Table Of ContentSpringer Proceedings in Mathematics & Statistics
Pankaj Jain
Hans-Jürgen Schmeisser Editors
Function
Spaces and
Inequalities
New Delhi, India, December 2015
Springer Proceedings in Mathematics & Statistics
Volume 206
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Pankaj Jain Hans-J rgen Schmeisser
(cid:129)
Editors
Function Spaces
and Inequalities
New Delhi, India, December 2015
123
Editors
PankajJain Hans-JürgenSchmeisser
Department ofMathematics Faculty of Mathematics andComputer
SouthAsianUniversity Science
NewDelhi, Delhi Friedrich Schiller University Jena
India Jena, Thuringia
Germany
ISSN 2194-1009 ISSN 2194-1017 (electronic)
SpringerProceedings in Mathematics& Statistics
ISBN978-981-10-6118-9 ISBN978-981-10-6119-6 (eBook)
DOI 10.1007/978-981-10-6119-6
LibraryofCongressControlNumber:2017947872
MathematicsSubjectClassification(2010): 26D10,43B20,42B25,42B35,46E30,46E35,47B38
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Preface
With the development in the last few decades, the theory of Function Spaces has
become a powerful tool in several areas of mathematical and physical sciences,
engineering etc. In particular, the concept of generalized functions (distributions)
enables to study Partial Differential Equations and Boundary Value Problems in a
much wider perspective. In order to deal withsuch problems, quite often, function
spaces and mapping properties of operators connected with corresponding norm
inequalities come into picture. With this in mind, an International Conference on
“Function Spaces and Inequalities” was organized under the leadership of Pankaj
JainatSouthAsianUniversity,NewDelhiduringDecember11–15,2015.Theaim
of the conference was to bring together experts and young researchers working in
the field of Function Spaces and Inequalities to share their latest
interests/investigations. The topics covered in the conference include
(Variable/Grand/Small) Lebesgue Spaces, Orlicz Spaces, Lorentz Spaces, Sobolev
Spaces, Morrey Spaces, Sequence spaces, Weight Theory, Integral Operators of
Hardy Type, Sobolev Type Imbeddings, Function Algebras, Banach Algebras,
Spaces&Algebrasof,AnalyticFunctions,GeometryofBanachSpaces,Isometries
of Function Spaces, (Weighted) Integral and Discrete Inequalities, Convexity
Theory, Harmonic Analysis.
Simultaneously, it was proposed to bring out an edited volume based on the
theme of the conference. This volume consists of original work as well as survey
articles on topics of Function Spaces and Inequalities by distinguished mathe-
maticiansworldwide.Thesurveyarticles areself-containedandprovideup-to-date
knowledgeintherespectivearea. Contributionsarealsofromthose who couldnot
or did not attend the conference. All the articles are thoroughly refereed.
We express our deep gratitude to all the authors for their valuable contribution.
Also,wearethankfultotheextremelytalentedmathematicianswhocontributedin
the reviewing process.
WeoweagreatdebtofgratitudetoDrKavitaASharma,President,SouthAsian
University for her encouragement and support towards the organization of the
conference. We record our gratitude to National Board for Higher Mathematics
v
vi Preface
(NBHM), Department of Science and Technology (DST) and South Asian
University for providing partial financial support for the conference.
