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SOBOLEV SPACESIN MATHEMATICS I
SOBOLEV TYPE INEQUALITIES
INTERNATIONAL MATHEMATICAL SERIES
Series Editor: Tamara RozhkovskayaNovosibirsk, Russia
1. Nonlinear Problems in Mathematical Physics and Related TopicsI. In Honor of Professor O.A. Ladyzhenskaya • M.Sh. Birman, S.Hildebrandt, V.A. Solonnikov, N.N. Uraltseva Eds. • 2002
2. Nonlinear Problems in Mathematical Physics and Related TopicsII. In Honor of Professor O.A. Ladyzhenskaya • M.Sh. Birman, S.Hildebrandt, V.A. Solonnikov, N.N. Uraltseva Eds. • 2003
3. Different Faces of Geometry • S. Donaldson, Ya. Eliashberg, M. Gro-mov Eds. • 2004
4. Mathematical Problems from Applied Logic I. Logics for theXXIst Century • D. Gabbay, S. Goncharov, M. Zakharyaschev Eds. •2006
5. Mathematical Problems from Applied Logic II. Logics for theXXIst Century • D. Gabbay, S. Goncharov, M. Zakharyaschev Eds. •2007
6. Instability in Models Connected with Fluid Flows. I • C. Bardos,A. Fursikov Eds. • 2008
7. Instability in Models Connected with Fluid Flows. II • C. Bardos,A. Fursikov Eds. • 2008
8. Sobolev Spaces in Mathematics I. Sobolev Type Inequalities •V. Maz’ya Ed. • 2009
9. Sobolev Spaces in Mathematics II. Applications in Analysis andPartial Differential Equations • V. Maz’ya Ed. • 2009
10. Sobolev Spaces in Mathematics III. Applications in Mathemat-ical Physics • V. Isakov Ed. • 2009
SOBOLEV SPACESIN MATHEMATICS I
Sobolev Type Inequalities
Editor: Vladimir Maz’yaOhio State University, USAUniversity of Liverpool, UKLinkoping University, SWEDEN
123Tamara Rozhkovskaya Publisher
Editor
Prof. Vladimir Maz’ya
Ohio State UniversityDepartment of MathematicsColumbus, USA
University of LiverpoolDepartment of Mathematical SciencesLiverpool, UK
Linkoping UniversityDepartment of MathematicsLinkoping, Sweden
This series was founded in 2002 and is a joint publication of Springer and “TamaraRozhkovskaya Publisher.” Each volume presents contributions from the Volume Editorsand Authors exclusively invited by the Series Editor Tamara Rozhkovskaya who also pre-pares the Camera Ready Manuscript. This volume is distributed by “Tamara RozhkovskayaPublisher” ([email protected]) in Russia and by Springer over all the world.
ISBN 978-0-387-85647-6 e-ISBN 978-0-387-85648-3ISBN 978-5-901873-24-3 (Tamara Rozhkovskaya Publisher)
ISSN 1571-5485
Library of Congress Control Number: 2008937494
c© 2009 Springer Science+Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part withoutthe written permission of the publisher (Springer Science+Business Media, LLC, 233 SpringStreet, New York, NY 10013, USA), except for brief excerpts in connection with reviewsor scholarly analysis. Use in connection with any form of information storage and retrieval,electronic adaptation, computer software, or by similar or dissimilar methodology nowknown or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms,even if they are not identified as such, is not to be taken as an expression of opinion as towhether or not they are subject to proprietary rights.
Printed on acid-free paper.
9 8 7 6 5 4 3 2 1
springer.com
To the memory of
Sergey L’vovich Sobolev
on the occasion of his centenary
Main Topics
Sobolev’s discoveries of the 1930’s have a strong influence on de-velopment of the theory of partial differential equations, analysis,mathematical physics, differential geometry, and other fields of math-ematics. The three-volume collection Sobolev Spaces in Mathematicspresents the latest results in the theory of Sobolev spaces and appli-cations from leading experts in these areas.
