Table Of ContentNonelliptic Partial Differential Equations
Developments in Mathematics
VOLUME 22
SeriesEditors:
KrishnaswamiAlladi,Universityof Florida
Hershel M.Farkas,HebrewUniversityofJerusalem
Robert Guralnick,UniversityofSouthernCalifornia
Forfurthervolumes:
http://www.springer.com/series/5834
David S. Tartakoff
Nonelliptic Partial
Differential Equations
Analytic Hypoellipticity and the Courage
to Localize High Powers of T
123
DavidS.Tartakoff
SMorganSt851
60607-7042ChicagoIllinois
USA
dst@uic.edu
ISSN1389-2177
ISBN978-1-4419-9812-5 e-ISBN978-1-4419-9813-2
DOI10.1007/978-1-4419-9813-2
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2011931713
(cid:2)c SpringerScience+BusinessMedia,LLC2011
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein
connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,
orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Contents
1 WhatThisBookIsandIsNot ............................................ 1
2 BriefIntroduction .......................................................... 5
3 OverviewofProofs ......................................................... 7
3.1 AFewPreliminaryDefinitions..................................... 7
3.2 EllipticEquationsandBoundaryValueProblems................. 8
3.3 TheSimplestSubellipticCase ..................................... 9
3.4 SubellipticEstimates ............................................... 11
3.5 LocalC1Regularity............................................... 13
3.6 ProvingC1Regularity............................................. 14
3.7 GevreyRegularity................................................... 15
3.8 EllipticOperators................................................... 16
3.8.1 SymmetrizationoftheEstimates......................... 16
3.8.2 ProofviaN theNormEstimate ............................. 19
3.9 NonellipticOperators............................................... 20
3.9.1 TheBaouendi–GoulaouicExample;Sharpness.......... 20
3.10 TheAnalyticityProblemandItsSolution ......................... 21
3.10.1 ObstructionstoProvingAnalyticity...................... 21
3.10.2 WhytheMostNaiveApproachFails..................... 21
3.10.3 TheFlavorofOurMethods............................... 23
3.10.4 TheConstructionof.Tp/ ................................ 25
'
3.11 TheRoleofStrictness .............................................. 28
3.12 Treves’Conjecture.................................................. 28
3.12.1 AParticularCase.......................................... 29
3.13 CounterexamplesintheComplexDomain ........................ 30
3.14 AHEforD2 Cx2kD2 (No.Tp/ Needed!)..................... 31
x1 1 x2 '
v
vi Contents
4 FullProoffortheHeisenbergGroup .................................... 33
4.1 TheModelOperator................................................ 33
4.1.1 T Derivatives.............................................. 34
4.1.2 Z Derivatives.............................................. 35
4.2 TheEndoftheProoffortheHeisenbergGroup................... 37
5 Coefficients ................................................................. 41
5.1 HowSpecialIstheHeisenbergModel?............................ 41
5.2 RigidCoefficients:T Derivatives.................................. 42
5.3 OurEstimatesandHowWeUseThem............................ 44
5.4 PureZandMixedDerivatives..................................... 46
5.5 FormalObservations................................................ 49
5.6 NonrigidCoefficients............................................... 57
6 PseudodifferentialProblems .............................................. 65
6.1 GeneralizationtoPseudodifferentialOperators ................... 65
6.2 TheMicrolocal.Tp/ ............................................ 67
' (cid:2)
6.3 BracketswithCoefficients.......................................... 68
7 GeneralSumsofSquaresofRealVectorFields ........................ 71
7.1 ALittleHistory ..................................................... 71
7.2 ProofforaSumofMonomials..................................... 72
7.3 PartialRegularity ................................................... 73
7.4 OtherSpecialCases,LeadingtoaGeneralConjecture........... 78
7.5 TheGeneralConjectureandResult................................ 79
8 The@-NeumannProblemandtheBoundaryLaplacian
onStrictlyPseudoconvexDomains ....................................... 81
8.1 StatementoftheTheorems......................................... 82
8.2 NotationandaPrioriEstimates .................................... 83
8.2.1 MaximalEstimate......................................... 84
8.3 TheHeatEquationfor(cid:2) .......................................... 88
b
8.4 WeaklyPseudoconvexDomains ................................... 89
8.5 GlobalRegularity................................................... 89
9 SymmetricDegeneracies .................................................. 91
