SOBOLEV SPACES IN MATHEMATICS I
SOBOLEV TYPE INEQUALITIES INTERNATIONAL MATHEMATICAL SERIES
Series Editor: Tamara Rozhkovskaya Novosibirsk, Russia
1. Nonlinear Problems in Mathematical Physics and Related Topics I. 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 Topics II. 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 the XXIst Century • D. Gabbay, S. Goncharov, M. Zakharyaschev Eds. • 2006 5. Mathematical Problems from Applied Logic II. Logics for the XXIst 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 and Partial Differential Equations • V. Maz’ya Ed. • 2009 10. Sobolev Spaces in Mathematics III. Applications in Mathemat- ical Physics • V. Isakov Ed. • 2009 SOBOLEV SPACES IN MATHEMATICS I
Sobolev Type Inequalities
Editor: Vladimir Maz’ya Ohio State University, USA University of Liverpool, UK Link¨oping University, SWEDEN
123 Tamara Rozhkovskaya Publisher Editor
Prof. Vladimir Maz’ya Ohio State University Department of Mathematics Columbus, USA
University of Liverpool Department of Mathematical Sciences Liverpool, UK
Link¨oping University Department of Mathematics Link¨oping, Sweden
This series was founded in 2002 and is a joint publication of Springer and “Tamara Rozhkovskaya Publisher.” Each volume presents contributions from the Volume Editors and Authors exclusively invited by the Series Editor Tamara Rozhkovskaya who also pre- pares the Camera Ready Manuscript. This volume is distributed by “Tamara Rozhkovskaya Publisher” ([email protected]) in Russia and by Springer over all the world.
ISBN 978-0-387-85647-6 e-ISBN 978-0-387-85648-3 ISBN 978-5-901873-24-3 (Tamara Rozhkovskaya Publisher) ISSN 1571-5485
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987654321 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 Mathematics presents the latest results in the theory of Sobolev spaces and appli- cations from leading experts in these areas.
I. Sobolev Type Inequalities In 1938, exactly 70 years ago, the original Sobolev inequality (an embed- ding theorem) was published in the celebrated paper by S.L. Sobolev “On a theorem of functional analysis.” By now, the Sobolev inequality and its numerous versions continue to attract attention of researchers because of the central role played by such inequalities in the theory of partial differ- ential equations, mathematical physics, and many various areas of analysis and differential geometry. The volume presents the recent study of different Sobolev type inequalities, in particular, inequalities on manifolds, Carnot– Carath´eodory 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 functions in Sobolev spaces, etc. Some unfamiliar settings of Sobolev type inequalities (for example, on graphs) are also discussed. The volume opens with the survey article “My Love Affair with the Sobolev Inequality” by David R. Adams.
II. Applications in Analysis and Partial Differential Equations Sobolev spaces become the established language of the theory of partial dif- ferential equations and analysis. Among a huge variety of problems where Sobolev spaces are used, the following important topics are in the focus of this volume: boundary value problems in domains with singularities, higher order partial differential equations, nonlinear evolution equations, local polynomial approximations, regularity for the Poisson equation in cones, harmonic func- tions, inequalities in Sobolev–Lorentz spaces, properties of function spaces in cellular domains, the spectrum of a Schr¨odinger operator with negative po- tential, the spectrum of boundary value problems in domains with cylindrical and quasicylindrical outlets to infinity, criteria for the complete integrability of systems of differential equations with applications to differential geome- try, some aspects of differential forms on Riemannian manifolds related to the Sobolev inequality, a Brownian motion on a Cartan–Hadamard manifold, etc. Two short biographical articles with unique archive photos of S.L. Sobolev are also included. viii Main Topics
III. Applications in Mathematical Physics The mathematical works of S.L. Sobolev were strongly motivated by particu- lar problems coming from applications. The approach and ideas of his famous book “Applications of Functional Analysis in Mathematical Physics” of 1950 turned out to be very influential and are widely used in the study of various problems 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 stochastic Ginzburg–Landau model with white noise simulating the phenomenon of su- perconductivity in materials under low temperatures, spectral asymptotics for the magnetic Schr¨odinger operator, the theory of boundary controllabil- ity for models of Kirchhoff plate and the Euler–Bernoulli plate with various physically meaningful boundary controls, asymptotics for boundary value problems in perforated domains and bodies with different type defects, the Finsler metric in connection with the study of wave propagation, the electric impedance tomography problem, the dynamical Lam´e system with residual stress, etc. Contents
