preliminary lower bound

In this section we prove some technical results used in the sequel. In particular, inProposition 2.1.3 below, we establish sufficient conditions in order for a Melin-typeinequality to hold for certain non classical pseudodifferential operators.

We deal with global symbols a(x, ξ) ∈ C∞(Rn×Rn), where “global” means thata(x, ξ) satisfies uniform estimates in x ∈ Rn (see (18.1.1)′′ of [30]). In particular,we denote by Smcl the class of global classical symbols (see Definition 18.1.5 of [30],with h = 1).

In Section 2.2 we will reduce matters to considering an operator P = P ∗ ∈OPNm,k(Rn,Σ) with

Σ = (x, ξ) ∈ T ∗Rn \ 0 | x′ = ξ′ = ξ′′ = 0, ξ′′′ 6= 0, (2.1)

where we set x = (x′, x′′, x′′′) ∈ Rνx′×Rl

x′′×Rn−ν−lx′′′ and, accordingly, ξ = (ξ′, ξ′′, ξ′′′).

Moreover, we will often use microlocal “cut and paste” arguments that require cut-off functions ψ(x, ξ) ∈ C∞(Rn × Rn) satisfying the following homogeneity property

ψ(x, tξ′, t1/2ξ′′, tξ′′′) = ψ(x, ξ), for t ≥ 1 and |ξ| ≥ 1. (2.2)

Therefore, we start off by defining a pseudodifferential calculus that allows to treatthis kind of cut-off functions together with the symbols in Nm,k(Rn × Rn,Σ). Tothis purpose consider the following metric in T ∗Rn

gx,ξ =|dx′|2

d(x, ξ)2+ |dx′′|2 + |dx′′′|2 +

|dξ′|2

〈ξ〉2d(x, ξ)2+|dξ′′|2

〈ξ〉+|dξ′′′|2

〈ξ〉2, (2.3)

25

26 PROOF OF THE GENERALIZATION OF HORMANDER’S INEQUALITY

and the weights

Λ(ξ) = 〈ξ〉, d(x, ξ) =

(|x′|2

〈x′〉2+|ξ′|2

〈ξ〉2+|ξ′′|2

〈ξ〉2+

1

〈ξ〉

) 12

. (2.4)

As shown in Proposition 2.A.1 and 2.A.2 of Appendix 2.A, gx,ξ is a Hormandermetric with Planck function

h = Λ−1 supd−2,Λ1/2

and the weights Λ and d are g-admissible.As claimed, it is easy to check that

Nm,k(Rn × Rn,Σ) ∩ Smcl (Rn × Rn) ⊂ S(Λmdk, g). (2.5)

The proof of (2.5) is the same as the one of Lemma 2.4 of Maniccia and Mughetti[40].

Remark 2.1.1. Notice that the global version of the classes Sm,k(Rn × Rn,Σ) in[4] correspond to the metric

|dx′|2

d(x, ξ)2+ |dx′′|2 + |dx′′′|2 +

|dξ′|2

〈ξ〉2d(x, ξ)2+

|dξ′′|2

〈ξ〉2d(x, ξ)2+|dξ′′′|2

〈ξ〉2, (2.6)

and to the weight function 〈ξ〉md(x, ξ)k. On the other hand, any function ψ(x, ξ) ∈C∞(Rn×Rn) satisfying (2.2), with compact support in x, belongs to S(1, g), but itis not necessarily a symbol of order zero with respect to Boutet de Monvel’s metric(2.6), which is the reason for introducing the metric (2.3).

Finally we observe that every symbol in Sm,k(Rn×Rn,Σ) (see [4] and Definition2.3.1 below) with compact support in x is in S(Λmdk, g) too.

Remark 2.1.2. Let p ∈ Nm,k(Rn ×Rn,Σ)∩ Smcl (Rn ×Rn) be transversally elliptic,with non negative principal part. Let us suppose that p does not depend on thevariables x′′, x′′′ for |x′′|+ |x′′′| large, and that pm(x, ξ) ≥ c|ξ|m, c > 0, uniformly for(x, ξ) ∈ Rn×Rn \0 with |x′| large. Then it easily seen that one can write p = p1 + p2

with p1 ≥ c1Λmdk, for some positive constant c1 > 0, and p2 ∈ S(Λm−1dk−2, g).

Let us denote by Sµ(Rlξ′′ ; R2ν

x′,ξ′) the class of smooth functions b(ξ′′, x′, ξ′) satisfy-ing

|∂β′′

ξ′′ ∂α′

x′ ∂β′

ξ′ b(ξ′′, x′, ξ′)| ≤ Cα′,β′,β′′(1+ |x′|+ |ξ′|+ |ξ′′|)µ−|α′|−|β′|, (ξ′′, x′, ξ′) ∈ Rl×R2ν .

A symbol b ∈ Sµ(Rlξ′′ ; R2ν

x′,ξ′) is called globally elliptic if |b(ξ′′, x′, ξ′)| ≥ c(1 + |x′| +|ξ′|+ |ξ′′|)µ.

2.1. A PRELIMINARY LOWER BOUND 27

In what follows, we shall write that a smooth function fλ(x, ξ), depending on aparameter λ ∈ R, belongs to a class of symbols, if fλ(x, ξ)λ∈R is a bounded familyin such a class.

We are now in a position to state the main result of this section.

Proposition 2.1.3. Let B = B∗ = Op(b) for a symbol b ∈ S(Λmdk, g) satisfyingthe following assumptions:

1) b = b1 + b2 with b1 ≥ cΛmdk for a suitable constant c > 0, and b2 ∈ S(hΛmdk, g);

2) there exists a function b(k) ∈ C∞(Σ,Sk(Rlξ′′ ; R2ν

x′,ξ′)), independent of x′′, x′′′ for|x′′|+ |x′′′| large and quasi-homogeneous, in the sense that

b(k)(t−1/2x′, x′′, x′′′, t1/2ξ′, t1/2ξ′′, tξ′′′) = tm−k/2b(k)(x′, x′′, x′′′, ξ′, ξ′′, ξ′′′), t > 0,

such that

2a) b(k)(ρ, .) is globally elliptic, uniformly when ρ varies in compact subsetsof Σ, and b− b(k) satisfies the estimates of the class S(Λmdk+1, g) in

ΓR := (x, ξ) ∈ Rn × Rn : |x′| < R, |ξ′|+ |ξ′′| < R|ξ′′′|, |ξ′′′| > 1/R,

for some R > 0 (b(k) is called the localized symbol of b),

2b) b(k)(ρ, ξ′′;x′, Dx′) : S(Rν) → S(Rν) is nonnegative and injective (henceinvertible), for every ρ ∈ Σ and every ξ′′ ∈ Rl.

Then, for every s < m/2− k/4 there exist constants c > 0, C > 0 such that

(Bu, u) ≥ c‖u‖2m/2−k/4 − C‖u‖2

s, ∀u ∈ S(Rn). (2.7)

Proof. From now on, without loss of generality, we assume m > k/2.We first prove that there exist two operators Cλ, Sλ ∈ L

(L2(Rn)

), smoothly depend-

ing on a parameter λ ∈ R, for which(B + λkI

) Cλ = I + Sλ,

where ‖Sλ‖L(L2(Rn)) ≤ c(λk+1)−1 for a positive c constant independent of λ. There-fore, if |λ| is chosen large enough, it turns out that B + λkI is invertible in L2(Rn),whence (

Bu, u)≥ −C‖u‖2

0, ∀u ∈ S(Rn).

Finally, a perturbation argument will complete the proof.Define the weight function in Rn × Rn

dλ(x, ξ) =

(|x′|2

〈x′〉2+|ξ′|2

〈ξ〉2+|ξ′′|2

〈ξ〉2+

1

〈ξ〉+

λ2

〈ξ〉2m/k

)1/2

, λ ∈ R, (2.8)


which is g-admissible uniformly in λ, as shown in Proposition 2.A.1 in Appendix2.A.It is easily seen that b+λk ∈ S(Λmdkλ, g) and qλ := (b1(x, ξ)+λ

k)−1 ∈ S(Λ−md−kλ , g),whence,

(b+ λk)#qλ − 1 ∈ S(hdkd−kλ , g).

As usual, for any N ∈ N, we can construct a symbol qN,λ ∈ S(Λ−md−kλ , g) such that

(b+ λk)#qN,λ = 1− rN,λ, (2.9)

where rN,λ ∈ S((dkd−kλ )N(Λ−Nd−2N + Λ−N/2), g), for h ≤ Λ−1d−2 + Λ−1/2.Denote by Sµpar(Rl+1; R2ν) the class of functions bλ(ξ

′′, x′, ξ′) ∈ C∞(Rl+1y:=(ξ′′,λ) ×

R2νx′,ξ′), smoothly depending on the parameter y = (ξ′′, λ), such that

|∂γy∂α′

x′ ∂β′

ξ′ bλ(x′, ξ′, ξ′′)| ≤ Cγ,α′,β′(1+ |x′|+ |ξ′|+ |ξ′′|+ |λ|)µ(1+ |x′|+ |ξ′|+ |ξ′′|)−|α′|−|β′|.

and observe thatbλ := b(k) + λk ∈ C∞(Σ; Skpar(Rl+1; R2ν)).

Moreover, |bλ| ≈ (1 + |x′| + |ξ′| + |ξ′′| + |λ|)k uniformly in ρ ∈ K ⊂⊂ Σ andthe corresponding operator b(k)(ρ, ξ′′, x′, Dx′) + λk is nonnegative and invertible forevery ρ ∈ Σ, (ξ′′, λ) ∈ Rl+1. Arguing as in [5], one can show that its inverse is apseudodifferential operator, with symbol tλ = tλ(ρ, ξ

′′;x′, ξ′) such that:

· tλ ∈ C∞(Σ; S−kpar(Rl+1; R2ν)) (see Lemma 2.B.1 in Appendix 2.B);

· tλ is quasi-homogeneous of degree −m+ k/2, i.e.

tsm/k−1/2λ(x′′, x′′′, sξ′′′, s1/2ξ′′; s−1/2x′, s1/2ξ′) = s−m+k/2tλ(x

′′, x′′′, ξ′′′, ξ′′;x′, ξ′);

· tλ does not depend on x′′, x′′′ for |x′′|+ |x′′′| large.

For any M ∈ R+, set

ΓM = (x, ξ) ∈ Rn × Rn : |x′| < M, |ξ′|+ |ξ′′| < M |ξ′′′|, |ξ′′′| > 1/M,

and choose a cut-off function χ ∈ S01,0(Rn×Rn) such that suppχ ⊂ ΓR/3 and χ ≡ 1

in ΓR/4, where R is the constant in 2a). As a consequence of the above facts,χtλ ∈ S(Λ−md−kλ , g), and our purpose is now to prove that

(b+ λk)#(χtλ) = χ+ sλ, with sλ ∈ S(d, g). (2.10)

In (2.10) we can replace b by χ′b where χ′ ∈ S01,0(Rn×Rn) satisfies suppχ′ ⊂ ΓR and

χ′ ≡ 1 in ΓR/2 (so that χ′ = 1 in a neighborhood of suppχ). In fact, (1−χ′)b#χtλ ∈

2.1. A PRELIMINARY LOWER BOUND 29

S−∞ uniformly with respect to λ.In the region where |x′| ≥ R/2 or |ξ′′′| ≤ 2/R, this immediately follows from thestandard composition formula because of h(x, ξ) ≈ 〈ξ〉−1/2. Assume that |x′| < R/2,|ξ′′′| > 2/R and, furthermore, |ξ′| + |ξ′′| < R|ξ′′′|/3 since χtλ is supported in ΓR/3;one has, as an oscillatory integral,

(1− χ′)b#χtλ(x, ξ) = (2π)−n∫∫

eizζ ((1− χ′)b) (x, ξ + ζ) (χtλ) (x− z, ξ) dz dζ.

(2.11)In the region |ζ| < c|ξ| the above integral vanishes if the constant c is small enough,because |ξ′ + ζ ′|+ |ξ′′ + ζ ′′| ≤ R|ξ′′′ + ζ ′′′|/2 and thus (x, ξ+ ζ) ∈ ΓR/2. On the otherhand, upon setting

L =1−∆2

z

1 + |ζ|2

one gets Leizζ = eizζ , whence repeated integrations by parts in (2.11), where |ζ| ≥c|ξ|, show that (1− χ′)b#χtλ ∈ S−∞ uniformly with respect to λ.By Hypothesis 2a) we have χ′(b + λk) − χ′bλ ∈ S(Λmdk+1, g), we are thus reducedto proving that

χ′bλ#χtλ − χ ∈ S(d, g). (2.12)

Since χ′bλ#χtλ ∈ S(d−kλ dk, g) and χ ∈ S01,0(Rn × Rn) ⊂ S(1, g), (2.12) is satisfied

away from ΓR for dλ(x, ξ) ≈ d(x, ξ) ≈ 1. Arguing as in 2.4.9 [21], it is seen that

χ′bλ#χtλ = χ′bλ#′χtλ + S(Λ−1/2, g) (2.13)

where #′ denotes the usual composition in S(Rν) with respect to the variables(x′, ξ′). Upon denoting by SΓR

(d, g) the set of symbols satisfying the estimates ofS(d, g) in ΓR, by (2.13) we have to check that χ′bλ#

′χtλ − χ ∈ SΓR(d, g). To this

aim, write χ = χ|x′=ξ′=0 + r, χ′ = χ′|x′=ξ′=0 + r′ and observe that

χ′|x′=ξ′=0bλ#′χ|x′=ξ′=0tλ = χ′χ|x′=ξ′=0bλ#

′tλ = χ|x′=ξ′=0,

which is equal to χ, modulo SΓR(d, g). Hence, it remains to show that the other

products χ′|x′=ξ′=0bλ#′rtλ, r

′bλ#′χ|x′=ξ′=0tλ and r′bλ#

′rtλ are in SΓR(d, g). To this

end, let us introduce the weights

d′(x, ξ) =

(|x′|2 +

|ξ′|2

〈(ξ′′, ξ′′′)〉2+

|ξ′′|2

〈(ξ′′, ξ′′′)〉2+

1

〈(ξ′′, ξ′′′)〉

) 12

, Λ′(ξ′′, ξ′′′) = 〈(ξ′′, ξ′′′)〉,

d′λ(x, ξ) =

(|x′|2 +

|ξ′|2

〈(ξ′′, ξ′′′)〉2+

|ξ′′|2

〈(ξ′′, ξ′′′)〉2+

1

〈(ξ′′, ξ′′′)〉+

λ2

〈(ξ′′, ξ′′′)〉2m/k

) 12

,

and the metric

g′x′,ξ′ =|dx′|2

d′(x, ξ)2+

|dξ′|2

〈(ξ′′, ξ′′′)〉2d′(x, ξ)2.


It turns out that g′ is a Hormander metric in R2ν uniformly with respect to(x′′, x′′′, ξ′′, ξ′′′) ∈ Rn−ν × Rn−ν , λ ∈ R, and the weights d′, d′λ, Λ′ are uniformlyg′-admissible.We have χ′|x′=ξ′=0bλ ∈ S(Λ′md′λ

k, g′) and rtλ ∈ S(Λ′−md′λ−kd′, g′) so that

χ′|x′=ξ′=0bλ#′rtλ ∈ S(d′, g′). This and similar estimates for the derivatives with re-

spect to the variables (x′′, x′′′, ξ′′, ξ′′′) prove that χ′|x′=ξ′=0bλ#′rtλ ∈ SΓR

(d, g). Thesame arguments show that r′bλ#

′χ|x′=ξ′=0tλ, r′bλ#

′rtλ ∈ SΓR(d, g). This concludes

the proof of (2.10).Since the weight function d(x, ξ) is equivalent to 1 on the support of 1− χ and

qλ := (b1(x, ξ) + λk)−1 ∈ S(Λ−md−kλ , g), we have (1− χ)qλ ∈ S(Λ−m, g) and

(b+ λk)#((1− χ)qλ) = 1− χ+ sλ, (2.14)

with sλ ∈ S(hdk, g) ⊂ S(d, g). Setting qλ := χtλ + (1− χ)qλ ∈ S(Λ−md−kλ , g), (2.10)and (2.14) give

(b+ λk)#qλ − 1 ∈ S(d, g),

whence, by standard arguments, for any M ∈ N there exists qM,λ ∈ S(Λ−md−kλ , g)such that

(b+ λk)#qM,λ = 1− rM,λ, (2.15)

with rM,λ ∈ S(dM , g). By virtue of (2.9) and (2.15) we have cλ := qN,λ#rM,λ+qM,λ ∈S(Λ−md−kλ , g) for suitable integers N > 0,M > 0 to be chosen later on, and

(b+ λk)#cλ = (b+ λk)#qN,λ︸︷︷︸1−rN,λ

#rM,λ + (b+ λk)#qM,λ︸︷︷︸1−rM,λ

= 1− rN,λ#rM,λ, (2.16)

whererN,λ#rM,λ ∈ S((dkd−kλ )N(Λ−Nd−2N+M + Λ−N/2dM), g).

Hence, since (dkd−kλ )N = dkd−kλ (dkd−kλ )N−1 ≤ dkd−kλ ≤ λ−kΛmdk, one has, uniformlyin λ,

λkrN,λ#rM,λ ∈ S(Λm−Nd−2N+k+M + Λm−N/2dk+M , g). (2.17)

For λ = 0 the above “parametrix” construction shows that B is self-adjoint asan unbounded operator in L2(Rn) with domain the Bony-Chemin’s Sobolev spaceH(Λmdk, g) (see [3] for details). Moreover, we see from (2.16) and (2.17) that, ifwe choose N,M such that m − N/2 ≤ 0 and −2N + k + M ≥ 0, B + λkI hasa right inverse for |λ| large enough, thus it is surjective. Since B + λkId is self-adjoint (with domain H(Λmdk, g)), it is actually invertible with continuous inverse(B + λkId)−1 : L2(Rn) → H(Λmdk, g) → L2(Rn). Hence the spectrum of B isbounded from below, which immediately gives

(Bu, u) ≥ −C‖u‖20, ∀u ∈ S(Rn), (2.18)

2.2. PROOF OF THEOREM 1.2.2 31

(see Kato [33], Section 5.3.10, page 278).To prove (2.7) we now argue as follows. We observe that the smooth map of sym-bols (ρ, ξ′′) 7→ (1 + |ξ′′|/|ξ′′′|1/2)ktλ=0(ρ, ξ

′′; ·) takes values in a bounded subset ofS0(R2ν) := a ∈ C∞(R2ν

z ) : |∂γz a(z)| ≤ Cγ〈z〉−|γ| when (x′′, x′′′) ∈ Rn−ν , ξ′′ ∈ Rl,|ξ′′′| = 1, so that the corresponding operators are uniformly bounded on L2(Rν). Byhomogeneity, we get

‖tλ=0(ρ, ξ′′;x′, Dx′)‖L(L2(Rν)) ≤ C|ξ′′′|−m+k/2

(1 +

|ξ′′||ξ′′′|1/2

)−k≤ C|ξ′′′|−m+k/2,

whence for every ρ ∈ Σ, ξ′′ ∈ Rl,

(b(k)(ρ, ξ′′;x′, Dx′)v, v) ≥ C−1|ξ′′′|m−k/2‖v‖20, ∀v ∈ S(Rν).

Hence, if 0 < τ ∈ R is small enough, the operator Bτ := B − τΛm−k/2 still satisfies

the hypotheses of Proposition 2.1.3 with b(k)τ := b(k) − τ |ξ′′′|m−k/2; therefore (2.18)

holds for Bτ and gives (2.7).

2.2 Proof of Theorem 1.2.2

In this section we prove Theorem 1.2.2 by reducing Inequality (1.14) to suitablemicrolocal estimates. More precisely, Theorem 2.1 of [53] shows that we are reducedto proving the following result.

Theorem 2.2.1. Assume that the hypotheses of Theorem 1.2.2 are satisfied. Then,for every ρ0 ∈ T ∗Rn \ 0, there exists a conic neighborhood Vρ0 of ρ0 such that forevery real symbol θ ∈ S0

1,0(Rn×Rn) supported in Vρ0, one has, for a suitable constantC > 0,

(PΘu,Θu) ≥ −C‖u‖2m2− k+2

4

∀u ∈ S(Rn), (2.19)

with Θ = θ(x,D).

A crucial point is that Inequality (2.19) is invariant under canonical changes ofvariables (as shown, for instance, in Lemma 2.7 of [53]). By means of a canonicalflattening we may thus suppose that P = P ∗ ∈ OPNm,k(Rn,Σ) with

Σ = (x, ξ) ∈ T ∗Rn \ 0 | x′ = ξ′ = ξ′′ = 0, ξ′′′ 6= 0,

with x = (x′, x′′, x′′′) ∈ Rνx′ × Rl

x′′ × Rn−ν−lx′′′ and ξ = (ξ′, ξ′′, ξ′′′). Furthermore, the

map χ = Id : T ∗Rn → T ∗Rn induces a trivialization (TΣ ∩ TΣσ)′ ' Σ × Rlξ′′ , with

coordinates ρ = (x′′, x′′′, ξ′′′) ∈ Σ, v = ξ′′.


As in the Introduction, we denote by p ∼∑

j≥0 pm−j the usual symbol of P ,i.e. P = p(x,D). Since Σ is given by the equation in (2.1), the localized operatorinduced by χ = Id can be written in terms of its symbol as follows:

p(k)(ρ, ξ′′;x′, Dx′) =∑

|α|+|β|+|γ|+2j=k

1

α!β!γ!∂αx′∂

βξ′∂

γξ′′pm−j(ρ)ξ

′′γx′αDβx′ , ρ ∈ Σ.

(2.20)We refer to Proposition 2.3 of [52] for the proof of this fact (of course, Weyl quan-tization is essential to have an invariant definition in the general non-flat case). Inthis setting we denote by λ(ρ, ξ′′) its lowest eigenvalue and by V (ρ, ξ′′) ⊂ S(Rν) thecorresponding eigenspace. It is easily seen that the above localized operator enjoysthe following homogeneity property

p(k)(x′′, x′′′, tξ′′′, t1/2ξ′′; t−1/2x′, t1/2Dx′) = tm−k/2Mtp(k)(x′′, x′′′, ξ′′′, ξ′′;x′, Dx′)M

−1t ,

(2.21)where (Mtf)(y) = tν/4f(t1/2y), t > 0 (such applications Mt define automorphismsof S(Rν), S ′(Rν) and, thanks to the factor tν/4, unitary transformations of L2(Rν)as well). As a consequence, we have

λ(x′′, x′′′, tξ′′′, t1/2ξ′′) = tm−k/2λ(x′′, x′′′, ξ′′′, ξ′′), t > 0. (2.22)

Since Theorem 2.2.1 is a local result near any given ρ0 ∈ T ∗Rn \ 0, we may supposethat p(x, ξ) ∼

∑j≥0 pm−j(x, ξ) satisfies uniform estimates with respect to x ∈ Rn.

Furthermore we need only prove Theorem 2.2.1 for ρ0 ∈ Σ, hence it is no restric-tion to assume that p(x, ξ) and the related terms pm−j(x, ξ) do not depend onthe variables x′′, x′′′ if |x′′| + |x′′′| large; for the same reason, we can assume thatpm(x, ξ) ≥ c|ξ|m, c > 0, uniformly in (x, ξ) ∈ T ∗Rn \ 0 with |x′| large.By virtue of Lemma 2.6 of [53] and (i) of Theorem 1.2.2, it suffices to consider ρ0 ∈ Σsuch that λ(ρ0, 0) = 0. In this case Inequality (2.19) will be obtained by patchingtogether two different estimates, obtained by microlocalizing “parabolically” near Σand “parabolically” away from Σ (but still conically near Σ).To be definite, let us fix the notation used throughout; for any U ⊂⊂ Σ, δ > 0, ε > 0,set

U±,δ = (x′′, x′′′, ξ′′, ξ′′′) ∈ R2n−2ν : (x′′, x′′′, ξ′′′) ∈ U,±(|ξ′′| − δ|ξ′′′|1/2) > 0,

Uε = (x, ξ) ∈ Rn : (x′′, x′′′, ξ′′′) ∈ U, |ξ′|+ |ξ′′| < ε|ξ′′′|, |x′| < ε, |ξ′′′| > 1/ε,

U±,δε = (x, ξ) ∈ Rn : (x, ξ) ∈ Uε, (x′′, x′′′, ξ′′, ξ′′′) ∈ U±,δ.

The following propositions are what we need in order to prove Theorem 2.2.1.


