Preface
Lisf of Spaces and Norms
1. PRELIMINARIES
Notation
Topological Vector Spaces
Normed Spaces
Spaces of Continuous Functions
The Lebesgue Measure in Rn
The Lebesgue Integral
Distributions and Weak Derivatives
2. THE LEBESGUE SPACES Lp■
Definition and Basic Properties
Completeness of Lp ■
Approximation by Continuous Functions
Convolutions and Young's Theorem
Mollifiers and Approximation by Smooth Functions
Precompact Sets in Lp ■
Uniform Convexity
The Normed Dual of LP■
Mixed-Norm Lp Spaces
The Marcinkiewicz Interpolation Theorem
3. THE SOBOLEV SPACES ■
Definitions and Basic Properties
Duality and the Spaces W-m,■
Approximation by Smooth Functions on■
Approximation by Smooth Functions on ■
Approximation by Functions in ■
Coordinate Transformations
4.THE SOBOLEV IMBEDDING THEOREM
Geometric Properties of Domains
Imbeddings by Potential Arguments
Imbeddings by Averaging
Imbeddings into Lipschitz Spaces
Sobolev's Inequality
Variations of Sobolev's Inequality
Wm■ as a Banach Algebra
Optimality of the Imbedding Theorem
Nonimbedding Theorems for Irregular Domains
Imbedding Theorems for Domains with Cusps
Imbedding Inequalities Involving Weighted Norms
Proofs of Theorems 4.51-4.53
5.INTERPOLATION, EXTENSION, AND APPROXIMATION
THEOREMS
Interpolation on Order of Smoothness
Interpolation on Degree of Sumability
Interpolation Involving Compact Subdomains
Extension Theorems
An Approximation Theorem
Boundary Traces
6.COMPACT IMBEDDINGS OF SOBOLEV SPACES
The Rellich-Kondrachov Theorem
Two Counterexamples
Unbounded Domains -- Compact Imbeddings of Wo■
An Equivalent Norm for W■
Unbounded Domains -- Decay at Infinity
Unbounded Domains -- Compact Imbeddings of W■
Hilbert-Schmidt Imbeddings
7.FRACTIONAL ORDER SPACES
Introduction
The Bochner Integral
Intermediate Spaces and Interpolation -- The Real Method
The Lorentz Spaces
Besov Spaces
Generalized Spaces of HNder Continuous Functions
Characterization of Traces
Direct Characterizations 9f Besov Spaces
Other Scales of Intermediate Spaces
Wavelet Characterizations
8.ORLICZ SPACES AND ORLICZ-SOBOLEV SPACES
Introduction
N-Functions
Orlicz Spaces
Duality in Orlicz Spaces
Separability and Compactness Theorems
A Limiting Case of the Sobolev Imbedding Theorem
Orlicz-Sobolev Spaces
Imbedding Theorems for Orlicz-Sobolev Spaces
References
Index