I. Sobolev Type Inequalities Vladimir Maz’ya Ed.
MyLoveAffairwiththeSobolevInequality ...... 1 David R. Adams
MaximalFunctionsinSobolevSpaces ...... 25 Daniel Aalto and Juha Kinnunen Hardy Type Inequalities Via Riccati and Sturm–Liouville Equations . . . . 69 Sergey Bobkov and Friedrich G¨otze Quantitative Sobolev and Hardy Inequalities, and Related SymmetrizationPrinciples ...... 87 Andrea Cianchi Inequalities of Hardy–Sobolev Type in Carnot–Carath´eodory Spaces . . . 117 Donatella Danielli, Nicola Garofalo, and Nguyen Cong Phuc SobolevEmbeddingsandHardyOperators ...... 153 David E. Edmunds and W. Desmond Evans Sobolev Mappings between Manifolds and Metric Spaces ...... 185 Piotr Hajlasz A Collection of Sharp Dilation Invariant Integral Inequalities forDifferentiableFunctions ...... 223 Vladimir Maz’ya and Tatyana Shaposhnikova Optimality of Function Spaces in Sobolev Embeddings ...... 249 LuboˇsPick On the Hardy–Sobolev–Maz’ya Inequality and Its Generalizations . . . . . 281 Yehuda Pinchover and Kyril Tintarev Sobolev Inequalities in Familiar and Unfamiliar Settings ...... 299 Laurent Saloff-Coste A Universality Property of Sobolev Spaces in Metric Measure Spaces . . 345 Nageswari Shanmugalingam Cocompact Imbeddings and Structure of Weakly Convergent Sequences ...... 361 Kiril Tintarev Index...... 377 x Sobolev Spaces in Mathematics I–III II. Applications in Analysis and Partial Differential Equations Vladimir Maz’ya Ed.
On the Mathematical Works of S.L. Sobolev in the 1930s ...... 1 Vasilii Babich SobolevinSiberia ...... 11 Yuri Reshetnyak
Boundary Harnack Principle and the Quasihyperbolic Boundary Condition ...... 19 Hiroaki Aikawa Sobolev Spaces and their Relatives: Local Polynomial ApproximationApproach ...... 31 Yuri Brudnyi Spectral Stability of Higher Order Uniformly Elliptic Operators ...... 69 Victor Burenkov and Pier Domenico Lamberti Conductor Inequalities and Criteria for Sobolev-Lorentz Two-WeightInequalities ...... 103 Serban Costea and Vladimir Maz’ya Besov Regularity for the Poisson Equation in Smooth and PolyhedralCones ...... 123 Stephan Dahlke and Winfried Sickel Variational Approach to Complicated Similarity Solutions of Higher Order Nonlinear Evolution Partial Differential Equations ...... 147 Victor Galaktionov, Enzo Mitidieri,andStanislav Pokhozhaev
Lq,p-Cohomology of Riemannian Manifolds with Negative Curvature . . .199 Vladimir Gol’dshtein and Marc Troyanov Volume Growth and Escape Rate of Brownian Motion on a Cartan–HadamardManifold ...... 209 Alexander Grigor’yan and Elton Hsu Sobolev Estimates for the Green Potential Associated with the Robin–Laplacian in Lipschitz Domains Satisfying a UniformExteriorBallCondition ...... 227 T¨unde Jakab, Irina Mitrea,andMarius Mitrea Properties of Spectra of Boundary Value Problems inCylindricalandQuasicylindricalDomains ...... 261 Sergey Nazarov Estimates for Completeley Integrable Systems of Differential OperatorsandApplications ...... 311 Yuri Reshetnyak Contents xi
Counting Schr¨odinger Boundstates: Semiclassics and Beyond ...... 329 Grigori Rozenblum and Michael Solomyak FunctionSpacesonCellularDomains ...... 355 Hans Triebel Index...... 387