Proposition 2.2.2. There exists a conic neighborhood U ⊂⊂ Σ of ρ0 ∈ Σ andδ > 0, ε > 0 such that, for any real symbol ψ ∈ S0

1/2,0(Rn × Rn) supported in U−,δε ,

there exists C > 0 for which

(PΨu,Ψu) ≥ −C‖u‖2m2− k+2

4

, ∀u ∈ S(Rn), (2.23)

with Ψ = ψ(x,D).

Proposition 2.2.3. For every open conic subset U ⊂⊂ Σ, for every δ > 0 andε > 0 and for every real symbol ψ ∈ S0

1/2,0(Rn × Rn) supported in U+,δε , one has

(PΨu,Ψu) ≥ c‖Ψu‖2m2− k

4

− C‖u‖2m2− k+2

4

∀u ∈ S(Rn), (2.24)

for suitable constants c, C > 0, with Ψ = ψ(x,D).

Proposition 2.2.2 is the most delicate and is proved in Section 2.3. It requires, inaccordance with our approach, the full strength of the hypotheses in Theorem 1.2.2,whereas the proof of Proposition 2.2.3 is easier and does not use Hypotheses (iii)and (iv). Before proving Proposition 2.2.3, we want to show that Theorem 2.2.1 isa consequence of the propositions above.

We need the following lemma (cf. Lemma 2.2 of [53]).

Lemma 2.2.4. Let Q ∈ OPS0(Rn) be an elliptic pseudodifferential operator, ands ∈ R. Suppose that there exists a constant C > 0 such that

(Q∗PQv, v) ≥ −C‖v‖2s, ∀v ∈ C∞

0 (Rn). (2.25)

Then the same lower bound holds for P .

Proof. Let S be a properly supported parametrix of Q. Upon setting v = Su in(2.25), we obtain

(Q∗PQSu, Su) = (PQSu,QSu) = (Pu, u) + (Tu, u)

for some T ∈ OPS−∞(Rn). The claim then follows by continuity.

Proof of Theorem 2.2.1. Take U, δ as in Proposition 2.2.2 and ε > 0. Let θ, θ′ ∈S0

1,0(Rn × Rn) be real valued symbols, supported in Uε, such that θ′ is classical andequals to 1 in a neighborhood of the support of θ. Let Θ = Op(θ), Θ′ = Op(θ′).

We choose ψ1(ξ′′, ξ′′′), ψ2(ξ

′′, ξ′′′) ∈ S01/2,0(Rn−ν × Rn−ν) quasi-homogeneous of

degree 0 with respect to the dilation (ξ′′, ξ′′′) 7→ (t1/2ξ′′, tξ′′′) for t ≥ 1 and |ξ′′| +|ξ′′′|1/2 large, such that ψ2

1 + ψ22 ≡ 1, with

suppψ1 ⊂ (ξ′′, ξ′′′) : |ξ′′| ≤ δ|ξ′′′|1/2, |ξ′′|+ |ξ′′′|1/2 > 1/δ,


and

ψ1 = 1 on (ξ′′, ξ′′′) : |ξ′′| ≤ δ/2|ξ′′′|1/2, |ξ′′|+ |ξ′′′|1/2 > 2/δ.

Now we set Ψj := (θ′ψj)(x,D) ∈ OPS01/2,0(Rn), for j = 1, 2. Observe that obviously

supp θ′ψ1 ⊂ U−,δε and moreover, if ε is small enough, supp θ′ψ2 ⊂ U

+,δ/2ε . Indeed, it

is clear that supp θ′ψ2 ⊂ Uε. On the other hand,|ξ′′| ≤ δ/2|ξ′′′|ψ2(ξ

′′, ξ′′′) 6= 0

=⇒ |ξ′′|+ |ξ′′′|1/2 ≤ 2/δ,

and therefore |ξ′′′| ≤ 4/δ2. This shows that if ε ≤ δ2/4 no point (x, ξ) with ξ′′′ = ξ′′′

belongs to Uε.Upon writing

Θ′ 2u =2∑j=1

Ψ∗jΨju+ (Θ′ 2 −

2∑j=1

Ψ∗jΨj)u

we obtain, for every s ∈ R (we may then choose s = m/2− (k + 2)/4),(PΘu,Θu

)= Re

(PΘ′ 2Θu,Θu

)+O(‖u‖2

s) =

= Re(P

2∑j=1

Ψ∗jΨjΘu,Θu

)+ Re

(P (Θ′ 2 −

2∑j=1

Ψ∗jΨj)Θu,Θu

)+O(‖u‖2

s) =

=2∑j=1

(PΨjΘu,ΨjΘu

)+

2∑j=1

Re([P,Ψ∗

j ]ΨjΘu,Θu)+

+ Re(P (Θ′ 2 −

2∑j=1

Ψ∗jΨj)Θu,Θu

)+O(‖u‖2

s) =

=2∑j=1

(PΨjΘu,ΨjΘu

)+(LΘu,Θu

)+O(‖u‖2

s)

where L = L∗ = L1 + L2, with

L1 :=1

2

2∑j=1

[[P,Ψj],Ψj] +1

2

2∑j=1

([P,Ψ∗

j −Ψj]Ψj + (Ψ∗j −Ψj)[Ψj, P ]

),

and

L2 := ReP (Θ′ 2 −2∑j=1

Ψ∗jΨj).


Now, by Lemma 2.2.4, it suffices to prove Theorem 2.2.1 for the operator (Id +µΛ−1)P (Id + µΛ−1), where µ is a large real constant to be chosen later on. Theabove computation then shows that

((Id + µΛ−1)P (Id + µΛ−1)Θu,Θu

)=

2∑j=1

(PΨjΘu,ΨjΘu

)+(RµΘu,Θu

)+O(‖u‖2

s), ∀s ∈ R,

withRµ = µΛ−1P + µPΛ−1 + µ2Λ−1PΛ−1 + L. (2.26)

In view of Proposition 2.2.2 and Proposition 2.2.3, the proof is complete if we showthat

(Rµu, u) ≥ −C‖u‖2m/2−(k+2)/4, ∀u ∈ S(Rn).

On the other hand, this estimate is an immediate consequence of Proposition 2.1.3applied to the operator B := Rµ+νΛm−k/2−1 (with m−1 in place of m). In fact, wehave to check that the hypotheses of Prop. 2.1.3 are satisfied by B provided µ andν are large enough. First of all, one easily sees that the operators L1, L2 belong toOPS(Λm−1dk, g). As a result, L = Op(l) = L1 + L2 ∈ OPS(Λm−1dk, g). Moreover,by (2.26) the symbol rµ(x, ξ) of Rµ is given by

rµ = 2µΛ−1p+ l modulo S(Λm−2dk−2, g).

Hence Hypothesis 1) in Proposition 2.1.3 is fulfilled by Remark 2.1.2, if µ is largeenough. As for 2), take

b(k) := 2µ|ξ′′′|−1p(k) + l(k) + ν|ξ′′′|m−k/2−1,

where l(k) is the localized symbol of l, i.e. l−l(k) satisfies the estimate of S(Λm−1dk+1, g)in ΓR for some R > 0. (1) An easy computation shows that b(k) satisfies Condition2a) of Prop. 2.1.3.In order to check 2b), let us define the following family of norms, depending on theparameters ξ′′′ ∈ Rn−(ν+l) \ 0, ξ′′ ∈ Rl,

‖f‖2µ;ξ′′′,ξ′′ :=

∑|α|+|β|≤k

(1 + |ξ′′|2)k−|α|−|β||ξ′′′||α|−|β|‖x′αDβx′f‖

20,

1This can be easily obtained in the following way: in the asymptotic expansion of the symbol ofL we replace p by p(k) and θ′ψj by the quasi-homogeneous function ψ0j(ξ′′, ξ′′′) which coincides withψj for |ξ′′|+ |ξ′′′|1/2 large; then we retain only the terms quasi-homogeneous of degree m− k/2− 1with respect to the dilation (x, ξ) 7→ (t−1/2x′, x′′, x′′′, t1/2ξ′, t1/2ξ′′, tξ′′′), t > 0. At any rate, we donot need its explicit expression here.


and observe that by (A2.6) in [53], there exist constants c, C > 0 such that, forevery ρ ∈ Σ, ξ′′ ∈ Rl and for every g ∈ S(Rν)

(2µ|ξ′′′|−1p(k)(ρ, ξ′′;x′, Dx′)g, g)L2(Rν) ≥ cµ|ξ′′′|m−k/2−1[‖g‖2k/2,ξ′′,ξ′′/|ξ′′′|1/2 − C‖g‖2

0].

On the other hand, arguing exactly as in the proof of Lemma 6.3 a) of [53], we seethat, for every ρ ∈ Σ, ξ′′ ∈ Rl,

|(l(k)(ρ, ξ′′;x′, Dx′)g, g)L2(Rν)| ≤ C ′|ξ′′′|m−k/2−1‖g‖2k/2,ξ′′,ξ′′/|ξ′′′|1/2 ∀g ∈ S(Rν).

for a suitable constant C ′ > 0. Hence, if we choose µ, ν such that cµ > C ′ andν > cCµ, the proof is complete.

Proof of Proposition 2.2.3. This proposition is a consequence of the following fact.The cut-off function ψ in Inequality (2.24) is supported “parabolically” away fromΣ (conically near ρ0). Hence, by virtue of Hypothesis (1.13), we can modify P awayfrom U+,δ

ε , in such a way that its localized operator is nonnegative and invertibleeverywhere (as an unbounded operator in L2(Rν)).More precisely, let U be a conic subset such that U ⊂⊂ U ⊂⊂ Σ and 0 ≤ ψ ∈S0

1/2,0(Rn ×Rn) be a real symbol with supp ψ ⊂ U+,δ/22ε , satisfying, for t ≥ 1 and |ξ|

large, the following homogeneity property

ψ(x, tξ′, t1/2ξ′′, tξ′′′) = ψ(x, ξ′, ξ′′, ξ′′′).

Consider the operator Q = Op(q) = Re Op(q) ∈ OPSm1,0(Rn) with

q(x, ξ) = 〈ξ〉m−k(|ξ′|2 +

|x′|2

〈x′〉2|ξ′′′|2 + |ξ′′|2

)k/2. (2.27)

Note that Q = Q∗ ∈ OPNm,k(Rn,Σ) is transversally elliptic with nonnegative in-vertible localized operator at every point of Σ× Rl. In fact, we have

q(k)(ρ, ξ′′, x′, Dx′) = q(k)(ρ, ξ′′, x′, Dx′) = |ξ′′′|m−k(|Dx′|2 + |x′|2|ξ′′′|2 + |ξ′′|2

)k/2,

and the lowest eigenvalue is given by |ξ′′′|m−k(|ξ′′′|+ |ξ′′|2)k/2.We now modify P as follows

B = b(x,D) := Ψ∗P Ψ + (Id− Ψ∗)Q(Id− Ψ),

and apply Proposition 2.1.3 to this operator. We get

(Bu, u) ≥ c‖u‖2m/2−k/4 − C‖u‖2

µ, ∀u ∈ S(Rn),

2.3. PROOF OF PROPOSITION 2.2.2 37

and Inequality (2.24) follows at once. Hence we have only to check that B satisfiesHypotheses 1) and 2) of Proposition 2.1.3. The first one is an immediate conse-quence of Remark 2.1.2. As for the second one, consider the quasi-homogeneousfunction ψ0 ∈ C∞(T ∗Rn−ν \ 0) such that ψ0 ≡ ψ|x′=ξ′=0 for |ξ| large (notice that ψ0

is supported in U+,δ/2) and define the “localized symbol” of B as follows

b(k) = ψ20p

(k) + (1− ψ0)2q(k).

In order to show that b(k) satisfies Hypothesis 2), it suffices to prove that the low-est eigenvalue of b(k)(ρ, ξ′′, x′, Dx′) is strictly positive in L2(Rν) for every (ρ, ξ′′) ∈T ∗Rn−ν \ 0. Since this trivially holds for the lowest eigenvalue of q(k)(ρ, ξ′′, x′, Dx′),it remains to see that the lowest eigenvalue of p(k)(ρ, ξ′′, x′, Dx′) satisfies, for every(ρ, ξ′′) ∈ U+,δ/2,

λ(ρ, ξ′′) ≥ cδ|ξ′′′|m−k/2. (2.28)

By virtue of Lemma 6.3 b1) of [53], Inequality (2.28) holds for every (ρ, ξ′′) ∈ U+,δ/2

if δ is chosen large enough, whereas in the region U+,δ/2 \ U+,δ/2 it follows from(2.22), (1.13), (i) and the continuity of λ(ρ, ξ′′).

This concludes the proof.

2.3 Proof of Proposition 2.2.2

Before starting with the proof, we want to give an idea about our strategy, whichfollows that of [53].As observed in the previous section, it is enough to consider those points ρ0 ∈ Σwhere λ(ρ0, 0) = 0. The crucial step is the construction of a suitable operator Π of or-der 0, whose localized operator Π(ρ, ξ′′) is, roughly speaking, the L2(Rν)−projectoronto the eigenspace V (ρ, ξ′′) corresponding to λ(ρ, ξ′′). By means of Π we reducethe estimate of (PΨu,Ψu) to studying the quadratic forms (PΠu,Πu), (P (Id −Π)u, (Id− Π)u), and (PΠu, (Id− Π)u). It turns out that

PΠ = (L+B)Π + negligible terms,

where L is a ψdo with symbol given by λ(ρ, ξ′′) near Σ, whereas B is a vector valuedpseudodifferential operator of order lower than L. Hypothesis (iv) in Theorem 1.2.2will allow us to apply the Sharp Garding inequality and get the desired estimatefor (PΠu,Πu). In order to treat the term (P (Id− Π)u, (Id− Π)u) we observe thatthe localized operator p(k)(ρ, ξ′′, x′, Dx′)(Id − Π(ρ, ξ′′)) of P (Id − Π) is invertiblein L2(Rν), since Id − Π(ρ, ξ′′) is the L2(Rν)−projector onto V (ρ, ξ′′)⊥. Therefore(P (Id − Π)u, (Id − Π)u) satisfies a better lower bound. Finally the rectangle term(PΠu, (Id− Π)u) is re-absorbed by the previous ones.


This completes our short sketch and we now start with the proof. First of all, we needto introduce some classes of operators considered, for instance, in [4],[5],[19],[53]. Werecall from [19] their definitions and their main properties. In the sequel all operatorswill be properly supported.

SetSµ(RN) = a ∈ C∞(RN

z ) : |∂γz a(z)| ≤ Cγ(1 + |z|)µ−|γ|

(see [22]), and Σ′ = (x′, x′′, x′′′, ξ′′, ξ′′′) : x′ = ξ′′ = 0.

Definition 2.3.1.

(1) By Sm,k(Rn ×Rn,Σ), (m, k ∈ R), we denote the class of symbols a(x, ξ) whichare in Sm(Rn×Rn) away from Σ and, microlocally near Σ, satisfy the followingestimates

|∂α′x′ ∂α′′

x′′ ∂α′′′

x′′′ ∂β′

ξ′ ∂β′′

ξ′′ ∂β′′′

ξ′′′ a(x, ξ)| .

. |ξ′′′|m−|β|(|x′|+ |ξ′|

|ξ′′′|+|ξ′′||ξ′′′|

+1

|ξ′′′|1/2

)k−|α′|−|β′|−|β′′|,

with α = (α′, α′′, α′′) and β = (β′, β′′, β′′).Moreover we set Hm(Rn × Rn,Σ) := ∩j≥0S

m−j,k−2j(Rn × Rn,Σ).

By Sm,khom(Rn×Rn,Σ) (resp. Hm−k/2hom (Rn×Rn,Σ) ) we denote the class of smooth

functions in C∞(Σ,Sk(R2ν+l)) (resp. C∞(Σ,S(R2ν+l))) such that

a(x′′, x′′′, tξ′′′; t−1/2x′, t1/2ξ′, t1/2ξ′′) = tm−k/2a(x′′, x′′′, ξ′′′;x′, ξ′, ξ′′), t > 0.

Microlocally near Σ, symbols in Sm,khom(Rn×Rn,Σ) (resp. Hm−k/2hom (Rn×Rn,Σ))

are in Sm,k(Rn×Rn,Σ) (resp.Hm−k/2(Rn×Rn,Σ)). We denote by OPSm,k(Rn,Σ)and by OPHm(Rn,Σ) the corresponding classes of ψdo’s.

(2) By Sm,k(Rn−ν × Rn−ν ,Σ), (with m, k ∈ R), we denote the class of symbolsb(x′′, x′′′, ξ′′, ξ′′′) which are in Sm(Rn−ν×Rn−ν) away from Σ (here Σ is regardedas a subset of T ∗Rn−ν) and, microlocally near Σ, satisfy the following estimates

|∂α′′x′′ ∂α′′′

x′′′ ∂β′′

ξ′′ ∂β′′′

ξ′′′ b(x′′, x′′′, ξ′′, ξ′′′)| . |ξ′′′|m−|β′′|−|β′′′|

(|ξ′′||ξ′′′|

+1

|ξ′′′|1/2

)k−|β′′|,

As usual, we set Hm(Rn−ν ×Rn−ν ,Σ) := ∩j≥0Sm−j,k−2j(Rn−ν ×Rn−ν ,Σ), and

OPSm,k(Rn−ν ,Σ), OPHm(Rn−ν ,Σ) for the corresponding classes of operators.

(3) We define the class Hm(Rn−ν × Rn−ν ,Σ′), (m ∈ R), of all smooth functionsh(x′′, x′′′, ξ′′, ξ′′′, y), where (x′′, x′′′, ξ′′, ξ′′′) ∈ Rn−ν × Rn−ν , y ∈ Rν, which are


in S−∞ away from Σ, and for (x′′, x′′′, ξ′′, ξ′′′) microlocally near points of Σand y ∈ Rl satisfy for any l ≥ 0 the following estimates

|∂α′′x′′ ∂α′′′

x′′′ ∂α′

y ∂β′′

ξ′′ ∂β′′′

ξ′′′ h(x′′, x′′′, ξ′′, ξ′′′; y)| .

. |ξ′′′|m+ν/4−l−|β′′|−|β′′′|(|y|+ |ξ′′|

|ξ′′′|+

1

|ξ′′′|1/2

)−2l−|α′|−|β′′|

.

For h ∈ Hm(Rn−ν × Rn−ν ,Σ′), the operator Op(h) : C∞0 (Rn−ν) → C∞(Rn) is

defined by

(Op(h)f) (y, x′′, x′′′) :=

= (2π)−(n−ν)∫∫

ei〈x′′,ξ′′〉+i〈x′′′,ξ′′′〉h(x′′, x′′′, ξ′′, ξ′′′; y)f(ξ′′, ξ′′′)dξ′′ dξ′′′,

and OPHm(Rn−ν ,Σ′) is the corresponding class of operators, modulo smooth-ing.Moreover, by OPH∗m(Rn−ν ,Σ′) we denote the class of operators C∞

0 (Rn) →C∞(Rn−ν) that, modulo smoothing, are given by

(Op(h)∗g)(x′′, x′′′) =

= (2π)−(n−ν)∫∫∫

ei〈x′′,ξ′′〉+i〈x′′′,ξ′′′〉h(x′′, x′′′, ξ′′, ξ′′′, y)g(y, ξ′′, ξ′′′)dξ′′dξ′′′dy,

with

g(y, ξ′′, ξ′′′) =

∫∫e−i〈x

′′,ξ′′〉−i〈x′′′,ξ′′′〉g(y, x′′, x′′′)dx′′dx′′′

and h ∈ Hm(Rn−ν × Rn−ν ,Σ′).

By Hmhom(Rn−ν×Rn−ν ,Σ′) we denote the class of the quasi-homogeneous func-

tionsh(x′′, x′′′, ξ′′′, ξ′′; y) ∈ C∞(Σ× Rl,S(Rν

y)) of degree m+ ν/4, i.e.

h(x′′, x′′′, tξ′′′, t1/2ξ′′; t−1/2y) = tm+ν/4h(x′′, x′′′, ξ′′′, ξ′′; y), t > 0.

Symbols in Hmhom(Rn−ν × Rn−ν ,Σ′) define, parabolically near Σ (i.e. where

|ξ′′| ≤ C|ξ′′′|1/2), symbols that are in Hm(Rn−ν × Rn−ν ,Σ′).

(4) We denote by Smanis(Rn×Rn) the class of symbols a(x, ξ) satisfying the followingestimates

|∂αx∂β′

ξ′ ∂β′′

ξ′′ ∂β′′′

ξ′′′ a(x, ξ)| . (1 + |ξ|)m−|β′|−12|β′′|−|β′′′|.


Similarly we denote by Smanis(Rn−ν×Rn−ν) the class of symbols a(x′′, x′′′, ξ′′, ξ′′′)satisfying the estimates

|∂α(x′′,x′′′)∂β′′

ξ′′ ∂β′′′

ξ′′′ a(x′′, x′′′, ξ′′, ξ′′′)| . (1 + |ξ′′|+ |ξ′′′|)m−

12|β′′|−|β′′′|.

Finally, OPSmanis(Rn) (OPSmanis(Rn−ν) denotes the class of operators with sym-bols in Smanis(Rn×Rn) (resp. Smanis(Rn−ν×Rn−ν)), modulo smoothing operators.

We recall that operators in OPHm(Rn,Σ) map H tcomp(Rn) → H t−m

comp(Rn) contin-uously for any t ∈ R.

Proposition 2.3.2.

(1) If A = Op(f0 + f−1) ∈ OPHm(Rn−ν ,Σ′), with f−s ∈ Hm−s/2(Rn−ν × Rn−ν ,Σ′),s = 0, 1, then A∗ ∈ OPH∗m(Rn−ν ,Σ′) with

A∗ = Op

(f0 + f−1 + i

l∑j=1

∂2f0

∂ξ′′j ∂x′′j

)∗

mod OPH∗m−1(Rn−ν ,Σ′).

(2) If A = Op(f0 + f−1) ∈ OPHm(Rn−ν ,Σ′) and B = Op(g0 + g−1)∗ ∈

OPH∗m′(Rn−ν ,Σ′), with f−s ∈ Hm−s/2(Rn−ν×Rn−ν ,Σ′), g−s ∈ Hm′−s/2(Rn−ν×

Rn−ν ,Σ′), s = 0, 1, then AB ∈ OPHm+m′(Rn,Σ) and BA ∈ OPHm+m′

(Rn−ν ,Σ),with

AB = Op(e−i〈y,η〉f0(ρ, ξ

′′; y)g0(ρ, ξ′′; η))

mod OPHm+m′−1/2(Rn,Σ),

and, modulo OPHm+m′−1(Rn−ν ,Σ),

BA = Op

(∫Rν

(f0g0 + f−1g0 + f0g−1 − i

l∑j=1

∂g0

∂ξ′′j

∂f0

∂x′′j

)dy

).

(3) If A = Op(h) ∈ OPHm(Rn−ν ,Σ′) and B = Op(b) ∈ OPHm′(Rn−ν ,Σ), or

B ∈ OPSm′

anis(Rn−ν), then AB ∈ OPHm+m′(Rn−ν ,Σ′) with

AB = Op

(hb− i

l∑j=1

∂h

∂ξ′′j

∂b

∂x′′j

)mod OPHm+m′−1(Rn−ν ,Σ′).

(4) If P = Op(p) ∈ OPNm,k(Rn,Σ), with classical asymptotic expansion p ∼∑j≥0 pm−j, and A = Op(f0+f−1) ∈ OPHm′

(Rn−ν ,Σ′) with f−s ∈ Hm′−s/2(Rn−ν×


Rn−ν ,Σ′), s = 0, 1, then PA ∈ OPHm+m′−k/2(Rn−ν ,Σ′) and microlocally nearΣ we have,

PA = Op(p(k)(ρ, ξ′′, y,Dy)(f0 + f−1) + p(k+1)(ρ, ξ′′, y,Dy)f0 +

− il∑

j=1

∂p(k)

∂ξ′′j(ρ, ξ′′, y,Dy)

∂f0

∂x′′j

)mod OPHm+m′−k/2−1(Rn−ν ,Σ′),

where ρ = (x′′, ξ′′, ξ′′′) and for s = 0, 1 we set

p(k+s)(ρ, ξ′′, y,Dy) :=∑

|α|+|β|+|γ|+2j=k+s

1


βξ′∂


′′γyαDβy .

(5) If C = Op(c) ∈ OPSmanis(Rn) for a symbol c(y, x′′, x′′′, η, ξ′′, ξ′′′) satisfying∂αη c(0, x

′′, x′′′, 0, ξ′′, ξ′′′) = 0 for |α| = 1, and A = Op(h) ∈ OPHm′(Rn−ν ,Σ′),

then CA ∈ OPHm+m′(Rn−ν ,Σ′), with

CA = Op

(c|η=0h− i

l∑j=1

∂c

∂ξ′′j

∣∣∣∣η=0

∂h

∂x′′j

)mod OPHm+m′−1(Rn−ν ,Σ′).

