Wave front set of solutions to sums of squares of vector fields

91 Pages·2013·0.8 MB·English

by  Paolo Albano, Antonio Bove

MEMOIRS
of the
American Mathematical Society
Volume 221 • Number 1039 (third of 5 numbers) • January 2013
Wave Front Set of Solutions
to Sums of Squares
of Vector Fields
Paolo Albano
Antonio Bove
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Number 1039
Wave Front Set of Solutions
to Sums of Squares
of Vector Fields
Paolo Albano
Antonio Bove
January2013 • Volume221 • Number1039(thirdof5numbers) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Albano,Paolo,1969-
Wavefrontsetofsolutionstosumsofsquaresofvectorfields/PaoloAlbano,AntonioBove.
p. cm. –(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;number1039)
“January2013,volume221,number1039(thirdof5numbers).”
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-7570-4(alk. paper)
1. Wavelets (Mathematics) 2. Summability theory. 3. Vector fields. I. Bove, Antonio.
II.Title.
QA403.3.A426 2013
515(cid:2).2433–dc23 2012035212
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Contents
Chapter 1. Introduction vii
Chapter 2. The Poisson–Treves Stratification 1
2.1. Analytic Stratification of an Analytic Set 1
2.2. Symplectic Stratification of an Analytic Submanifold 2
2.3. Poisson Stratification 4
2.4. Poisson Stratification Associated to Vector Fields 5
Chapter 3. Standard Forms for a System of Vector Fields 7
3.1. The Symplectic Case of Depth >1 7
3.2. The Symplectic Case of Depth 1 10
3.3. The Nonsymplectic Case of Depth >1 16
3.4. The Nonsymplectic Case of Depth 1 19
Chapter 4. Nested Strata 27
Chapter 5. Bargman Pseudodifferential Operators 29
5.1. The FBI Transform 29
5.2. Pseudodifferential Operators 30
5.3. Some Pseudodifferential Calculus 31
Chapter 6. The “A Priori” Estimate on the FBI Side 33
6.1. Proof of Theorem 6.1 34
6.2. First Part of the Estimate: Estimate from Below 34
6.3. Second Part of the Estimate: Estimate from Above 35
Chapter 7. A Single Symplectic Stratum 41
7.1. dimΣ=2 and X ,...,X Quasi-homogeneous 44
1 N
7.2. codimΣ>2 46
7.3. One Symplectic Stratum of Depth 1 47
Chapter 8. A Single Nonsymplectic Stratum 51
(cid:2)
8.1. The Case rankσ(cid:2) =2 and X Quasi-homogeneous 52
i
Char(P)
8.2. The Transversally Elliptic Case 53
8.3. A Class of Nontransversally Elliptic Operators 54
Chapter 9. Microlocal Regularity in Nested Strata 55
9.1. Symplectic Stratifications 55
9.2. A Case of Nonsymplectic Stratification 57
9.3. A Case of Two Strata 60
iii
iv CONTENTS
Chapter 10. Known Cases and Examples 61
10.1. The Case of codimΣ=2 61
10.2. Okaji’s Theorem 62
Appendix A. A Bracket Lemma 63
Appendix B. Nonsymplectic Strata Do Not Have the Reproducing Bracket
Property 69
Bibliography 71
Index 73
Abstract
We study the (micro)hypoanalyticity and the Gevrey hypoellipticity of sums
of squares of vector fields in terms of the Poisson–Treves stratification. The FBI
transform is used. We prove hypoanalyticity for several classes of sums of squares
and show that our method, though not general, includes almost every known hy-
poanalyticity result. Examples are discussed.
ReceivedbytheeditorNovember2,2010.
ArticleelectronicallypublishedonMay21,2012;S0065-9266(2012)00663-0.
2010 MathematicsSubjectClassification. Primary35A18;Secondary35H10,35H20.
Keywordsandphrases. AnalyticHypoellipticity,FBITransform,WaveFrontSet,Canonical
Forms.
(cid:2)c2012 American Mathematical Society
v

CHAPTER 1
Introduction
This paper is concerned with the problem of the analytic regularity of the
solutions to sums of squares operators. More precisely let X (x,D),...,X (x,D)
1 N
be N vector fields in Rn with coefficients belonging to Cω(U), the space of all real
analytic functions defined in the open subset U ⊂Rn.
