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” (tamara@mathbooks.ru) 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.

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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

vlmaz@mai.liu.se

vlmaz@math.ohio-state.edu

Victor Isakov

Wichita State UniversityWichita, KS 67206USA

victor.isakov@wichita.edu

ContributorsAuthors

Daniel AaltoInstitute of MathematicsHelsinki University of TechnologyP.O. Box 1100, FI-02015FINLAND

e-mail: daniel.aalto@tkk.fi

David R. AdamsUniversity of KentuckyLexington, KY 40506-0027USA

e-mail: dave@ms.uky.edu

Hiroaki AikawaHokkaido UniversitySapporo 060-0810JAPAN

e-mail: aik@math.sci.hokudai.ac.jp

Vasili BabichSteklov Mathematical InstituteRussian Academy of Sciences27 Fontanka Str., St.-Petersburg 191023RUSSIA

e-mail: babich@pdmi.ras.ru

Mikhail BelishevSteklov Mathematical InstituteRussian Academy of Sciences27 Fontanka Str., St.-Petersburg 191023RUSSIA

e-mail: belishev@pdmi.ras.ru

Sergey BobkovUniversity of MinnesotaMinneapolis, MN 55455USA

e-mail: bobkov@math.umn.edu

xvi Sobolev Spaces in Mathematics I–III

Yuri BrudnyiTechnion – Israel Institute of TechnologyHaifa 32000ISRAEL

e-mail: ybrudnyi@math.technion.ac.il

Victor BurenkovUniversita degli Studi di Padova63 Via Trieste, 35121 PadovaITALY

e-mail: burenkov@math.unipd.it

Andrea CianchiUniversita di FirenzePiazza Ghiberti 27, 50122 FirenzeITALY

e-mail: cianchi@unifi.it

Serban CosteaMcMaster University1280 Main Street WestHamilton, Ontario L8S 4K1CANADA

e-mail: secostea@math.mcmaster.ca

Stephan DahlkePhilipps–Universitat MarburgFachbereich Mathematik und InformatikHans Meerwein Str., Lahnberge 35032 MarburgGERMANY

e-mail: dahlke@mathematik.uni-marburg.de

Donatella DanielliPurdue University150 N. University Str.West Lafayette, IN 47906USA

e-mail: danielli@math.purdue.edu

David E. EdmundsSchool of Mathematics Cardiff UniversitySenghennydd Road CARDIFFWales CF24 4AGUK

e-mail: davideedmunds@aol.com

W. Desmond EvansSchool of Mathematics Cardiff UniversitySenghennydd Road CARDIFFWales CF24 4AGUK

e-mail: EvansWD@cf.ac.uk

Contributors. Authors xvii

Andrei FursikovMoscow State UniversityVorob’evy Gory, Moscow 119992RUSSIA

e-mail: fursikov@mtu-net.ru

Victor GalaktionovUniversity of BathBath, BA2 7AYUK

e-mail: vag@maths.bath.ac.uk

Nicola GarofaloPurdue University150 N. University Str.West Lafayette, IN 47906USA

e-mail: garofalo@math.purdue.edu

Friedrich GotzeBielefeld UniversityBielefeld 33501GERMANY

e-mail: goetze@math.uni-bielefeld.de

Vladimir Gol’dshteinBen Gurion University of the NegevP.O.B. 653, Beer Sheva 84105ISRAEL

e-mail: vladimir@bgu.ac.il

Alexander Grigor’yanBielefeld UniversityBielefeld 33501GERMANY

e-mail: grigor@math.uni-bielefeld.de

Max GunzburgerFlorida State UniversityTallahassee, FL 32306-4120USA

e-mail: gunzburg@scs.fsu.edu

Piotr Haj�laszUniversity of Pittsburgh301 Thackeray Hall, Pittsburgh, PA 15260USA

e-mail: hajlasz@pitt.edu

xviii Sobolev Spaces in Mathematics I–III

Elton HsuNorthwestern University2033 Sheridan Road, Evanston, IL 60208-2730USA

e-mail: ehsu@math.northwestern.edu

Victor IsakovWichita State UniversityWichita, KS 67206USA

e-mail: victor.isakov@wichita.edu

Victor IvriiUniversity of Toronto40 St.George Str., Toronto, Ontario M5S 2E4CANADA