(6) If A = Op(h) ∈ OPHm(Rn−ν ,Σ) and B = Op(b) ∈ OPSm′

anis(Rn−ν), thenAB ∈ OPHm+m′

(Rn−ν ,Σ) and

AB = Op(hb) mod OPHm+m′−1/2(Rn−ν ,Σ).

Proof. All the statements but the point (5) can be found in [19].In proving (5), we work microlocally near a point of Σ; hence we may suppose

that the symbols c and h are compactly supported with respect to x, and also that|ξ′| < C|ξ′′′| on the support of c. The symbol of CA is then given by

(2π)−n∫eizζc(x, ζ ′, ζ ′′ + ξ′′, ζ ′′′ + ξ′′′)h(x− z, ξ′′, ξ′′′)dz dζ = F−1

ζ′→x′b(x, ζ′, ξ′′, ξ′′′),

with

b(x, ζ ′, ξ′′, ξ′′′) = (2π)−n+ν

∫eiz

′′ζ′′+iz′′′ζ′′′c(x, ζ ′, ζ ′′ + ξ′′, ζ ′′′ + ξ′′′)

× h(ζ ′, x′′ − z′′, x′′′ − z′′′, ξ′′, ξ′′′) dz′′ dz′′′ dζ ′′dζ ′′′ =

(c(x′, ζ ′, ·)#h(ζ ′, ·))(x′′, x′′′, ξ′′, ξ′′′). (2.29)


Now it is easy to see that for any l ≥ 0 and ζ ′ ∈ Rν the Fourier transform of hsatisfies the following estimates:

|∂β′

ζ′ ∂αx′′,x′′′∂

β′′

ξ′′ ∂β′′′

ξ′′′ h(ζ′, x′′, x′′′, ξ′′, ξ′′′)| .

. C|ξ′′′|m′−ν/4− 12|β′|− 1

2|β′′|−|β′′′|

(1 +

|ζ ′||ξ′′′|1/2

+|ξ′′||ξ′′′|1/2

)−l, (2.30)

with β = (β′, β′′, β′′′). Moreover, we have

|∂αx∂βζ′,ξ′′,ξ′′′c(x, ζ

′, ξ′′, ξ′′′)| ≤ C|ξ′′′|m−|β′|−12|β′′|−|β′′′|, (2.31)

as it easily follows from the support properties of c if we Taylor expand c at ξ′′ = 0,ξ′′′ = 0, until to have a remainder term of order zero. Hence it follows from (2.29)that b satisfies the following estimates:

|∂αx∂βζ′,ξ′′,ξ′′′b(x, ζ

′, ξ′′, ξ′′′) ≤ C|ξ′′′|m+m′−ν/4− 12|β′|− 1

2|β′′|−|β′′′|

(1 +

|ζ ′||ξ′′′|1/2

+|ξ′′||ξ′′′|1/2

)−l,

whence the wanted estimates for the symbol of CA follow. This shows that the firstassertion in the statement holds, more in general, if in the symbol estimates for c thegain for any ξ′-derivative is just 〈ξ〉−1/2. To have the principal part of the symbol ofCA as given in (5) of Proposition 2.3.2, it is instead essential a gain 〈ξ〉−1 for anyξ′ derivative. Indeed, to obtain such a formula, we perform a Taylor expansion ofc at ζ ′ = ζ ′′ = ζ ′′′ = 0 in (2.29). By the previous part of the proof we see that theonly terms which at the end do not produce necessarily Hermite operators of orderm+m′ − 1 are given by

c(x, 0, ξ′′, ξ′′′) +l∑

j=1

ζ ′′j∂c

∂ξ′′j(x, 0, ξ′′, ξ′′′) +

ν∑r=1

ζ ′r∂c

∂ξ′r(x, 0, ξ′′, ξ′′′). (2.32)

On the other hand, by the hypothesis we have

∂c

∂ξ′r(x, 0, ξ′′, ξ′′′) =

ν∑k=1

x′kcrk(x, ξ′′, ξ′′′)

for suitable symbols crk of order m− 1. Since each factor x′k yields a gain of 1/2 inorder, the last sum in (2.32) is actually irrelevant. This concludes the proof.

Let us now come to the proof of Proposition 2.2.2. The following two lemmaswill be crucial in the proof.


Lemma 2.3.3. There exist a conic neighborhood U ⊂⊂ Σ of ρ0, δ > 0, and func-tions f0,j ∈ C∞(U−,δ,S(Rν)), j = 1, . . . , d, quasi-homogeneous of degree ν/4 withrespect to the dilation (x, ξ′′, ξ′′′) 7→ (t−1/2x′, x′′, x′′′, t1/2ξ′′, tξ′′′), t > 0, such thatf0,j(ρ, ξ

′′, ·)j=1,...,d is an orthonormal basis of V (ρ, ξ′′) and

either f0,j(ρ,−ξ′′,−y) = f0,j(ρ, ξ′′, y) ∀(ρ, ξ′′) ∈ U−,δ, ∀j = 1, . . . , d,

or f0,j(ρ,−ξ′′,−y) = −f0,j(ρ, ξ′′, y) ∀(ρ, ξ′′) ∈ U−,δ, ∀j = 1, . . . , d.

(2.33)

Proof. In view of Hypothesis (ii) in the statement of Theorem 1.2.2 we can finda neighborhood W of (ρ0, 0) and d smooth functions φj(ρ, ξ

′′, y) ∈ C∞(W,S(Rν)),j = 1, . . . , d, defining a basis for V (ρ, ξ′′) for every (ρ, ξ′′) ∈ W . We then restrictthe functions φj to the surface |ξ′′′| = |ξ′′′ρ0| and extend them to quasi-homogeneousfunctions f0,j ∈ C∞(U−,δ,S(Rν)) of degree ν/4, where U ⊂⊂ Σ is a suitable conicneighborhood of ρ0, and δ > 0 is small enough. By (2.21), these functions define abasis of V (ρ, ξ′′) for every (ρ, ξ′′) ∈ U−,δ.

We now show that replacing them with a suitable linear combination allows usto satisfy the desired orthogonality and parity properties. To this end, consider theinvolution j : L2(Rν) 3 f(y) 7→ f(−y) ∈ L2(Rν). It easily seen that

p(k)(ρ,−ξ′′, y,Dy) j = j p(k)(ρ, ξ′′, y,Dy),

which shows that the operators p(k)(ρ, ξ′′, y,Dy) and p(k)(ρ,−ξ′′, y,Dy) have the samespectrum. In particular

λ(ρ, ξ′′) = λ(ρ,−ξ′′) is even as a function of ξ′′. (2.34)

As another consequence, j induces a map

I(ρ, ξ′′) : V (ρ, ξ′′) → V (ρ,−ξ′′)

such that, by Hypothesis (iii) in the statement of Theorem 1.2.2,

either I(ρ0, 0) = Id or I(ρ0, 0) = −Id

accordingly to the parity of the eigenfunction in V (ρ0, 0). Therefore, given any basisf0,j(ρ, ξ

′′; y) ∈ C∞(U−,δ,S(Rν)), j = 1, . . . , d, of V (ρ, ξ′′), if U−,δ is small enoughthe functions

1

2(f0,j(ρ, ξ

′′; y)± f0,j(ρ,−ξ′′;−y))

(according to the sign of Id) gives rise to a basis for V (ρ, ξ′′) satisfying (2.33).Finally, we easily get the required orthonormal basis of V (ρ, ξ′′), by the Gram-

Schmidt procedure, which preserves parity. In fact, the entries of the matrix whichdescribes the change of basis are smooth functions of (ρ, ξ′′), even with respect toξ′′.


Remark 2.3.4. By (iii) of Theorem 1.2.2 the function λ(ρ, ξ′′) is smooth in U−,δ.

Consider now the equation

(p(k)(ρ, ξ′′, y,Dy)− λ(ρ, ξ′′))f−1,j = Bf0,j −d∑

k=1

〈Bf0,j, f0,k〉L2(Rνy)f0,k (2.35)

where

B := −p(k+1)(ρ, ξ′′, y,Dy) + i

l∑r=1

∂(p(k)(ρ, ξ′′, y,Dy)− λ(ρ, ξ′′))

∂ξ′′r

∂

∂x′′r.

Lemma 2.3.5. Possibly after shrinking U−,δ, there exist quasi-homogeneous smoothfunctions f−1,j ∈ C∞(U−,δ,S(Rν)) of degree ν/4− 1/2, solving Equations 2.35.

Proof. Since the right hand side in (2.35) is orthogonal to the kernel of p(k)(ρ, ξ′′, y,Dy)−λ(ρ, ξ′′)Id, the statement follows by the Fredholm theorem by using the homogeneityproperties of pk, pk+1, and λ(ρ, ξ′′).

In general, the functions f−1,j do not have any parity property.Let now ψ ∈ S0

1/2,0(Rn × Rn) be supported in U−,δε ; we can suppose that there

exists a real symbol ψ′(ρ, ξ′′) ∈ S0anis(Rn−ν × Rn−ν) supported in U−,δ, with the

following properties: ψ′ is quasi-homogeneous for |ξ′′| + |ξ′′′| large, 0 ≤ ψ′ ≤ 1,ψ′ = 1 in a neighborhood of suppψ as a function on Rn × Rn, and ψ′ = 0 for|ξ′′′| < 1/(2ε).We then define

Hj := Op(ψ′(f0,j + f−1,j)) ∈ OPH0(Rn−ν ,Σ′),

and

Π :=d∑j=1

HjH∗j ∈ OPH0(Rn,Σ).

Upon writingΨu = ΠΨu+ (Id− Π)Ψu,

one has(PΨu,Ψu

)=(PΠΨu,ΠΨu

)+ 2Re

(PΠΨu, (Id− Π)Ψu

)+

+(P (Id− Π)Ψu, (Id− Π)Ψu

). (2.36)

Now we separately estimate the three terms in the right hand side of (2.36).


2.3.1 Estimate of (PΠΨu,ΠΨu)

Let χ(ξ) ∈ S01,0(Rn × Rn) be a classical symbol such that

suppχ ⊂ξ ∈ Rn : |ξ′|+ |ξ′′| < 2 maxε, δ/

√2ε|ξ′′′|

,

and

χ = 1 inξ ∈ Rn : |ξ′|+ |ξ′′| < maxε, δ/

√2ε|ξ′′′| and |ξ′′′| > 1/(4ε)

.

We see that, in particular, χ = 1 in a neighborhood of suppψ, and χ(0, ξ′′, ξ′′′) = 1and ∂χ

∂ξ′j(0, ξ′′, ξ′′′) = 0 in a neighborhood of suppψ′.

By Proposition 2.3.2 we have, modulo Hm−k/2−1(Rn−ν × Rn−ν ,Σ′),

σ(PHj) ≡ ψ′(p(k)(ρ, ξ′′, y,Dy)f0,j + p(k)(ρ, ξ′′, y,Dy)f−1,j + p(k+1)(ρ, ξ′′, y,Dy)f0,j

− il∑

r=1

∂p(k)

∂ξ′′r(ρ, ξ′′, y,Dy)

∂f0,j

∂x′′r)− i

l∑r=1

∂ψ′

∂x′′r

∂p(k)

∂ξ′′r(ρ, ξ′′, y,Dy)f0,j

≡ ψ′λ(f0,j + f−1,j)− iψ′l∑

r=1

∂λ

∂ξ′′r

∂f0,j

∂x′′r−

d∑k=1

ψ′bjkf0,k

− il∑

r=1

∂ψ′

∂x′′r

∂p(k)

∂ξ′′r(ρ, ξ′′, y,Dy)f0,j

≡ σ(LHj −d∑

k=1

BjkHk) + r,

(2.37)

where

L = Op(χλΣ) ∈ OPSm−k/2anis (Rn), with λΣ = ψ′λ ∈ Sm−k/2anis (Rn−ν × Rn−ν),

Bjk = Op(χψ′bjk) ∈ OPSm−k/2−1/2anis (Rn), with bjk = 〈Bf0,j, f0,k〉L2(Rν

y),

and

r = (ψ′−ψ′2)

(λ(f0,j + f−1,j)− i

l∑r=1

∂λ

∂ξ′′r

∂f0,j

∂x′′r−

d∑k=1

bjkf0,k

)+ iλf0,j

l∑r=1

∂ψ′

∂ξ′′r

∂ψ′

∂x′′r

+ iλψ′l∑

r=1

∂ψ′

∂ξ′′r

∂f0,j

∂x′′r+ iψ′f0,j

l∑r=1

∂λ

∂ξ′′r

∂ψ′

∂x′′r− i

l∑r=1

∂ψ′

∂x′′r

∂p(k)

∂ξ′′r(ρ, ξ′′, y,Dy)f0,j.


As already observed, the function λ(ρ, ξ′′) is even with respect to the variables ξ′′;similarly one sees that the functions bjk(ρ, ξ

′′) above are odd. These parity propertieswill be crucially used in the sequel.

From (1) and (2) of Prop. 2.3.2 it follows that

σ(H∗jHk) ≡ δjkψ

′2 modulo H−1/2(Rn−ν ,Σ), (2.38)

and, since ψ′ ≡ 1 in a neighborhood of the support of ψ, one has

(PΠΨu,ΠΨu

)=(LΠΨu,ΠΨu

)−

d∑j,k=1

(BjkHkH

∗jΨu,ΠΨu

)+O

(‖u‖2

m/2−(k+2)/4

)=

=(LΠΨu,ΠΨu

)−

d∑j,k=1

(BjkHkH

∗jΠΨu,ΠΨu

)+O

(‖u‖2

m/2−(k+2)/4

)=

= Re(LΠΨu,ΠΨu

)− Re

d∑j,k=1

(BjkHkH

∗jΠΨu,ΠΨu

)+O

(‖u‖2

m/2−(k+2)/4

).

We have thus to prove the lower bound

Re

((L−

d∑j,k=1

BjkHkH∗j )ΠΨu,ΠΨu

)≥ −C‖u‖2

m/2−(k+2)/4, (2.39)

and this leads us to consider L −∑d

j,k=1BjkHkH∗j as a pseudodifferential operator

with operator-valued symbol in L(H), where H := L2(Rν) (L(H) stands for thespace of all linear bounded operators in H). Precisely, from Proposition 2.3.2 itfollows that

Re BjkHkH∗j = Op

(1

2ψ′

3(bjke

−i〈x′,ξ′〉f0,kf 0,j + bjke−i〈x′,ξ′〉f0,j f 0,k)

)mod OPHm−k/2−1(Rn,Σ), (2.40)

(for any operator Q, Re Q denotes its real part, i.e. Re Q = (Q∗ +Q)/2). Considernow the operator-valued symbol U−,δ 3 (ρ, ξ′′) 7−→ G(ρ, ξ′′) ∈ L(H) defined by

G(ρ, ξ′′)v =1

2

d∑j,k=1

(bjk(ρ, ξ

′′)〈v, f0,j(ρ, ξ′′; ·)〉Hf0,k(ρ, ξ

′′; ·)

+bjk(ρ, ξ′′)〈v, f0,k(ρ, ξ′′; ·)〉Hf0,j(ρ, ξ

′′; ·))∈ V (ρ, ξ′′), ∀v ∈ H,

and note that

G(x′′, x′′′, tξ′′′, t1/2ξ′′) = tm−k/2−1/2MtG(ρ, ξ′′)M−1t ,


where (Mtf)(y) = tν/4f(t1/2y). As a consequence, it is readily seen that

ψ′G ∈ Sm−k/2−1/21/2,0 (Rn−ν × Rn−ν ;L(H)). Hence (2.40) yields

Red∑

j,k=1

BjkHkH∗j = Op(ψ′

3G) mod OPHm−k/2−1(Rn,Σ),

= Re Op(ψ′3G) mod OPHm−k/2−1(Rn,Σ).

Since χ = 1, ψ′ = 1 in a neighborhood of suppψ, we have

Re

((L−

d∑j,k=1

BjkHkH∗j )ΠΨu,ΠΨu

)=

= Re(Op(ψ′(λIdH −G)

)ΠΨu,ΠΨu

)+O(‖u‖2

m/2−(k+2)/4).

(2.41)Therefore, inequality (2.39) follows if we prove that

Re(Op(ψ′(λIdH −G)

)ΠΨu,ΠΨu

)≥ −C‖u‖2

m/2−(k+2)/4. (2.42)

This will be accomplished in several steps. First of all, we shall show that, by (1.13)and by Hypothesis (iv) of Theorem 1.2.2, the operator-valued symbol ψ′(λIdH −G) ∈ S

m−k/21/2,0 (Rn−ν × Rn−ν ;L(H)) is nonnegative as a self-adjoint operator in H,

up to negligible terms (Step 1). Hence, since the operator is of type (1/2, 0), ifwe could apply the Fefferman-Phong inequality, estimate (2.42) would immediatelyfollow. Unfortunately, the Fefferman-Phong inequality is in general false for vector-valued operators as pointed out by Hormander [29] and Brummelhuis [6] (see alsoParmeggiani [58] and the references therein for a precise discussion about this fact).However, the symbol ψ′(λIdH − G) is actually of type (1, 0) with respect to thevariables x′′′, ξ′′′. Thus, by virtue of Hypothesis (iv) of Theorem 1.2.2, we will beable to apply the Morse lemma to the nonnegative function λ (with respect to thevariables ξ′′) and get

ψ′(λIdH −G) = Q+ “negligible”terms,

where Q ∈ Sm−k/21,0 (Rn−ν ×Rn−ν ;L(H)) is a non negative self-adjoint operator (Step2). Finally, an application of the Sharp Garding inequality (Theorem 18.6.14 of [30])to the operator Q will complete the proof of (2.42) (Step 3).

Step 1. Lower bound for the operator-valued symbol

Here we show that there exists C > 0 such that, for every (ρ, ξ′′) ∈ U−,δ,

〈(λ(ρ, ξ′′)IdH −G(ρ, ξ′′)

)v, v〉H ≥ −C|ξ′′′|m−k/2−1‖v‖2

H, ∀v ∈ H, (2.43)


provided U and δ are chosen sufficiently small. To this aim, we point out someproperties of the ground energy λ(ρ, ξ′′).Set ρ = (x′′, x′′′, ξ′′′/|ξ′′′|), ζ = ξ′′/|ξ′′′|1/2, and let A(ρ) := Hessξ′′λ(ρ, 0). In view ofHypothesis (iv) of Theorem 1.2.2, there exists a neighborhood V of ρ0 such that, forany ρ ∈ V , A(ρ) is invertible and, hence, positive definite because of (1.13). Sincethe lowest eigenvalue λ(ρ, ζ) is a nonnegative function which is even with respect toζ (see (2.34)), if δ is chosen small enough, one has

λ(ρ, ζ) ≥ c|ζ|2, ∀(ρ, ζ) ∈ V ×B(0, δ). (2.44)

As regards G(ρ, ξ′′) in (2.43), recall that bjk = 〈Bf0,j, f0,k〉H with j, k = 1, ..., d, andobserve that, for any given (x′′, x′′′, ξ′′′, ξ′′) ∈ U−,δ and any given t > 0,

bjk(x′′, x′′′, tξ′′′, t1/2ξ′′) = tm−k/2−1/2bjk(x

′′, x′′′, ξ′′′, ξ′′). (2.45)

Since f0,k(ρ, ξ′′; ·)j=1,...,d is an orthonormal basis of V (ρ, ξ′′), we have, for every

(ρ, ξ′′) ∈ U−,δ and every v ∈ H,

|ξ′′′|−(m−k/2−1/2)|〈G(ρ, ξ′′)v, v〉H| ≤d∑

j,k=1

|bjk(ρ, ζ)|‖v‖2H. (2.46)

For all j, k = 1, ..., d, one has bjk(ρ,−ζ) = −bjk(ρ, ζ), whence it turns out thatbjk(ρ, 0) = 0 and an application of the Taylor formula yields

bjk(ρ, ζ) =l∑

s=1

bjk,s(ρ, ζ)ζs, (2.47)

for suitable smooth functions bjk,s ∈ C∞(V ×B(0, δ)). By using (2.47) in (2.46) weobtain

|ξ′′′|−(m−k/2−1/2)|〈G(ρ, ξ′′)v, v〉H| ≤ C|ζ|‖v‖2H. (2.48)

Finally, from (2.44) and (2.48) it follows that, for any (ρ, ξ′′) ∈ U−,δ and everyv ∈ H,

|ξ′′′|−(m−k/2)(λ(ρ, ξ′′)‖v‖2

H − 〈G(ρ, ξ′′)v, v〉H)≥(c|ζ|2 − C|ξ′′′|−1/2|ζ|

)‖v‖2

H

≥ −C2

4c|ξ′′′|−1‖v‖2

H.

This proves (2.43).

Step 2. Application of the Morse lemma


In view of Hypothesis (iv) of Theorem 1.2.2, we can apply the Morse Lemma tothe function λ(ρ, ξ′′) (see, for example, Lemma C.6.1 of [30]); as a result, thereexists a neighborhood V ⊂ Σ (not conic in general) of ρ0 = (x′′ρ0 , x

′′′ρ0, ξ′′′ρ0) and

φ ∈ C∞(V,B(0, r)), B(0, r) ⊂ Rl, with φ(ρ0) = 0, such that

λ(ρ, ξ′′) = λ(ρ, φ(ρ)) +l∑

j=1

cj(ρ, ξ′′)2, ∀ρ ∈ V, ∀ξ′′ ∈ B(0, r), (2.49)

for suitable independent real functions cj ∈ C∞(V ×B(0, r)), j = 1, . . . , l. By virtueof (2.22), if we replace the functions φ and cj by

(|ξ′′′|/|ξ′′′ρ0|)1/2φ(x′′, x′′′, |ξ′′′ρ0|ξ

′′′/|ξ′′′|)

(|ξ′′′|/|ξ′′′ρ0|)m/2−k/4cj(x

′′, x′′′, |ξ′′′ρ0|ξ′′′/|ξ′′′|, |ξ′′′ρ0|

1/2ξ′′/|ξ′′′|1/2),

possibly after shrinking U−,δ, we get (2.49) for every (ρ, ξ′′) ∈ U−,δ, where φ and cjare now homogeneous of degree 1/2 and quasi-homogeneous of degree m/2 − k/4,respectively.

From (2.49) it follows that cj(ρ, ξ′′) = 0, j = 1, ..., l, are local equations of the

manifold ξ′′ = φ(ρ). Therefore an application of the Taylor formula at ξ′′ = φ(ρ)yields, for every (ρ, ξ′′) ∈ U−,δ,

bjk(ρ, ξ′′) = bjk(ρ, φ(ρ)) + 2

l∑j=1

cj(ρ, ξ′′)b′jk,s(ρ, ξ

′′) (2.50)

where by (2.45) we can assume that b′jk,s ∈ C∞(U−,δ) are quasi-homogeneous ofdegree m/2− k/4− 1/2 for any j, k = 1, ..., d and any s = 1, ..., l. Upon defining

Gs(ρ, ξ′′)v =

1

2

d∑j,k=1

(b′jk,s(ρ, ξ

′′)〈v, f0,j(ρ, ξ′′; ·)〉Hf0,k(ρ, ξ

′′; ·)

+b′jk,s(ρ, ξ′′)〈v, f0,k(ρ, ξ

′′; ·)〉Hf0,j(ρ, ξ′′; ·)), ∀v ∈ H,

it turns out that ψ′Gs ∈ Sm/2−k/4−1/21/2,0 (Rn−ν × Rn−ν ;L(H)) and that, for every

(ρ, ξ′′) ∈ U−,δ,

G(ρ, ξ′′) = G(ρ, φ(ρ)) + 2l∑

s=1

cs(ρ, ξ′′)Gs(ρ, ξ

′′), (2.51)


whence, for (ρ, ξ′′) ∈ U−,δ, one has

λ(ρ, ξ′′)IdH −G(ρ, ξ′′) = λ(ρ, φ(ρ))IdH −G(ρ, φ(ρ))︸︷︷︸A1(ρ)

+

+l∑

s=1

(cs(ρ, ξ

′′)IdH −Gs(ρ, ξ′′))2

︸︷︷︸A2(ρ,ξ′′)

−l∑

s=1

Gs(ρ, ξ′′)2

︸︷︷︸A3(ρ,ξ′′)

. (2.52)

Step 3. End of the proof of (2.42)

Without loss of generality, we assume that m ≥ k/2 + 1. Since ψ′A1, ψ′A2 ∈

Sm−k/21/2,0 (Rn−ν × Rn−ν ;L(H)) and ψ′2A3 ∈ S

m−k/2−11/2,0 (Rn−ν × Rn−ν ;L(H)), by (2.52)

we obtain


)ΠΨu,ΠΨu

)= Re

(Op(ψ′A1)ΠΨu,ΠΨu

)+Re

(Op(ψ′

2A2)ΠΨu,ΠΨu

)+O(‖u‖2

m/2−(k+2)/4) (2.53)

where Π,Ψ = ψ(x,D) are here regarded as operators in OPS01/2,0(Rn−ν ;L(H)).

