GRADUATE STUDIES IN MATHEMATICS 181 A First Course in Sobolev Spaces Second Edition Giovanni Leoni American Mathematical Society 10.1090/gsm/181 A First Course in Sobolev Spaces Second Edition GRADUATE STUDIES IN MATHEMATICS 181 A First Course in Sobolev Spaces Second Edition Giovanni Leoni American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 46E35; Secondary 26A27, 26A30, 26A42, 26A45, 26A46, 26A48, 26B30, 30H25. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-181 Library of Congress Cataloging-in-Publication Data Names: Leoni, Giovanni, 1967– Title: A first course in Sobolev spaces / Giovanni Leoni. Description: Second edition. | Providence, Rhode Island : American Mathematical Society, [2017] | Series: Graduate studies in mathematics; volume 181 | Includes bibliographical references and index. Identifiers: LCCN 2017009991 | ISBN 9781470429218 (alk. paper) Subjects: LCSH: Sobolev spaces. Classification: LCC QA323 .L46 2017 | DDC 515/.782–dc23 LC record available at https://lccn.loc.gov/2017009991 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2017 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 222120191817 In memory of my Ph.D. advisor, James Serrin Contents Preface xiii Preface to the Second Edition xiii Preface to the First Edition xv Acknowledgments xxi Second Edition xxi First Edition xxii Part 1. Functions of One Variable Chapter 1. Monotone Functions 3 §1.1. Continuity 3 §1.2. Differentiability 9 Chapter 2. Functions of Bounded Pointwise Variation 29 §2.1. Pointwise Variation 29 §2.2. Continuity 34 §2.3. Differentiability 40 §2.4. Monotone Versus BPV 44 §2.5. The Space BPV (I; Y )47 §2.6. Composition in BPV (I; Y )55 §2.7. Banach Indicatrix 59 Chapter 3. Absolutely Continuous Functions 67 §3.1. AC(I; Y )VersusBPV (I; Y )67 §3.2. The Fundamental Theorem of Calculus 71 vii viii Contents §3.3. Lusin (N) Property 84 §3.4. Superposition in AC(I; Y )91 §3.5. Chain Rule 95 §3.6. Change of Variables 100 §3.7. Singular Functions 103 Chapter 4. Decreasing Rearrangement 111 §4.1. Definition and First Properties 111 §4.2. Function Spaces and Decreasing Rearrangement 126 Chapter 5. Curves 133 §5.1. Rectifiable Curves 133 §5.2. Arclength 143 §5.3. Length Distance 146 §5.4. Curves and Hausdorff Measure 149 §5.5. Jordan’s Curve Theorem 152 Chapter 6. Lebesgue–Stieltjes Measures 157 §6.1. Measures Versus Increasing Functions 157 §6.2. Vector-valued Measures Versus BPV (I; Y ) 168 §6.3. Decomposition of Measures 177 Chapter 7. Functions of Bounded Variation and Sobolev Functions 183 §7.1. BV (Ω) Versus BPV (Ω) 183 §7.2. Sobolev Functions Versus Absolutely Continuous Functions 188 §7.3. Interpolation Inequalities 196 Chapter 8. The Infinite-Dimensional Case 205 §8.1. The Bochner Integral 205 §8.2. Lp Spaces on Banach Spaces 212 §8.3. Functions of Bounded Pointwise Variation 220 §8.4. Absolute Continuous Functions 224 §8.5. Sobolev Functions 229 Part 2. Functions of Several Variables Chapter 9. Change of Variables and the Divergence Theorem 239 §9.1. Directional Derivatives and Differentiability 239 §9.2. Lipschitz Continuous Functions 242 §9.3. The Area Formula: The C1 Case 249 Contents ix §9.4. The Area Formula: The Differentiable Case 262 §9.5. The Divergence Theorem 273 Chapter 10. Distributions 281 §10.1. The Spaces DK (Ω), D(Ω), and D (Ω) 281 §10.2. Order of a Distribution 288 §10.3. Derivatives of Distributions and Distributions as Derivatives 290 §10.4. Rapidly Decreasing Functions and Tempered Distributions 298 §10.5. Convolutions 302 §10.6. Convolution of Distributions 305 §10.7. Fourier Transforms 309 §10.8. Littlewood–Paley Decomposition 316 Chapter 11. Sobolev Spaces 319 §11.1. Definition and Main Properties 319 §11.2. Density of Smooth Functions 325 §11.3. Absolute Continuity on Lines 336 §11.4. Duals and Weak Convergence 344 §11.5. A Characterization of W 1,p(Ω) 349 Chapter 12. Sobolev Spaces: Embeddings 355 §12.1. Embeddings: mp < N 356 §12.2. Embeddings: mp = N 372 §12.3. Embeddings: