Table Of ContentSpringer Proceedings in Mathematics & Statistics
Luigi G. Rodino
Joachim Toft Editors
Mathematical
Analysis and
Applications—
Plenary Lectures
ISAAC 2017, Växjö, Sweden
Springer Proceedings in Mathematics & Statistics
Volume 262
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Luigi G. Rodino Joachim Toft
(cid:129)
Editors
Mathematical Analysis
—
and Applications Plenary
Lectures
ä ö
ISAAC 2017, V xj , Sweden
123
Editors
Luigi G.Rodino Joachim Toft
Dipartimento di Matematica Department ofMathematics
Universitàdi Torino Linnaeus University
Turin, Italy Växjö,Sweden
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-3-030-00873-4 ISBN978-3-030-00874-1 (eBook)
https://doi.org/10.1007/978-3-030-00874-1
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MathematicsSubjectClassification(2010): 35-XX,46-XX,60-XX,32-XX,47-XX,65-XX
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Preface
This volume is a collection of articles devoted to current research topics in
Mathematical Analysis, with emphasis on Fourier analysis and general theory of
partial differential equations. It originates from plenary lectures given at the 11th
InternationalISAACCongress,heldduring14–18August2017attheUniversityof
Växjö, Sweden.
The papers are authored by six eminent specialists and aim at presenting to a
largeaudiencesomechallengingandattractivethemesofthemodernMathematical
Analysis, namely as follows:
– The contribution of Nils Dencker is devoted to the local solvability of sub-
principal type operators. Dencker proved in 2006 the so-called Nirenberg–
Treves conjecture, expressing a necessary and sufficient condition for the
solvability of the operators of principal type. In this paper, he addresses to
operators with multiple characteristics. Namely, the principal symbol of the
operatorisassumedtovanishofatleastsecondorderonaninvolutivemanifold.
Precise necessary conditions are expressed for solvability, in terms of the sub-
principal symbol, involving lower order terms.
– In these last years, the fractional-order Laplacian was frequently used in
Mathematical Physics, Differential Geometry, Probability and Finance. The
paper of Gerd Grubb is devoted to this operator and its generalizations, with
attentiontohomogeneousandnon-homogeneousboundaryvalueproblemsand
correspondingheatequations.Thepaperrepresentsthefirstgeneralandrigorous
approachtonon-localproblemsofthistype,inthesettingofpseudo-differential
operators.
– The paper of Abdelhamid Meziani is devoted to degenerate complex vector
fields in the plane, basic example being the Mizohata operator. The corre-
sponding equations share many properties with the Cauchy–Riemann equation.
Combining with the theory of the hypo-analytic structures of Treves, Meziani
presents an elegant treatment of the subject, including generalizations of the
Riemann–Hilbert problem.
v
vi Preface
– The paper of Alberto Parmeggiani gives a survey on the problem of the lower
bounds. Does the positivity of the symbol imply the positivity of the corre-
sponding operator? A precise answer to this question is largely open. Main
reference is the classical theorem of Fefferman–Phong, stating positivity of the
operator modulo small errors. Here Parmeggiani addresses also to the case of
systems of operators; for them, precise lower bounds represent a largely
unexplored area.
– Following the original idea of Morrey [1], Peetre and others introduced the
function spaces which are nowadays called Morrey spaces. They have a large
numberofapplicationsindifferentcontexts,andtheirdefinitionwasextendedin
different directions. Yoshihiro Sawano, in his contribution to the volume, pre-
sents a review addressed to non-experts, including the interpolation theory and
the weighted version.
– Localization operators were defined by Berezin [2] in the frame of Quantum
Mechanics, and later used by Daubechies [3] and others in Signal Theory. The
paper of Nenad Teofanov is devoted to this important topic. The definition of
localizationoperatorisgivenhereintermsoftheGrossmann–Royertransform,
whichsimplifiestheproofofseveralknownresults.Particularemphasisisgiven
to the action on Gelfand–Shilov and modulation spaces.
Besides plenary talks, about 250 scientific communications were delivered during
the Växjö ISAAC Congress. Their texts are published in an independent volume.
On the whole, the Congress demonstrated, in particular, the relevant role of the
Nordic European countries in several research areas of Mathematical Analysis.