We now observe that

Op(ψ′2A2) =

l∑s=1

K∗sKs + iQs +Rs,

withKs = Op(ψ′csIdH − ψ′Gs) ∈ OPS

m/2−k/41/2,0 (Rn−ν ;L(H)),

Rs ∈ OPSm−k/2−11/2,0 (Rn−ν ;L(H)) and Qs = Q∗

s ∈ OPSm−k/2−1/21/2,0 (Rn−ν ;L(H)) for any

s = 1, ..., l. As a result, we get

Re(Op(ψ′(λIdH−G)

)ΠΨu,ΠΨu

)≥ Re

(Op(ψ′A1)ΠΨu,ΠΨu

)+O(‖u‖2

m/2−(k+2)/4).

Now we choose a positive scalar microlocalizer χ′ ∈ S01,0(Rn−ν ×Rn−ν) supported in

(ρ, ξ′′) : ρ ∈ U, |ξ′′| < R|ξ′′′|, |ξ′′′| ≥ 1/R, for some large R > 0, and such thatχ′ = 1 in a neighborhood of suppψ′. Then, we get


)ΠΨu,ΠΨu

)≥ Re

(Op(χ′ψ′A1)ΠΨu,ΠΨu

)+O(‖u‖2

m/2−(k+2)/4)

= Re(Op(χ′A1)ΠΨu,ΠΨu

)+O(‖u‖2

m/2−(k+2)/4).

(2.54)


Finally, from (2.43) there exists a positive constant C > 0 such that

0 ≤ χ′A1 + C〈(ξ′′, ξ′′′)〉m−k/2−1IdH ∈ Sm−k/21,0 (Rn−ν × Rn−ν ;L(H))

as a self-adjoint operator, hence an application of the Sharp Garding inequalityfor vector valued pseudodifferential operators completes the proof of (2.42)(see forexample [30], Theorem 18.6.14).

2.3.2 Estimate of (P (Id− Π)Ψu, (Id− Π)Ψu)

In this section we prove that (P (Id − Π)Ψu, (Id − Π)Ψu) satisfies a better lowerbound that will be crucial in treating the last term Re (PΠΨu, (Id− Π)Ψu).

Let T := Op((1 + |ξ′′|2 + |ξ′′′|2)m/4−k/8

)∈ OPS

m/2−k/41,0 (Rn−ν) and set Hj := HjT ∈

OPHm/2−k/4(Rn−ν ,Σ′) (hence H∗j = T ∗H∗

j ), Π :=∑d

k=1 HkH∗k ∈ OPHm−k/2(Rn,Σ).

We start off by showing that(P (Id− Π)Ψu, (Id− Π)Ψu

)=(

(P + Π)(Id− Π)Ψu, (Id− Π)Ψu)

+O(‖u‖2

m/2−(k+2)/4

). (2.55)

Indeed, upon denoting Kkj := δkjT∗ψ′2(x,D)− H∗

kHj ∈ OPHm/2−k/4−1/2(Rn−ν ,Σ),one has

(Id− Π)Π(Id− Π)Ψ =d∑

k=1

(Id− Π)HkH∗k(Id−

d∑j=1

HjH∗j )Ψ

=d∑

k=1

(Id− Π)(HkH∗k −

d∑j=1

HkH∗kHjH

∗j )Ψ

=d∑

k,j=1

(Id− Π)(HkKkjH∗j )Ψ mod OPHm−k/2−1(Rn,Σ)

=d∑

k,j,l=1

(HkKkjH

∗j −HlH

∗l HkKkjH

∗j

)Ψ mod OPHm−k/2−1(Rn,Σ),

whence (2.55) readily follows, for

H∗l Hk = δlkψ

′2(x,D)T ∗ mod OPHm/2−k/4−1/2(Rn−ν ,Σ).

Since the “localized operator” of Π is a sort of projection onto V (ρ, ξ′′), the localizedoperator of P +Π is strictly positive in L2(Rν), hence, it turns out that, for suitableconstants c, C > 0,

((P + Π)(Id−Π)(Ψu), (Id−Π)(Ψu)) ≥ c‖(Id−Π)(Ψu)‖2m/2−k/4 −C‖u‖2

m/2−(k+2)/4.(2.56)


This inequality is easily proved by applying Proposition 2.1.3 to the following oper-ator

B := P + Π + Re(Op(χ′(1− ψ′))Λm−k/2

).

Indeed, in view of (2) of Proposition 2.3.2, b(k)(ρ, ξ′′, x′, Dx′) is given by

p(k)(ρ, ξ′′;x′, Dx′)+ψ′0(ρ, ξ

′′)|ξ′′′|m−k/2Opx′,ξ′

(d∑j=1

e−i〈x′,ξ′〉f0,j(ρ, ξ

′′;x′)f0,j(ρ, ξ′′; ξ′)

)+

+(1−ψ′0(ρ, ξ′′))|ξ′′′|m−k/2IdL2(Rν)

= p(k)(ρ, ξ′′;x′, Dx′)+ψ′0(ρ, ξ

′′)|ξ′′′|m−k/2ProjV (ρ,ξ′′)+(1−ψ′0(ρ, ξ′′))|ξ′′′|m−k/2IdL2(Rν).

where ψ′0 ∈ C∞(Rn−ν × Rn−ν \ 0) is the quasi-homogeneous function equal to ψ′

when |ξ′′|+ |ξ′′′| is large.It is straightforward to check that b(k)(ρ, ξ′′;x′, Dx′) is a sum of nonnegative opera-tors. Moreover, since the operators p(k)(ρ, ξ′′;x′, Dx′) and ProjV (ρ,ξ′′) have orthogonal

kernels, b(k)(ρ, ξ′′;x′, Dx′) is injective for every (ρ, ξ′′) ∈ Σ × Rl. Therefore we canapply Proposition 2.1.3 to the operator B and finally get (2.56). In view of (2.55)we hence obtain

(P (Id− Π)Ψu, (Id− Π)Ψu) ≥ c‖(Id− Π)Ψu‖2m/2−k/4 − C‖u‖2

m/2−(k+2)/4.

2.3.3 Estimate of 2Re (PΠΨu, (Id− Π)Ψu)

Let LΣ := Op(λΣ) ∈ OPSm−k/2anis (Rn−ν) (where, recall, λΣ = ψ′λ). The computations

in (2.37) show that

(PΠΨu, (Id− Π)Ψu

)=

d∑k=1

((Id− Π)(HkLΣ +Rk)H

∗kΨu,Ψu

),

with Rk ∈ OPHm−k/2−1/2(Rn−ν ,Σ′). Now by (1) and (2) of Proposition 2.3.2 wehave

σ(H∗jHk) = ψ′

2(δjk + qjk) + rjk mod H−1(Rn−ν × Rn−ν ,Σ), (2.57)

where the symbols rjk ∈ H−1/2(Rn−ν ×Rn−ν ,Σ) vanish in a neighborhood of suppψ(when regarded as functions on Rn × Rn), and the functions

qjk :=

∫Rν

(f−1,jf0,k + f 0,jf−1,k − if0,k

l∑r=1

∂2f 0,j

∂ξ′′r∂x′′r

− il∑

r=1

∂f 0,j

∂ξ′′r

∂f0,k

∂x′′r

)dy


are quasi-homogeneous of degree −1/2 and odd with respect to ξ′′. Hence, by (2.57)and by (6) of Proposition 2.3.2 it turns out that(PΠΨu, (Id− Π)Ψu

)=

=d∑

k=1

((HkLΣH

∗k −

d∑j=1

HjH∗jHkLΣH

∗k)Ψu,Ψu

)+

d∑k=1

(RkH

∗kΨu, (Id− Π)Ψu

)=

= −d∑

k,j=1

(HjCjkH

∗kΨu,Ψu

)+

d∑k=1

(RkH


)+O

(‖u‖2

m/2−(k+2)/4

),

where Cjk = Op(ψ′3λqjk) ∈ OPHm−k/2−1/2(Rn−ν ,Σ).

Setting Cjk = Op(χψ′3cjk) ∈ OPSm−k/2−1/2anis (Rn), by (5) of Proposition 2.3.2 and by

(2.57), we thus get(PΠΨu, (Id− Π)Ψu

)=

= −d∑

k,j=1

(CjkHjH

∗kΨu,Ψu

)+

d∑k=1

(RkH

∗kΨu, (Id−Π)Ψu

)+O

(‖u‖2

m/2−(k+2)/4

)=

= −d∑

k,j=1

(CjkHjH

∗kΠΨu,Ψu

)+

d∑k=1

(RkH


)+O

(‖u‖2

m/2−(k+2)/4

)=

= −d∑

k,j=1

(CjkHjH

∗kΠΨu,ΠΨu

)︸︷︷︸

(I)

+(RΨu, (Id− Π)Ψu

)︸︷︷︸(II)

+O(‖u‖2

m/2−(k+2)/4

),

(2.58)

where R := −∑d

k,j=1 CjkHjH∗kΠ +

∑dk=1RkH

∗k ∈ OPHm−k/2−1/2(Rn,Σ). In order

to estimate the term (I) in (2.58), we can replace Bjk by Bjk + Ckj in (2.39) andrepeat the same arguments used in Section 2.3.1 (Step 1− 3). In fact, the functionsλ(ρ, ξ′′)qjk(ρ, ξ

′′), as well as the bjk(ρ, ξ′′) considered above, are quasi-homogeneous

of degree m− k/2− 1/2 (see (2.45)) and odd with respect to ξ′′.As for (II), we observe that, for any ε > 0, one has

|(RΨu, (Id− Π)Ψu

)| ≤ ε‖(Id− Π)Ψu‖2

m/2−k/4 +1

ε‖RΨu‖2

−m/2+k/4 =

= ε‖(Id− Π)Ψu‖2m/2−k/4 +O

(‖u‖2

m/2−(k+2)/4

).

(2.59)

Hence, if ε > 0 is small enough, this term can be re-absorbed in view of (2.55) and(2.56). This concludes the proof of Proposition 2.2.2.


2.4 Concluding remarks

2.4.1 Some further remarks on localized operators

As promised in the Introduction, we show that, in checking the hypotheses of The-orem 1.2.2, we do not need at all to know explicitly a canonical flattening of Σ nearevery ρ0 ∈ Σ.

Fixed any ρ0 ∈ Σ, starting from the local equations of Σ we construct a dif-ferential operator Pρ,ζ′′ , smoothly depending on ρ in a neighborhood V ⊂ Σ of ρ0

and ζ ′′ ∈ Rl, unitarily equivalent to a localized operator Pχ,χ(ρ),ζ′′ for a particularcanonical flattening χ of Σ near ρ0. Precisely, we get a suitable parametrizationψ : V × R2ν+l

z′,ζ′;ζ′′ −→ NΣ|V of NΣ|V and define Pρ,ζ′′ as the Weyl quantization of

p(k) ψ in the variables (z′, ζ ′) (see (1.6)).In what follows we assume that Σ has codimension 3 with local equations of V givenby φ1 = φ2 = φ3 = 0. Similar arguments work as well if Σ has higher codimension.In view of Hypotheses (H1) and (H2) in Section 1.2, the rank of the symplectic2−form is constant on Σ and is equal to 0 or to 2. In the first case, Σ is an invo-lutive manifold and we refer to [53] for the explicit form of the localized operator.Here we treat the case rkσ|Σ = 2. Hence, without loss of generality, we can assumethat φ1, φ2(ρ) 6= 0 for any ρ ∈ V . For every ρ ∈ V , we perform the followingchange of local equations of Σ

φ′1 =1

φ1, φ2(ρ)φ1, φ′2 = φ2, φ′3 =

φ2, φ3(ρ)φ1, φ2(ρ)

φ1 +φ3, φ1(ρ)φ1, φ2(ρ)

φ2 + φ3

and define the following vector field in TT ∗Rn (smoothly depending on the parameterρ ∈ V )

X =σ(∇φ′3, Hφ′2

)(ρ)

‖∇φ′3(ρ)‖2Hφ′1

+σ(Hφ′1

,∇φ′3)(ρ)

‖∇φ′3(ρ)‖2Hφ′2

− 1

‖∇φ′3(ρ)‖2∇φ′3,

(here Hf and ∇f denote the Hamiltonian and the gradient of the smooth functionf , respectively). An easy check shows that

σ(Hφ′1

, Hφ′2

)(ρ) = σ

(Hφ′3

, X)(ρ) = 1,

σ(Hφ′3

, Hφ′1

)(ρ) = σ

(Hφ′3

, Hφ′2

)(ρ) = σ

(Hφ′1

, X)(ρ) = σ

(Hφ′2

, X)(ρ) = 0.

We then setψ(ρ, z1, ζ1, ζ2) = [z1Hφ′1

(ρ)− ζ1Hφ′2(ρ)− ζ2X(ρ)],

([v] being the residue class of v ∈ TρT ∗X in NρΣ) so that

p(k)(ψ(ρ, z1, ζ1, ζ2)

)=

∑α+β+γ+2j=k

1

α!β!γ!

(Hαφ′1Hβφ′2Xγqm−j

)(ρ)zα1 (−ζ1)β(−ζ2)γ.

2.4. CONCLUDING REMARKS 55

Upon defining Pρ,ζ2 as the Weyl quantization of p(k)(ψ(ρ, z1, ζ1, ζ2)

)with respect to

the variables (z1, ζ1), one can easily show that Pρ,ζ2 is unitarily equivalent to thelocalized operator Pχ,χ(ρ),ζ2 , for a suitable canonical flattening χ of Σ near ρ0 ∈ Σ.

2.4.2 Two counterexamples

We want to show, by means of two simple examples, that each of Hypotheses (iii)and (iv) is essential in Theorem 1.2.2.

First of all, we assume that Σ is given by the equations x′ = ξ′ = ξ′′ = 0, ξ′′′ 6= 0.The following result is a necessary condition that will be crucial in the sequel.

Proposition 2.4.1. Suppose that the total symbol p(x, ξ) of P does not depend onthe (x′′, x′′′)-variables and that P satisfies the following inequality: For every ε > 0,µ < m/2− (k + 1)/4, K ⊂⊂ Rn there exists Cε,µ,K such that

(Pu, u) ≥ −ε‖u‖2m/2−(k+1)/4 − Cε,µ,K‖u‖2

µ, ∀u ∈ C∞0 (K). (2.60)

Then, one has

λ(ρ, ξ′′) = 0 =⇒ (p(k+1)(ρ, ξ′′;x′, Dx′)u, u) ≥ 0, ∀u ∈ Ker p(k)(ρ, ξ′′;x′, Dx′),(2.61)

where, as usual,

p(k+1)(ρ, ξ′′;x′, Dx′) =∑

|α|+|β|+|γ|+2j=k+1

1


βξ′∂


′′γx′αDβx′ .

Proof. Let χ ∈ C∞0 (Rn) be a real valued function, identically equal to 1 in a neigh-

borhood of (0, x′′0, x′′′0 ), with ρ0 = (0, x′′0, x

′′′0 , 0, 0, ξ

′′0 ). We then observe that the

operator χPχ satisfies the following global version of (2.60):For every ε > 0, µ < m/2− (k + 1)/4, there exists Cε,µ such that

(χPχu, u) ≥ −ε‖u‖2m/2−(k+1)/4 − Cε,µ‖u‖2

µ, ∀u ∈ S(Rn). (2.62)

Let now v ∈ Ker p(k)(ρ0, ξ′′0 , x

′, Dx′) and let φ ∈ S(Rn−ν) such that ‖φ‖0 = 1. Weset u = φv ∈ S(Rn) and ut(x) = eit

2〈x,ξ0〉u(tx′, t1/2(x′′ − x′′0), t1/2(x′′′ − x′′′0 )), x =

(x′, x′′, x′′′) ∈ Rν × Rl × Rn−ν−l.A direct computation yields, for any s ∈ R

‖ ut ‖2s= t4s−(n+ν)/2

(‖ v ‖2

L2(Rν) +o(1)), as t→ +∞.

Moreover, since the total symbol p does not depend on the (x′′, x′′′)-variables, onegets

Put(x) = eit2〈x,ξ0〉ψt

(tx′, t1/2(x′′ − x′′0), t

1/2(x′′′ − x′′′0 ))


with

ψt(y) = (2π)−n∫ei〈y,η〉p

(y′t, x′′0, x

′′′0 , tη

′, t1/2η′′ + t2ξ′′0 , t1/2η′′′ + t2ξ′′′0

)u(η)dη.

An application of the Taylor expansion gives

p(y′t, x′′0, x

′′′0 , tη

′, t1/2η′′ + t2ξ′′0 , t1/2η′′′ + t2ξ′′′0

)= t4(m/2−k/4)p(k)(ρ0, ξ

′′0 , y

′, η′)

+t4(m/2−(k+1)/4)p(k+1)(ρ0, ξ′′0 , y

′, η′) + o(t4(m/2−(k+1)/4)).

Hence,

(χPχut, ut) = t4(m/2−k/4)−(n+ν)/2(p(k)(ρ0, ξ′′0 , x

′, Dx′)v︸︷︷︸=0

, v)

+ t4(m/2−(k+1)/4)−(n+ν)/2(p(k+1)(ρ0, ξ′′0 , x

′, Dx′)v, v) + o(t4(m/2−(k+1)/4)−(n+ν)/2)

≥ −εt4(m/2−(k+1)/4)−(n+ν)/2(‖v‖20 + o(1))− Cε,µt

4µ−(n+ν)/2(‖v‖20 + o(1)).

Upon dividing by t4(m/2−(k+1)/4)−(n+ν)/2 and letting t → +∞, ε → 0+, we concludethe proof.

Remark 2.4.2. If the total symbol of P depends on all the x-variables, Proposition2.4.1 is, in general, false. However, by using the localization functionut(x) = eit

2〈x,ξ0〉u(t(x − x0)) with ρ0 = (x0, ξ0) ∈ Σ and by arguing as above, itturns out that the following differential operator in Rn

∑|α|+|β|+2j=k+1

1

α!β!(∂αx∂

βξ pm−j)(ρ0)x

αDβ

is nonnegative if it is restricted to the space Ker p(k)(ρ0, ξ′′0 , x

′, Dx′)⊗ S(Rn−ν).

We can now construct the two counterexamples. Recall that k in an even integer.As regards Hypothesis (iv), consider the differential operator P = P1 + P2 in R3,where

P1 = (D2x1

+ x21D

2x3−Dx3)

k/2, P2 = Dkx2−Dk/2−1

x3Dx2 ,

and note that P = P ∗ ∈ OPNk,k(R3,Σ) with Σ = x1 = ξ1 = ξ2 = 0, ξ3 6= 0.Furthermore, the localized operator of P at ρ = (x2, x3, ξ3) is given by

p(k)(ρ, ξ2;x1, Dx1) = (D2x1

+ x21ξ

23 − ξ3)

k/2 + ξk2 ,

whence λ(ρ, ξ2) = (|ξ3|−ξ3)k/2 +ξk2 ≥ 0 and λ(ρ, ξ2) = 0 if and only if ξ2 = 0, ξ3 > 0.Thus, Hypotheses (1.13) and (i) − (iii) are satisfied, whereas Hypothesis (iv) does


not hold if k > 2. Finally, P does not verify (2.61) (and therefore (1.14)) since

p(k+1)(ρ, ξ2;x1, Dx1) = −ξk/2−13 .

As for Hypothesis (iii), set k = 4k1 + 2k2 for suitable k1 ∈ N, k2 ∈ 0, 1, andconsider the differential operator P = P1 +P2 in R4 where P1 = Dk

x3+D2

x3Dk−2x4

and

P2 = p2(x,D) = (D2x1

+x21D

2x4

+D2x2

+x22D

2x4

)k2(2(D2x1

+x21D

2x4

)+D2x2

+x22D

2x4−7Dx4)

2k1

+ x1x22D

k/2+1x4

.

We have P = P ∗ ∈ OPNk,k(R4,Σ) with Σ = x1 = ξ1 = x2 = ξ2 = ξ3 = 0, ξ4 6= 0.Fixed ρ = (x3, x4, ξ4), it turns out that

p(k)(ρ, ξ3;x1, x2, Dx1 , Dx2) = ξk3 + ξk−24 ξ2

3

+ (D2x1

+ x21ξ

24 +D2

x2+ x2

2ξ24)k2(2(D2

x1+ x2

1ξ24) +D2

x2+ x2

2ξ24 − 7ξ4)

2k1 ,

and thatp(k+1)(ρ, ξ3;x1, x2, Dx1 , Dx2) = x1x

22ξk/2+14 ,

as a multiplication operator.Now, it is easily seen that the spectrum of p(k)(ρ, ξ3, x

′, Dx′) is given by

Spec p(k)(ρ, ξ3, x′, Dx′) = F (ρ, ξ3, β); β ∈ Z2

+,

with

F (ρ, ξ3, β) := ξ23 + ξk−2

4 ξ23 + |ξ4|k2(2|β|+ 2)k2((4β1 + 2β2 + 3)|ξ4| − 7ξ4)

2k1 .

Setting

φh(t) := π−1/4(2hh!)−1/2

(d

dt− t

)he−t

2/2, h = 0, 1, . . . ,

for the h-th Hermite function, it is immediate to check that the function

φβ1(|ξ4|1/2x1)φβ2(|ξ4|1/2x2)

is an eigenfunction for p(k)(ρ0, ξ3, x′, Dx′) corresponding to the eigenvalue F (ρ0, ξ3, β).

Since tensor products of Hermite functions define a Hilbert basis for L2(R2), we ob-tain in this way the whole spectrum of p(k)(ρ0, ξ3, x

′, Dx′). In particular, we seethat when ξ3 6= 0 or ξ4 < 0 the localized operator p(k)(ρ0, ξ3, x

′, Dx′) is invertible,whereas when ξ3 = 0 and ξ4 > 0 its lowest eigenvalue is zero. In this case we haveKer p(k)(ρ0, 0, x

′, Dx′) = span u1, u2 where u1(x1, x2) = φ0(ξ1/24 x1)φ2(ξ

1/24 x2) and

u2(x1, x2) = φ1(ξ1/24 x1) φ0(ξ

1/24 x2), so that (1.13) is satisfied, as well as Hypotheses

(i), (ii) and (iv). However, (iii) is not fulfilled because u1 is even and u2 is odd.


Let us now verify that estimate (2.61) (and therefore (1.14)) is not satisfied.By taking u := u1 + u2, we have

(p(k+1)(ρ0, 0, x′, Dx′)u, u) = 2ξ

k/2+14

∫x1x

22u1(x1, x2)u2(x1, x2) dx1dx2 =

= 2ξk/2−3/24

∫x1φ0(x1)φ1(x1) dx1

∫x2

2φ0(x2)φ2(x2) dx2 < 0,

since ∫x1φ0(x1)φ1(x1) dx1 = −π−1/221/2

∫x2

1e−x2

1 dx1 < 0,

and∫x2

2φ0(x2)φ2(x2) dx2 = (2π)−1/2

∫x2

2(2x22 − 1)e−x

22 dx2 =

= 2(2π)−1/2

∫e−x

22 dx2 > 0.

This contradicts (2.60).

2.4.3 Lower bounds in absence of Levi-type conditions

Here we study the lower bounds (Is) in the Introduction, namely

(Pu, u)L2(Rn) ≥ −CK‖u‖2m−s

2, ∀u ∈ C∞

0 (K), (2.63)

s ≥ 1, for a classical operator P = P ∗ ∈ OPSm(X) with multiple characteristicswhich does not satisfy necessarily the Levi-type conditions (1.4). We assume thatthe characteristic set Σ is a smooth submanifold, but we do not suppose that σ hasconstant rank on Σ. Moreover the principal symbol is assumed to vanish exactly toeven order k > 2 on Σ.

We emphasize that when k > 2 we have Tr+(Fx,ξ) ≡ 0, and therefore the secondcondition in (1.3) reduces to

psm−1 ≥ 0 on Σ. (2.64)

(Do not confuse the “s” in psm−1 with the gain of derivatives!).Let us begin with the following result ([51]).

