the term "weakly differentiable functions" in the title refers to those inte grable functions defined on an open subset of rn whose partial derivatives in the sense of distributions are either lr functions or (signed) measures with finite total variation. the former class of functions comprises what is now known as sobolev spaces, though its origin, traceable to the early 1900s, predates the contributions by sobolev. both classes of functions, sobolev spaces and the space of functions of bounded variation (bv functions), have undergone considerable development during the past 20 years. from this development a rather complete theory has emerged and thus has provided the main impetus for the writing of this book. since these classes of functions play a significant role in many fields, such as approximation theory, calculus of variations, partial differential equations, and non-linear potential theory, it is hoped that this monograph will be of assistance to a wide range of graduate students and researchers in these and perhaps other related areas. some of the material in chapters 1-4 has been presented in a graduate course at indiana university during the 1987-88 academic year, and i am indebted to the students and colleagues in attendance for their helpful comments and suggestions.
目录
Preface 1 Preliminaries 1.1 Notation Inner product of vectors Support of a function Boundary of a set Distance from a point to a set Characteristic function of a set Multi-indices Partial derivative operators Function spaces--continuous， HSlder continuous， HSlder continuous derivatives 1.2 Measures on Rn Lebesgue measurable sets Lebesgue measurability of Borel sets Suslin sets 1.3 Covering Theorems Hausdorff maximal principle General covering theorem Vitali covering theorem Covering lemma, with n-balls whose radii vary in Lips hitzian way Besicovitch covering lemma Besicovitch differentiation theorem 1.4 Hausdorff Measure Equivalen e of Hausdorff and Lebesgue measures Hausdorff dimension 1.5 LP-Spaces Integration of a function via its distribution function Young's inequality Holder's and Jensen's inequality 1.6 Regularization LP-spaces and regularization 1.7 Distributions Functions and measures, as distributions Positive distributions Distributions determined by their lo al behavior Convolution of distributions Differentiation of distributions 1.8 Lorentz Spaces Non-in reasing rearrangement of a fun tion Elementary properties of rearranged functions Lorentz spaces O'Neil's inequality, for rearranged functions Equivalence of LP-norm and (p,p)-norm Hardy's inequality Inclusion relations of Lorentz spaces Exercises Historical Notes Sobolev Spaces and Their Basic Properties 2.1 Weak Derivatives Sobolev spaces Absolute continuity on lines LP-norm of difference quotients Truncation of Sobolev functions Composition of Sobolev functions 2.2 Change of Variables for Sobolev functions Radema her's theorem Bi-Lipschitzian change of variables 2.3 Approximation of Sobolev functions by Smooth functions Partition of unity Smooth functions are dense in Wk'p 2.4 Sobolev Inequalities Sobolev's inequality 2.5 The Relli h-Kondrachov compactness Theorem Extension domains 2.6 Bessel Potentials and apacity Riesz and Bessel kernels Bessel potentials Bessel apacity Basic properties of Bessel apacity Capa itability of Suslin sets Minimax theorem and alternate formulation of Bessel apacity …… 3 Pointwise Behavior of Sobolev Functions 4 Poincare Inequalities 5 Functions of Bounded Variation Bibliography List of Symbols Index
