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Zbl 0692.46022
Ziemer, William P.
Weakly differentiable functions. Sobolev spaces and functions of bounded variation. (English)
[B] Graduate Texts in Mathematics, 120. Berlin etc.: Springer-Verlag. xvi, 308 p. DM 108.00 (1989). ISBN 0-387-97017-7

``Weakly differentiable functions'' refers to those integrable functions defined on an open subset of $R\sp n$ whose partial derivatives in the sense of distributions are either $L\sp p$ functions or signed measures with finite total variation. The main purpose of this book is the analysis of pointwise behaviour of Sobolev and bounded variation functions. \par The book has 5 chapters. 1. Preliminaries (covering lemmas, Hausdorff measure, distributions, Lorentz spaces). 2. Sobolev spaces and their basic properties (basic facts, including Sobolev inequalities and Bessel capacities). \par The chapters 3-5 are the heart of the book. 3. Pointwise behaviour of Sobolev functions (continuity, Lebesgue points, approximate and fine continuity, exceptional sets are expressed in terms of capacities). 4. Poincaré inequalities - a unified approach (several concrete and abstract versions, involving Hausdorff measures and capacities; measures belonging to the dual of Sobolev spaces). 5. Functions of bounded variation (similar problems as in chapter 3, now involving geometric measure theory). \par The book is a good complement to the existing literature on the theory of Sobolev spaces concentrating on those recent topics which are often somewhat neglected.
[H.Triebel]

MSC 2000:: *46E35 Sobolev spaces and generalizations
28A75 Geometric measure theory
46-02 Research monographs (functional analysis)

Keywords: inequalities; analysis of pointwise behaviour of Sobolev and bounded variation functions; covering lemmas; Hausdorff measure; distributions; Lorentz spaces; Sobolev spaces; Bessel capacities; Lebesgue points; approximate and fine continuity; exceptional sets; Poincaré inequalities; Functions of bounded variation; geometric measure theory

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