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Softcover ISBN:  9781470477028 
Product Code:  GSM/181.S 
List Price:  $89.00 
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AMS Member Price:  $71.20 
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Softcover ISBN:  9781470477028 
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AMS Member Price:  $139.20 $105.20 
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Book DetailsGraduate Studies in MathematicsVolume: 181; 2017; 734 ppMSC: Primary 46; Secondary 26; 30
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.
ReadershipGraduate students and researchers interested in Sobolev spaces, particularly their applications to PDEs.

Table of Contents

Part 1. Functions of one variable

Monotone functions

Functions of bounded pointwise variation

Absolutely continuous functions

Decreasing rearrangement

Curves

Lebesgue–Stieltjes measures

Functions of bounded variation and Sobolev functions

The infinitedimensional case

Part 2. Functions of several variables

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

Appendix A. Functional analysis

Appendix B. Measures

Appendix C. The Lebesgue and Hausdorff measures

Appendix D. Notes

Appendix E. Notation and list of symbols


Additional Material

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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.
Graduate students and researchers interested in Sobolev spaces, particularly their applications to PDEs.

Part 1. Functions of one variable

Monotone functions

Functions of bounded pointwise variation

Absolutely continuous functions

Decreasing rearrangement

Curves

Lebesgue–Stieltjes measures

Functions of bounded variation and Sobolev functions

The infinitedimensional case

Part 2. Functions of several variables

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

Appendix A. Functional analysis

Appendix B. Measures

Appendix C. The Lebesgue and Hausdorff measures

Appendix D. Notes

Appendix E. Notation and list of symbols