I. Sobolev Type InequalitiesIn 1938, exactly 70 years ago, the original Sobolev inequality (an embed-ding theorem) was published in the celebrated paper by S.L. Sobolev “Ona theorem of functional analysis.” By now, the Sobolev inequality and itsnumerous versions continue to attract attention of researchers because ofthe central role played by such inequalities in the theory of partial differ-ential equations, mathematical physics, and many various areas of analysisand differential geometry. The volume presents the recent study of differentSobolev type inequalities, in particular, inequalities on manifolds, Carnot–Caratheodory spaces, and metric measure spaces, trace inequalities, inequal-ities with weights, the sharpness of constants in inequalities, embedding theo-rems in domains with irregular boundaries, the behavior of maximal functionsin Sobolev spaces, etc. Some unfamiliar settings of Sobolev type inequalities(for example, on graphs) are also discussed. The volume opens with the surveyarticle “My Love Affair with the Sobolev Inequality” by David R. Adams.
II. Applications in Analysis and Partial Differential EquationsSobolev spaces become the established language of the theory of partial dif-ferential equations and analysis. Among a huge variety of problems whereSobolev spaces are used, the following important topics are in the focus of thisvolume: boundary value problems in domains with singularities, higher orderpartial differential equations, nonlinear evolution equations, local polynomialapproximations, regularity for the Poisson equation in cones, harmonic func-tions, inequalities in Sobolev–Lorentz spaces, properties of function spaces incellular domains, the spectrum of a Schrodinger operator with negative po-tential, the spectrum of boundary value problems in domains with cylindricaland quasicylindrical outlets to infinity, criteria for the complete integrabilityof systems of differential equations with applications to differential geome-try, some aspects of differential forms on Riemannian manifolds related to theSobolev inequality, a Brownian motion on a Cartan–Hadamard manifold, etc.Two short biographical articles with unique archive photos of S.L. Sobolevare also included.
viii Main Topics
III. Applications in Mathematical PhysicsThe mathematical works of S.L. Sobolev were strongly motivated by particu-lar problems coming from applications. The approach and ideas of his famousbook “Applications of Functional Analysis in Mathematical Physics” of 1950turned out to be very influential and are widely used in the study of variousproblems of mathematical physics. The topics of this volume concern mathe-matical problems, mainly from control theory and inverse problems, describ-ing various processes in physics and mechanics, in particular, the stochasticGinzburg–Landau model with white noise simulating the phenomenon of su-perconductivity in materials under low temperatures, spectral asymptoticsfor the magnetic Schrodinger operator, the theory of boundary controllabil-ity for models of Kirchhoff plate and the Euler–Bernoulli plate with variousphysically meaningful boundary controls, asymptotics for boundary valueproblems in perforated domains and bodies with different type defects, theFinsler metric in connection with the study of wave propagation, the electricimpedance tomography problem, the dynamical Lame system with residualstress, etc.
Contents
I. Sobolev Type InequalitiesVladimir Maz’ya Ed.
My Love Affair with the Sobolev Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1David R. Adams
Maximal Functions in Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25Daniel Aalto and Juha Kinnunen
Hardy Type Inequalities Via Riccati and Sturm–Liouville Equations . . . . 69Sergey Bobkov and Friedrich Gotze
Quantitative Sobolev and Hardy Inequalities, and RelatedSymmetrization Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Andrea Cianchi
Inequalities of Hardy–Sobolev Type in Carnot–Caratheodory Spaces . . . 117Donatella Danielli, Nicola Garofalo, and Nguyen Cong Phuc
Sobolev Embeddings and Hardy Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153David E. Edmunds and W. Desmond Evans
Sobolev Mappings between Manifolds and Metric Spaces . . . . . . . . . . . . . . .185Piotr Haj�lasz
A Collection of Sharp Dilation Invariant Integral Inequalitiesfor Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Vladimir Maz’ya and Tatyana Shaposhnikova
Optimality of Function Spaces in Sobolev Embeddings . . . . . . . . . . . . . . . . .249Lubos Pick
On the Hardy–Sobolev–Maz’ya Inequality and Its Generalizations . . . . . 281Yehuda Pinchover and Kyril Tintarev
Sobolev Inequalities in Familiar and Unfamiliar Settings . . . . . . . . . . . . . . . 299Laurent Saloff-Coste
A Universality Property of Sobolev Spaces in Metric Measure Spaces . . 345Nageswari Shanmugalingam
Cocompact Imbeddings and Structure of Weakly ConvergentSequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Kiril Tintarev
