This monograph presents an introductory study of of the properties of certain Banach spaces of weakly differentiable functions of several real variables that arise in connection with numerous problems in the theory of partial differential equations,approximation theory, and many other areas of pure and applied mathematics.

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