Theorem 2.4.3. Let P = P ∗ ∈ OPSm(X) be a classical pseudodifferential operatorwith pm vanishing exactly to even order k > 2 on Σ smooth manifold. Inequality(2.63) holds with s = k/(k − 1) if and only if

pm(x, ξ) ≥ 0 ∀(x, ξ) ∈ T ∗X \ 0,

psm−1(x, ξ) ≥ 0 ∀(x, ξ) ∈ Σ.(2.65)


Since conditions (1.3) are necessary for estimate (2.63) to hold with s > 1, itis clear that conditions (2.65) in Theorem 2.4.3 are necessary when k > 2. Thefact that they are also sufficient will follow from a more general result (Theorem2.4.4 below) where also the contribution of the higher order terms in the Taylorexpansion of psm−1 at Σ is taken into account. The proof is based on the Fefferman-Phong inequality ([15],[30],[2]). For k = 2, our argument will give that (2.65) issufficient for the maximum gain s = 2; with respect to Hormander’s inequality, theassumptions on Σ are weaker, but (2.65) is of course stronger than (1.3) in this case.At the end of this section we give two examples: the first shows that our resultsare sharp, in particular the values of s in Theorem 2.4.3 and Theorem 2.4.4 belowcannot be improved in general; the second example outlines possible applications tothe Cauchy problem. Finally we refer to [12] for a generalization of Theorem 2.4.3to symbols with low regularity.

Let us now state the above mentioned result.Consider the vector bundle TT ∗X|Σ on Σ, and ρ ∈ Σ. Given U ∈ TρT

∗X, weconsider any vector field Ξ on T ∗X with Ξ(ρ) = U . Then we define the smoothmaps

Ij(ρ, U) =1

j!Ξ · · ·Ξ︸︷︷︸j times

psm−1(ρ) : TT ∗X|Σ → R. (2.66)

These maps have an invariant meaning for 0 ≤ j ≤ k− 2, see for example [39], [42].

Theorem 2.4.4. Let P = P ∗ ∈ OPSm(X) be a classical pseudodifferential operatorwith pm vanishing exactly to even order k ≥ 2 on Σ smooth manifold. Let J ∈0, 1, . . . , k/2− 1. Let us suppose

pm ≥ 0 on T ∗X \ 0, (2.67)

and that for every ρ ∈ Σ there exists a neighborhood V ⊂ TT ∗X|Σ of (ρ, 0) such that

J∑j=0

Ij(ρ, U) ≥ 0, ∀(ρ, U) ∈ V. (2.68)

Then for every K ⊂⊂ X there exists a constant CK > 0 such that

(Pu, u) ≥ −CK‖u‖2m−s

2, ∀u ∈ C∞

0 (K), (2.69)

with s = k/(k − J − 1).

Remark 2.4.5. For J = 0 conditions (2.67) and (2.68) reduce to (2.65), so that weobtain Theorem 2.4.3 as a particular case. We have 1 < s ≤ 2 and the maximumgain of derivatives s = 2 is obtained for J = k/2 − 1. Moreover, s decreases andtends to 1 as k → +∞, for fixed J . Also, observe that (2.68) is certainly satisfied ifpsm−1 vanishes to the order J + 1 on Σ, in fact in this case Ij = 0, for j = 0, . . . , J .


For the proof of Theorem 2.4.4, we will need the following elementary result.

Lemma 2.4.6. Let p(ζ) =∑

|α|=k cαζα be a homogeneous real polynomial in Rl,

with p(ζ) > c|ζ|k for ζ 6= 0, c > 0, and q(ζ) a homogeneous real polynomial of degrees ∈ 1, . . . , k − 1. Then there exists a constant C > 0 such that for every t > 0 wehave

minζ∈Rl

p(ζ) + tq(ζ) ≥ −Ctk

k−s .

If the coefficients of p and q vary in a bounded subset of R but the constant c isuniform, so is C.

Proof of Lemma 2.4.6. We have

p(ζ) + tq(ζ) ≥ c|ζ|k − C1t|ζ|s

= |ζ|s(c|ζ|k−s − C1t);

so p(ζ) + tq(ζ) ≥ 0 when |ζ| ≥ (C1t/c)1

k−s . Hence, since p(0) = q(0) = 0,

minζ∈Rl

p(ζ) + tq(ζ) = min|ζ|≤(C1t/c)

1k−s

p(ζ) + tq(ζ)

≥ −C1t(C1t/c)s

k−s

= −Ctk

k−s .


Proof of Theorem 2.4.4. Standard arguments, see the proof of Theorem 22.3.2 in[30], show that it suffices to prove the following microlocal estimates:For every ρ ∈ T ∗X \0 there exists a conic open neighborhood Γ of ρ in T ∗X \0 suchthat for every compact K ⊂ X and every symbol χ ∈ S0(X × Rn) with suppχ ⊂ Γwe have

(Pχu, χu) ≥ −C‖u‖2m−s

2, s = k/(k − J − 1), ∀u ∈ C∞

0 (K), (2.70)

(where χu = χ(x,D)u).If ρ 6∈ Σ, this follows from the Garding inequality ([20]), so we only have to

consider the case when ρ ∈ Σ.Consider then a conic neighborhood V of ρ, with coordinates (u, v) such that u

(respectively v) are homogeneous functions of degree 0 (respectively 1) and Σ∩V =u = 0, v 6= 0. Let pw ∼ pm + psm−1 + . . . be the Weyl symbol of P . With abuseof notation, we denote by pw(u, v), pm(u, v), etc., the Weyl symbol, the principal


symbol, etc., expressed in the new coordinates (u, v). It suffices to prove that forsuitable ε > 0, C > 0, it turns out

pw(u, v) ≥ −C|v|m−s, (2.71)

on Γε := |u| < ε, |v/|v| − vρ/|vρ|| < ε, |v| > 1. Indeed, granted (2.71), we maywrite pw = p + r where p and r are classical symbols, respectively of order m andm− s, with p ≥ 0. Since s ≤ 2 an application of the Fefferman-Phong inequality tothe operator with Weyl symbol p gives at once (2.70).

By Taylor’s formula we can write

pm(u, v) =∑|α|=k

aα(v)uα + r(u, v),

psm−1(u, v) =∑|α|≤J

bα(v)uα +

∑|α|=J+1

cα(u, v)uα,

with r(u, v) vanishing on Σ to order k + 1 and |cα(u, v)| ≤ C1|v|m−1 in Γε. Weobserve that Hypothesis (2.68) here reads∑

|α|≤J

bα(v)uα ≥ 0 in Γε,

since taking as Ξ in (2.66) the scalar product a∂u we get Ij = (j!)−1(a∂u)j psm−1|u=0 =∑

|α|=j(α!)−1∂αupsm−1(0, v)a

α.The transversal ellipticity assumption gives

|r(u, v)| ≤ 1

2

∑|α|=k

aα(v)uα in Γε,

if ε is small enough.Hence in Γε we have

pm(u, v) + psm−1(u, v) ≥1

2

∑|α|=k

aα(v)uα +

∑|α|=J+1

cα(u, v)uα =

= |v|m1

2

∑|α|=k

aα(v)

|v|muα +

1

|v|∑

|α|=J+1

cα(u, v)

|v|m−1uα

≥ −C2|v|m−s,

where we applied Lemma 2.4.6 with t = |v|−1 and s = J + 1. Since obviously|pw(u, v) − pm(u, v) − psm−1(u, v)| ≤ C3|v|m−2, we obtain (2.71) and this concludesthe proof.


Example 2.4.7. For differential operators condition (2.68) actually reads Ij = 0for j = 0, . . . , J . In particular, with J ∈ 0, 1, . . . , k/2− 1, the constant coefficientdifferential operator in R2

P = Dkx1−Dk−J−2

x2DJ+1x1

, (x1, x2) ∈ R2, (2.72)

verifies the assumptions of Theorem 2.4.4. Let us check directly that there exists aconstant C > 0 such that

(Pu, u) ≥ −C‖u‖2k−s2

, ∀u ∈ C∞0 (R2), (2.73)

if and only if s ≤ k/(k − J − 1). In fact the symbol ξk1 − ξk−J−22 ξJ+1

1 is nonnegative

in a conic neighborhood of ξ2 = 0. For ξ2 6= 0 we set ζ = ξ1/|ξ2|k−1

k ; then

ξk1 − ξk−J−22 ξJ+1

1 ≥|ξ2|k−1(ζk − |ξ2|−

J+1k |ζ|J+1

)≥|ξ2|k−1

(−C|ξ2|−

J+1k−J−1

)=− C|ξ2|k−k/(k−J−1).

Moreover, if s > k/(k − J − 1) we can find s such that (k − J − 1)/(k − J − 2) <s < (J + 1)/(J + 2− s). Then, for every C > 0,

ξk1 − ξk−J−22 ξJ+1

1 + C〈ξ〉k−s → −∞

as ξ1 → +∞ on the curve of equation ξs1 = ξ2, ξ1 > 0. This gives the conclusion.

Example 2.4.8. Consider the operator with Weyl symbol

q(x, ξ) = a(x)n∑

j,r=1

ajk(x)ξjξr + b(x)n∑j=1

cj(x)ξj, (x, ξ) ∈ T ∗X,

for real C∞ functions ajr, a, b, cj, j, r = 1, . . . , n, with (ajr) positive definitematrix and such that a−1(0) is a C∞ manifold where a vanishes exactly to order kand b(x)2 ≤ CKa(x), x ∈ K ⊂⊂ X. Since p1 vanishes on Σ = a−1(0) × (Rn \ 0)to order k/2, we have then from Theorem 2.4.4 a lower bound with a gain of twoderivatives. By following the arguments of Hormander [30], this can apply to theCauchy problem for the operator A = D2

t − q(x,D), which is known otherwise tobe well-posed in C∞ under the preceding assumption, cf. Ivrii-Petkov [32]. We havealso from Theorem 2.4.4 that if instead b(x) vanishes to order J + 1 ≤ k/2 on Σ,then (2.69) is valid with s = k/(k−J −1). Under the same hypotheses, the Cauchyproblem for A = D2

t − q(x,D) is well-posed in suitable Gevrey classes, see Ivrii [31],Dreher and Reissig [13]. We do not give details here, but observe that seeminglyour bounds may lead to precise estimates of the corresponding solutions in Sobolevand Gevrey-Sobolev norms, respectively.

APPENDICES

2.A A temperate calculus

In this appendix, we report on from Mughetti [45] some technical results concerningthe metric and the weights introduced in Section 2.1 (cf. Maniccia and Mughetti[40] as well). First we recall some basic definitions concerning the Weyl-Hormandercalculus; see Chapter XVIII of [30] for details.

A metric is a measurable function g : (x, ξ) 7→ gx,ξ of R2n into the set of positivedefinite quadratic forms on R2n. With any metric gx,ξ it is associated the so-calledPlanck function h(x, ξ), defined by

h(x, ξ) :=

(sup(t,τ)

gx,ξ(t, τ)

gσx,ξ(t, τ)

)1/2

,

where gσ is the dual quadratic form:

gσx,ξ(t, τ) := supgx,ξ(y,η)=1

σ((t, τ); (y, η)

)2,

with respect to the standard symplectic 2–form σ =∑n

i=1 dξi ∧ dxi in R2n.A Hormander metric is a metric which is

· slowly varying, i.e. there exists C > 0 such that, with X, Y ∈ R2n,

gX(Y −X) ≤ C−1 =⇒ C−1gY (Z) ≤ gX(Z) ≤ CgY (Z) ∀Z ∈ R2n;

· σ-temperate, i.e. there exist constants C > 0, N ∈ N such that

gX(Z) ≤ CgY (Z)(1 + gσ(Y −X))N ∀X,Y, Z ∈ R2n;

· satisfying the uncertainty principle, namely

h(x, ξ) ≤ 1 ∀(x, ξ) ∈ R2n.

63


A g-admissible weight is a positive measurable function m : R2n → R+, which is

· g-continuous, i.e. there exists C > 0 such that

gX(Y −X) ≤ C−1 =⇒ C−1m(Y ) ≤ m(X) ≤ Cm(Y );

· (σ, g)-temperate, i.e. there exist constant C > 0, N ∈ N such that

m(X) ≤ Cm(Y )(1 + gσ(Y −X))N ∀X,Y ∈ R2n.

We denote by S(m, g) the set of the smooth functions a : R2n → C satisfying

sup(x,ξ)

|a|gk(x, ξ)m(x, ξ)

<∞, for all k ∈ N,

where |a|g0(x, ξ) := |a(x, ξ)| and

|a|gk(x, ξ) := supTj∈R2n

|a(k)((x, ξ);T1, . . . , Tk

)|

gx,ξ(T1)1/2 · · · gx,ξ(Tk)1/2, for k ≥ 1,

where a(k)(X, ·) denotes the k-multi-linear form given by the differential of order kof a at X ∈ R2n.

The space S(m, g) is equipped with the Frechet topology given by the seminorms

|a|k;S(m,g) := sup(x,ξ)

‖a‖gk(x, ξ)m(x, ξ)

, (k ∈ N),

where‖a‖gk(x, ξ) := sup

j≤k|a|gj (x, ξ).

Proposition 2.A.1. The metric gx,ξ defined in (2.3) is a Hormander metric.

Proof. The dual metric gσx,ξ is given by

gσx,ξ(y, η) = 〈ξ〉2d(x, ξ)2|y′|2 + 〈ξ〉|y′′|2 + 〈ξ〉2|y′′′|2 + d(x, ξ)2|η′|2 + |η′′|2 + |η′′′|2,

and the Planck function is

h(x, ξ) = max〈ξ〉−1d(x, ξ)−2, 〈ξ〉−1/2, 〈ξ〉−1 = max〈ξ〉−1d(x, ξ)−2, 〈ξ〉−1/2.

Hence the uncertainty principle h ≤ 1 is satisfied, because d(x, ξ) ≥ 〈ξ〉−1.

2.A. A TEMPERATE CALCULUS 65

The metric gx,ξ is slowly varying.Let X = (x, ξ), Y = (y, η), Z = (z, ζ), with

gX(Y −X) ≤ 1

C. (2.74)

We have to prove that if C is large enough, there exists c > 0 such that

c−1gX(Z) ≤ gY (Z) ≤ cgX(Z) (2.75)

It suffices to prove that(2)

〈ξ〉 ≈ 〈η〉 (2.76)

andd(X) ≈ d(Y ) (2.77)

Since trivially d(x, ξ) ≥ 4, by (2.74) we have

1

4

|η′ − ξ′|2

〈ξ〉2≤ |η′ − ξ′|2

〈ξ〉2d(x, ξ)2≤ 1

C,

so that

|η′ − ξ′|2 ≤ 4

C〈ξ〉2.

Yet from (2.74) it also follows

|η′′ − ξ′′|2 ≤ 〈ξ〉C

≤ 〈ξ〉2

Cand |η′′′ − ξ′′′|2 ≤ 〈ξ〉2

C.

Hence, if C > 24,

|η − ξ|2 ≤ 6

C〈ξ〉2 ≤ 1

4〈ξ〉2,

which yields

〈η〉2 ≤ 5

2〈ξ〉2 and 〈η〉2 ≥ 1

4〈ξ〉2,

namely (2.76).Let us verify (2.77). From (2.74) we have

|η′|2

〈η〉2≤ |η′|2

〈ξ〉2.|ξ′|2

〈ξ〉2+|η′ − ξ′|2

〈ξ〉2.|ξ′|2

〈ξ〉2+d(X)2

C. d(X)2, (2.78)

and similarly

|ξ′|2

〈ξ〉2≤ |ξ′|2

〈η〉2.|η′|2

〈η〉2+|η′ − ξ′|2

〈η〉2. d(Y )2 +

d(X)2

C. (2.79)

2Contrary to the previous convention, in this section we use the notation f . g for two realfunction f and g satisfying f(z) ≤ Cg(z) globally for every z ∈ R2n. Similarly we use & and ≈.


On the other hand, again from (2.74) it follows

|x′ − y′|2 ≤ d(X)2

C≤ 1

C

(|x′|2

〈x′〉2+ 3

)≤ 1

C

(|x′|2 + 3

)≤ 1

4〈x′〉2,

if C ≥ 12, so that

〈y′〉2 ≤ 5

2〈x′〉2 and 〈y′〉2 ≥ 1

4〈x′〉2. (2.80)

From (2.78) and (2.80) we obtain

|y′|2

〈y′〉2.|y′|2

〈x′〉2.|x′|2

〈x′〉2+|x′ − y′|2

〈x′〉2≤ |x′|2

〈x′〉2+d(X)2

C. d(X)2, (2.81)

and|x′|2

〈x′〉2.|x′|2

〈x′〉2.|x′|2

〈y′〉2+|x′ − y′|2

〈y′〉2≤ |y′|2

〈y′〉2+d(X)2

C. (2.82)

Similarly, we have

|η′′|2

〈η〉2.|η′′|2

〈ξ〉2.|ξ|2

〈ξ〉2+|η′′ − ξ′′|2

〈ξ〉2≤ |ξ′′|2

〈ξ〉2+

1

C〈ξ〉. d(X)2, (2.83)

and|ξ′′|2

〈ξ〉2.|ξ′′|2

〈η〉2.|η′′|2

〈η〉2+|η′′ − ξ′′|2

〈η〉2≤ |ξ′′|2

〈η〉2+d(X)2

C. d(X)2. (2.84)

From (2.76),(2.78),(2.81),(2.83), we obtain d(Y ) . d(X), and from (2.76), (2.79),

(2.82),(2.84) we obtain d(X)2 ≤ c′d(Y )2+ c′d(X)2

C, which gives d(X) . d(Y ) if C > c′.

Hence (2.77) is proved.

The metric gx,ξ is temperate.It is equivalent to proving that there exist N > 0 such that

gσY (Z) . gσX(Z)(1 + gσX(Y −X))N .

Hence it will suffice to prove the following:

〈η〉 . 〈ξ〉(1 + gσX(Y −X)) (2.85)

d(Y )2 . d(X)2(1 + gσX(Y −X)) (2.86)

〈η〉2d(Y )2 . 〈ξ〉2d(X)2(1 + gσX(Y −X))2 (2.87)

〈η〉2 . 〈ξ〉2(1 + gσX(Y −X))2. (2.88)

Of course, (2.85) implies (2.88), whereas (2.86) and (2.88) imply (2.87). Hence, weare reduced to prove (2.85) and (2.86).

2.A. A TEMPERATE CALCULUS 67

Proof of (2.85)We have

〈ξ〉(1 + gσX(Y −X)) ≥ 〈ξ〉(1 + gσ(Y −X))1/2 ≥≥ 〈ξ〉(1 + d(X)2|η′ − ξ′|2 + |η′′ − ξ′′|2 + |η′′′ − ξ′′′|2)1/2 =

= (〈ξ〉2 + 〈ξ〉︸︷︷︸≥1

〈ξ〉d(X)2︸︷︷︸≥1

|η′ − ξ′|2 + 〈ξ〉2︸︷︷︸≥1

|η′′ − ξ′′|2 + 〈ξ〉2︸︷︷︸≥1

|η′′′ − ξ′′′|2)1/2 ≥

≥ (1 + |ξ|2 + |η − ξ|2)1/2 & 〈η〉.

Proof of (2.86)Assume |ξ − η| ≥ 1

3|ξ|. Then

d(X)2(1 + gσX(Y −X)) ≥ d(X)2(1 + d(X)2|η − ξ|2) &

& d(X)4(1 + |η − ξ|2) &

& 〈ξ〉2d(X)4 ≥ 1 & d(Y )2,

which is (2.86).Let then |ξ − η| ≤ 1

3|ξ|, so that |ξ| ≈ |η|. We have

d(X)2(1 + gσX(Y −X)) ≥ d(X)2 ≥ 1

〈ξ〉&

1

〈η〉, (2.89)

d(X)2(1 + gσX(Y −X)) ≥ d(X)2 + d(X)4|η′ − ξ′|2 ≥ |ξ′|2

〈ξ〉2+

1

〈ξ〉2|η′ − ξ′|2 &

&|η′|2

〈ξ〉2&|η′|2

〈η〉2, (2.90)

and

d(X)2(1 + gσX(Y −X)) ≥ d(X)2 + d(X)2|η′′ − ξ′′|2 &|ξ′′|2

〈ξ〉2+

1

〈ξ〉2|η′′ − ξ′′|2 &

&|η′′|2

〈ξ〉2&|η′′|2

〈η〉2. (2.91)

Finally, we also have

d(X)2(1 + gσX(Y −X)) &|y′|2

〈y′〉2. (2.92)

Indeed, when |x′ − y′| ≤ 13|y′|, so that |x′| ≈ |y′|, it turns out

d(X)2(1 + gσX(Y −X)) ≥ d(X)2 ≥ |x′|2

〈x′〉2&|y′|2

〈y′〉2,


whereas when |x′ − y′| ≥ 13|y′| we have

d(X)2(1 + gσX(Y −X)) ≥ d(X)2〈ξ〉2|y′ − x′|2 & |y′|2 ≥ |y′|2

〈y′〉2.

Hence,(2.89), (2.90), (2.91) and (2.92) give (2.86).

Proposition 2.A.2. The weights 〈ξ〉 and d(x, ξ) defined in (2.4) are g-admissible.

Proof. We already proved in Proposition 2.A.1 that

gX(Y −X) ≤ 1

C=⇒ 〈ξ〉 ≈ 〈η〉 and d(X) ≈ d(Y ).

Moreover, we can write

〈ξ〉 = gX(Z0)−1/2 and d(X) = gX(Z1)

−1/2,

for any

Z0 = (0, 0, 0, 0, 0, ζ ′′′), with |ζ ′′′| = 1

Z1 = (z′, 0, 0, 0, 0, 0), with |ζ ′| = 1.

Hence, by the proof of Proposition 2.A.1, we have

〈ξ〉〈η〉

. (1 + gσX(Y −X))

andd(X)

d(Y ). (1 + gσX(Y −X)).

This the concludes the proof.

Proposition 2.A.3. The weight dλ in (2.8) is g-admissible uniformly with respectto λ ∈ R.

Proof. We observe that

dλ(x, ξ) ≈ d(x, ξ) +|λ|

〈ξ〉m/k.

If gX(Y − X) ≤ 1C, with C large enough, it follows from the proof of Proposition

2.A.1 that 〈ξ〉 ≈ 〈η〉 and d(X) ≈ d(Y ), so that

dλ(X) ≈ dλ(Y ).

2.B. A TECHNICAL RESULT 69

It is also immediate to see that

dλ(X)

dλ(Y ). (1 + gσX(Y −X)). (2.93)

Indeed,

dλ(X)

dλ(Y )≈ d(X)

d(Y ) + |λ|〈η〉m/k

+

|λ|〈ξ〉m/k

d(Y ) + |λ|〈η〉m/k

.

.d(X)

d(Y )+

|λ|〈ξ〉m/k

|λ|〈η〉m/k

=d(X)

d(Y )+〈η〉m/k

〈ξ〉m/k.

This gives (2.93), for the weights 〈·〉 and d were already shown to be g-admissible.

2.B A technical result

This appendix is devoted to a result which is used in the proof of Proposition2.1.3. Hence we adopt the notation introduced there. Precisely, we show thatif a pseudodifferential operator has a parametrix and is L2-invertible then, undersuitable hypotheses, its inverse belongs to the same class of its parametrix.

Proposition 2.B.1. Assume the same hypotheses of Proposition 2.1.3. Then theoperator bλ(ρ, ξ

′′, x′, Dx′) has pseudodifferential inverse in L2(Rν) with symbol tλ ∈C∞(Σ; S−kpar(Rl+1; R2ν)).

We start off by showing the following result.

Lemma 2.B.2. There exists b′λ ∈ C∞(Σ,S−kpar(Rl+1; R2ν)) such that

bλ#′b′λ = 1 + rλ, (2.94)

with rλ ∈ C∞(Σ,S(Rl+1+2νξ′′,λ,x′,ξ′)).

Proof of Lemma 2.B.2. First of all, b−1λ ∈ C∞(Σ,S−kpar(Rl+1; R2ν)). Indeed, since

bλ ∈ C∞(Σ,Skpar(Rl+1; R2ν)), this follows at once from the ellipticity assumption if

we observe that any derivative ∂γb−1λ , γ ∈ Zn+1

+ , is a finite sum of terms of the formb−1−rλ ∂α1bλ · · · ∂αrbλ, with 0 ≤ r ≤ |γ| and α1 + . . .+ αr = γ.