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
x Sobolev Spaces in Mathematics I–III
II. Applications in Analysis andPartial Differential EquationsVladimir Maz’ya Ed.
On the Mathematical Works of S.L. Sobolev in the 1930s . . . . . . . . . . . . . . . . 1Vasilii Babich
Sobolev in Siberia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Yuri Reshetnyak
Boundary Harnack Principle and the Quasihyperbolic BoundaryCondition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Hiroaki Aikawa
Sobolev Spaces and their Relatives: Local PolynomialApproximation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Yuri Brudnyi
Spectral Stability of Higher Order Uniformly Elliptic Operators . . . . . . . . . 69Victor Burenkov and Pier Domenico Lamberti
Conductor Inequalities and Criteria for Sobolev-LorentzTwo-Weight Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Serban Costea and Vladimir Maz’ya
Besov Regularity for the Poisson Equation in Smooth andPolyhedral Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
Stephan Dahlke and Winfried Sickel
Variational Approach to Complicated Similarity Solutions ofHigher Order Nonlinear Evolution Partial Differential Equations . . . . . . . 147
Victor Galaktionov, Enzo Mitidieri, and Stanislav Pokhozhaev
Lq,p-Cohomology of Riemannian Manifolds with Negative Curvature . . . 199Vladimir Gol’dshtein and Marc Troyanov
Volume Growth and Escape Rate of Brownian Motion ona Cartan–Hadamard Manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Alexander Grigor’yan and Elton Hsu
Sobolev Estimates for the Green Potential Associated withthe Robin–Laplacian in Lipschitz Domains Satisfyinga Uniform Exterior Ball Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227
Tunde Jakab, Irina Mitrea, and Marius Mitrea
Properties of Spectra of Boundary Value Problemsin Cylindrical and Quasicylindrical Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Sergey Nazarov
Estimates for Completeley Integrable Systems of DifferentialOperators and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Yuri Reshetnyak
Contents xi
Counting Schrodinger Boundstates: Semiclassics and Beyond . . . . . . . . . . 329Grigori Rozenblum and Michael Solomyak
Function Spaces on Cellular Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355Hans Triebel