III. Applications in Mathematical Physics Victor Isakov Ed.
Preface ...... 1 Victor Isakov Geometrization of Rings as a Method for Solving Inverse Problems ...... 5 Mikhail Belishev The Ginzburg–Landau Equations for Superconductivity with RandomFluctuations ...... 25 Andrei Fursikov, Max Gunzburger,andJanet Peterson Carleman Estimates with Second Large Parameter for Second OrderOperators ...... 135 Victor Isakov and Nanhee Kim SharpSpectralAsymptoticsforDiracEnergy ...... 161 Victor Ivrii Linear Hyperbolic and Petrowski Type PDEs with Continuous Boundary Control → Boundary Observation Open Loop Map: Implication on Nonlinear Boundary Stabilization with OptimalDecayRates ...... 187 Irena Lasiecka and Roberto Triggiani Uniform Asymptotics of Green’s Kernels for Mixed and Neumann ProblemsinDomainswithSmallHolesandInclusions ...... 277 Vladimir Maz’ya and Alexander Movchan Finsler Structures and Wave Propagation ...... 317 Michael Taylor Index...... 335 Contributors Editors
Vladimir Maz’ya
Ohio State University Columbus, OH 43210 USA University of Liverpool Liverpool L69 7ZL UK Link¨oping University Link¨oping SE-58183 SWEDEN [email protected] [email protected]
Victor Isakov
Wichita State University Wichita, KS 67206 USA [email protected] Contributors Authors
Daniel Aalto Institute of Mathematics Helsinki University of Technology P.O. Box 1100, FI-02015 FINLAND e-mail: [email protected] David R. Adams University of Kentucky Lexington, KY 40506-0027 USA e-mail: [email protected] Hiroaki Aikawa Hokkaido University Sapporo 060-0810 JAPAN e-mail: [email protected] Vasili Babich Steklov Mathematical Institute Russian Academy of Sciences 27 Fontanka Str., St.-Petersburg 191023 RUSSIA e-mail: [email protected] Mikhail Belishev Steklov Mathematical Institute Russian Academy of Sciences 27 Fontanka Str., St.-Petersburg 191023 RUSSIA e-mail: [email protected] Sergey Bobkov University of Minnesota Minneapolis, MN 55455 USA e-mail: [email protected] xvi Sobolev Spaces in Mathematics I–III
Yuri Brudnyi Technion – Israel Institute of Technology Haifa 32000 ISRAEL e-mail: [email protected] Victor Burenkov Universit`a degli Studi di Padova 63 Via Trieste, 35121 Padova ITALY e-mail: [email protected] Andrea Cianchi Universit`a di Firenze Piazza Ghiberti 27, 50122 Firenze ITALY e-mail: [email protected] Serban Costea McMaster University 1280 Main Street West Hamilton, Ontario L8S 4K1 CANADA e-mail: [email protected] Stephan Dahlke Philipps–Universit¨at Marburg Fachbereich Mathematik und Informatik Hans Meerwein Str., Lahnberge 35032 Marburg GERMANY e-mail: [email protected] Donatella Danielli Purdue University 150 N. University Str. West Lafayette, IN 47906 USA e-mail: [email protected] David E. Edmunds School of Mathematics Cardiff University Senghennydd Road CARDIFF Wales CF24 4AG UK e-mail: [email protected] W. Desmond Evans School of Mathematics Cardiff University Senghennydd Road CARDIFF Wales CF24 4AG UK e-mail: [email protected] Contributors. Authors xvii
