Weakly Differentiable Functions（簡體書）

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.

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
Youngs inequality
Holders and Jensens 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
ONeils inequality, for rearranged functions
Equivalence of LP-norm and (p,p)-norm
Hardys 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 hers theorem
Bi-Lipschitzian change of variables
2.3 Approximation of Sobolev functions by Smooth functions
Partition of unity
Smooth functions are dense in Wkp
2.4 Sobolev Inequalities
Sobolevs 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