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
III. Applications in Mathematical PhysicsVictor Isakov Ed.
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Victor Isakov
Geometrization of Rings as a Method for Solving Inverse Problems . . . . . . .5Mikhail Belishev
The Ginzburg–Landau Equations for Superconductivity withRandom Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
Andrei Fursikov, Max Gunzburger, and Janet Peterson
Carleman Estimates with Second Large Parameter for SecondOrder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Victor Isakov and Nanhee Kim
Sharp Spectral Asymptotics for Dirac Energy . . . . . . . . . . . . . . . . . . . . . . . . . .161Victor Ivrii
Linear Hyperbolic and Petrowski Type PDEs with ContinuousBoundary Control → Boundary Observation Open Loop Map:Implication on Nonlinear Boundary Stabilization withOptimal Decay Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187
Irena Lasiecka and Roberto Triggiani
Uniform Asymptotics of Green’s Kernels for Mixed and NeumannProblems in Domains with Small Holes and Inclusions . . . . . . . . . . . . . . . . . 277
Vladimir Maz’ya and Alexander Movchan
Finsler Structures and Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317Michael Taylor
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
ContributorsEditors
Vladimir Maz’ya
Ohio State UniversityColumbus, OH 43210USA
University of LiverpoolLiverpool L69 7ZLUK
Linkoping UniversityLinkoping SE-58183
SWEDEN
Victor Isakov
Wichita State UniversityWichita, KS 67206USA
ContributorsAuthors
Daniel AaltoInstitute of MathematicsHelsinki University of TechnologyP.O. Box 1100, FI-02015FINLAND
e-mail: [email protected]
David R. AdamsUniversity of KentuckyLexington, KY 40506-0027USA
e-mail: [email protected]
Hiroaki AikawaHokkaido UniversitySapporo 060-0810JAPAN
e-mail: [email protected]
Vasili BabichSteklov Mathematical InstituteRussian Academy of Sciences27 Fontanka Str., St.-Petersburg 191023RUSSIA
e-mail: [email protected]
Mikhail BelishevSteklov Mathematical InstituteRussian Academy of Sciences27 Fontanka Str., St.-Petersburg 191023RUSSIA
e-mail: [email protected]
Sergey BobkovUniversity of MinnesotaMinneapolis, MN 55455USA
e-mail: [email protected]
xvi Sobolev Spaces in Mathematics I–III
Yuri BrudnyiTechnion – Israel Institute of TechnologyHaifa 32000ISRAEL
e-mail: [email protected]
Victor BurenkovUniversita degli Studi di Padova63 Via Trieste, 35121 PadovaITALY
e-mail: [email protected]
Andrea CianchiUniversita di FirenzePiazza Ghiberti 27, 50122 FirenzeITALY
e-mail: [email protected]
Serban CosteaMcMaster University1280 Main Street WestHamilton, Ontario L8S 4K1CANADA
e-mail: [email protected]
Stephan DahlkePhilipps–Universitat MarburgFachbereich Mathematik und InformatikHans Meerwein Str., Lahnberge 35032 MarburgGERMANY
e-mail: [email protected]
Donatella DanielliPurdue University150 N. University Str.West Lafayette, IN 47906USA
e-mail: [email protected]
David E. EdmundsSchool of Mathematics Cardiff UniversitySenghennydd Road CARDIFFWales CF24 4AGUK
e-mail: [email protected]
W. Desmond EvansSchool of Mathematics Cardiff UniversitySenghennydd Road CARDIFFWales CF24 4AGUK
e-mail: [email protected]
Contributors. Authors xvii
Andrei FursikovMoscow State UniversityVorob’evy Gory, Moscow 119992RUSSIA
e-mail: [email protected]
Victor GalaktionovUniversity of BathBath, BA2 7AYUK
e-mail: [email protected]
Nicola GarofaloPurdue University150 N. University Str.West Lafayette, IN 47906USA
e-mail: [email protected]
Friedrich GotzeBielefeld UniversityBielefeld 33501GERMANY
e-mail: [email protected]
Vladimir Gol’dshteinBen Gurion University of the NegevP.O.B. 653, Beer Sheva 84105ISRAEL
e-mail: [email protected]
Alexander Grigor’yanBielefeld UniversityBielefeld 33501GERMANY
e-mail: [email protected]
Max GunzburgerFlorida State UniversityTallahassee, FL 32306-4120USA