Andrei Fursikov Moscow State University Vorob’evy Gory, Moscow 119992 RUSSIA e-mail: [email protected] Victor Galaktionov University of Bath Bath, BA2 7AY UK e-mail: [email protected] Nicola Garofalo Purdue University 150 N. University Str. West Lafayette, IN 47906 USA e-mail: [email protected] Friedrich G¨otze Bielefeld University Bielefeld 33501 GERMANY e-mail: [email protected] Vladimir Gol’dshtein Ben Gurion University of the Negev P.O.B. 653, Beer Sheva 84105 ISRAEL e-mail: [email protected] Alexander Grigor’yan Bielefeld University Bielefeld 33501 GERMANY e-mail: [email protected] Max Gunzburger Florida State University Tallahassee, FL 32306-4120 USA e-mail: [email protected] Piotr Hajlasz University of Pittsburgh 301 Thackeray Hall, Pittsburgh, PA 15260 USA e-mail: [email protected] xviii Sobolev Spaces in Mathematics I–III
Elton Hsu Northwestern University 2033 Sheridan Road, Evanston, IL 60208-2730 USA e-mail: [email protected] Victor Isakov Wichita State University Wichita, KS 67206 USA e-mail: [email protected] Victor Ivrii University of Toronto 40 St.George Str., Toronto, Ontario M5S 2E4 CANADA e-mail: [email protected] T¨unde Jakab University of Virginia Charlottesville, VA 22904 USA e-mail: [email protected] Nanhee Kim Wichita State University Wichita, KS 67206 USA e-mail: [email protected] Juha Kinnunen Institute of Mathematics Helsinki University of Technology P.O. Box 1100, FI-02015 FINLAND e-mail: [email protected] Pier Domenico Lamberti Universit´a degli Studi di Padova 63 Via Trieste, 35121 Padova ITALY e-mail: [email protected] Irena Lasiecka University of Virginia Charlottesville, VA 22904 USA e-mail: [email protected] Vladimir Maz’ya Ohio State University Columbus, OH 43210 USA Contributors. Authors xix
University of Liverpool Liverpool L69 7ZL UK Link¨oping University Link¨oping SE-58183 SWEDEN e-mail: [email protected] e-mail: [email protected] Enzo Mitidieri Universit`adiTrieste Via Valerio 12/1, 34127 Trieste ITALY e-mail: [email protected] Irina Mitrea University of Virginia Charlottesville, VA 22904 USA e-mail: [email protected] Marius Mitrea University of Missouri Columbia, MO USA e-mail: [email protected] Alexander Movchan University of Liverpool Liverpool L69 3BX UK e-mail: [email protected] Sergey Nazarov Institute of Problems in Mechanical Engineering Russian Academy of Sciences 61, Bolshoi pr., V.O., St.-Petersburg 199178 RUSSIA e-mail: [email protected] Janet Peterson Florida State University Tallahassee FL 32306-4120 USA e-mail: [email protected] Nguyen Cong Phuc Purdue University 150 N. University Str. West Lafayette, IN 47906 USA e-mail: [email protected] xx Sobolev Spaces in Mathematics I–III
LuboˇsPick Charles University Sokolovsk´a 83, 186 75 Praha 8 CZECH REPUBLIC e-mail: [email protected] Yehuda Pinchover Technion – Israel Institute of Technology Haifa 32000 ISRAEL e-mail: [email protected] Stanislav Pokhozhaev Steklov Mathematical Institute Russian Academy of Sciences 8, Gubkina Str., Moscow 119991 RUSSIA e-mail: [email protected] Yuri Reshetnyak Sobolev Institute of Mathematics Siberian Branch Russian Academy of Sciences 4, Pr. Koptyuga, Novosibirsk 630090 RUSSIA Novosibirsk State University 2, Pirogova Str., Novosibirsk 630090 RUSSIA e-mail: [email protected] Grigori Rozenblum University of Gothenburg S-412 96, Gothenburg SWEDEN e-mail: [email protected] Laurent Saloff-Coste Cornell University Mallot Hall, Ithaca, NY 14853 USA e-mail: [email protected] Nageswari Shanmugalingam University of Cincinnati Cincinnati, OH 45221-0025 USA e-mail: [email protected] Tatyana Shaposhnikova Ohio State University Columbus, OH 43210 USA Contributors. Authors xxi