e-mail: ivrii@math.toronto.edu

Tunde JakabUniversity of VirginiaCharlottesville, VA 22904USA

e-mail: tj8y@virginia.edu

Nanhee KimWichita State UniversityWichita, KS 67206USA

e-mail: kim@math.wichita.edu

Juha KinnunenInstitute of MathematicsHelsinki University of TechnologyP.O. Box 1100, FI-02015FINLAND

e-mail: juha.kinnunen@tkk.fi

Pier Domenico LambertiUniversita degli Studi di Padova63 Via Trieste, 35121 PadovaITALY

e-mail: lamberti@math.unipd.it

Irena LasieckaUniversity of VirginiaCharlottesville, VA 22904USA

e-mail: il2v@virginia.edu

Vladimir Maz’yaOhio State UniversityColumbus, OH 43210USA

Contributors. Authors xix

University of LiverpoolLiverpool L69 7ZLUK

Linkoping UniversityLinkoping SE-58183SWEDENe-mail: vlmaz@mai.liu.se

e-mail: vlmaz@math.ohio-state.edu

Enzo MitidieriUniversita di TriesteVia Valerio 12/1, 34127 TriesteITALY

e-mail: mitidier@units.it

Irina MitreaUniversity of VirginiaCharlottesville, VA 22904

USA

e-mail: im3p@virginia.edu

Marius MitreaUniversity of MissouriColumbia, MOUSA

e-mail: marius@math.missouri.edu

Alexander MovchanUniversity of LiverpoolLiverpool L69 3BXUK

e-mail: abm@liverpool.ac.uk

Sergey NazarovInstitute of Problems in Mechanical EngineeringRussian Academy of Sciences61, Bolshoi pr., V.O., St.-Petersburg 199178RUSSIA

e-mail: serna@snark.ipme.ru

Janet PetersonFlorida State UniversityTallahassee FL 32306-4120USA

e-mail: peterson@scs.fsu.edu

Nguyen Cong PhucPurdue University150 N. University Str.West Lafayette, IN 47906USA

e-mail: pcnguyen@math.purdue.edu

xx Sobolev Spaces in Mathematics I–III

Lubos PickCharles UniversitySokolovska 83, 186 75 Praha 8CZECH REPUBLIC

e-mail: pick@karlin.mff.cuni.cz

Yehuda PinchoverTechnion – Israel Institute of TechnologyHaifa 32000ISRAEL

e-mail: pincho@techunix.technion.ac.il

Stanislav PokhozhaevSteklov Mathematical InstituteRussian Academy of Sciences8, Gubkina Str., Moscow 119991RUSSIA

e-mail: pokhozhaev@mi.ras.ru

Yuri ReshetnyakSobolev Institute of MathematicsSiberian BranchRussian Academy of Sciences4, Pr. Koptyuga, Novosibirsk 630090RUSSIA

Novosibirsk State University2, Pirogova Str., Novosibirsk 630090RUSSIA

e-mail: Reshetnyak@math.nsc.ru

Grigori RozenblumUniversity of GothenburgS-412 96, GothenburgSWEDEN

e-mail: grigori@math.chalmers.se

Laurent Saloff-CosteCornell UniversityMallot Hall, Ithaca, NY 14853USA

e-mail: lsc@math.cornell.edu

Nageswari ShanmugalingamUniversity of CincinnatiCincinnati, OH 45221-0025USA

e-mail: nages@math.uc.edu

Tatyana ShaposhnikovaOhio State UniversityColumbus, OH 43210USA

Contributors. Authors xxi

Linkoping UniversityLinkoping SE-58183SWEDEN

e-mail: tasha@mai.liu.se

Winfried SickelFriedrich-Schiller-Universitat JenaMathematisches InstitutErnst–Abbe–Platz 2, D-07740 JenaGERMANY