We are concerned with the analytic regularity of the solutions for the operator
(cid:3)N
(1.1) P(x,D)= X (x,D)a (x,D)X (x,D)
i ij j
j=1
(cid:3)N
+ b (x,D)X (x,D)+c(x,D),
j j
j=1
where a , b , c are real analytic pseudodifferential operators of order zero and
ij j
moreover
(1.2) a (x,ξ)+a (x,ξ)>0,
ij ji
where a (x,ξ) denotes the principal symbol of a (x,D).
ij ij
In some cases it is not difficult to see that the vector fields may be replaced by
first order pseudodifferential operators, but we will in general stick to the vector
fields case, which is the most popular and rich in structure.
Condition (1.2) just implies that the characteristic set for the operator P,
Char(P), is the characteristic set associated to the vector fields
Char(X)={(x,ξ) | X (x,ξ)=0, j =1,...,N},
j
X (x,ξ) denoting the symbol of the vector field X .
j j
WearealwaysassumingthatthevectorfieldsX ,...,X satisfytheH¨ormander
1 N
condition
(H) TheLie algebrageneratedby thevectorfields andtheir commutatorshas
dimension n, equal to the dimension of the ambient space.
TheimportanceofCondition(H)isduetothepaper[15],byL.Ho¨rmander,where
it is proved that (H) is sufficient for C∞ hypoellipticity. This means that if Pu =
f ∈C∞(U), for a certain distribution u on U, then u is actually smooth on U, i.e.
u∈C∞(U).
Using Nagano’s Theorem it has been shown by Derridj ([9]) that Condition
(H) is also necessary for C∞ hypoellipticity, provided the coefficients of the vector
fields have real analytic regularity.
AsafurtherstepinstudyingthehypoellipticitypropertiesofP onemay ask if
it is analytic hypoelliptic, i.e. if Pu=f ∈Cω for a certain distribution u∈D(cid:3)(U)
implies that actually u∈Cω(U).
vii
viii 1. INTRODUCTION
In this case the answer is difficult and in fact not much is known. We briefly
mention the most important results on this problem, but we stress the fact that
the list is by no means exhaustive or complete.
First ofallintheearlyseventiesM.S.BaouendiandC.Goulaouic, [2], showed
thatthereareoperatorswithrealanalyticcoefficients, satisfying(H),andthusC∞
hypoelliptic, that are not analytic hypoelliptic, thus wreaking havoc in this part of
the theory.
Here is a short list of the major positive results proved since then.
In 1978 D. S. Tartakoff [25] and F. Treves [28] independently proved that if P
vanishesexactlyofthesecondorderonChar(P)andifChar(P)isasymplecticCω
submanifold of the cotangent bundle, then P is analytic hypoelliptic. A different
proof has been given in 1983 by J. Sj¨ostrand in [22], using FBI methods.
We recall in passing that a submanifold is symplectic if the restriction of the
symplectic form, σ =dξ∧dx, to the submanifold in question has rank equal to its
dimension. Anotherway of statingthe latter factis tosay that the 2-form induced
by σ on Char(P) is non degenerate. Moreover we say that P vanishes exactly of
the second order on Char(P)if dimkerF (ρ)=dimChar(P),for every ρ∈W, W
p
conic open set of Char(P). Here F (ρ)=dH (ρ) is the Hamilton matrix of P at ρ
p p
and H (ρ)=∂ p(ρ)∂ −∂ p(ρ)∂ denotes the usual Hamilton field of the principal
p ξ x x ξ
symbol p of P.
We note that the operator studied by Baouendi and Goulaouic has a charac-
teristic set which is a nonsymplectic manifold.
In 1980 M´etivier, following the ideas of F. Treves, showed that if P is a
(pseudo)differentialoperatorofsymbolp(x,ξ)=pm(x,ξ)+pm−1(x,ξ)+···,where
pm vanishesexactlytotheordermonasymplecticrealanalyticmanifoldandpm−j
vanishes atleast totheorder (m−2j) 1 on the same manifold, thenP is analytic
+
hypoelliptic. The vanishing conditions on the lower order terms are also called the
Levi conditions.
A few years later Okaji extended M´etivier’s results to the case when p ∼
m
(|ξ |+|x|μ)m for a certain positive integer μ. He also needed Levi conditions,
whichareslightlymoreinvolved; wedonotstatethemhere,sinceweareinterested
in the second order case with no given lower order terms.