Link¨oping University Link¨oping SE-58183 SWEDEN e-mail: [email protected] Winfried Sickel Friedrich-Schiller-Universit¨at Jena Mathematisches Institut Ernst–Abbe–Platz 2, D-07740 Jena GERMANY e-mail: [email protected] Michael Solomyak The Weizmann Institute of Science Rehovot, 76100 ISRAEL e-mail: [email protected] Michael Taylor University of North Carolina Chapel Hill, NC 27599 USA e-mail: [email protected] Kyril Tintarev Uppsala University P.O. Box 480, SE-751 06 Uppsala SWEDEN e-mail: [email protected] Hans Triebel Mathematisches Institut Friedrich-Schiller-Universit¨at Jena D-07737 Jena GERMANY e-mail: [email protected] Roberto Triggiani University of Virginia Charlottesville, VA 22904 USA e-mail: [email protected] Marc Troyanov Institute of Geometry, Algebra, and Topology Ecole´ Polytechnique F´ed´eraledeLausanne 1015 Lausanne SWITZERLAND e-mail: [email protected] Sobolev Type Inequalities
Vladimir Maz’ya Ed. Contents
My Love Affair with the Sobolev Inequality ...... 1 David R. Adams 1 TheTraceInequality ...... 4 2 AMixedNormInequality...... 7 3 AMorrey–SobolevInequality...... 9 4 AMorrey–BesovInequality...... 12 5 Exponential Integrability ...... 13 6 Vanishing Exponential Integrability ...... 18 7 ConcludingRemarks...... 20 References...... 21 Maximal Functions in Sobolev Spaces...... 25 Daniel Aalto and Juha Kinnunen 1 Introduction...... 25 2 Maximal Function Defined on the Whole Space ...... 27 2.1 Boundedness in Sobolev spaces ...... 27 2.2 Acapacitaryweaktypeestimate...... 32 3 MaximalFunctionDefinedonaSubdomain...... 33 3.1 Boundedness in Sobolev spaces ...... 33 3.2 Sobolev boundary values ...... 38 4 PointwiseInequalities...... 41 4.1 Lusin type approximation of Sobolev functions . . . . 45 5 HardyInequality...... 49 6 MaximalFunctionsonMetricMeasureSpaces ...... 54 6.1 Sobolev spaces on metric measure spaces ...... 55 6.2 Maximal function defined on the whole space ...... 57 6.3 Maximal function defined on a subdomain ...... 62 6.4 Pointwise estimates and Lusin type approximation . 64 References...... 65
xxv xxvi Contents
Hardy Type Inequalities via Riccati and Sturm–Liouville Equations...... 69 Sergey Bobkov and Friedrich G¨otze 1 Introduction...... 69 2 RiccatiEquations...... 71 3 Transition to Sturm–Liouville Equations ...... 75 4 HardyTypeInequalitieswithWeights ...... 77 5 Poincar´eTypeInequalities ...... 83 References...... 85 Quantitative Sobolev and Hardy Inequalities, and Related Symmetrization Principles ...... 87 Andrea Cianchi 1 Introduction...... 87 2 SymmetrizationInequalities...... 89 2.1 Rearrangements of functions and function spaces . . . 89 2.2 TheHardy–Littlewoodinequality...... 92 2.3 The P´olya–Szeg¨oinequality...... 96 3 SobolevInequalities ...... 101 3.1 Functions of Bounded Variation ...... 101 3.2 The case 1 6 Approximation Numbers of Embeddings on Generalized RidgedDomains ...... 181 References...... 182 Sobolev Mappings between Manifolds and Metric Spaces ..... 185 Piotr Hajlasz 1 Introduction...... 185 2 SobolevMappingsbetweenManifolds...... 187 3 SobolevMappingsintoMetricSpaces...... 197 3.1 Density...... 202 4 SobolevSpacesonMetricMeasureSpaces...... 205 4.1 Integrationonrectifiablecurves...... 205 4.2 Modulus...... 207 4.3 Uppergradient...... 208 4.4 Sobolev spaces N 1,p ...... 208 4.5 Doublingmeasures...... 209 4.6 OtherspacesofSobolevtype...... 