e-mail: [email protected]
Piotr Haj�laszUniversity of Pittsburgh301 Thackeray Hall, Pittsburgh, PA 15260USA
e-mail: [email protected]
xviii Sobolev Spaces in Mathematics I–III
Elton HsuNorthwestern University2033 Sheridan Road, Evanston, IL 60208-2730USA
e-mail: [email protected]
Victor IsakovWichita State UniversityWichita, KS 67206USA
e-mail: [email protected]
Victor IvriiUniversity of Toronto40 St.George Str., Toronto, Ontario M5S 2E4CANADA
e-mail: [email protected]
Tunde JakabUniversity of VirginiaCharlottesville, VA 22904USA
e-mail: [email protected]
Nanhee KimWichita State UniversityWichita, KS 67206USA
e-mail: [email protected]
Juha KinnunenInstitute of MathematicsHelsinki University of TechnologyP.O. Box 1100, FI-02015FINLAND
e-mail: [email protected]
Pier Domenico LambertiUniversita degli Studi di Padova63 Via Trieste, 35121 PadovaITALY
e-mail: [email protected]
Irena LasieckaUniversity of VirginiaCharlottesville, VA 22904USA
e-mail: [email protected]
Vladimir Maz’yaOhio State UniversityColumbus, OH 43210USA
Contributors. Authors xix
University of LiverpoolLiverpool L69 7ZLUK
Linkoping UniversityLinkoping SE-58183SWEDENe-mail: [email protected]
e-mail: [email protected]
Enzo MitidieriUniversita di TriesteVia Valerio 12/1, 34127 TriesteITALY
e-mail: [email protected]
Irina MitreaUniversity of VirginiaCharlottesville, VA 22904
USA
e-mail: [email protected]
Marius MitreaUniversity of MissouriColumbia, MOUSA
e-mail: [email protected]
Alexander MovchanUniversity of LiverpoolLiverpool L69 3BXUK
e-mail: [email protected]
Sergey NazarovInstitute of Problems in Mechanical EngineeringRussian Academy of Sciences61, Bolshoi pr., V.O., St.-Petersburg 199178RUSSIA
e-mail: [email protected]
Janet PetersonFlorida State UniversityTallahassee FL 32306-4120USA
e-mail: [email protected]
Nguyen Cong PhucPurdue University150 N. University Str.West Lafayette, IN 47906USA
e-mail: [email protected]
xx Sobolev Spaces in Mathematics I–III
Lubos PickCharles UniversitySokolovska 83, 186 75 Praha 8CZECH REPUBLIC
e-mail: [email protected]
Yehuda PinchoverTechnion – Israel Institute of TechnologyHaifa 32000ISRAEL
e-mail: [email protected]
Stanislav PokhozhaevSteklov Mathematical InstituteRussian Academy of Sciences8, Gubkina Str., Moscow 119991RUSSIA
e-mail: [email protected]
Yuri ReshetnyakSobolev Institute of MathematicsSiberian BranchRussian Academy of Sciences4, Pr. Koptyuga, Novosibirsk 630090RUSSIA
Novosibirsk State University2, Pirogova Str., Novosibirsk 630090RUSSIA
e-mail: [email protected]
Grigori RozenblumUniversity of GothenburgS-412 96, GothenburgSWEDEN
e-mail: [email protected]
Laurent Saloff-CosteCornell UniversityMallot Hall, Ithaca, NY 14853USA
e-mail: [email protected]
Nageswari ShanmugalingamUniversity of CincinnatiCincinnati, OH 45221-0025USA
e-mail: [email protected]
Tatyana ShaposhnikovaOhio State UniversityColumbus, OH 43210USA
Contributors. Authors xxi
Linkoping UniversityLinkoping SE-58183SWEDEN
e-mail: [email protected]
Winfried SickelFriedrich-Schiller-Universitat JenaMathematisches InstitutErnst–Abbe–Platz 2, D-07740 JenaGERMANY
e-mail: [email protected]
Michael SolomyakThe Weizmann Institute of ScienceRehovot, 76100ISRAEL
e-mail: [email protected]
Michael TaylorUniversity of North CarolinaChapel Hill, NC 27599USA
e-mail: [email protected]
Kyril TintarevUppsala UniversityP.O. Box 480, SE-751 06 UppsalaSWEDEN
e-mail: [email protected]
Hans TriebelMathematisches InstitutFriedrich-Schiller-Universitat JenaD-07737 JenaGERMANY
e-mail: [email protected]
Roberto TriggianiUniversity of VirginiaCharlottesville, VA 22904USA
e-mail: [email protected]
Marc TroyanovInstitute of Geometry, Algebra, and TopologyEcole Polytechnique Federale de Lausanne1015 LausanneSWITZERLAND
e-mail: [email protected]
Sobolev Type Inequalities
Vladimir Maz’ya Ed.