e-mail: sickel@minet.uni-jena.de

Michael SolomyakThe Weizmann Institute of ScienceRehovot, 76100ISRAEL

e-mail: michail.solomyak@weizmann.ac.il

Michael TaylorUniversity of North CarolinaChapel Hill, NC 27599USA

e-mail: met@email.unc.edu

Kyril TintarevUppsala UniversityP.O. Box 480, SE-751 06 UppsalaSWEDEN

e-mail: kyril.tintarev@math.uu.se

Hans TriebelMathematisches InstitutFriedrich-Schiller-Universitat JenaD-07737 JenaGERMANY

e-mail: triebel@minet.uni-jena.de

Roberto TriggianiUniversity of VirginiaCharlottesville, VA 22904USA

e-mail: rt7u@virginia.edu

Marc TroyanovInstitute of Geometry, Algebra, and TopologyEcole Polytechnique Federale de Lausanne1015 LausanneSWITZERLAND

e-mail: marc.troyanov@epfl.ch

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

My Love Affair with the SobolevInequality

David R. Adams

Abstract Reminiscence about different versions of the Sobolev inequalityobtained by the author and others.

Due to the fact that the Sobolev Inequality isso central to much of mathematical analysis,especially to partial differential equations, it isnot surprising that there are by now, 70 yearsafter Sobolev’s original paper, many differentversions of the Sobolev Inequality and by manydifferent authors. This paper is a tribute toS.L. Sobolev.

David R. Adams

On a cold December morning shortly after Christmas 1969, a small group ofpeople were huddled against the cold near a limousine type bus parked in themiddle of the main street of a very small South Dakota farming communityin the central plains of the USA – my ancestral hometown. The bus, run bythe Greyhound Company, was the “east-west connector bus” and the middleof Main Street was the usual passenger pick-up and drop-off spot in town.Here Main Street consisted of just one block of store front businesses and it

David R. AdamsUniversity of Kentucky, Lexington, KY 40506-0027, USA, e-mail: dave@ms.uky.edu

V. Maz’ya (ed.), Sobolev Spaces in Mathematics I,International Mathematical Series.

doi: 10.1007/978-0-387-85648-3, c© Springer Science + Business Media, LLC 2009

1

2 D. R. Adams

was wide enough for angular parking on each side and still with plenty ofroom to accommodate car traffic on either side of the parked bus and thewaiting people. Most of these people were my relatives, gathered in town forChristmas and now taking the opportunity to see me off. This bus will meet alarger north-south bus at some highway crosswords out on the nearby prairie.I am expecting to tranfer to that bus for the next stage of my journey.

When I boarded the small bus, there was only one other passenger, anelderly woman. And as I sat down and waved a final farewell to my family,the woman turned to me and asked, “where are you going, young man?” Icould have been very dramatic and responded, “into History!”, but I did not,nor did it even cross my mind to say such a thing. I just said, “to Rome,Italy”, which I am sure was dramatic enough. “My land”, she responded, “Iam only going to see my sister in Minnesota, you are going a long ways.”

And I guess it was, both physically and psychologically for all concerned.And now I can confess that I was not very well prepared for my Italiansojourn. Though I did eventually adjust and adapt to life in Rome (January–August 1970), and even began to thrive there toward the end of my stay.However my budding Italian speech never broke away from my American-midwestern accent. Though if I kept my mouth shut, I eventually could passfor Italian at least in dress and demeanor. Once an American tourist stoppedme on the street during my last days to ask, in English, where some place-street was located. I responded by telling her where it was and how to getthere. “Wow, you speak good English!” she said. “Thank you” I replied andwalked away, leaving her with the illusion.

Thus with my initial bus trip, I began my mathematical odyssey – first tothe CNR in Rome as a Post Doc under the direction of Guido Stampacchia,later as an instructor at Rice University, an acting Assistant Professor at theUniversity of California, San Diego, a visiting professor at Indiana University,and finally a Professor at the University of Kentucky – for the past 30+ years.When I left for Italy, I had just days earlier received my Ph.D. degree fromthe University of Minnesota under the direction of N.G. Meyers. And I beganmy Post Doc studies by tackling a question posed earlier by him and thenat the CNR under the watchful eye of Stampacchia. As it turned out, itwas this question of Meyers’ that essentially started me down the road oflooking at variations of the now classical Sobolev Inequality. For I did notconsciously look to that direction, but as it all transpired, I have over theyears returned again and again to this theme, eventually giving five or sixversions of the Sobolev Inequality during my career. And due to the factthat the Sobolev Inequality is so central to much of mathematical analysis,especially to partial differential equations, it is not surprising that there areby now, 70 years after Sobolev’s original paper, many different versions of theSobolev Inequality and by many different authors. This paper is a tribute toS.L. Sobolev. I do not pretend to review all of this literature, only at best,part of my role in it. This is after all the story of my romance with theSobolev Inequality.