9.1 WeaklyPseudoconvexDomains ................................... 91
10 DetailsofthePreviousChapter .......................................... 99
10.1 StatementoftheTheorems......................................... 99
10.2 TheVectorFieldM andtheLocalization ......................... 100
10.3 BehavioroftheLocalizedOperators............................... 108
Contents vii
10.4 ProofofTheorem10.1.............................................. 114
10.5 ProofofTheorem10.2.............................................. 118
10.6 EndoftheProofofTheorem10.2 ................................. 122
11 NonsymplecticStrataandGermAnalyticHypoellipticity ............. 131
11.1 ProofforHanges’Operator(11.1)................................. 132
11.2 AMoreComplicatedExample..................................... 134
11.3 TheGeneralScheme................................................ 135
11.4 TheVectorFieldM andtheLocalization ......................... 136
11.5 TheCommutationRelationsforRp ............................... 137
'
11.6 TheBracketŒX ;Rp(cid:3)............................................... 137
2 '
11.7 TheBracketŒX ;Rp(cid:3)............................................... 138
1 '
11.8 ProofofTheorem11.1.............................................. 144
11.9 NonclosedBicharacteristicswithNontrivialLimitSet ........... 150
12 OperatorsofKohnTypeThatLoseDerivatives ........................ 153
12.1 ObservationsandSimplifications .................................. 154
12.2 TheLocalizationofHighPowersofT ............................ 154
12.3 TheRecurrence ..................................................... 155
12.4 ConclusionoftheProof ............................................ 156
12.5 GevreyRegularityforKohn–OleinikOperators................... 157
13 NonlinearProblems ........................................................ 159
13.1 GlobalRegularity................................................... 159
13.2 SomeNotationandDefinitions..................................... 160
13.3 MaximalEstimates.................................................. 162
13.4 HighPowersoftheVectorFieldT ................................ 165
13.5 MixedDerivatives:TheCaseofGlobalX ........................ 171
13.6 LocallyDefinedX ................................................. 172
j
13.7 HighPowersofXINewLocalizingFunctions.................... 174
13.8 TheLocalizingFunctions........................................... 175
13.9 TakingaLocalizingFunctionoutoftheNorm.................... 176
13.10 LocalRegularity.................................................... 179
13.11 Results............................................................... 179
13.12 Proof................................................................. 180
13.13 PassingtoAnotherLocalizingFunction.......................... 184
13.14 ReducingtheOrderbyHalf;theEndoftheProof................ 186
13.15 TheWidthoftheCriticalBand..................................... 187
13.16 TheFaa` diBrunoFormula......................................... 188
14 Treves’Approach .......................................................... 191
15 Appendix .................................................................... 195
15.1 ADiscussionoftheLocalizingFunctions......................... 195
viii Contents
15.2 TheAnalyticalMaterialUsed...................................... 197
15.2.1 SomeFourierAnalysisandSobolevSpaces............. 197
15.2.2 TheHeisenbergGroup.................................... 197
15.2.3 PseudodifferentialOperators ............................. 197
References ........................................................................ 199
Chapter 1
What This Book Is and Is Not
The question of high smoothness of solutions to partial differential equations, in
particulartheequationsstudiedhereandcalledthe“@-Neumannproblem”andthe
complexboundaryLaplacian,lieattheinterfacebetweenrealandcomplexanalysis
andhasdeeprepercussionsinboth.
TheC1regularityresultswereestablishedin1963byJ.J.Kohn[K1]andshortly
afterward by others [KN], [Ho¨1] and only fifteen years later did the local real
analytic regularity find resolution, independently by F. Treves [Tr4] and by the
presentauthorin[T4],[T5].G. Me´tiviergeneralizedtheresultsin1980following
Treves[Me´2], and then in 1983 J. Sjo¨strand [Sj2] used a still differenttechnique,
that of the so-called FBI transform, to re-prove these results. These proofs are
radicallydifferentfromoneanother:Treves’constructsaparametrixforaso-called
Grusˇin operator and then treats the generalcase as a perturbation,mine explicitly
and directly localizes a high power of a vector field and then successively and
naturallycorrectsthecommutationerrorsthatarriveinusingL2estimates,andthat
ofSjo¨strand usesthe powerfulbutsomewhatrigidFBI transform,a clevervariant
oftheFouriertransformthatpermitsonetoavoidlocalizationdirectly.Seealsothe
workofOkaji[Ok]from1985.WewillcommentonTreves’approachinmoredetail
below.