Contents
My Love Affair with the Sobolev Inequality . . . . . . . . . . . . . . . . . . 1David R. Adams
1 The Trace Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 A Mixed Norm Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 A Morrey–Sobolev Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 A Morrey–Besov Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Exponential Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Vanishing Exponential Integrability . . . . . . . . . . . . . . . . . . . . . . 187 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Maximal Functions in Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 25Daniel Aalto and Juha Kinnunen
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Maximal Function Defined on the Whole Space . . . . . . . . . . . 27
2.1 Boundedness in Sobolev spaces . . . . . . . . . . . . . . . . . . 272.2 A capacitary weak type estimate . . . . . . . . . . . . . . . . . 32
3 Maximal Function Defined on a Subdomain . . . . . . . . . . . . . . . 333.1 Boundedness in Sobolev spaces . . . . . . . . . . . . . . . . . . 333.2 Sobolev boundary values . . . . . . . . . . . . . . . . . . . . . . . . 38
4 Pointwise Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.1 Lusin type approximation of Sobolev functions . . . . 45
5 Hardy Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 Maximal Functions on Metric Measure Spaces . . . . . . . . . . . . 54
6.1 Sobolev spaces on metric measure spaces . . . . . . . . . . 556.2 Maximal function defined on the whole space . . . . . . 576.3 Maximal function defined on a subdomain . . . . . . . . 626.4 Pointwise estimates and Lusin type approximation . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
xxv
xxvi Contents
Hardy Type Inequalities via Riccati and Sturm–LiouvilleEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69Sergey Bobkov and Friedrich Gotze
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692 Riccati Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713 Transition to Sturm–Liouville Equations . . . . . . . . . . . . . . . . . 754 Hardy Type Inequalities with Weights . . . . . . . . . . . . . . . . . . . 775 Poincare Type Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Quantitative Sobolev and Hardy Inequalities, and RelatedSymmetrization Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87Andrea Cianchi
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872 Symmetrization Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
2.1 Rearrangements of functions and function spaces . . . 892.2 The Hardy–Littlewood inequality . . . . . . . . . . . . . . . . 922.3 The Polya–Szego inequality . . . . . . . . . . . . . . . . . . . . . 96
3 Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.1 Functions of Bounded Variation . . . . . . . . . . . . . . . . . . 1013.2 The case 1 < p < n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.3 The case p > n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4 Hardy Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.1 The case 1 < p < n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.2 The case p = n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Inequalities of Hardy–Sobolev Type in Carnot–CaratheodorySpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117Donatella Danielli, Nicola Garofalo, and Nguyen Cong Phuc
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213 Pointwise Hardy Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274 Hardy Inequalities on Bounded Domains . . . . . . . . . . . . . . . . . 1395 Hardy Inequalities with Sharp Constants . . . . . . . . . . . . . . . . . 145References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Sobolev Embeddings and Hardy Operators . . . . . . . . . . . . . . . . . . . 153David E. Edmunds and W. Desmond Evans
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1532 Hardy Operators on Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1543 The Poincare Inequality, α(E) and Hardy Type Operators . . 1584 Generalized Ridged Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625 Approximation and Other s-Numbers of Hardy Type
Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Contents xxvii
6 Approximation Numbers of Embeddings on GeneralizedRidged Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Sobolev Mappings between Manifolds and Metric Spaces . . . . . 185Piotr Haj�lasz
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1852 Sobolev Mappings between Manifolds . . . . . . . . . . . . . . . . . . . . 1873 Sobolev Mappings into Metric Spaces . . . . . . . . . . . . . . . . . . . . 197
3.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2024 Sobolev Spaces on Metric Measure Spaces . . . . . . . . . . . . . . . . 205
4.1 Integration on rectifiable curves . . . . . . . . . . . . . . . . . . 2054.2 Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.3 Upper gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.4 Sobolev spaces N1,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.5 Doubling measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2094.6 Other spaces of Sobolev type . . . . . . . . . . . . . . . . . . . . 2114.7 Spaces supporting the Poincare inequality . . . . . . . . . 214