My Love Affair with the Sobolev Inequality 3

The classical Sobolev Inequality that I refer to can take on one or twoequivalent forms. For example, if u(x) is a smooth function of compact sup-port, then for 1 < p < n/m, there is a constant c depending only on n, mand p such that

(∗) ||u||Lq(IRn) � c||Dmu||Lp(Rn),

where 1/q = 1/p − m/n1 Here Dmu denotes the vector of all mth orderderivatives of u. Or, if we use Riesz potentials, an equivalent form of (∗) is

(∗∗) ||Iαf ||Lq(Rn) � c′||f ||Lp(Rn)

with again q = np/(n− αp), where α = m. Here I have written

Iαf(x) =∫

Rn

|x− y|α−nf(y) dy,

0 < α < n, 1 < p < n/α.On the connection between the two, simply note |u(x)| � cIm (|Dmu|)(x)

and singular integrals obtained by differentiating Im f m-times.

So now I begin the story of my involvement with the Sobolev Inequality –from my first struggle at the CNR to prove a trace inequality to my morerecent work on a vanishing mean exponential integrability condition withR. Hurri-Syrjanen.

Some time lines are:

1971 – The trace inequality (Sect. 1)

1973 – An exponential trace inequality (Sect. 6)

1974 – A mixed norm inequality; with R. Bagby (Sect. 2)

1975 – A Morrey–Sobolev Inequality (Sect. 3)

1976 – A trace inequality with CSI, q = p (Sect. 1)

1982 – A Morrey–Besov inequality; with J. Lewis (Sect. 4)

1988 – Exponential integrability (Sect. 5)

1998 – Estimates for Mαf (Sect. 7, (4))

2003 – Vanishing exponential integrability; with R. Hurri-Syrjanen (Sect. 6)

2004 – Trace estimates for Morrey–Sobolev functions; with J. Xiao (Sect. 3)

1 Inequality (∗) also holds for p = 1 by the Gagliardo–Nirenberg estimates (see [55]).However, (∗∗) does not hold for p = 1.

4 D. R. Adams

1 The Trace Inequality

The struggle alluded to above was the question of finding necessary and suffi-cient conditions on a Borel measures μ defined on subsets of R

n that insures,in the language of N.G. Meyers, that μ has positive capacity. In the late1960’s, Meyers wrote a paper (unpublished, to this date) titled: Capacities,extremal length and traces of strongly differentiable functions. Here, to unifythe ideas of capacity and extremal length, among other ideas, he defined acapacity of a set of measures on R

n. The usual capacity of a standard sub-set K ⊂ R

n then reduced to taking the sets of Dirac measures {δx}x∈K . Asimplified version of this might be

Cα,p,q(K) = inf {||f ||pLp(Rn) : ||Iαf ||Lq(ν) � 1 ∀ν ∈ K and f � 0},

where K ⊂ M+ = all Borel measures on Rn. The question posed by Meyers

was to characterize all measures μ with positive capacity Cα,p,q({μ}) > 0.This is clearly equivalent to the trace estimate

||Iαf ||Lq(μ) � c1||f ||Lp(Rn). (1.1)

What I eventually proved, in [1], was the simple and elegant necessary andsufficient condition

μ(B(x, r)) � Ard (1.2)

for all r > 0 and all x ∈ Rn. Here B(x, r) is an open ball centered at x

and of radius r > 0. The conditions of equivalency are: 0 < d � n, q =dp/(n−αp), 1 < p < q <∞. The coefficient A is a constant independent ofx and r. I worked this out in the Spring of 1970. At the time the only resultI was aware of along these lines was the trace estimate of Il’in [37] which is(1.1) with μ = Lebesgue measure on a hyperplane. Later in [2], I found amuch simpler proof of the equivalency of (1.1) and (1.2). There, the resultfollowed easily from the weak type estimate

μ([Iαf > t]) �(ct||f ||Lp(Rn)

)q(1.3)

for f � 0, followed by an application of the Marcinkiewicz InterpolationTheorem, since here 1/q < 1/p (see [11, Theorem 7.2.2] or [41, Theorem 1,p. 52] or even [55, Theorem 4.7.2]).