211 4.7 Spaces supporting the Poincar´einequality...... 214 5 SobolevMappingsbetweenMetricSpaces...... 215 5.1 Lipschitzpolyhedra...... 218 References...... 219 A Collection of Sharp Dilation Invariant Integral Inequalities for Differentiable Functions ...... 223 Vladimir Maz’ya and Tatyana Shaposhnikova 1 Introduction...... 223 2 Estimate for a Quadratic Form of the Gradient ...... 226 3WeightedG˚arding Inequality for the Biharmonic Operator. . 230 4 Dilation Invariant Hardy’s Inequalities with Remainder Term...... 233 5 Generalized Hardy–Sobolev Inequality with Sharp Constant...... 241 6 Hardy’s Inequality with Sharp Sobolev Remainder Term . . . 244 References...... 245 Optimality of Function Spaces in Sobolev Embeddings ...... 249 LuboˇsPick 1 Prologue...... 249 2 Preliminaries...... 256 3 ReductionTheorems...... 258 4 Optimal Range and Optimal Domain of Rearrangement- InvariantSpaces ...... 261 5 Formulas for Optimal Spaces Using the Functional f ∗∗ − f ∗ ...... 264 6 Explicit Formulas for Optimal Spaces in Sobolev Embeddings...... 267 xxviii Contents 7 CompactnessofSobolevEmbeddings...... 270 8 Boundary Traces ...... 275 9 GaussianSobolevEmbeddings...... 276 References...... 278 On the Hardy–Sobolev–Maz’ya Inequality and Its Generalizations ...... 281 Yehuda Pinchover and Kyril Tintarev 1 Introduction...... 281 2 Generalization of the Hardy–Sobolev–Maz’ya Inequality . . . . 284 D1,2 3 The Space V (Ω) and Minimizers for the Hardy–Sobolev– Maz’yaInequality...... 292 4 Convexity Properties of the Functional Q for p>2 ...... 293 References...... 296 Sobolev Inequalities in Familiar and Unfamiliar Settings ...... 299 Laurent Saloff-Coste 1 Introduction...... 299 2 Moser’sIteration...... 300 2.1 Thebasictechnique...... 300 2.2 Harnackinequalities...... 302 2.3 Poincar´e, Sobolev, and the doubling property ...... 303 2.4 Examples...... 310 3 AnalysisandGeometryonDirichletSpaces...... 312 3.1 Firstordercalculus ...... 312 3.2 Dirichletspaces...... 312 3.3 Local weak solutions of the Laplace and heat equations...... 314 3.4 HarnacktypeDirichletspaces ...... 316 3.5 Imaginary powers of −A and the wave equation . . . . 318 3.6 Roughisometries ...... 320 4 FlatSobolevInequalities...... 322 4.1 HowtoproveaflatSobolevinequality?...... 322 4.2 Flat Sobolev inequalities and semigroups of operators...... 324 4.3 TheRozenblum–Cwikel–Liebinequality...... 326 4.4 Flat Sobolev inequalities in the finite volume case . . 329 4.5 Flat Sobolev inequalities and topology at infinity . . 330 5 SobolevInequalitiesonGraphs...... 330 5.1 Graphs of bounded degree ...... 331 5.2 Sobolev inequalities and volume growth ...... 332 5.3 Randomwalks...... 333 5.4 Cayleygraphs...... 335 References...... 339 Contents xxix A Universality Property of Sobolev Spaces in Metric Measure Spaces ...... 345 Nageswari Shanmugalingam 1 Introduction...... 345 2 Background ...... 347 3 Dirichlet Forms and N 1,2(X) ...... 349 4 Axiomatic Sobolev Spaces and N 1,p(X)...... 356 References...... 358 Cocompact Imbeddings and Structure of Weakly Convergent Sequences ...... 361 Kiril Tintarev 1 Introduction...... 361 2 Dislocation Space and Weak Convergence Decomposition . . . 363 3 CocompactnessandMinimizers...... 368 4 Flask Subspaces ...... 372 5 CompactImbeddings...... 373 References...... 375 Index ...... 377