Table of Contents

1 Preliminaries.- 1.1 Notation.- 1.2 Measures on Rn.- 1.3 Covering Theorems.- 1.4 Hausdorff Measure.- 1.5 Lp-Spaces.- 1.6 Regularization.- 1.7 Distributions.- 1.8 Lorentz Spaces.- Exercises.- Historical Notes.- 2 Sobolev Spaces and Their Basic Properties.- 2.1 Weak Derivatives.- 2.2 Change of Variables for Sobolev Functions.- 2.3 Approximation of Sobolev Functions by Smooth Functions.- 2.4 Sobolev Inequalities.- 2.5 The Rellich-Kondrachov Compactness Theorem.- 2.6 Bessel Potentials and Capacity.- 2.7 The Best Constant in the Sobolev Inequality.- 2.8 Alternate Proofs of the Fundamental Inequalities.- 2.9 Limiting Cases of the Sobolev Inequality.- 2.10 Lorentz Spaces, A Slight Improvement.- Exercises.- Historical Notes.- 3 Pointwise Behavior of Sobolev Functions.- 3.1 Limits of Integral Averages of Sobolev Functions.- 3.2 Densities of Measures.- 3.3 Lebesgue Points for Sobolev Functions.- 3.4 LP-Derivatives for Sobolev Functions.- 3.5 Properties of Lp-Derivatives.- 3.6 An Lp-Version of the Whitney Extension Theorem.- 3.7 An Observation on Differentiation.- 3.8 Rademacher’s Theorem in the Lp-Context.- 3.9 The Implications of Pointwise Differentiability.- 3.10 A Lusin-Type Approximation for Sobolev Functions.- 3.11 The Main Approximation.- Exercises.- Historical Notes.- 4 Poincaré Inequalities—A Unified Approach.- 4.1 Inequalities in a General Setting.- 4.2 Applications to Sobolev Spaces.- 4.3 The Dual of WM,p(?).- 4.4 Some Measures in (W0M,p(?))*.- 4.5 Poincaré Inequalities.- 4.6 Another Version of Poincaré’s Inequality.- 4.7 More Measures in (WM,p(?))*.- 4.8 Other Inequalities Involving Measures in (WM,p)*.- 4.9 The Case p= 1.- Exercises.- Historical Notes.- 5 Functions of Bounded Variation.- 5.1 Definitions.- 5.2 Elementary Properties of BV Functions.- 5.3Regularization of BV Functions.- 5.4 Sets of Finite Perimeter.- 5.5 The Generalized Exterior Normal.- 5.6 Tangential Properties of the Reduced Boundary and the Measure-Theoretic Normal.- 5.7 Rectifiability of the Reduced Boundary.- 5.8 The Gauss-Green Theorem.- 5.9 Pointwise Behavior of BV Functions.- 5.10 The Trace of a BV Function.- 5.11 Sobolev-Type Inequalities for BV Functions.- 5.12 Inequalities Involving Capacity.- 5.13 Generalizations to the Case p> 1.- 5.14 Trace Defined in Terms of Integral Averages.- Exercises.- Historical Notes.- List of Symbols.

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