Weakly differentiable functions. Sobolev spaces and functions of bounded variation. (English) Zbl 0692.46022

Ziemer, William P.

Graduate Texts in Mathematics, 120. Berlin etc.: Springer-Verlag. xvi, 308 p. DM 108.00 (1989).

“Weakly differentiable functions” refers to those integrable functions defined on an open subset of \(R^ n\) whose partial derivatives in the sense of distributions are either \(L^ p\) 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.
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).
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).
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.

Reviewer: H.Triebel

MSC:

46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
28A75	Length, area, volume, other geometric measure theory
46-02	Research exposition (monographs, survey articles) pertaining to 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