Turin, Italy Luigi G. Rodino
Växjö, Sweden Joachim Toft
July 2018
References
1. MorreyJr.,C.B.:Onthesolutionsofquasi-linearellipticpartialdifferentialequations.Trans.
Am.Math.Soc.43(1),126–166(1938)
2. Berezin, F.A.: Wick and anti-wick symbols of operators. Mat. Sb. (N.S.) 86(128), 578–610
(1971)
3. Daubechies,I.:Time-frequencylocalizationoperators:ageometricphasespaceapproach.IEEE
Trans.Inform.Theory34(4),605–612(1988)
Contents
Solvability of Subprincipal Type Operators . . . . . . . . . . . . . . . . . . . . . . 1
Nils Dencker
Fractional-Order Operators: Boundary Problems,
Heat Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Gerd Grubb
A Class of Planar Hypocomplex Vector Fields: Solvability
and Boundary Value Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
A. Meziani
Almost-Positivity Estimates of Pseudodifferential Operators . . . . . . . . . 109
Alberto Parmeggiani
Morrey Spaces from Various Points of View . . . . . . . . . . . . . . . . . . . . . 139
Yoshihiro Sawano
The Grossmann–Royer Transform, Gelfand–Shilov Spaces,
and Continuity Properties of Localization Operators
on Modulation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Nenad Teofanov
vii
Contributors
Nils Dencker Centre for Mathematical Sciences, Lund University, Lund, Sweden
Gerd Grubb Department of Mathematical Sciences, Copenhagen University,
Copenhagen, Denmark
A. Meziani Department of Mathematics, Florida International University, Miami,
FL, USA
Alberto Parmeggiani Department of Mathematics, University of Bologna,
Bologna, Italy
Yoshihiro Sawano Department of Mathematics, Tokyo Metropolitan University,
Hachioji Tokyo, Japan
NenadTeofanov DepartmentofMathematicsandInformatics,UniversityofNovi
Sad, Novi Sad, Serbia
ix
Solvability of Subprincipal Type
Operators
NilsDencker
Abstract Inthispaperweconsiderthesolvabilityofpseudodifferentialoperatorsin
thecasewhentheprincipalsymbolvanishesoforderk ≥2atanonradialinvolutive
manifold(cid:2) .Weshallassumethattheoperatorisofsubprincipaltype,whichmeans
2
that the kth inhomogeneous blowup at (cid:2) of the refined principal symbol is of
2
principaltypewithHamiltonvectorfieldparalleltothebase(cid:2) ,buttransversalto
2
thesymplecticleavesof(cid:2) atthecharacteristics.Whenk =∞thisblowupreduces
2
tothesubprincipalsymbol.Wealsoassumethattheblowupisessentiallyconstant
on the leaves of (cid:2) , and does not satisfying the Nirenberg–Treves condition ((cid:3)).
2
We also have conditions on the vanishing of the normal gradient and the Hessian
oftheblowupatthecharacteristics.Undertheseconditions,weshowthat P isnot
solvable.
1 Introduction
We shall consider the solvability for a classical pseudodifferential operator P ∈
(cid:3)m(X)onaC∞ manifold X ofdimensionn.Thismeansthat P hasanexpansion
cl
pm + pm−1+...wherepj ∈ Shjomishomogeneousofdegree j,∀ j,andpm =σ(P)
is the principal symbol of the operator. A pseudodifferential operator is said to be
ofprincipaltypeiftheHamiltonvectorfield H oftheprincipalsymboldoesnot
havetheradialdirectionξ·∂ on p−1(0),inparptmicular H (cid:6)=0.Weshallconsider
ξ m pm
thecasewhentheprincipalsymbolvanishesofatleastsecondorderataninvolutive
manifold(cid:2) ,thus P isnotofprincipaltype.
2
P islocallysolvableatacompactset K ⊆ X iftheequation
Pu =v (1.1)
B
N.Dencker( )
CentreforMathematicalSciences,LundUniversity,Box118,22100Lund,Sweden
e-mail:[email protected]
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L.G.RodinoandJ.Toft(eds.),MathematicalAnalysisandApplications—Plenary
Lectures,SpringerProceedingsinMathematics&Statistics262,
https://doi.org/10.1007/978-3-030-00874-1_1