Of course, it easily becomes clear that (1.1) and (1.2) are no longer equiv-alent when q = p. Indeed, if d = n−αp in (1.2), then there is an f ∈ Lp+(Rn)such that Iαf = +∞ on a set of positive finite μ measure, or to say it another

My Love Affair with the Sobolev Inequality 5

way, Cα,p(K) ≡ Cα,p({δx : x ∈ K}) can be zero for a compact set K ⊂ Rn

with positive Hausdorff capacity (content) Hn−αp∞ (K) > 0. Here

Hd∞(K) = inf

⎧⎨⎩∑i

rdi : K is covered by a countablenumber of balls of radius ri > 0

⎫⎬⎭ .

Left to right: V.P. Havin, V.G. Maz’ya, and D.R. Adams. Maz’yalecturing on the finer points of Potential Theory on the wall of abuilding in Leningrad. Summer 1974.

All of my attempts to find a simple substitute for condition (1.2) when q =p failed – until I made a pilgrimage to Leningrad (with L. Hedberg andJ. Brennan) to meet with V. Maz’ya (and V. Havin) in the summer of 1974.Discussions with Maz’ya profoundly changed my view of the trace question.The limiting case q = p needs the Maz’ya type capacity inequality

∞∫

0

Cα,p([Iαf > t]) dtp � c||f ||pLp(Rn) (1.4)

for f � 0 and 1 < p < n/α. In [4], I dubbed (1.4) a capacity strong typeinequality (or CSI) in analogy to the weak and strong type estimates forsingular integrals in Harmonic Analysis; note that the weak type estimate

Cα,p([Iαf > t]) � t−p||f ||pLp(Rn)

6 D. R. Adams

is trivial. Maz’ya was the first to establish (1.4); the paper [39] treats the casesα = 1 and 2. And then after my Leningrad meeting, I managed to prove (1.4)for α a positive integer (see [4]). Later, Dahlberg [26] noted how to get (1.4)in the remaining fractional α cases, and finally Hansson [34] gave a generalargument that extended (1.4) considerably (see also [11, p. 187f ]). Maz’ya[40] proved a CSI for Besov spaces also around this time. The connection of(1.4) to (1.1) is of course

μ(K) � c Cα,p(K) (1.5)

for all compact sets K ⊂ Rn. A similar condition: μ(K) � c Cα,p(K)q/p for

all compact K works to give (1.1) when 1 < p < q < ∞, but the advantageof checking (1.2) for only balls is enormous. But as we have observed above(1.1) does not hold simply by having (1.5) for K = B(x, r) only. There is aball condition in [38], but it is no longer so simple and a famous ball conditionof Fefferman–Phong in [27], but that one is only sufficient.

Another interesting sufficient condition for (1.5) is the boundedness of thenonlinear potential Uμα,p(x) = Iα(Iαμ)p

′−1(x) or equivalently, for the Wolffpotential, where αp < n

Wμα,p(x) ≡

∞∫

0

[rαp−nμ(B(x, r))]p′−1 dt

t.

This last statement is a consequence of the Wolff inequality

||Iαμ||p′

Lp′(Rn)� c∫

Rn

Wμα,p(x)dμ(x) (1.8)

andμ(K) �

∫Iαfdμ

K � ||f ||Lp(Rn)||IαμK ||Lp′(Rn)

� ||f ||Lp(Rn) c||Wμα,p||1/p

′

L∞ μ(K)1/p′

orμ(K) � c||Wμ

α,p||p−1L∞ Cα,p(K).

Here I have written μK = μ∠K. Notice also that

||Iαμ||p′

Lp′(Rn)=∫Uμα,p dμ

and so a lower bound matching (1.8) for p′ norm of the potential Iαμ holds –it is a simple estimate, whereas the Wolff inequality requires much heavierwork (see [11, p. 109] or [6]).