5 Sobolev Mappings between Metric Spaces . . . . . . . . . . . . . . . . 2155.1 Lipschitz polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
A Collection of Sharp Dilation Invariant Integral Inequalitiesfor Differentiable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223Vladimir Maz’ya and Tatyana Shaposhnikova
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2232 Estimate for a Quadratic Form of the Gradient . . . . . . . . . . . 2263 Weighted Garding Inequality for the Biharmonic Operator . . 2304 Dilation Invariant Hardy’s Inequalities with Remainder
Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2335 Generalized Hardy–Sobolev Inequality with Sharp
Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2416 Hardy’s Inequality with Sharp Sobolev Remainder Term . . . 244References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
Optimality of Function Spaces in Sobolev Embeddings . . . . . . . . 249Lubos Pick
1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2492 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2563 Reduction Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2584 Optimal Range and Optimal Domain of Rearrangement-
Invariant Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2615 Formulas for Optimal Spaces Using
the Functional f∗∗ − f∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2646 Explicit Formulas for Optimal Spaces in Sobolev
Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
xxviii Contents
7 Compactness of Sobolev Embeddings . . . . . . . . . . . . . . . . . . . . 2708 Boundary Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2759 Gaussian Sobolev Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . 276References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
On the Hardy–Sobolev–Maz’ya Inequality and ItsGeneralizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281Yehuda Pinchover and Kyril Tintarev
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2812 Generalization of the Hardy–Sobolev–Maz’ya Inequality . . . . 2843 The Space D1,2
V (Ω) and Minimizers for the Hardy–Sobolev–Maz’ya Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
4 Convexity Properties of the Functional Q for p > 2 . . . . . . . . 293References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Sobolev Inequalities in Familiar and Unfamiliar Settings . . . . . . 299Laurent Saloff-Coste
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2992 Moser’s Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
2.1 The basic technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3002.2 Harnack inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3022.3 Poincare, Sobolev, and the doubling property . . . . . . 3032.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
3 Analysis and Geometry on Dirichlet Spaces . . . . . . . . . . . . . . . 3123.1 First order calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3123.2 Dirichlet spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3123.3 Local weak solutions of the Laplace and heat
equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3143.4 Harnack type Dirichlet spaces . . . . . . . . . . . . . . . . . . . 3163.5 Imaginary powers of −A and the wave equation . . . . 3183.6 Rough isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
4 Flat Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3224.1 How to prove a flat Sobolev inequality? . . . . . . . . . . . 3224.2 Flat Sobolev inequalities and semigroups of
operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3244.3 The Rozenblum–Cwikel–Lieb inequality . . . . . . . . . . . 3264.4 Flat Sobolev inequalities in the finite volume case . . 3294.5 Flat Sobolev inequalities and topology at infinity . . 330
5 Sobolev Inequalities on Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 3305.1 Graphs of bounded degree . . . . . . . . . . . . . . . . . . . . . . 3315.2 Sobolev inequalities and volume growth . . . . . . . . . . . 3325.3 Random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3335.4 Cayley graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Contents xxix
A Universality Property of Sobolev Spaces in Metric MeasureSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345Nageswari Shanmugalingam
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3452 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3473 Dirichlet Forms and N1,2(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3494 Axiomatic Sobolev Spaces and N1,p(X) . . . . . . . . . . . . . . . . . . 356References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
Cocompact Imbeddings and Structure of Weakly ConvergentSequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361Kiril Tintarev
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612 Dislocation Space and Weak Convergence Decomposition . . . 3633 Cocompactness and Minimizers . . . . . . . . . . . . . . . . . . . . . . . . . 3684 Flask Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3725 Compact Imbeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377