New Delhi, India Pankaj Jain
Hans-Jürgen Schmeisser
Contents
TheFundamentalFunctionofCertainInterpolationSpacesGenerated
by N-Tuples of Rearrangement-Invariant Spaces.. .... .... ..... .... 1
Fernando Cobos and Luz M. Fernández-Cabrera
Sobolev Embeddings for Herz-Type Triebel-Lizorkin Spaces ..... .... 15
Douadi Drihem
Order Sharp Estimates for Monotone Operators on Orlicz–Lorentz
Classes... .... .... .... .... ..... .... .... .... .... .... ..... .... 37
Mikhail L. Goldman
Complex Interpolation of Morrey Spaces .... .... .... .... ..... .... 85
Denny Ivanal Hakim and Yoshihiro Sawano
Gagliardo-Nirenberg Inequalities for Spaces with Dominating Mixed
Derivatives.... .... .... .... ..... .... .... .... .... .... ..... .... 117
Dorothee D. Haroske and Hans-Jürgen Schmeisser
Recent Trends in Grand Lebesgue Spaces ... .... .... .... ..... .... 137
Pankaj Jain, Monika Singh and Arun Pal Singh
On Certain New Method to Construct Weighted Hardy-Type
Inequalities and Its Application to the Sharp Hardy-Poincaré
Inequalities ... .... .... .... ..... .... .... .... .... .... ..... .... 161
Agnieszka Kałamajska and Iwona Skrzypczak
Intrinsic Characterization and the Extension Operator in Variable
Exponent Function Spaces on Special Lipschitz Domains ... ..... .... 175
Henning Kempka
The Boundedness of Sublinear Operators in Weighted Morrey
Spaces Defined on Spaces of Homogeneous Type.. .... .... ..... .... 193
Vakhtang Kokilashvili and Alexander Meskhi
vii
viii Contents
Essentially Algebraic Composition Operators on Lorentz Sequence
Spaces with a Weight .. .... ..... .... .... .... .... .... ..... .... 213
Romesh Kumar, Ajay K. Sharma, Sumit Dubey and Shagoon Wasir
Higher Dimensional Hardy-Type Inequalities. .... .... .... ..... .... 225
Santosh Kumari
Recent Advances on Generalized Trigonometric Systems in Higher
Dimensions.... .... .... .... ..... .... .... .... .... .... ..... .... 241
Jan Lang and Osvaldo Méndez
Pointwise Multipliers on Musielak-Orlicz-Morrey Spaces ... ..... .... 257
Eiichi Nakai
The Fatou Property of Function Spaces, Heat Kernels, Admissible
Norms and Mapping Properties.... .... .... .... .... .... ..... .... 283
Hans Triebel
A Survey on Some Variable Function Spaces. .... .... .... ..... .... 299
Dachun Yang, Wen Yuan and Ciqiang Zhuo
The Fundamental Function of Certain
Interpolation Spaces Generated by N-Tuples
of Rearrangement-Invariant Spaces
FernandoCobosandLuzM.Fernández-Cabrera
Abstract Inthispaperwedeterminethefundamentalfunctionofthespaceobtained
byapplyinganexactinterpolationfunctorofexponentθtoanN-tupleofrearrange-
ment-invariant function spaces. Results apply to the extension of the real method
studiedbyYoshikawaandSparr,andtotheextensionofthecomplexmethodinves-
tigatedbyLionsandFavini.Moreover,wealsoconsiderthecaseofthegeneralreal
methodforcouplesofspaces.
· ·
Keywords Rearrangement-invariant function spaces Fundamental function
·
Interpolationmethodsforn-tuples Generalrealmethod
1 Introduction
A number of important function spaces have the property that any two equimea-
surablefunctionshavethesamenorm.Thesefunctionspacesarecalledrearrange-
ment-invariant and form a class which includes Lebesgue spaces, Lorentz spaces,
OrliczspacesandMarcinkiewiczspaces,amongothers(see[4, 12, 17, 18]).They
haveacloseconnectionwithinterpolationtheory.Infact,theclassofrearrangement-
invariant spaces coincides with the class of exact interpolation spaces between L
1
and L∞.See[7]or[4,TheoremIII.2.12].
Thefundamentalfunctionϕ ofarearrangement-invariantspace X isanuseful
X
tool to study X. The function ϕ allows to characterize certain duality and sepa-
X
rabilitypropertiesof X,aswellastocompare X withLorentzandMarcinkiewicz
spaces.See[4,12,17].Thefundamentalfunctionisalsoconnectedwiththegrowth
B
F.Cobos( )
FacultaddeMatemáticas,DepartamentodeAnálisisMatemático,UniversidadComplutensede
Madrid,PlazadeCiencias3,28040Madrid,Spain
e-mail:[email protected]
L.M.Fernández-Cabrera
FacultaddeEstudiosEstadísticos,SecciónDepartamentaldeMatemáticaAplicada,Universidad
ComplutensedeMadrid,28040Madrid,Spain
e-mail:[email protected]
©SpringerNatureSingaporePteLtd.2017 1
P.JainandH.-J.Schmeisser(eds.),FunctionSpacesandInequalities,
SpringerProceedingsinMathematics&Statistics206,
DOI10.1007/978-981-10-6119-6_1
