Table Of ContentTrends in Mathematics
M. W. Wong
Hongmei Zhu
Editors
Pseudo-
Differential
Operators:
Groups, Geometry
and Applications
Trends in Mathematics
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M.W. Wong • Hongmei Zhu
Editors
Pseudo-Differential
Operators: Groups,
Geometry and Applications
Editors
M.W.Wong HongmeiZhu
DepartmentofMathematicsandStatistics DepartmentofMathematicsandStatistics
YorkUniversity YorkUniversity
Toronto Toronto
Ontario,Canada Ontario,Canada
ISSN2297-0215 ISSN2297-024X (electronic)
TrendsinMathematics
ISBN978-3-319-47511-0 ISBN978-3-319-47512-7 (eBook)
DOI10.1007/978-3-319-47512-7
LibraryofCongressControlNumber:2017931964
MathematicsSubjectClassification(2010):35J05,35P30,35S35,42A30,42A38,42A45,42C40,47F05,
47G30,53C21,60F10,60H10,94A12
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Preface
Two major congresses in mathematics took place back to back in the summer of
2015. The Tenth Congress of the InternationalSociety for Analysis, Applications
and Computation (ISAAC) was hosted by the University of Macau in China on
August3–8,2015,andthis was followedby the EighthInternationalCongresson
IndustrialandAppliedMathematics(ICIAM)heldattheChinaNationalConvention
CenterinBeijingonAugust10–14,2015.Presentedatthesetwocongresseswere,
respectively, the special session “Pseudo-Differential Operators” and the mini-
symposium “Pseudo-Differential Operators in Industries and Technologies”. The
formerwasbroaderinnatureembracingallaspectsofpseudo-differentialoperators
in analysis, applications and computations, and the latter was skewed towards
“real-life” applications. The two camps of participants in these two events had
a significant and viable intersection. The editors of this volume belonged to this
intersectionanddecidedtopresentavolumereflectingthecoreofthevibrantevents
with papers most appropriately classified as Groups, Geometry and Applications.
Theoverarchingthemesare ofcourseonpseudo-differentialoperatorsunderstood
byusasalwaysin thebroadestsense. Thesepapersarecompletepapersby either
theparticipantsorinvitedcontributors.Theyarepeer-reviewedusingthestandards
wellestablishedinpureandappliedmathematics.
Categorized under Groups are two papers in ascending order of complexities
of the groups being considered. They are the affine group and the non-isotropic
Heisenberg group with multidimensional centre. Classified under Geometry are
also three papers with one on the curvaturesof the Heisenberg group, the second
one on ellipticity on compact manifolds with boundary or edge and the final one
onlocalizationoperatorsrelatedtothequaternionFouriertransforms.Lastbutnot
least, the section that is best named as Applicationsin this volume containsmore
than half of the papers. We begin with a time-frequency approach to the study
of the relationship of the Langevin equation and the simple harmonic oscillator.
This is then followed by two papers on the role of pseudo-differential operators
in probability with the first one in quantum probability and the second one in
mathematical probability. A paper on nonlinear systems of integro-differential
equationswithanomalousdiffusionmodelledbyfractionalpowersofthenegative
v
vi Preface
of the Laplacian is presented. The volume ends with one paper on the wavelet
transformsandtwopapersontheStockwelltransforms.
The aim of this volume is to show that the developmentof pseudo-differential
operators, which have been studied since the 1960s, is gaining momentum in its
ramificationsnot only on differentsettings such as groupsand geometry,but also
in diverse applications such as quantum physics, probability and statistics and
time-frequencyanalysis.Thisresonateswellwiththecontemporarysocietalneeds
in basic research in mathematics relevant to the proliferation of interdisciplinary
sciences. This trend fits in well with previously published volumes on pseudo-
differentialoperatorsbyBirkhäuserinBasel,anditisourvisionthatthisendeavour
willresultinquantumleapsintheyearstocome.
Toronto,Canada M.W.Wong
Toronto,Canada HongmeiZhu
Contents
Pseudo-DifferentialOperatorsontheAffineGroup ......................... 1
AparajitaDasguptaandM.W.Wong
Pseudo-differentialOperatorsonNon-isotropicHeisenberg
GroupswithMulti-dimensionalCenters ...................................... 15
ShahlaMolahajloo
CurvatureoftheHeisenbergGroup ........................................... 37
BartekEwertowskiandM.W.Wong
EllipticitywithGlobalProjectionConditions................................. 47
B.-W.Schulze
MultilinearLocalizationOperatorsAssociatedtoQuaternion
FourierTransforms.............................................................. 107
GuangshengMaandJimanZhao
A Time-Frequency Relationship Between the Langevin
EquationandtheHarmonicOscillator........................................ 119
LorenzoGalleani
Are There Quantum Operators and Wave Functions
inStandardProbabilityTheory................................................ 133
LeonCohen
A ClassofNon-MarkovianPseudo-differentialOperators
ofLévyType...................................................................... 149
RémiLéandre
OntheSolvabilityofSomeSystemsofIntegro-Differential
EquationswithAnomalousDiffusion.......................................... 161
VitaliVougalterandVitalyVolpert
vii
viii Contents
Visualizing Discrete Complex-Valued Time-Frequency
Representations .................................................................. 181
YusongYanandHongmeiZhu
TheReassignedSpectrogramoftheStockwellTransform.................. 203
ChengLiuandHongmeiZhu
ContinuousMultiwaveletTransformforBlindSignalSeparation ......... 219
RyuichiAshino,TakeshiMandai,andAkiraMorimoto
Pseudo-Differential Operators on the Affine
Group
AparajitaDasguptaandM.W.Wong
Abstract Pseudo-differential operators are defined on the affine group using the
FourierinversionformulafortheFouriertransformonthe affinegroup.The Weyl
transform on the affine group is given and so are the L2-Lp estimates for pseudo-
differentialoperatorsontheaffinegroup.
Keywords Affine group • SL(2,R) • Semi-directproduct • Fouriertransform •
Plancherel formula • Fourier inversion formula • Pseudo-differential operator •
Weyltransform • L2-Lpestimates
MathematicsSubjectClassification(2000). Primary47G30
1 Introduction
It is a well-known fact from [16] that pseudo-differential operators on Rn are
based on the Plancherel formula for the Fourier transform on Rn. The Plancherel
formula gives rise to the Fourier inversion formula, which says that the identity
operator for L2.Rn/ can be expressed in terms of the Fourier transform on Rn.
The Fourier inversion formula, albeit useful in many situations, gives a perfect
ThisresearchhasbeensupportedbytheNaturalSciencesandEngineeringResearchCouncilof
CanadaunderDiscoveryGrant0008562.
A.Dasgupta
EPFLSBMATHAAPDE,MAC1657(BâtimentMA),Station8,CH-1015,
Lausanne,Switzerland
e-mail:aparajita.dasgupta@epfl.ch
M.W.Wong((cid:2))
DepartmentofMathematicsandStatistics,YorkUniversity,4700KeeleStreet,
Toronto,ON,M3J1P3,Canada
e-mail:[email protected]
©SpringerInternationalPublishingAG2017 1
M.W.Wong,HongmeiZhu(eds.),Pseudo-DifferentialOperators:Groups,
GeometryandApplications,TrendsinMathematics,
DOI10.1007/978-3-319-47512-7_1
