Analysis of Pseudo-Differential Operators

259 Pages·2019·3.36 MB·English

by  Shahla Molahajloo, M. W. Wong

Trends in Mathematics
Shahla Molahajloo
M. W. Wong
Editors
Analysis
of Pseudo-
Differential
Operators

Trends in Mathematics
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Shahla Molahajloo • M. W. Wong
Editors
Analysis
of Pseudo-Differential
Operators
Editors
ShahlaMolahajloo M.W.Wong
DepartmentofMathematics DepartmentofMathematicsandStatistics
InstituteforAdvancedStudiesinBasic YorkUniversity
Sciences Toronto,ON,Canada
Zanjan,Iran
ISSN2297-0215 ISSN2297-024X (electronic)
TrendsinMathematics
ISBN978-3-030-05167-9 ISBN978-3-030-05168-6 (eBook)
https://doi.org/10.1007/978-3-030-05168-6
LibraryofCongressControlNumber:2019935501
MathematicsSubjectClassification(2010):47-XX,46-XX
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Preface
Sincethe2003ISAACCongressatYorkUniversity,ithasbecomeatraditionthata
volumebasedonthespecialsessiononpseudo-differentialoperatorsbepublished.
Itisnotonlyintendedtodocumenttheevent,butalsotoprovideguidanceforfuture
researchonpseudo-differentialoperatorsandrelatedtopics.
The11thISAACCongresswasheldatLinnæusUniversityinSwedenonAugust
14–18,2018.Thisvolume,asasequeltoitspredecessors,isbasedontalksgivenat
thecongressandinvitedarticlesbyexpertsinthefield.
There are ten chapters in this volume, titled “Analysis of Pseudo-Differential
Operators.” The first four chapters address the functional analysis of pseudo-
differential operators in a broad range of settings, from Z to Rn, to compact and
Hausdorffgroups.Chapters5and6focusonoperatorsonLiegroupsandmanifolds
with edge. The next two chapters discuss topics in probability, while the last two
chapterscovertopicsindifferentialequations.
It is hoped that these volumes on pseudo-differential operators published by
Birkhäuser in Basel over a span of fifteen years have served and will continue
to serve as useful reference guides for young mathematicians aspiring to explore
newdirectionsin pseudo-differentialoperators.Itisalso ourfirm beliefthatthese
volumes on pseudo-differential operators will continue to grow and develop in
unforeseendirections,thankstotheinputofnewgenerationsofmathematicians.
Zanjan,Iran ShahlaMolahajloo
Toronto,ON,Canada M.W.Wong
v
Contents
DiscreteAnalogsofWignerTransformsandWeylTransforms ............ 1
ShahlaMolahajlooandM.W.Wong
Characterization and Spectral Invariance of Non-Smooth
PseudodifferentialOperatorswithHölderContinuousCoefficients....... 21
HelmutAbelsandChristinePfeuffer
FredholmnessandEllipticityof(cid:2)DOs onBs (Rn)andFs (Rn)........ 63
pq pq
PedroT.P.Lopes
CharacterizationsofSelf-Adjointness,Normality,Invertibility,
and Unitarity of Pseudo-DifferentialOperatorson Compact
andHausdorffGroups........................................................... 79
MajidJamalpourbirganiandM.W.Wong
MultilinearCommutatorsinVariableLebesgueSpacesonStratified
Groups............................................................................. 97
DongliLiu,JianTan,andJimanZhao
VolterraOperatorswithAsymptoticsonManifoldswithEdge............. 121
M.HedayatMahmoudiandB.-W.Schulze
Bismut’s Way of the Malliavin Calculus for Non-Markovian
Semi-groups:AnIntroduction.................................................. 157
RémiLéandre
OperatorTransformationofProbabilityDensities........................... 181
LeonCohen
TheTime-FrequencyInterferenceTermsoftheGreen’sFunction
fortheHarmonicOscillator..................................................... 215
LorenzoGalleani
On the Solvability in the Sense of Sequences for Some
Non-FredholmOperatorsRelatedtotheAnomalousDiffusion ............ 229
VitaliVougalterandVitalyVolpert
vii
Discrete Analogs of Wigner Transforms
and Weyl Transforms
ShahlaMolahajlooandM.W.Wong
Abstract WefirstintroducethediscreteFourier–Wignertransformandthediscrete
Wigner transform acting on functions in L2(Z). We prove that properties of the
standardWignertransformof functionsin L2(Rn) such as the Moyalidentity,the
inversionformula,time-frequencymarginalconditions,andtheresolutionformula
hold for the Wigner transforms of functions in L2(Z). Using the discrete Wigner
transform,wedefinethediscreteWeyltransformcorrespondingtoasuitablesymbol