We now verify that

b(k)#′b−1λ = bkb−1

λ − cλ, with cλ ∈ C∞(Σ,S−1par(Rl+1; R2ν)). (2.95)


To this end, define the metric g = (1 + |x′|2 + |ξ′|2 + |ξ′′|2)−1/2(|dx′|2 + |dξ′|2) andthe weights m(x′, ξ′, ξ′′) = (1 + |x′|2 + |ξ′|2 + |ξ′′|2)1/2, mλ(x

′, ξ′, ξ′′) = (1 + |x′|2 +|ξ′|2 + |ξ′′|2 + λ2)1/2. One can see that g is a Hormander’s metric in R2ν uniformlywith respect to ξ′′ ∈ Rl, and that the weights m and m are g-admissible uniformlywith respect to (ξ′′, λ) ∈ Rl+1. Then, when ρ is in a bounded subset of Σ and(ξ′′, λ) ∈ Rl+1, the symbols bλ and b−1

λ vary in bounded subsets of S(mk, g) andS(m−1

λ , g) respectively, so that cλ is in a bounded subset of S(m−1λ , g) as well. By

the Leibniz formula, the same claim holds for the derivatives of cλ with respect to(ρ, ξ′′, λ), i.e., cλ ∈ C∞(Σ,S−1

par(Rl+1; R2ν)). We then have (2.95), whence

bλ#′b−1λ = 1− cλ.

Moreover the same argument above shows that

C∞(Σ,Sµpar(Rl+1; R2ν))#′C∞(Σ,Sµ′par(Rl+1; R2ν)) ⊂ C∞(Σ,Sµ+µ′

par (Rl+1; R2ν)),(2.96)

and therefore c#jλ ∈ C∞(Σ,S−jpar(Rl+1; R2ν)), for any j ∈ Z+. We now claim that

there exists c′λ ∈ C∞(Σ,S0par(Rl+1; R2ν)) such that c′λ ∼

∑j≥0 c

#jλ , in the sense that

c′λ −N∑j≥0

c#jλ ∈ C∞(Σ,S−N−1par (Rl+1; R2ν)).

Indeed, we take φ ∈ C∞(R) such that φ(t) = 0 for t ≤ 1 and φ(t) = 1 for t ≥ 2.Then standard arguments show that c′λ :=

∑j≥0 φ(εjmλ)c

#jλ is the required function,

provided the sequence (εj)j∈N decreases to zero rapidly enough.Hence, one sees at once that b′λ := b−1

λ #′c′λ is the parametrix we were looking for.

Proof of Proposition 2.B.1. Let us start off by verifying that, for any fixed ρ0,tλ(ρ0, ξ

′′, x′, ξ′) ∈ S−kpar(Rl+1; R2ν). Indeed, we have from Theorem 4.1 of [5] thattλ(ρ0, ξ

′′, x′, ξ′) ∈ C∞(Rl+1y ;S−k(R2ν)), where we set y := (ξ′′, λ) and, recall, Sµ(R2ν) =

a ∈ C∞(R2νz ) : |∂γz a(z)| ≤ Cγ(1 + |z|)µ−|γ|. Notice that if |y| is bounded, the es-

timates required by S−kpar(Rl; R2ν) are exactly the ones satisfied by the symbols inC∞(Rl+1

y ;S−k(R2ν)). Therefore, in what follows we can assume |y| large.Now, for any real M ,

‖rλ‖L(L2(Rν),L2(Rν)) ≤ CM(1 + |y|)−M .

Hence, if |y| ≥ c for some constant c > 0 large enough, rλ(ρ0, ξ′′, x′, Dx′) has norm

less than 1 as a bounded operator in L2(Rν) and the operator Id + rλ(ρ0, ξ′′, x′, Dx′)

is invertible.At this point we would like to apply Lemma 4.2 of [5] to Id + rλ(ρ0, ξ

′′, x′, Dx′)

2.C. ON THE COMPOSITION OF PSEUDODIFFERENTIAL OPERATORS 71

and get that σ((Id + rλ(ρ0, ξ

′′, x′, Dx′))−1 − Id

)∈ S(Rl+1+2ν). Unfortunately, we

cannot assume Σ = ρ0, N = Rl+1+2ν with the notation in Lemma 4.2 of [5], sinceId + rλ(ρ0) is L2−invertible only for |y| ≥ c. In order to overcome this difficulty,choose a cut-off function ψ ∈ C∞(Rl+1), with 0 ≤ ψ ≤ 1, such that ψ(y) ≡ 0 if|y| ≤ c and ψ(y) ≡ 1 if |y| ≥ 2c. Now,

‖ψ(y)rλ‖L(L2(Rν),L2(Rν)) = ψ(y)‖rλ‖L(L2(Rν),L2(Rν)),

whence Id + ψ(y)rλ(ρ0, ξ′′, x′, Dx′) is invertible for every y ∈ Rn in L2(Rν) (and

hence in S(Rν)). As a consequence of Theorem 3.1 and Lemma 4.2 of [5], withΣ = ρ0, N = Rl+1+2ν we have that its inverse has the form Id + s(ρ0, y, x

′, Dx′)with

s(ρ0, y, x′, ξ′) ∈ S−∞par (Rl+1,R2ν).

Hence, for large |y|, we have tλ(ρ0, ξ′′, ·) = b′λ(ρ0, ξ

′′, ·)#′(1 + s(ρ0, y, ·)) and (2.96)gives the claim.

To prove that tλ depends smoothly on ρ ∈ Σ, we work locally near any point ρ0.One considers the new parametrix b(ρ, y, ·) := bλ(ρ, ξ

′′, ·)− bλ(ρ0, ξ′′, ·)+ tλ(ρ0, ξ

′′, ·),which is then in C∞(Σ,S−kpar(Rl+1; R2ν)). We have

bλ#′b = 1 + r,

with r ∈ C∞(Σ;S(Rl+1+2ν)) and r(ρ0, ·) = 0, for b(ρ0, y, ·) = tλ(ρ0, ξ′′, ·). Since,

in particular, the map Σ 3 ρ 7→ r(ρ, ·) ∈ S(Rl+1+2ν) is continuous, the operatorId + r(ρ0, y, x

′, Dx′) is invertible for every y ∈ Rl+1 for ρ in a neighborhood U ofρ0. Yet an application of Theorem 3.1 and Lemma 4.2 of [5] (now with Σ = U andN = Rl+1+2ν), concludes the proof.

2.C On the composition of pseudodifferential op-

erators

For the sake of completeness, here we report on a property concerning the compo-sition of two pseudodifferential operators which is often used in the present thesis.First of all, we need the following elementary result.

Lemma 2.C.1. Let H,K be two closed subsets of Rn with dist(H,K) = δ > 0.Then there exists a smooth function φ ∈ C∞

b (Rn) such that φ ≡ 1 in H and φ ≡ 0 inK (here C∞

b (Rn) denotes the class of the smooth functions with bounded derivativesof any order).


Proof. Let ψ ∈ C∞0 (Rn) such that 0 ≤ ψ ≤ 1, suppψ ⊂ x ∈ Rn : |x| ≤ 1, and∫

ψ(x)dx = 1. Denote by χH, ε the characteristic function of the set

Hε = x ∈ Rn | ∃y ∈ H : |x− y| ≤ ε

with ε = δ/4 and define

φ(x) =

∫χH(y)ψ

(y − x

ε

)dyεn, x ∈ Rn.

An easy check shows that φ is the cut-off function we were looking for.

Proposition 2.C.2. Let a ∈ Smρ,δ(Rn × Rn), b ∈ Sm′

ρ,δ(Rn × Rn), ρ > 0, δ < 1, andsuppose that for some open conic subset V = Ω× Γ ⊂ T ∗Rn of product type, one ofthe symbols a or b is in S(V ) (i.e., all its derivatives are dominated by 〈ξ〉−N , forany N ≥ 0, when (x, ξ) ∈ V ). Then for any open subset Ω′ ⊂ Ω with dist(Ω′,c Ω) > 0and for any open conic subset Γ′ ⊂⊂ Γ the product symbol a#b is in S(Ω′ × Γ′).

We observe that the condition dist(Ω′,c Ω) > 0 is certainly satisfied if Ω′ ⊂⊂ Ωalthough, in general, unbounded sets Ω′ are allowed as well.

Proof. The product symbol is given by the oscillatory integral

(2π)−n∫∫

eizζa(x, ξ + ζ)b(x− z, ξ) dz dζ. (2.97)

Since derivatives with respect to x, ξ pass under the integral sign, we have only toprove the 0th order estimates.

Let us suppose a in S(V ). Repeated integrations by parts show that the integral(2.97) is dominated by the integral∫∫

(1+ |ζ|2)−N |(1−∆ζ)Na(x, ξ+ζ)(1−∆z)

N [(1+ |z|2)−Nb(x−z, ξ)]| dz dζ, (2.98)

which is absolutely convergent if N is large enough. Now we look at this integralfirst in the region where |ζ| > |ξ|/K and then where |ζ| ≤ |ξ|/K. The first con-tribution defines a rapidly decreasing function, since 1 + |ζ|2 ≈ 1 + |ζ|2 + |ξ|2 onthe domain of integration. The second integral instead defines a function which israpidly decreasing in Ω × Γ′, if K is large enough, because ξ + ζ ∈ Γ when ξ ∈ Γ′,in that case. Hence a#b is in S(Ω× Γ′).

Let us now suppose b in S(V ). By Lemma 2.C.1, there exists a function φ(x) ∈C∞b (Rn) with φ ≡ 1 on Ω′ and suppφ ⊂ Ω.

We writea#b = a#(φb) + a#(1− φ)#b.

2.C. ON THE COMPOSITION OF PSEUDODIFFERENTIAL OPERATORS 73

Let x ∈ Ω′. By Taylor’s formula we have, for any N ≥ 0,

(a#(1− φ))(x, ξ) = (2π)−n∫∫

eizζa(x, ξ + ζ)(1− φ)(x− z) dz dζ =

=∑|α|=N

∫∫eizζ

(∂αξ a)(x, ξ + ζ)rα(x, z) dz dζ,

for suitable functions rα(x, z) ∈ C∞b (R2n), which yields a function rapidly decreasing

in Ω′ × Rn. Arguing similarly for the derivatives we obtain that a#(1 − φ) is inS(Ω′ ×Rn) and therefore, from the first part of this proof, we have a#(1− φ)#b ∈S(Ω′ × Rn).As it concerns a#φb, we observe that φb is in S(Rn × Γ). Then estimating a#φbby an integral like (2.98) gives at once that a#φb is rapidly decreasing in Rn × Γ.Hence a#b ∈ S(Ω′ × Γ), and this concludes the proof.

Chapter III

AN APPROXIMATE CONSTRUCTION OF

POSITIVE PARTS FOR SECOND ORDER

OPERATORS

In this chapter we prove Theorem 1.3.1 in Section 1.3. First of all we recall somedefinitions and properties about the theory of Hermite operators when the charac-teristic manifold is symplectic and flat, namely in the form (3.1) below; we addressthe reader to [4, 21, 5, 53, 54] for details.

3.1 Preliminaries on the theory of Hermite-opera-

tors

As above, all the operators are properly supported.Define Sµ(R2ν) = a ∈ C∞(R2ν

z ) : |∂γz a(z)| ≤ Cγ(1 + |z|)µ−|γ| (see [22]), andassume that

Σ = (x, ξ) = (x′, x′′, ξ′, ξ′′) : x′ = ξ′ = 0, ξ′′ 6= 0, (3.1)

where we set x = (x′, x′′) ∈ Rn = Rνx′×Rn−ν

x′′ and accordingly ξ = (ξ′, ξ′′). Notice thatthis is actually the microlocal model for any symplectic sub-manifold of codimension2ν. Moreover we set Σ′ = (x′, x′′, ξ′′) : x′ = 0.

Definition 3.1.1. By Sm,k(Rn×Rn,Σ), (m, k ∈ R), we denote the class of symbolsa(x, ξ) which are in Sm(Rn × Rn) away from Σ and which satisfy the followingestimates

|∂α′x′ ∂α′′

x′′ ∂β′

ξ′ ∂β′′

ξ′′ a(x, ξ)| ≤ C|ξ′′|m−|β|(|x′|+ |ξ′|

|ξ′′|+

1

|ξ′′|1/2

)k−|α′|−|β′|, (3.2)

microlocally near Σ (with α = (α′, α′′) and β = (β′, β′′)).Moreover we setHm(Rn×Rn,Σ) := ∩j≥0S

m−j,k−2j(Rn×Rn,Σ). As usual OPSm,k(Rn,Σ)and OPHm(Rn,Σ) denote the classes of the corresponding properly supported ψdo’s.

75

76 AN APPROXIMATE CONSTRUCTION OF POSITIVE PARTS...

Definition 3.1.2. For an open conic subset Γ ⊂ Σ, we denote by Sm,khom(Γ,Σ)

(resp. Hm−k/2hom (Γ,Σ) ) the class of all smooth functions in C∞(Γ,Sk(R2ν)) (resp.

C∞(Γ,S(R2ν))) such that

a(x′′, tξ′′; t−1/2x′, t1/2ξ′) = tm−k/2a(x′′, ξ′′;x′, ξ′), t > 0.

We recall that operators in OPHm(Rn,Σ) map H tcomp(Rn) → H t−m

comp(Rn) contin-uously for every t ∈ R.

Remark 3.1.3. Microlocally near Σ, symbols in Sm,khom(Γ,Σ) (resp. Hm−k/2hom (Γ,Σ))

are in Sm,k(Rn × Rn,Σ) (resp. Hm−k/2(Rn × Rn,Σ)).

We now define a class of symbols with an asymptotic expansion in quasi-homogene-ous terms. (We take Γ = Σ.)

Definition 3.1.4. We say that a symbol a ∈ Sm,k(Rn × Rn,Σ) (resp. in Hm(Rn ×Rn,Σ)) has an asymptotic expansion in quasi-homogeneous terms, if there exist

functions a(k+j) ∈ Sm,k+jhom (Σ,Σ), j ≥ 0, (resp. a(j) ∈ Hm−j/2hom (Σ,Σ)) such that a −∑N

j=0 a(k+j) ∈ Sm,k+N+1(Rn×Rn,Σ) (resp. a−

∑Nj=0 a

(j) ∈ Hm−(N+1)/2(Rn×Rn,Σ))microlocally near Σ.

The symbol a(k) (resp. a(0)) is called the localized symbol of the operator Op(a)and a(k)(ρ, y,Dy) ∈ OPSk(Rν) (resp. a(0)(ρ, y,Dy) ∈ OPS−∞(Rν)) is called thelocalized operator at ρ = (x′′, ξ′′) ∈ Σ.

Remark 3.1.5. In the sequel, we use the Sjostrand classes OPNm,k(Rn,Σ) of theclassical pseudodifferential operators P with symbol Sm(R2n) 3 p ∼

∑j≥0 pm−j such

that, for every j < k/2, pm−j vanishes on Σ to order k − 2j. We point out thatOPNm,k(Rn,Σ) is a very important subclass of OPSm,k(Rn,Σ). Hence, accordingto Definition 3.1.4, for P we have

p(k+j)(ρ, x′, ξ′) :=∑

|α|+|β|+2l=k+j

1

α!β!∂αx′∂

βξ′pm−l(ρ)x

′αξ′β.

Actually, the techniques we are recalling in this section were introduced by Boutetde Monvel [4], Boutet de Monvel, Grigis and Helffer [5] in order to study the hy-poellipticity of the operators in OPNm,k(Rn,Σ).

The following composition properties (see [21, 53]) have several applications inthe sequel.

Proposition 3.1.6. (1) If A = Op(a) ∈ OPSm,k(Rn,Σ), and B = OP(b) ∈OPSm

′,k′(Rn,Σ), then AB = Op(c) ∈ OPSm+m′,k+k′(Rn,Σ). Moreover, if

3.1. PRELIMINARIES ON THE THEORY OF HERMITE-OPERATORS 77

a ∼∑

j≥0 a(k+j), and b ∼

∑j≥0 b

(k′+j), in the sense of Definition 3.1.4, with

a(k+j) ∈ Sm,khom(Σ,Σ), b(k′+j) ∈ Sm

′,k′

hom (Σ,Σ), then c ∼∑

r≥0 c(k+k′+r), with

c(k+k′+r) =

∑i+j+2|α|=r

1

α!∂αξ′′a

(k+i)#Dαx′′b

(k′+j), (3.3)

( ∂αξ′′a(k+i)#Dα

x′′b(k′+j) being the symbol of the composition in S(Rν) of the op-

erators (∂αξ′′a(k+i))(ρ, y,Dy) and (Dα

x′′b(k′+j))(ρ, y,Dy)).

(2) If A = Op(a) ∈ OPHm(Rn,Σ), and B = Op(b) ∈ OPHm′(Rn,Σ), then AB =

Op(c) ∈ OPHm+m′(Rn,Σ). Moreover, if a ∼

∑j≥0 a

(j), and b ∼∑

j≥0 b(j), in

the sense of Definition 3.1.4, with a(j) ∈ Hm−j/2(Σ,Σ), b(j) ∈ Hm′−j/2(Σ,Σ),then c ∼

∑r≥0 c

(r), with

c(r) =∑

i+j+2|α|=r

1

α!∂αξ′′a

(i)#Dαx′′b

(j). (3.4)

(3) If A = Op(a) ∈ OPHm(Rn,Σ) then A∗ = Op(a∗) ∈ OPHm(Rn,Σ). Moreover,

if a ∼∑

j≥0 a(j) in the sense of Definition 3.1.4, with a(j) ∈ Hm−j/2

hom (Σ,Σ),

then a∗ ∼∑

r≥0 a∗(r), with

a∗(r) =∑

j+2|α|=r

1

α!Dαξ′′∂

αx′′(a

(j))∗,

where (a(j))∗ denotes the symbol of the operator a(j)(ρ, y,Dy)∗ on S(Rν

y).

We also need the following classes of Hermite operators.

Definition 3.1.7. We denote by Hm(Rn−ν × Rn−ν ,Σ′), m ∈ R, the class of allsmooth functions h(x′′, ξ′′, y), where (x′′, ξ′′) ∈ Rn−ν × Rn−ν , y ∈ Rν, which satisfyfor any l ≥ 0 the following estimates

|∂α′′x′′ ∂α′

y ∂β′′

ξ′′ h(x′′, ξ′′; y)| ≤ C|ξ′′|m+ν/4−l−|β′′|

(|y|+ 1

|ξ′′|1/2

)−2l−|α′|

,

uniformly in x′′ ∈ K ′′ ⊂⊂ Rn−ν, |ξ′′| ≥ 1, y ∈ Rν.For h ∈ Hm(Rn−ν×Rn−ν ,Σ′), the operator Op(h) : C∞

0 (Rn−ν) → C∞(Rn) is definedby

(Op(h)f) (y, x′′) := (2π)−(n−ν)∫ei〈x

′′,ξ′′〉h(x′′, ξ′′; y)f(ξ′′) dξ′′,


and OPHm(Rn−ν ,Σ′) is the corresponding class of operators, modulo smoothing op-erators.Moreover, by OPH∗m(Rn−ν ,Σ′) we denote the class of operators C∞

0 (Rn) → C∞(Rn−ν)that, modulo smoothing operators, are given by

(Op(h)∗g)(x′′) = (2π)−(n−ν)∫∫

ei〈x′′,ξ′′〉h(x′′, ξ′′, y)g(y, ξ′′)dξ′′dy,

with g(y, ξ′′) =∫e−i〈x

′′,ξ′′〉g(y, x′′)dx′′ and h ∈ Hm(R2(n−ν),Σ′).Finally, for an open conic subset Γ ⊂ Σ, we denote by Hm

hom(Γ,Σ′) the class ofthe quasi-homogeneous functions h(x′′, ξ′′, y) ∈ C∞(Γ,S(Rν

y)) of degree m+ν/4, i.e.

h(x′′, tξ′′, t−1/2y) = tm+ν/4h(x′′, ξ′′, y), t > 0.

Remark 3.1.8. Functions in Hmhom(Γ,Σ′) give rise, microlocally near Σ, to symbols

that are in Hm(Rn−ν × Rn−ν ,Σ′).

Proposition 3.1.9.

(1) If A = Op(f0 + f−1) ∈ OPHm(Rn−ν ,Σ′), with f−s ∈ Hm−s/2(Rn−ν × Rn−ν ,Σ′),s = 0, 1, then A∗ ∈ OPH∗m(Rn−ν ,Σ′) with

A∗ = Op (f0 + f−1)∗ mod OPH∗m−1(Rn−ν ,Σ′).

(2) If A = Op(f0 + f−1) ∈ OPHm(Rn−ν ,Σ′) and B = Op(g0 + g−1)∗ ∈

OPH∗m′(Rn−ν ,Σ′), with f−s ∈ Hm−s/2(Rn−ν×Rn−ν ,Σ′), g−s ∈ Hm′−s/2(Rn−ν×

Rn−ν ,Σ′), s = 0, 1, then AB ∈ OPHm+m′(Rn,Σ) and BA ∈ OPHm+m′

(Rn−ν ,Σ),with

AB = Op(e−i〈y,η〉f0(ρ, y)g0(ρ, η)

)mod OPHm+m′−1/2(Rn,Σ),

and

BA = Op

(∫Rν

(f0g0 + f−1g0 + f0g−1) dy

)mod OPHm+m′−1(Rn−ν ,Σ).

(3) If A = Op(h) ∈ OPHm(Rn−ν ,Σ′) and B ∈ OPSm′(Rn−ν), then AB ∈

OPHm+m′(Rn−ν ,Σ′) with

AB = Op (hb) mod OPHm+m′−1(Rn−ν ,Σ′).

As a consequence of Remark 3.1.5 and Proposition 3.1.6, we obtain the followingresults.

3.1. PRELIMINARIES ON THE THEORY OF HERMITE-OPERATORS 79

Proposition 3.1.10.

(1) Let P = Op(p) ∈ OPNm,k(Rn,Σ) and B = Op(b) ∈ OPHm′(Rn,Σ) with

b ∼∑

j≥0 b(j) in the sense of Definition 3.1.4, with b(j) ∈ Hm′−j/2

hom (Σ,Σ).

Then PB and BP are in OPHm+m′−k/2(Rn,Σ) and have localized symbolsgiven by p(k)(ρ, ·)#b(0)(ρ, ·) and by b(0)(ρ, ·)#p(k)(ρ, ·), respectively. As a result,one gets σ(PB) − p(k)(ρ, ·)#b(0)(ρ, ·) ∈ Hm+m′−k/2−1/2(Rn,Σ) and σ(BP ) −b(0)(ρ, ·)#p(k)(ρ, ·) ∈ Hm+m′−k/2−1/2(Rn,Σ) near Σ.

(2) Let P = Op(p) ∈ OPNm,k(Rn,Σ), with classical asymptotic expansion p ∼∑j≥0 pm−j, and A = Op(f0+f−1) ∈ OPHm′

(Rn−ν ,Σ′) with f−s ∈ Hm′−s/2(Rn−ν×Rn−ν ,Σ′), s = 0, 1, then PA ∈ OPHm+m′−k/2(Rn−ν ,Σ′) and, microlocally nearΣ, we have

PA = Op(p(k)(ρ; y,Dy)(f0 + f−1) + p(k+1)(ρ; y,Dy)f0

)mod OPHm+m′−k/2−1(Rn−ν ,Σ′).

We end this section by recalling from [53] the following lower bound, whichgeneralizes the Melin inequality (see [43]) for operators with multiple characteristicsdue to Mohamed [44]. In the sequel we will apply this result for m = k = 2.

We fix the following notation. For any given open conic subset Γ ⊂ Σ and anygiven positive constant ε, we define

Γε = (x, ξ) ∈ Rn : (x′′, ξ′′) ∈ Γ, |x′|+ |ξ′|/|ξ′′| < ε, |ξ′′| ≥ 1/ε. (3.5)

Proposition 3.1.11. Let P = P ∗ ∈ OPNm,k(Rn,Σ) be transversally elliptic, withnonnegative principal symbol, and let B = B∗ ∈ OPHm−k/2(Rn,Σ) with localizedoperator b(0)(ρ, x′, Dx′) ∈ OPS−∞(Rν) at any ρ ∈ Σ. Fix ρ0 ∈ Σ and suppose that:

(1)((p(k)(ρ0, x

′, Dx′) + b(0)(ρ0, x′, Dx′)

)f, f)≥ 0, ∀f ∈ S(Rν);

(2) p(k)(ρ0, x′, Dx′) + b(0)(ρ0, x

′, Dx′) : S(Rν) → S(Rν) is invertible.