It is the main purpose of this book, however,to familiarize the reader with the
techniqueandconstructionsthatIhavedeveloped,which,whileutterlyelementary
intheiressence,requireacertainamountoftimefortheirexposition,andIfeltthat
alongerformat,suchasabook,wouldprovidethematrixforthisnarrative.
Allthreemethodsalludedtoabovehavestoodthetestoftime.Theonepresented
herenotonlyiselementaryinnature,butalsoseemstobethemostflexibleandopen
toperturbations.Anditmaybeapproachedthroughthesimpleexampleofsumsof
squaresofrealvectorfieldsofthemostelementary(nontrivial)sort.
Andwhilethetechniqueiselementaryinnature—itusesnothingbeyondagood
first-yeargraduatecourseinanalysis—itdoesnotreplacethatcourse.
D.S.Tartakoff,NonellipticPartialDifferentialEquations:AnalyticHypoellipticity 1
andtheCouragetoLocalizeHighPowersofT,DevelopmentsinMathematics22,
DOI10.1007/978-1-4419-9813-2 1,©SpringerScience+BusinessMedia,LLC2011
2 1 WhatThisBookIsandIsNot
Thusinordertofacilitatereadabilityforthosewhoknow,orhaveheardof,this
technique,I havechosen notto start off with the definitionsof Lebesguemeasure
and integration,the Fourier transform,normed vector spaces, Sobolev spaces and
the Sobolev embedding theorem, left-invariant vector fields on the Heisenberg
group, and some elementary theory of pseudodifferential operators, but have
includedakindof“referencesection”intheappendix,whichcontainsthenecessary
definitionsandbasicresultstowhichthereadermaywishtoreferfromtimetotime.
However,forthosewhowanttoassessrightawaytheirpreparednessforthemain
text, here are some of the facts thatare developedfurtherin the Appendix.If you
arecomfortablewiththeseresultsandarecontenttoproceedwithoutproofsatthis
point,byallmeanscontinuetothenextchapter.
• The Fourier transform fO.(cid:4)/ of a function f.x/ is an isometry of L2.Rn/ and
takes@f=@x to.1=i/(cid:4) fO:
j j
• The Sobolev space Hs consists of all functions (tempered distributions) f for
which .1Cj(cid:4)j2/s=2fO 2 L2: Thusif s isa nonnegativeinteger,f 2 Hs if and
onlyif@˛f=@x˛ 2L2forallmulti-indices˛withj˛j(cid:2)s:
• (theSobolevembeddingtheorem)f.x/ iscontinuousiff 2 Hs forsomes >
n=2:Itfollowsthatf 2C1,providedf 2Hs 8s:
• Distributions on Rn; denoted by D.Rn/; are elements of the topological dual
space to C1.Rn/: They may be differentiated and multiplied by smooth func-
0
tions.EverydistributionbelongslocallytosomeHs:
• Thevectorfields
@ y @ @ x @
j j
X D (cid:3) ; Y D C
j j
@x 2 @t @y 2 @t
j j
aretheso-calledleft-invariantvectorfieldsontheHeisenberggroup.Theysatisfy
ŒX ;Y (cid:3)Dı T;
j k j;k
ŒX ;X (cid:3)DŒY ;Y (cid:3)DŒX ;T(cid:3)DŒY ;T(cid:3)D0
j k j k j j
(WewillusenothingelseabouttheHeisenberggroup;sufficeittosaythatthese
vector fields play the same role vis-a`-vis the Heisenberg group (they commute
with left translationin the group)that the coordinatepartialderivativesdo vis-
a`-vistheusualEuclideangroupstructureinRn:)Theywillprovidethesimplest
modelforthevectorfieldswewillstudy.
• The simplest pseudodifferentialoperators, as introduced by K.O. Friedrichs in
the 1960s, provide the algebraic tool needed to invert many partial differential
operators modulo (infinitely, or analytically) smoothing operators. Just as a
partialdifferentialoperator
X
1 @
P.x;D/D a D˛; D D ;
˛
i @x
j˛j(cid:3)m