On the other hand Oleinik and Radkeviˇc, in [19] and [20] and later Christ,
[7], Bove and Tartakoff, [3], showed that there are operators, whose characteristic
variety is a symplectic manifold, which are not analytic hypoelliptic.
Moreover in 1991 Hanges and Himonas, [12], and Christ, [6], considered the
operator
(cid:4) (cid:5)
∂2+ ∂ −xm−1∂ 2 in R3.
x y t
If m = 2 the above operator is known to be analytic hypoelliptic, due to the
Tartakoff–Treves theorem. In [12] it was shown that for m = 3,5,7,... the above
operator fails to be analytic hypoelliptic and in [6] it was proved that the operator
fails to be analytic hypoelliptic for every integer m ≥ 3. Note that the above
operator corresponds to the Kohn’s Laplacian for a degenerate three dimensional
CR manifold.
In an attempt to elucidate and unify the theory F. Treves in 1999, [29], intro-
duced the concept of stratification associated toa number of vector fields withreal
1k+ heredenotesk ifk≥0and0ifk<0.
1. INTRODUCTION ix
analytic coefficients. The stratification, henceforth called Poisson–Treves stratifi-
cation, is a disjoint union of real analytic manifolds, called the strata.
Below(seeChapter2)wegivemoredetailsonthePoisson–Trevesstratification
following the approach of [5].
In [29] the author formulated the
(cid:6)
Conjecture 1.1. The operator P = N X2(x,D), where the X are vector
j=1 j j
fields with real analytic coefficients satisfying Ho¨rmander’s condition, is analytic
hypoelliptic if and only if each stratum in the stratification associated to P is a
symplectic manifold.
TrevesformulatedalsosomevariantsoftheaboveConjecturein[30],butclearly
the above conjecture is microlocal rather than local.
Conjecture 1.1, to our knowledge, is neither proved nor disproved. We must
however say that it agrees with all the known theorems.
Inthispaperwe consider mainly operatorswithasingle stratum, althoughthe
techniquemayapplytomulti-strataoperatorsaswell. Letusconsidersinglestrata
first.
Insteadof provinglocal analytichypoellipticitywe focusonamicrolocalstate-
ment of the form:
Assume that P as given in (1.1) has a characteristic set whose stratification
is the (disjoint) union of just single strata. This means that in a microlocal neigh-
borhood of a given point (x ,ξ ) the characteristic set is actually a real analytic
0 0
manifold, Σ, where the symplectic form has constant rank. Denote by F the
(x0,ξ0)
Hamiltonleafgoing throughthepoint(x ,ξ ). Denote by U a(conic) neighborhood
0 0 0
of (x ,ξ ) in Σ. Then if (x ,ξ ) ∈/ WF (Pu) and ∂U ∩F ∩WF (u) = ∅,
0 0 0 0 a 0 (x0,ξ0) a
we can conclude that (x ,ξ ) ∈/ WF (u). Here ∂U denotes the boundary of U as
0 0 a 0 0
a set.
In particular if Σ is a symplectic manifold, i.e. if the rank of the restriction of
the symplectic form σ to Σ is maximal, then the Hamilton leaf F ={(x ,ξ )},
(x0,ξ0) 0 0
so that ∂U ∩F ∩WF (u) is always empty and thus (x ,ξ )∈/ WF (u).
0 (x0,ξ0) a 0 0 a
Actually,fortechnicalreasons,wehavetomakeanassumptiononthefoliation,
in the case of a non symplectic stratum: we assume that, roughly speaking, the
foliation has an injective projection from Σ onto the base of T∗Rn. This rules out
sums of squares like the M´etivier operator:
D2+(x2+y2)D2.
x y
Basically the method of proof relies on deducing an a priori estimate for the op-
erator once we take an FBI transform of it. Thus the a priori estimate has to
be in spaces suitable for the FBI transform, such as those given by weighted L2
norms of holomorphic functions defined on open subsets of Cn. A general estimate
of this kind, being the counterpart of the subelliptic estimate of H¨ormander and
Rothschild–Stein, has been proved in [1]. To make the paper more readable we
shall give below an idea of the proof of such an estimate.
Unfortunately this type of estimate is not enough since any estimate leading
to a (propagation of) regularity theorem has to be a uniqueness estimate, i.e. an
estimate ensuring that, when the data are zero in some sense, the solution is zero
in the same sense.
This isthereasonwhy weuse adeformationargument resembling amicrolocal
version of Holmgren theorem. Such an argument has been used first in the FBI