Then there exist a conic neighborhood Γ ⊂ Σ of ρ0 and a constant ε > 0 such that,for every symbol ψ ∈ S0(Rn × Rn) supported in Γε, there exist constants c, C > 0for which

((P +B)Ψu,Ψu) ≥ c‖Ψu‖2m/2−k/4 − C‖u‖2

m/2−k/4−1/2, ∀u ∈ S(Rn), (3.6)

with Ψ = ψ(x,D).


3.2 Construction of the positive and negative parts

In this Section we prove Theorem 1.3.1 by heavily using the technical machinerydeveloped in the previous section.It is now important to point out that Theorem 1.3.1 is invariant with respect tocanonical changes of variables. To be definite, let χ : T ∗X \ 0 −→ T ∗Y \ 0 bea canonical symplectomorphism (homogeneous of degree 1 in the fibres) and F ∈I0(Y ×X,Λχ) be a Fourier integral operator of order 0 associated with χ and suchthat

F ∗F ≡ IdX , FF ∗ ≡ IdY .

Then Theorem 1.3.1 applies to P if and only if it applies to P := FPF ∗.Since Σ is symplectic, from Theorem 21.2.4 of [30] there exists a conic neighborhoodΓ ⊂ T ∗X \ 0 of ρ0, an open conic set Γ ⊂ T ∗(Rν × Rn−ν) \ 0, (2ν = codim Σ), anda smooth homogeneous symplectomorphism χ : Γ −→ Γ such that

χ(Γ ∩ Σ) = (x′, x′′, ξ′, ξ′′) ∈ Γ : x′ = ξ′ = 0.

Therefore, in the sequel we may suppose that P = P ∗ ∈ OPS2(Rn) is a classicaloperator, with principal symbol transversally elliptic with respect to

Σ = (x, ξ) ∈ Rn × Rn : x′ = ξ′ = 0, ξ′′ 6= 0.

In what follows, we simply write ρ = (x′′, ξ′′) for any point ρ = (0, x′′, 0, ξ′′) ∈ Σ.Since p2 vanishes exactly to 2−nd order on Σ, it is easily seen that Ker Fρ = TρΣ =(x, ξ) : x′ = ξ′ = 0, where Fρ denotes the fundamental matrix defined in (1.1). Asa consequence, according to Remark 3.1.5, the localized symbol of P is given by

p(2)(ρ, x′, ξ′) = σ( [x′ξ′

], Fρ

[x′

ξ′

] )+ p1(ρ),

where ρ = (x′′, ξ′′) ∈ Σ and the vector (x′, ξ′) is identified with (x′, 0, ξ′, 0); sinceP ∗ = P , the corresponding localized operator

Pρ = p(2)(ρ, y,Dy) : S(Rν) −→ S(Rν)

is self-adjoint when regarded as an unbounded operator in L2(Rν) with domain

B2(Rν) = f ∈ S ′(Rν) : yαDβy f ∈ L2(Rν) for |α|+ |β| ≤ 2.

Furthermore, in view of the transversal ellipticity of P , Pρ is globally elliptic in thesense of Helffer [22]. Hence the spectrum of Pρ consists of a sequence of eigenvaluesof finite multiplicity, diverging to +∞, and the corresponding eigenfunctions are

3.2. CONSTRUCTION OF THE POSITIVE AND NEGATIVE PARTS 81

Schwartz functions. Indeed, in [43] Melin explicitly computed the spectrum of Pρshowing that

Spec(Pρ) = mα(ρ), α ∈ Zν+,

where the functions mα are defined in (1.17). It is readily seen that the lowesteigenvalue of Pρ is obtained for α = 0, i.e. m0(ρ) = Tr+Fρ + ps1(ρ).

Hormander [29] proved that P satisfies (1.23) if and only if the lowest eigenvalueTr+Fρ + ps1(ρ) is nonnegative for every ρ ∈ Σ (Theorem 1.1.5) and, hence, if andonly if the spectrum of Pρ is nonnegative for every ρ ∈ Σ (see also Parenti andParmeggiani [53]). Therefore, the operators Π+ and Π− in Theorem 1.3.1 will beconstructed starting from the exact projectors onto the eigenspaces correspondingto the positive and the negative part of Spec(Pρ), respectively. More precisely, inorder to give an idea of the proof of Theorem 1.3.1, we point out that two differentcases can occur: either mα(ρ0) 6= 0 for every α ∈ Zν

+ (Case I) or there existsβ ∈ Zν

+ such that mβ(ρ0) = 0 (Case II).Case I is the easiest to study because the condition mα(ρ) 6= 0 remains true forevery ρ in a small conic neighborhood Γ ⊂ Σ of ρ0. Hence the sum of the negativeeigenspaces E−(ρ) =

⊕λ≤0 Ker[Pρ−λId] is a vector space with finite constant dimen-

sion as ρ = (x′′, ξ′′) varies in Γ ⊂ T ∗Rn−ν . As a consequence, the orthogonal projec-tor π−(ρ) onto E−(ρ) turns out to be a pseudodifferential operator with smooth sym-bol σ(π−(ρ))(x′, ξ′) ∈ C∞(Γ,S(R2ν)

). Then, the approximate negative “projector”

Π− in Theorem 1.3.1 will be obtained by quantizing σ(π−(ρ))(x′, ξ′) ∈ H0(R2n,Σ)in all the variables (x, ξ) = (x′, x′′, ξ′, ξ′′), whereas the approximate positive “pro-jector” will be defined by setting Π+ = Id− Π−.Case II is more delicate since the eigenvalues mβ(ρ) that vanish at ρ = ρ0 can, ingeneral, change sign near ρ0; we thus have to investigate the influence of the positiveand the negative parts of such eigenvalues upon the lower bounds satisfied by P .At this point Hypotheses (h2), (h3) play a crucial role and allow us to apply theconstruction of Fujiwara [17].According to the notation of the Introduction, we denote by Aρ0 the set α ∈ Zν

+ :mα(ρ0) = 0. With this definition, Case I reads as Aρ0 = ∅, whereas Case II asAρ0 6= ∅.

3.2.1 Case I: Aρ0= ∅

There exists a small constant δ > 0 and a conic neighborhood Γ ⊂ Σ of ρ0 suchthat, upon denoting by S∗Γ = (x′′, ξ′′/|ξ′′|) : (x′′, ξ′′) ∈ Γ,

Spec (Pρ) ∩ ((−∞,−δ−1] ∪ [−δ, δ]) = ∅, ∀ρ ∈ S∗Γ.


We then consider, for ρ ∈ S∗Γ, the orthogonal projector

π−(ρ) =1

2πi

∮γ

(ζ − Pρ)−1dζ, (3.7)

where γ is a closed path contained in the domain Re z < δ/2, around the real interval[−δ−1, 0]. For every ρ = (x′′, ξ′′) ∈ Γ we define

π−(x′′, ξ′′) = M|ξ′′|π−(x′′, ξ′′/|ξ′′|)M−1

|ξ′′|, (3.8)

whereMt, t > 0, is the unitary operator in L2(Rν) defined by (Mtf)(y) = tν/4f(t1/2y).It thus follows that, for every ρ ∈ Γ, π−(ρ) is the orthogonal projector on the finitedimensional subspace

⊕λ≤0 Ker[Pρ−λId]. In fact, this is an immediate consequence

of the following homogeneity property of Pρ

P(x′′,tξ′′) = tMtP(x′′,ξ′′)M−1t , ∀t > 0, ∀(x′′, ξ′′) ∈ Σ.

Moreover, an application of Theorem 4.1 of [5] shows that π−(x′′, ξ′′) : S(Rν) →S(Rν) is a pseudodifferential operator with symbol σ

(π−(x′′, ξ′′)

)(x′, ξ′) ∈

C∞(Γ,S(R2ν)).

Choose now as conic neighborhood V in Theorem 1.1 any subset of the type Γε(see (3.5) for this notation), for some conic neighborhood Γ ⊂⊂ Γ of ρ0. Fix twocut-off functions ψ, ψ ∈ S0(R2n) such that ψ is supported in Γε and ψ ∼

∑j≥0 ψj is

a classical symbol with

supp ψ ⊂ Γ2ε, ψ = 1 on Γε. (3.9)

From (3.8) and Theorem 4.1 of [5], it turns out that ψσ(π−(ρ)) ∈ H0(R2n,Σ), witha localized operator given by ψ0(ρ)π

−(ρ). In particular, for every ρ ∈ Γ one hasψ0(ρ) = 1, hence such a localized operator coincides with π−(ρ) and it turns outthat

ψσ(π−)− σ(π−) ∈ S−∞(Γε). (3.10)

Finally the operators Π+ and Π− in Theorem 1.3.1 are defined by

Π− = Op(ψσ(π−)) ∈ OPH0(Rn,Σ), Π+ = Id− Π−. (3.11)

Remark 3.2.1. Actually, with this definition, the operators Π− and Π+ are not self-adjoint. However they are self-adjoint modulo OPH−1(Rn,Σ), namely S := Π− −(Π−+Π−∗)/2 ∈ OPH−1(Rn,Σ) and similarly for Π+. By virtue of (1) in Proposition3.1.10 with m = k = 2 and m′ = −1, one gets PS ∈ OPH0(Rn,Σ) ⊂ L

(L2(Rn)

).

Hence we can limit ourselves to proving (1.18) and (1.19) by setting Π+ and Π− asin (3.11).


Before starting with the proof of (1.20) and (1.21), we put in evidence the fol-lowing useful properties of the operator Π−.

Lemma 3.2.2. One has

Π−2Ψ = Π−Ψ mod OPH−1(Rn,Σ), (3.12)

andΠ−∗P = PΠ− +R, with R ∈ OPH1/2(Rn,Σ). (3.13)

Proof. Formula (3.12) follows from (2) of Proposition 3.1.6. Indeed, by virtue of(3.10) one has that σ(Π−) = σ(π−) + S−∞(Γε) in the conic region Γε, whence (2) ofProp. 3.1.6 yields, in Γε,

σ(Π−2)

= σ(π−

2)+H−1

hom(Γ,Σ) = σ(π−)

+H−1hom(Γ,Σ).

Since supp ψ ⊂ Γε, (3.12) immediately follows from Remark 3.1.3.As regards (3.13), it suffices to study what happens in a neighborhood of a point

of Σ, because Π− and Π∗ are regularizing outside Σ. Now, in such a neighborhoodboth the localized operators of Π−∗P and PΠ− coincide with ψ0(ρ)Pρπ

−(ρ) (where,as above, ψ0 is the principal symbol of ψ, and Pρ is the localized operator of P atρ). Hence, an application of (1) of Proposition 3.1.10 concludes the proof.

Proof of (1.20) and (1.21) (Case I).

Let us prove the estimate

−Re(PΠ−Ψu,Ψu) ≥ c‖Π−Ψu‖21/2 − C‖u‖2

0, ∀u ∈ C∞0 (K). (3.14)

As a consequence of Lemma 3.2.2 and of (1) of Proposition 3.1.10, one has

−(PΠ−Ψu,Ψu) = −(PΠ−2Ψu,Ψu) +O(‖u‖2

0) =

= −(PΠ−Ψu,Π−Ψu) + (Π−Ψu,R∗u) +O(‖u‖20),

= −(PΠ−2Ψu,Π−2

Ψu) + (Π−Ψu,R∗u) +O(‖u‖20), (3.15)

where R∗ ∈ OPH1/2(Rn,Σ) by (3) of Proposition 3.1.6.In order to treat the first term in the r.h.s. of (3.15), we consider the Grushinoperator

G =ν∑j=1

D2xj

+ |x′|2n∑

j=ν+1

D2xj∈ OPN2,2(Rn,Σ)

and observe that, again by (3.12) and by (1) of Proposition 3.1.10, we have

−(PΠ−2Ψu,Π−2

Ψu) = (((Id−Π−∗)G(Id−Π−)−Π−∗PΠ−)Π−Ψu,Π−Ψu)+O(‖u‖20).


Now we are going to apply Proposition 3.1.11 (with m = k = 2) to the operator

(Id− Π−∗)G(Id− Π−)− Π−∗PΠ−.

Indeed, one readily sees that its localized operator at ρ = (x′′, ξ′′) ∈ Γ is given by

(Id− π−(x′′, ξ′′))(|Dy|2 + |y|2|ξ′′|2)(Id− π−(x′′, ξ′′))− P(x′′,ξ′′)(y,Dy)π−(x′′, ξ′′),

which is nonnegative and injective at ρ0 = (x′′0, ξ′′0 ). From Proposition 3.1.11 it thus

follows that, possibly after shrinking Γε, there exist constants c > 0, C > 0 for which

−Re(PΠ−2Ψu,Π−2

Ψu) ≥ c‖Π−Ψu‖21/2 − C‖u‖2

0. (3.16)

As regards the second term in the r.h.s of (3.15), we observe that, for every ε > 0

|(Π−Ψu,R∗u)| ≤ ε‖Π−Ψu‖21/2 + Cε‖R∗u‖2

−1/2

= ε‖Π−Ψu‖21/2 +O(‖u‖2

0).

Hence, if ε is small enough, this term can be re-absorbed by virtue of (3.16).This concludes the proof of (3.14).

Let us now prove (1.20), namely

Re(P (Id− Π−)Ψu,Ψu) ≥ c‖(Id− Π−)Ψu‖21/2 − C‖u‖2

0, ∀u ∈ C∞0 (K). (3.17)

As a consequence of Lemma 3.2.2 and (1) of Proposition 3.1.10 we have

(P (Id− Π−)Ψu,Ψu)

= (P (Id− Π−)Ψu, (Id− Π−)Ψu) + ((Id− Π−)Ψu,R∗Ψu) +O(‖u‖20), (3.18)

with R∗ ∈ OPH1/2(Rn,Σ). Again by (3.12), for every constant ω, we have

(P (Id− Π−)Ψu, (Id− Π−)Ψu)

= ((P + ωΠ−∗〈Dx〉Π−)(Id− Π−)Ψu, (Id− Π−)Ψu) +O(‖u‖20),

where, as usual, 〈Dx〉 = Op((1 + |ξ|2)1/2

). If we choose ω ∈ R such that

ω|ξ′′0 | > −(Tr+Fρ0 + ps1(ρ0)

)= −min

v 6=0

(Pρ0v, v)

‖v‖20

,

with ρ0 = (x′′0, ξ′′0 ), we can then apply Proposition 3.1.11 to the operator P +

ωΠ−∗〈Dx〉Π−. In fact, its localized operator at ρ0 is given by

Pρ0(y,Dy) + ω|ξ′′0 |π−(ρ0),

which is nonnegative and injective. Hence, possibly after shrinking Γε, there existconstants c, C > 0 such that

Re(P (Id− Π−)Ψu, (Id− Π−)Ψu) ≥ c‖(Id− Π−)Ψu‖21/2 − C‖u‖2

0. (3.19)

By arguing as above, one takes advantage of this stronger lower bounds to reabsorbthe term ((Id− Π−)Ψu,R∗Ψu) in the r.h.s. of (3.18).The proof of (3.17) is thus complete.


3.2.2 Case II: Aρ06= ∅

Since the spectrum of Pρ0 consists of eigenvalues with finite multiplicity, Aρ0 is afinite set with cardinality d := #Aρ0 ; hence Aρ0 = α1, . . . , αd ⊂ Zν

+. We canchoose a constant δ > 0 and a conic neighborhood Γ ⊂ Σ of ρ0 such that

±δ 6∈ Spec(Pρ), ∀ρ ∈ S∗Γ,

and

dimW (ρ) = d, ∀ρ ∈ S∗Γ,

where we set

W (ρ) :=⊕|λ| < δ

λ ∈ Spec(Pρ)

Ker[Pρ − λId], ∀ρ ∈ S∗Γ. (3.20)

For any ρ ∈ S∗Γ, we define π−(ρ) by (3.7), where γ is here a closed path containedin the region Re z ≤ −δ, around the real interval [−ω,−δ], ω 1; then, by using(3.8), we extend this definition to all ρ ∈ Γ. It turns out that π−(ρ) is the orthogonalprojector onto the negative eigenspaces of Pρ, except for the ones contained in W (ρ).Finally we set

Π−1 := Op(ψσ(π−)) ∈ OPH0(Rn,Σ), (3.21)

where ψ ∈ S0(R2n) is a classical symbol satisfying (3.9) above.Similarly we define

π(ρ) = M|ξ′′|1

2πi

∮γ

(ζ − Pρ)−1dζ M−1

|ξ′′|, ρ = (x′′, ξ′′) ∈ Γ,

with γ taking values in the region Re z ≤ δ and enclosing the real interval [−ω, δ],ω 1, and we set

Π+1 := Id−Op(ψσ(π)), (3.22)

where Op(ψσ(π)) ∈ OPH0(Rn,Σ).The next result is a consequence of Hypotheses (h2) and (h3).

Proposition 3.2.3.(1) Pρ is regular at ρ0 in the sense of [54] (Appendix 1). Namely, possibly aftershrinking Γ, there exists a smooth morphism of hermitian vector bundles

S∗Γ× Cd U−−−→ W :=⊔ρ∈S∗ΓW (ρ)y y

S∗Γ Id−−−→ S∗Γ


such that, for every ρ ∈ S∗Γ,

(U(ρ)ζ, U(ρ)ζ ′)L2(Rν) = 〈ζ, ζ ′〉Cd ∀ζ, ζ ′ ∈ Cd,

and

U(ρ)∗PρU(ρ) = diag[mα1(ρ), . . . ,mαd(ρ)],

where diag[mα1(ρ), . . . ,mαd(ρ)] is the diagonal matrix with the smooth functions

mαj(ρ), j = 1, . . . , d, as diagonal entries.

(2) KerPρ0 has an orthonormal basis φ1(y), . . . , φd(y) where the φj’s are either alleven or all odd in y ∈ Rν.

Proof. By virtue of (h2), from the invariance of the Weyl calculus ([30], Thm. 18.5.9)and from Hormander’s theorem on the classification of semi-definite quadratic forms([30], Thm. 21.5.3), it follows that there exists a conic neighborhood Γ and a smoothfamily of metaplectic transformations S∗Γ 3 ρ 7→ U(ρ) in L2(Rν), associated withlinear symplectic maps, such that

U(ρ)∗PρU(ρ) =ν∑j=1

µj(ρ)(D2yj

+ y2j ) + ps1(ρ) =: P ′

ρ, ∀ρ ∈ S∗Γ.

Let

hk(t) := π−1/4(2kk!)−1/2

(d

dt− t

)ke−t

2/2, k = 0, 1, . . . ,

be the k−th Hermite function, and define

ϕβ(y) = Πνj=1hβj

(yj), β = (β1, . . . , βν) ∈ Zν+.

We have Spec(P ′ρ) = mα(ρ) : α ∈ Zν

+ and P ′ρϕα(y) = mα(ρ)ϕα(y). In particular,

KerP ′ρ = Spanϕα; α ∈ Aρ. Therefore it suffices to define

U : S∗Γ× Cd 3 (ρ, ζ) 7→ (ρ, U(ρ)(ζ1ϕα1 + . . .+ ζdϕαd)) ∈ W (ρ).

As regards (2), we observe that the statement is trivially true for P ′ρ0

in view of (h3).On the other hand, KerPρ0 = U(ρ0)KerP ′

ρ0and any metaplectic transformation

associated with a linear symplectic map is parity preserving. This concludes theproof.

The regularity and parity properties established in Proposition 3.2.3 ((1) and(2) respectively) allow us to apply a construction due to Parenti and Parmeggiani[53, 54] in the context of lower bounds estimates, that here reads as follows.


Lemma 3.2.4. Possibly after shrinking Γ, there exist functions fj, gj ∈ C∞(Γ,S(Rν)),j = 1, . . . , d, such that

a) fj (resp. gj) are quasi-homogeneous of degree ν/4, (resp. ν/4− 1/2), i.e.

fj(x′′, tξ′′; t−1/2y) = tν/4fj(x

′′, ξ′′; y), ∀t > 0, ∀(x′′, ξ′′) ∈ Γ, ∀y ∈ Rν ,

gj(x′′, tξ′′; t−1/2y) = tν/4−1/2gj(x

′′, ξ′′; y), ∀t > 0, ∀(x′′, ξ′′) ∈ Γ, ∀y ∈ Rν ;

b) fj(ρ, ·)j=1,...,d is an orthonormal basis of W (ρ);

c) the eigenfunctions f0,j satisfies the following parity property

either fj(ρ,−y) = fj(ρ, y), ∀ρ ∈ Γ, ∀y ∈ Rν , ∀j = 1, . . . , d,

or fj(ρ,−y) = −fj(ρ, y), ∀ρ ∈ Γ, ∀y ∈ Rν , ∀j = 1, . . . , d;(3.23)

d) the functions gj, j = 1, ..., d, solve the equations

Pρ(fj(ρ, ·)+gj(ρ, ·))+Rρfj(ρ, ·) = mαj(ρ)(fj(ρ, ·)+gj(ρ, ·)), ∀ρ ∈ Γ, (3.24)

where

Rρ := p(3)(ρ, y,Dy) =∑

|α|+|β|+2j=3

1

α!β!∂αx′∂

βξ′p2−j(ρ)y

αDβy .

We now take a classical symbol ψ′ ∈ S0(Rn−ν×Rn−ν) supported in Γ∩|ξ′′| ≥ 1and satisfying ψ′ = 1 in Γ ∩ |ξ′′| ≥ 2. We then define the properly supportedoperators

Hj = Op(ψ′(fj + gj)) ∈ OPH0(Rn−ν ,Σ′),

and we set

Lj = Op(ψ′mαj) ∈ OPS1(Rn−ν), j = 1, . . . , d.

Remark 3.2.5. It follows from d) of Lemma 3.2.4, from (2) of Prop. 3.1.10 and (3)of Prop. 3.1.9, that

PHj = HjLj mod OPH0(Rn−ν ,Σ′). (3.25)

Property (3.25) will be crucially used in what follows.

Now we observe that the operators Lj ∈ OPS1(Rn−ν) are formally self-adjointmodulo OPS0(Rn−ν). Hence, an application of Theorem 1 of Fujiwara [17] yieldsthe following result.


Proposition 3.2.6. For every j = 1, . . . , d, using microlocalization, one can con-struct three bounded linear self-adjoint operators π+

j , π−j and Tj in L2(Rν) such that

i) for j = 1, . . . , d,π+j + π−j = Id + Tj; (3.26)

ii) there exists a positive constant C such that

Re(π+j Ljv, v) ≥ −C‖v‖2

0, ∀v ∈ S(Rν), (3.27)

−Re(π−j Ljv, v) ≥ −C‖v‖20, ∀v ∈ S(Rν), (3.28)

and‖TjLj‖+ ‖LjTj‖+ ‖[π+

j , Lj]‖+ ‖[π−j , Lj]‖ ≤ C, (3.29)

where [A,B] denotes the commutator of the operators A,B, and ‖A‖ representsthe norm of A as a bounded linear operator in L2(Rν).

Upon denoting

Π−0 =

d∑j=1

Hjπ−j H

∗j , Π+

0 =d∑j=1

Hj(π+j − Tj)H

∗j ,

we define the operators Π+, Π− of Theorem 1.3.1 by

Π+ = Π+0 + Π+

1 , Π− = Π−0 + Π−

1 ,

where Π−1 and Π+

1 are defined in (3.21) and (3.22). Notice that Remark 3.2.1 alsoapplies to this case.

It follows that the localized operator of the pseudodifferential operator Π++Π− isjust the identity operator in L2(Rν) so that R := Id− (Π+ +Π−) ∈ OPH−1/2(Rn,Σ)(see Definition 3.1.4). By virtue of (1) of Proposition 3.1.10, one has PR, RP ∈OPH1/2(Rn,Σ).

Proof of (1.18) and (1.19) (Case II)

We start by observing that Lemma 3.2.2 holds true for the operators Π−1 and Π+

1 .Inequality (1.19), i.e.

−Re(P (Π−0 + Π−

1 )Ψu,Ψu) ≥ −C‖u‖20, (3.30)

is a consequence of the following lower bounds

−Re(PΠ−1 Ψu,Ψu) ≥ −C1‖u‖2

0, (3.31)


and−Re(PΠ−

0 Ψu,Ψu) ≥ −C2‖u‖20. (3.32)

Since the proof of (3.31) goes exactly as the one of (3.14), here we focus only on(3.32). In view of (3.25) we have

−Re(PΠ−0 Ψu,Ψu) = −

d∑j=1

Re(PHjπ−j H

∗jΨu,Ψu) =

= −d∑j=1

Re(Ljπ−j H

∗jΨu,H

∗jΨu) +O(‖u‖2

0)

≥ −C2‖u‖20,

where, for the last inequality, we used the properties (3.28) and (3.29) of Proposition3.2.6, and the fact that H∗

j : L2(Rn) → L2(Rn−ν) is a linear bounded operator.This concludes the proof of (3.32) and therefore (3.30) is verified.

