A First Course in Sobolev Spaces: Second Edition (Graduate Studies in Mathematics) (Graduate Studies in Mathematics, 181)

Monotone functions -- Functions of bounded pointwise variation -- Absolutely continuous functions -- Decreasing rearrangement -- Curves -- Lebesgue-Stieltjes measures -- Functions of bounded variation and Sobolev functions -- The infinite-dimensional case -- Change of variables and the divergence theorem -- Distributions -- Sobolev spaces -- Sobolev spaces: embeddings -- Sobolev spaces: further properties -- Functions of bounded variation -- Sobolev spaces: symmetrization -- Interpolation of Banach spaces -- Besov spaces -- Sobolev spaces: traces.;This book is about differentiation of functions. It is divided into two parts, which can be used as different textbooks, one for an advanced undergraduate course in functions of one variable and one for a graduate course on Sobolev functions. The first part develops the theory of monotone, absolutely continuous, and bounded variation functions of one variable and their relationship with Lebesgue-Stieltjes measures and Sobolev functions. It also studies decreasing rearrangement and curves. The second edition includes a chapter on functions mapping time into Banach spaces. The second part of the book studies functions of several variables. It begins with an overview of classical results such as Rademacher's and Stepanoff's differentiability theorems, Whitney's extension theorem, Brouwer's fixed point theorem, and the divergence theorem for Lipschitz domains. It then moves to distributions, Fourier transforms and tempered distributions. The remaining chapters are a treatise on Sobolev functions. The second edition focuses more on higher order derivatives and it includes the interpolation theorems of Gagliardo and Nirenberg. It studies embedding theorems, extension domains, chain rule, superposition, Poincaré's inequalities and traces. A major change compared to the first edition is the chapter on Besov spaces, which are now treated using interpolation theory. Title Page......Page 4 Copyright......Page 5 Dedication......Page 6 Contents......Page 8 Preface to the Second Edition......Page 14 Preface to the First Edition......Page 16 Acknowledgments......Page 22 Part 1. Functions of One Variable......Page 24 1. Monotone Functions......Page 26 2. Functions of Bounded Pointwise Variation......Page 52 3. Absolutely Continuous Functions......Page 90 4. Decreasing Rearrangement......Page 134 5. Curves......Page 156 6. Lebesgue–Stieltjes Measures......Page 180 7. Functions of Bounded Variation and Sobolev Functions......Page 206 8. The Infinite-Dimensional Case......Page 228 Part 2. Functions of Several Variables......Page 260 9. Change of Variables and the Divergence Theorem......Page 262 10. Distributions......Page 304 11. Sobolev Spaces......Page 342 12. Sobolev Spaces: Embeddings......Page 378 13. Sobolev Spaces: Further Properties......Page 434 14. Functions of Bounded Variation......Page 482 15. Sobolev Spaces: Symmetrization......Page 520 16. Interpolation of Banach Spaces......Page 540 17. Besov Spaces......Page 562 18. Sobolev Spaces: Traces......Page 614 Appendix A. Functional Analysis......Page 658 Appendix B. Measures......Page 674 Appendix C. The Lebesgue and Hausdorff Measures......Page 704 Appendix D. Notes......Page 726 Appendix E. Notation and List of Symbols......Page 734 Bibliography......Page 740 Index......Page 752 This book is about differentiation of functions. It is divided into two parts, which can be used as different textbooks, one for an advanced undergraduate course in functions of one variable and one for a graduate course on Sobolev functions. The first part develops the theory of monotone, absolutely continuous, and bounded variation functions of one variable and their relationship with Lebesgue-Stieltjes measures and Sobolev functions. It also studies decreasing rearrangement and curves. The second edition includes a chapter on functions mapping time into Banach spaces.The second part of the book studies functions of several variables. It begins with an overview of classical results such as Rademacher's and Stepanoff's differentiability theorems, Whitney's extension theorem, Brouwer's fixed point theorem, and the divergence theorem for Lipschitz domains. It then moves to distributions, Fourier transforms and tempered distributions. The remaining chapters are a treatise on Sobolev functions. The second edition focuses more on higher order derivatives and it includes the interpolation theorems of Gagliardo and Nirenberg. It studies embedding theorems, extension domains, chain rule, superposition, Poincare's inequalities and traces. A major change compared to the first edition is the chapter on Besov spaces, which are now treated using interpolation theory. This book is about differentiation of functions. It is divided into two parts, which can be used as different textbooks, one for an advanced undergraduate course in functions of one variable and one for a graduate course on Sobolev functions. The first part develops the theory of monotone, absolutely continuous, and bounded variation functions of one variable and their relationship with Lebesgue-Stieltjes measures and Sobolev functions. It also studies decreasing rearrangement and curves. The second edition includes a chapter on functions mapping time into Banach spaces. The second part of the book studies functions of several variables. It begins with an overview of classical results such as Rademacher's and Stepanoff's differentiability theorems, Whitney's extension theorem, Brouwer's fixed point theorem, and the divergence theorem for Lipschitz domains. It then moves to distributions, Fourier transforms and tempered distributions. [source : 4ème de couv.] Examines differentiation of functions. The first part of the book develops the theory of monotone, absolutely continuous, and bounded variation functions of one variable and their relationship with Lebesgue-Stieltjes measures and Sobolev functions. The second part studies functions of several variables.