mp > N 378 §12.4. Superposition 387 §12.5. Interpolation Inequalities in RN 399 Chapter 13. Sobolev Spaces: Further Properties 411 §13.1. Extension Domains 411 §13.2. Poincar´e Inequalities 430 §13.3. Interpolation Inequalities in Domains 449 Chapter 14. Functions of Bounded Variation 459 §14.1. Definition and Main Properties 459 §14.2. Approximation by Smooth Functions 462 §14.3. Bounded Pointwise Variation on Lines 468 §14.4. Coarea Formula for BV Functions 478 §14.5. Embeddings and Isoperimetric Inequalities 482 x Contents §14.6. Density of Smooth Sets 489 §14.7. A Characterization of BV (Ω) 493 Chapter 15. Sobolev Spaces: Symmetrization 497 §15.1. Symmetrization in Lp Spaces 497 §15.2. Lorentz Spaces 502 §15.3. Symmetrization of W 1,p and BV Functions 504 §15.4. Sobolev Embeddings Revisited 510 Chapter 16. Interpolation of Banach Spaces 517 §16.1. Interpolation: K-Method 517 §16.2. Interpolation: J-Method 526 §16.3. Duality 530 §16.4. Lorentz Spaces as Interpolation Spaces 535 Chapter 17. Besov Spaces 539 s,p §17.1. Besov Spaces Bq 539 §17.2. Some Equivalent Seminorms 545 §17.3. Besov Spaces as Interpolation Spaces 551 §17.4. Sobolev Embeddings 561 s,p + − §17.5. The Limit of Bq as s → 0 and s → m 565 §17.6. Besov Spaces and Derivatives 571 §17.7. Yet Another Equivalent Norm 578 §17.8. And More Embeddings 585 Chapter 18. Sobolev Spaces: Traces 591 §18.1. The Trace Operator 592 §18.2. Traces of Functions in W 1,1(Ω) 598 §18.3. Traces of Functions in BV (Ω) 605 §18.4. Traces of Functions in W 1,p(Ω), p>1 606 §18.5. Traces of Functions in W m,1(Ω) 621 §18.6. Traces of Functions in W m,p(Ω), p>1 626 §18.7. Besov Spaces and Weighted Sobolev Spaces 626 Appendix A. Functional Analysis 635 §A.1. Topological Spaces 635 §A.2. Metric Spaces 638 §A.3. Topological Vector Spaces 639 Contents xi §A.4. Normed Spaces 643 §A.5. Weak Topologies 645 §A.6. Hilbert Spaces 648 Appendix B. Measures 651 §B.1. Outer Measures and Measures 651 §B.2. Measurable and Integrable Functions 655 §B.3. Integrals Depending on a Parameter 662 §B.4. Product Spaces 663 §B.5. Radon–Nikodym’s and Lebesgue’s Decomposition Theorems 665 §B.6. Signed Measures 666 §B.7. Lp Spaces 668 §B.8. Modes of Convergence 673 §B.9. Radon Measures 676 §B.10. Covering Theorems in RN 678 Appendix C. The Lebesgue and Hausdorff Measures 681 §C.1. The Lebesgue Measure 681 §C.2. The Brunn–Minkowski Inequality 683 §C.3. Mollifiers 687 §C.4. Maximal Functions 694 §C.5. BMO Spaces 695 §C.6. Hardy’s Inequality 698 §C.7. Hausdorff Measures 699 Appendix D. Notes 703 Appendix E. Notation and List of Symbols 711 Bibliography 717 Index 729 Preface The Author List, I: giving credit where credit is due. The first author: Senior grad student in the project. Made the figures. — Jorge Cham, www.phdcomics.com Preface to the Second Edition There are a lot of changes in the second edition. In the first part of the book, starting from Chapter 2, instead of considering real-valued functions, I treat functions u : I → Y ,whereI ⊆ R is an interval and Y is a metric space. This change is motivated by the addition of a new chapter, Chapter 8, where I introduce the Bochner integral and study functions mapping time into Banach spaces. This type of functions plays a crucial role in evolution equations. Another important addition in the first part of the book is Section 7 in Chapter 7, which begins the study of interpolation inequalities for Sobolev functions of one variable. One wants to estimate some appropriate norm of an intermediate derivative in terms of the norms of the function and the highest-order derivative. Except for Chapter 8, the first part of the book is meant as a text- book for an advanced undergraduate course or beginning graduate course on real analysis or functions of one variable. One should sim- ply take Y to be the real line R so that H1 reduces to the Lebesgue measure L1. All the results needed from measure theory are listed in the appendices at the end of the book.