onZ×S1.Wegiveanecessaryandsufficientconditionfortheself-adjointnessof
thediscreteWeyltransform.Moreover,wegiveanecessaryandsufficientcondition
foradiscreteWeyltransformtobeaHilbert–Schmidtoperator.Thenweshowhow
we can reconstruct the symbol from its correspondingWeyl transform. We prove
thatthe productoftwo Weyltransformsis againa Weyltransformandan explicit
formulaforthesymboloftheproductoftwoWeyltransformsisgiven.Thisresult
givesanecessaryandsufficientconditionfortheWeyltransformtobeinthetrace
class.
Keywords Fourier–Wignertransform · Wignertransform · Weyltransform ·
Moyalidentity · Time-frequencymarginalconditions · Wignerinversion
formula · Weylinversionformula · Kernels · Hilbert–Schmidtoperators · Trace
classoperators · Twistedconvolution · Weylcalculus
MathematicsSubjectClassification(2000) 47F05,47G30
This research has been supported by the Natural Sciences and Engineering Research Council
ofCanadaunderDiscoveryGrant0008562.
S.Molahajloo
DepartmentofMathematics,InstituteforAdvancedStudiesinBasicSciences,Zanjan,Iran
M.W.Wong((cid:2))
DepartmentofMathematicsandStatistics,YorkUniversity,Toronto,ON,Canada
e-mail:[email protected]
©SpringerNatureSwitzerlandAG2019 1
S.Molahajloo,M.W.Wong(eds.),AnalysisofPseudo-DifferentialOperators,
TrendsinMathematics,https://doi.org/10.1007/978-3-030-05168-6_1
2 S.MolahajlooandM.W.Wong
1 Introduction
Toputthispaperinperspective,wefirstrecalltheWignertransformandtheWeyl
transform mapping functions in L2(Rn) into functions on, respectively, Rn ×Rn
andRn.
Let σ ∈ L2(Rn × Rn). Then the Weyl transform W : L2(Rn) → L2(Rn)
σ
correspondingtothesymbolσ isdefinedby
(cid:2) (cid:2)
(Wσf,g)L2(Rn) =(2π)−n/2 σ(x,ξ)W(f,g)(x,ξ)dxdξ
Rn Rn
for all f and g in L2(Rn), where W(f,g) is the Wigner transform of f and g
definedby
(cid:2) (cid:3) (cid:4) (cid:3) (cid:4)
p p
W(f,g)(x,ξ) =(2π)−n/2 e−iξ·pf x+ g x− dp, x,ξ ∈Rn.
Rn 2 2
Closely related to the Wigner transform W(f,g) of f and g in L2(Rn) is the
Fourier–WignertransformV(f,g)givenby
(cid:2) (cid:3) (cid:4) (cid:3) (cid:4)
p p
V(f,g)(q,p)=(2π)−n/2 eiq·yf y+ g y− dy, q,p∈Rn.
Rn 2 2
Weyl transforms and Wigner transforms on Rn have been extensively studied in
[5,13]amongothers.
WeyltransformsongroupssuchastheHeisenberggroup,theupperhalfplane,
and the Poincaré unit disk are investigated in [8, 10–12]. Closely related to Weyl
transformsare pseudo-differentialoperatorson groups.See, for instance, [4, 7,9,
15].
The strategy that we use to developthe Weyl transform on Z is to have a look
atthecase ofRn,wherethesymbolσ isa functiononRn ×Rn.Recentworksin
pseudo-differentialoperatorsandWeyltransformsontopologicalgroupsGsuggest
thatthecorrectphasespacetoworkinisG×G(cid:5),whereG(cid:5)isthedualgroupofG.
ThatthedualgroupofRn isthesameasRn isthereasonwhythephasespaceon
whichsymbolsaredefinedisRn×Rn.
In the case of the group Z in this paper, the dual group is the unit circle S1
centeredattheoriginandthephasespaceG×G(cid:5)isthenZ×S1.
For1≤p<∞,thesetofallmeasurablefunctionsF onZsuchthat
(cid:6)
(cid:6)F(cid:6)p = |F(n)|p <∞
Lp(Z)
n∈Z
DiscreteAnalogsofWignerTransformsandWeylTransforms 3
isdenotedbyLp(Z).WedefineLp(S1)tobethesetofallmeasurablefunctionsf
ontheunitcircleS1withcenterattheoriginforwhich
(cid:2)
1 π
(cid:6)f(cid:6)p = |f(θ)|pdθ <∞.
Lp(S1) 2π −π
WedefinetheFouriertransformFZF ofF ∈L1(Z)tobethefunctiononS1by
(cid:6)
(FZF)(θ)= einθF(n), θ ∈[−π,π].
n∈Z
Iff isasuitablefunctiononS1,thenwedefinetheFouriertransformFS1f off to
bethefunctiononZby
(cid:2)
(cid:7) (cid:8) 1 π
FS1f (n)= e−inθf(θ)dθ, n∈Z.
2π −π
NotethatFZ :L2(Z)→L2(S1)isasurjectiveisomorphism.Infact,
F =F−1 =F∗
Z S1 S1
and
(cid:6)FZF(cid:6)L2(S1) =(cid:6)F(cid:6)L2(Z), F ∈L2(Z).
Let H be a suitable function on S1 × Z. Then we define the Fourier transform
FS1×ZH ofH tobethefunctiononZ×S1by
(cid:2)
(cid:7) (cid:8) 1 π (cid:6)
FS1×ZH (m,θ)= e−imφ+inθH(φ,n)dφ, (m,θ)∈Z×S1.
2π −π n∈Z
Similarly,forallsuitable functionsK onZ×S1, we define theFouriertransform
FZ×S1K ofK tobethefunctiononS1×Zby
(cid:2)
1 π (cid:6)
(FZ×S1K)(θ,m)= e−imφ+inθK(n,φ)dφ, (θ,m)∈S1×Z.
2π −π n∈Z
For1≤p <∞,wedefineLp(Z×S1)tobethespaceofallmeasurablefunctions
honZ×S1 suchthat
(cid:2)
1 (cid:6) π
(cid:6)h(cid:6)p = |h(n,θ)|pdθ <∞.
Lp(Z×S1) 2π n∈Z −π