It remains to show (1.18), i.e.

Re(P (Π+0 + Π+

1 )Ψu,Ψu) ≥ −C‖u‖20. (3.33)

If we repeat the proof of (3.17) with Π+1 in place of Id− Π− we obtain

Re(PΠ+1 Ψu,Ψu) ≥ −C‖u‖2

0. (3.34)

On the other hand, by Proposition 3.2.6 and by (3.25) it turns out that

Re(PΠ+0 Ψu,Ψu) =

d∑j=1

Re(PHj(π+j − Tj)H

∗jΨu,Ψu) =

=d∑j=1

Re(Ljπ+j H

∗jΨu,H

∗jΨu) +O(‖u‖2

0)

≥ −C‖u‖20.

This concludes the proof of (3.33) and therefore Theorem 1.3.1 is proved.

Chapter IV

PROOF OF THE WEAK-HORMANDER

INEQUALITY

In this chapter we prove Theorem 1.4.1.The necessity of (1) of Theorem 1.4.1 in order to have (1.28) is a consequence of

Theorem 4.1 of [54], since that condition is necessary for Melin’s inequality to hold,which is weaker than (1.28).Hence we have to prove (1) =⇒ (2) in Theorem 1.4.1. Moreover it is easily seen thatit suffices to prove the microlocal version of (1.28), that is, the following result.

Theorem 4.0.7. Assume the hypothesis of Theorem 1.4.1. Suppose λ0(ρ) ≥ 0 forevery ρ ∈ Σ. Then for every ρ0 ∈ Σ there exists a conic neighborhood V of ρ0 suchthat for every compact K ⊂ X and for every real scalar symbol ψ ∈ S0(Rn × Rn)supported in V there exists C = C(K,V, ψ) > 0 such that

(PΨu,Ψu) ≥ −C‖u‖2(m−3/2)/2, ∀u ∈ C∞

0 (K; CN), (4.1)

with Ψ = ψ(x,D)⊗ IdN×N .

When ρ0 6∈ Σ, estimate (4.1) follows at once from the Garding inequality (see[20]), which applies to microlocally elliptic systems. Hence from now on we willsuppose ρ0 ∈ Σ.

By a known decoupling argument due to Taylor (see [65], page 195, and alsoTheorem 7.1 of [54]), under our hypotheses there exists an elliptic intertwiner E ∈OPS0(X; CN) such that

E∗PE =

[A 00 B

]+ smoothing,

where A = A∗ ∈ OPNm,2(X,Σ; Cl), i.e. the entries of A belong to Sjostrand’sclass OPNm,2(X,Σ) (see [4, 5, 63], and Appendix A below) whereas B = B∗ ∈OPSm(X; CN−l) is positive elliptic. Now, estimate (4.1) holds for P if and only if it

91

92 PROOF OF THE WEAK-HORMANDER INEQUALITY

holds for A. On the other hand, in view of Lemma 4.2 of [54] the localized operatorsof P and A have the same spectrum, and therefore the same lowest eigenvalue. If wetake into account the invariance with respect to similarities given by scalar FIO’s,we are therefore reduced to considering an operator P ∈ OPNm,2(Rn,Σ; CN), with

Σ = (x, ξ); x′ = ξ′ = 0, ξ′′ 6= 0,

where we set x = (x′, x′′) ∈ Rνx′ × Rn−ν

x′′ and accordingly ξ = (ξ′, ξ′′) for the dualvariables.

Now Ker pm(ρ) = CN for every ρ ∈ Σ, and the localized operator L(χ,v)P (y′′, η′′; t, τ)

associated with the map χ = Id and the standard basis v of CN is defined by

LP (x′′, ξ′′; t, τ) =∑

|α|+|β|=2

1

α!β!

(∂αx′∂

βξ′pm

)(0, x′′, 0, ξ′′)tατβ + psm−1(0, x

′′, 0, ξ′′).

A direct computation shows that its Weyl-quantization is given by

Pρ := Opw (LP ) (ρ; t,Dt)

=∑

|α|+|β|=2

1

α!β!

(∂αx′∂

βξ′pm

)(0, x′′, 0, ξ′′)tαDβ

t + pm−1(0, x′′, 0, ξ′′),

with ρ = (x′′, ξ′′) ∈ Σ.Following a strategy due to Parenti and Parmeggiani [54], we are now going

to construct a convenient approximate spectral projector Π that will allow us toobtain the desired lower bound for the quadratic form (PΨu,Ψu) in (4.1), via thedecomposition

(PΨu,Ψu)

= (PΠΨu,ΠΨu) + 2Re(PΠΨu, (Id− Π)Ψu) + (P (Id− Π)Ψu, (Id− Π)Ψu). (4.2)

The construction of Π goes as follows.Let Γ ⊂ Σ be a conic neighborhood of ρ0 and δ > 0 such that (upon writing

S∗Γ = (x′′, ξ′′) ∈ Γ : |ξ′′| = 1),

δ 6∈ Spec(Pρ), ∀ρ ∈ S∗Γ,

anddimW (ρ) = d = dim (KerPρ0) , ∀ρ ∈ S∗Γ,

withW (ρ) :=

⊕0 ≤ λ < δ

λ ∈ Spec(Pρ)

Ker[Pρ − λId], ∀ρ ∈ S∗Γ. (4.3)

93

We then consider, for every ρ ∈ S∗Γ, the operator

π(ρ) =1

2πi

∮|λ|=δ

(ζ − Pρ)−1dζ,

and for ρ = (x′′, ξ′′) ∈ Γ we set

π(x′′, ξ′′) = M|ξ′′|π(x′′, ξ′′/|ξ′′|)M−1|ξ′′|, (4.4)

where Mt, (t > 0), are the unitary operators in L2(Rν) defined by Mtf(y) =tν/4f(t1/2y).

We have the following result.

Proposition 4.0.8.

(1) For every ρ ∈ Γ, the operator π(ρ) is the orthogonal projector on W (ρ);

(2) The symbol σ(π) of the parameter-dependent operator π(ρ) is inH0hom(Γ,Σ; CN)

(see Definition 4.A.4).

Proof. (1) The statement follows from the very definition of π(ρ) when ρ ∈ S∗Γ.For general ρ ∈ Γ it is a consequence of the following homogeneity property of thelocalized operator

Px′′,tξ′′ = tm−1MtPx′′,ξ′′M−1t , ∀t > 0, ∀(x′′, ξ′′) ∈ Σ,

which holds in view of (14) of [54].(2) The required homogeneity property follows at once from (4.4). Let us now

prove that σ(π) ∈ C∞(Γ,S(R2ν ; MatN×N(C))). To this end, let

Sµ(R2ν ; CN) = a ∈ C∞(R2ν ; MatN×N(C)) : |∂γz a(z)|MatN×N (C) ≤ Cγ(1 + |z|)µ−|γ|,

and let C = λ ∈ C : |λ| = δ.The operator λ − Pρ has a symbol in C∞(C × S∗Γ,Sk(R2ν ; CN)). Moreover it isglobally elliptic (see [22]) and also invertible for ρ ∈ S∗Γ, λ ∈ C. As a con-sequence of Theorem 4.1 of [5], the inverse operator is pseudodifferential with asymbol in C∞(C × S∗Γ;S−k(R2ν ; CN)). The operator π(ρ), ρ ∈ S∗Γ, is there-fore pseudodifferential, with a symbol in C∞(S∗Γ;S−k(R2ν ; CN)). Since π(ρ) =π(ρ)n, for every n = 1, 2, . . ., as an operator, say, on S(Rν ; CN), it follows thatits symbol σ(π) ∈ C∞(S∗Γ,S−nk(R2ν ; CN)) for every n ∈ Z+, and therefore it isin C∞(S∗Γ,S(R2ν ; MatN×N(C))). This fact and the above-mentioned homogeneityproperty conclude the proof.


Consider now any open conic neighborhood Γ ⊂⊂ Γ and ε > 0, and take

V := (x, ξ) : (x′′, ξ′′) ∈ Γ, |x′|+ |ξ′|/|ξ′′| < ε

as conic neighborhood in Theorem 4.0.7. Let therefore ψ ∈ S0(Rn×Rn) be any realsymbol supported in V .

Consider now a classical symbol ψ ∈ S0(Rn×Rn) supported in (x, ξ) : (x′′, ξ′′) ∈Γ, |x′|+ |ξ′|/|ξ′′| < 2ε, |ξ′′| > 1/ε, and satisfying ψ = 1 on V ∩ |ξ′′| > 2/ε.We define the properly supported operator (see Definition 4.A.2)

Π = Op(ψσ(π)) ∈ OPH0(Rn,Σ; CN).

Here are some properties of the operator Π.

Lemma 4.0.9. We have

Π2Ψ = ΠΨ mod OPH−1(Rn,Σ; CN); (4.5)

Π∗ΠΨ = ΠΨ mod OPH−1(Rn,Σ; CN); (4.6)

Proof. These formulas easily follow from Proposition 4.A.7, for the symbol of Πcoincides with the one of π in V ∩ |ξ′′| > 2/ε.

We can now separately estimate the three terms in (4.2).

Estimating (PΠΨu,ΠΨu).Let us observe that, by (4.5), Proposition 4.A.8 (with k = 2, m′ = −1) and thecontinuity properties of Hermite operators (Proposition 4.A.3 with m − 2 in placeof m) we have

(PΠΨu,ΠΨu) = (PΠ2Ψu,ΠΨu) +O(‖u‖2(m−2)/2).

Then, by Proposition 4.A.8 (with k = 2, m′ = 0) we obtain

(PΠΨu,ΠΨu) = (Op(G)ΠΨu,ΠΨu) +O(‖u‖2(m−3/2)/2),

whereG(x′′, ξ′′) = ψ(0, x′′, 0, ξ′′)Px′′,ξ′′π(x′′, ξ′′) ∈ L(L2(Rν ; CN)),

and “Op” is understood in the vector valued sense. Indeed, it follows from Remark4.A.5 that

G ∈ Sm−11,0 (Rn−ν × Rn−ν ;L(L2(Rν ; CN))).

Moreover, for every (x′′, ξ′′) ∈ Rn−ν×Rn−ν the operator G(x′′, ξ′′) is self-adjoint andnonnegative in L2(Rν ; CN).

95

Without loss of generality we may assume m ≥ 2. Then it follows from theSharp-Garding inequality for vector valued symbols (Theorem 18.6.14 of [30]) thatthere exists a constant C > 0 such that

Re (Op(G)ΠΨu,ΠΨu) ≥ −C‖u‖2(m−2)/2, ∀u ∈ C∞

0 (Rn).

Hence we obtain the estimate

(PΠΨu,ΠΨu) ≥ −C‖u‖2(m−3/2)/2.

Estimating (P (Id− Π)Ψu, (Id− Π)Ψu).Define the operator

Π = Π∗ = Π∗Λm−1Π ∈ OPHm−1(Rn,Σ; CN),

where we set Λs = Op((1 + |ξ|2)s/2IdMatN×N (C)

).

We claim that

(P (Id−Π)Ψu, (Id−Π)Ψu) = ((P+Π)(Id−Π)Ψu, (Id−Π)Ψu)+O(‖u‖2(m−2)/2). (4.7)

Indeed, from (4.5) it follows that

Π(Id− Π)Ψ ∈ OPHm−2(Rn,Σ; CN).

Now we observe that, according to Definition 4.A.6, the localized operator of P + Πat ρ0 = (x′′0, ξ

′′0 ) is given by

Px′′0 ,ξ′′0 + |ξ′′0 |m−1π(x′′0, ξ′′0 ),

which is nonnegative and clearly invertible as an operator on S(Rν ; CN). Hence itfollows from (4.7) and Proposition 4.A.9 (with k = 2) that possibly after shrinkingV , there exist constants c > 0, C > 0 such that

(P (Id− Π)Ψu, (Id− Π)Ψu) ≥ c‖u‖2(m−1)/2 − C‖u‖2

(m−2)/2, ∀u ∈ C∞0 (Rn). (4.8)

Estimating Re(PΠΨu, (Id− Π)Ψu).We have

(PΠΨu, (Id− Π)Ψu) = ((Id− Π∗)ΠPΨu,Ψu) + (RΨu, (Id− Π)Ψu), (4.9)

where RΨ = [P,Π]Ψ ∈ OPHm−3/2(Rn,Σ; CN) by Proposition 4.A.8 and Definition4.A.6. Indeed, the localized operator of R, regarded in OPHm−1(Rn,Σ; CN), is


exactly Pρπ(ρ)− π(ρ)Pρ = 0 when ρ ∈ Γ.Now, by (4.6) we have

((Id− Π∗)ΠPΨu,Ψu) = O(‖u‖2(m−2)/2),

whereas for every ε > 0 it turns out that

|(RΨu, (Id− Π)Ψu)| ≤ ‖RΨu‖−(m−1)/2‖(Id− Π)Ψu‖(m−1)/2

≤ ε‖(Id− Π)Ψu‖2(m−1)/2 +

1

ε‖u‖2

(m−2)/2.

Then we can take advantage of (4.8) to reabsorb this term if ε is chosen smallenough.


Remark 4.0.10. In estimating the rectangle term Re(PΠΨu, (Id−Π)Ψu) one couldignore the second expression in the right hand-side of (4.9), because it is indeedO(‖u‖2

(m−3/2)/2). However we preferred to put in evidence that the only error which

is O(‖u‖2(m−3/2)/2) and not a-priori better (namely O(‖u‖2

(m−2)/2)) comes from the

term (PΠΨu,ΠΨu).

APPENDICES

4.A Hermite operators with matrix valued sym-

bols

We recall the definition and the main properties of Sjostrand’s classes OPNm,k(X,Σ; CN)and of the so-called Hermite operators with matrix valued symbols; we address thereader to [4, 5, 21, 54, 63] for details.

All the operators are properly supported.

Definition 4.A.1. Let Σ ⊂ T ∗X \ 0 be a smooth closed conic manifold. We denoteby OPNm,k(X,Σ; CN), m ∈ R, k ∈ N, the space of N ×N systems of classical pseu-dodifferential operators with symbol p ∼

∑j≥0 pm−j satisfying, near Σ, the following

vanishing conditions

|pm−j(x, ξ)|MatN×N (C) ≤ C|ξ|m−jdistΣ(x, ξ/|ξ|)k−2j, for j ≤ k/2.

From now on we will take

Σ = (x, ξ) = (x′, x′′, ξ′, ξ′′) : x′ = ξ′ = 0, ξ′′ 6= 0,

where we set x = (x′, x′′) ∈ Rn = Rνx′ × Rn−ν

x′′ and accordingly ξ = (ξ′, ξ′′).

Definition 4.A.2. By Sm,k(Rn × Rn,Σ; CN), (m, k ∈ R), we denote the space ofsmooth functions Rn ×Rn 3 (x, ξ) 7−→ a(x, ξ) ∈ MatN×N(C) which are in Sm awayfrom Σ and satisfy the following estimates

|∂α′x′ ∂α′′

x′′ ∂β′

ξ′ ∂β′′

ξ′′ a(x, ξ)|MatN×N (C) ≤ C|ξ′′|m−|β|(|x′|+ |ξ′|

|ξ′′|+

1

|ξ′′|1/2

)k−|α′|−|β′|,

(4.10)(with α = (α′, α′′) and β = (β′, β′′)) microlocally near Σ.Moreover we set Hm(Rn × Rn,Σ; CN) := ∩j≥0S

m−j,k−2j(Rn × Rn,Σ; CN). As usualOPSm,k(Rn,Σ; CN) and OPHm(Rn,Σ; CN) denote the classes of the correspondingpseudodifferential operators.

97


Proposition 4.A.3. Operators in OPHm(Rn,Σ; CN) map

H tcomp(Rn; CN) → H t−m

comp(Rn; CN)

continuously for any t ∈ R.

Set

Sµ(R2ν ; CN) = a ∈ C∞(R2νz ; MatN×N(C)) : |∂γz a(z)|MatN×N (C) ≤ Cγ(1 + |z|)µ−|γ|,

(see [22]).

Definition 4.A.4. For an open conic subset Γ ⊂ Σ, we denote by Sm,khom(Γ,Σ; CN)

(resp. Hm−k/2hom (Γ,Σ; CN) ) the class of all smooth functions in C∞(Γ,Sk(R2ν ; CN))

(resp. C∞(Γ,S(R2ν ; MatN×N(C)))) such that

a(x′′, tξ′′; t−1/2x′, t1/2ξ′) = tm−k/2a(x′′, ξ′′;x′, ξ′), t > 0.

Microlocally near Σ, symbols in Sm,khom(Γ,Σ; CN) (resp. Hm−k/2hom (Γ,Σ; CN)) are in

Sm,k(Rn × Rn,Σ; CN) (resp. Hm−k/2(Rn × Rn,Σ; CN)).In

Remark 4.A.5. Often times it is useful to quantize symbols inHmhom(Γ,Σ; CN) with

respect to the variables x′, ξ′ alone, and regard them as operator valued symbols.More precisely, if a ∈ Hm

hom(Γ,Σ; CN), with Γ = Ω×(Rn−ν \0), for an open subsetΩ ⊂ Rn−ν , then for every α′′, β′′ ∈ Zn−ν

+ , Ω′ ⊂⊂ Ω, there exists C > 0 such that

‖∂α′′x′′ ∂β′′

ξ′′ a(x′′, ξ′′; y,Dy)‖L(L2(Rν ;CN )) ≤ C|ξ′′|m−|β′′|,

for any (x′′, ξ′′) ∈ Ω′ × (Rn−ν \ 0). This follows from the following homogeneityproperty (

∂α′′

x′′ ∂β′′

ξ′′ a)

(x′′, tξ′′; y,Dy) = tm−|β′′|Mta(x

′′, ξ′′; y,Dy)M−1t , t > 0,

where Mt is the unitary operator in L2(Rν ; CN) defined by (Mtf)(y) = tν/4f(t1/2y),

and from the fact that the map (x′′, ξ′′) 7−→ ∂α′′

x′′ ∂β′′

ξ′′ a(x′′, ξ′′; ·) takes values in a

bounded subset of S0(R2ν ; CN) when (x′′, ξ′′) ∈ Ω′ × (Rn−ν \ 0) and |ξ′′| = 1.

We now define a class of symbols with an asymptotic expansion in quasi-homogene-ous terms. (We take Γ = Σ.)

4.A. HERMITE OPERATORS WITH MATRIX VALUED SYMBOLS 99

Definition 4.A.6. We say that a symbol a ∈ Sm,k(Rn × Rn,Σ; CN) (resp. inHm(Rn × Rn,Σ; CN)) has an asymptotic expansion in quasi-homogeneous terms, if

there exist functions a(k+j) ∈ Sm,k+jhom (Σ,Σ; CN), j ≥ 0, (resp. a(j) ∈ Hm−j/2hom (Σ,Σ; CN))

such that a −∑N

j=0 a(k+j) ∈ Sm,k+N+1(Rn × Rn,Σ; CN) (resp., a −

∑Nj=0 a

(j) ∈Hm−(N+1)/2(Rn × Rn,Σ; CN)) microlocally near Σ.

The symbol a(k) ∈ Sm,khom(Σ,Σ; CN) (resp. a(0) ∈ Hmhom(Σ,Σ; CN)) is called the

localized symbol of the operator Op(a) and a(k)(ρ, y,Dy) ∈ OPSk(Rν ; CN) (resp.a(0)(ρ, y,Dy) ∈ OPS−∞(Rν ; CN)) is called the localized operator at ρ = (x′′, ξ′′) ∈Σ.

For example, if P = Op(p) ∈ OPNm,k(Rn,Σ; CN), then P ∈ OPSm,k(Rn,Σ; CN),and p admits an asymptotic expansion

∑j≥0 p

(k+j) with

p(k+j)(x′′, ξ′′;x′, ξ′) :=∑

|α|+|β|+2l=k+j

1

α!β!∂αx′∂

βξ′pm−l(0, x

′′, 0, ξ′′)x′αξ′β.

Here are some properties of these classes of operators (see [21]).

Proposition 4.A.7.(1) If A = Op(a) ∈ OPSm,k(Rn,Σ; CN), and B = Op(b) ∈ OPSm

′,k′(Rn,Σ; CN),then AB = Op(c) ∈ OPSm+m′,k+k′(Rn,Σ; CN). Moreover, if a ∼

∑j≥0 a

(k+j), and

b ∼∑

j≥0 b(k′+j), in the sense of Definition 4.A.6, with a(k+j) ∈ Sm,khom(Σ,Σ; CN),

b(k′+j) ∈ Sm

′,k′

hom (Σ,Σ; CN), then c ∼∑

r≥0 c(k+k′+r), with

c(k+k′+r) =

∑i+j+2|α|=r

1

α!∂αξ′′a

(k+i)#Dαx′′b

(k′+j). (4.11)

(we denoted by ∂αξ′′a(k+i)#Dα

x′′b(k′+j) the symbol of the product of the operators

(∂αξ′′a(k+i))(ρ, y,Dy) and (Dα

x′′b(k′+j))(ρ, y,Dy) on S(Rν

y ; CN)).

(2) If A = Op(a) ∈ OPHm(Rn,Σ; CN), and B = Op(b) ∈ OPHm′(Rn,Σ; CN),

then AB = Op(c) ∈ OPHm+m′(Rn,Σ; CN). Moreover, if a ∼

∑j≥0 a

(j), and

b ∼∑

j≥0 b(j), in the sense of Definition 4.A.6, with a(j) ∈ Hm−j/2

hom (Σ,Σ; CN),

b(j) ∈ Hm′−j/2hom (Σ,Σ; CN), then c ∼

∑r≥0 c

(r), with

c(r) =∑

i+j+2|α|=r

1

α!∂αξ′′a

(i)#Dαx′′b

(j). (4.12)

(3) If A = Op(a) ∈ OPHm(Rn,Σ; CN) then A∗ = Op(a∗) ∈ OPHm(Rn,Σ; CN).

Moreover, if a ∼∑

j≥0 a(j) in the sense of Definition 4.A.6, with a(j) ∈ Hm−j/2

hom (Σ,Σ; CN),


then a∗ ∼∑

r≥0 a∗(r), with

a∗(r) =∑

j+2|α|=r

1

α!Dαξ′′∂

αx′′(a

(j))∗,

where (a(j))∗ denotes the symbol of the operator a(r)(ρ, y,Dy)∗ on S(Rν

y ; CN).

In particular we have the following

Proposition 4.A.8. Let P = Op(p) ∈ OPNm,k(Rn,Σ; CN) and B = Op(b) ∈OPHm′

(Rn,Σ; CN) with b ∼∑

j≥0 b(j) in the sense of Definition 4.A.6, with b(j) ∈

OPHm′−j/2hom (Σ,Σ; CN). Then PB and BP are in OPHm+m′−k/2(Rn,Σ; CN) and have

localized symbols p(k)(ρ, ·)#b(0)(ρ, ·) and b(0)(ρ, ·)#p(k)(ρ, ·) respectively.

The following result is due to Parenti and Parmeggiani [54]; it can be regardedas a generalization of Mohamed’s Melin inequality [44].

Fix the following notation: for an open conic subset Γ ⊂ Σ and ε > 0 we set

Γε = (x, ξ) ∈ Rn : (x′′, ξ′′) ∈ Γ, |x′|+ |ξ′|/|ξ′′| < ε.

Proposition 4.A.9. Let P = P ∗ ∈ OPNm,k(Rn,Σ; CN) satisfy (1.25) and thesecond condition in (1.26) (with l = N), and let B = B∗ ∈ OPHm−k/2(Rn,Σ; CN)with localized operator b(0)(ρ, x′, Dx′) ∈ OPS−∞(Rν ; CN). Fix ρ0 ∈ Σ and suppose:

(1)((p(k)(ρ0, x

′, Dx′) + b(k)(ρ0, x′, Dx′)

)f, f)≥ 0, ∀f ∈ S(Rν ; CN);

(2) p(k)(ρ0, x′, Dx′) + b(k)(ρ0, x

′, Dx′) : S(Rν ; CN) → S(Rν ; CN) is invertible.

Then there exist a conic neighborhood Γ ⊂ Σ of ρ0 and ε > 0 such that for everyscalar symbol ψ ∈ S0(Rn × Rn) supported in Γε there exist constants c, C > 0 forwhich

((P +B)Ψu,Ψu) ≥ c‖Ψu‖2m/2−k/4 − C‖u‖2

m/2−k/4−1/2, ∀u ∈ S(Rn; CN), (4.13)

with Ψ = ψ(x,D)⊗ IdN×,N .

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