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, H?lder continuous, H?lder 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 Lipschitzian way
Besicovitch covering lemma
Besicovitch differentiation theorem
1.4 Hausdorff Measure
Equivalence 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 local behavior
Convolution of distributions
Differentiation of distributions
1.8 Lorentz Spaces
Non-increasing rearrangement of a function
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
2 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
Rademacher'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 Rellich-Kondrachov Compactness Theorem
Extension domains
2.6 Bessel Potentials and Capacity
Riesz and Bessel kernels
Bessel potentials
Bessel capacity
Basic properties of Bessel capacity
Capacitability of Suslin sets
Minimax theorem and alternate formulation of Bessel capacity
Metric properties of Bessel capacity
2.7 The Best Constant in the Soboley Inequality Co-area formula
Sobolev's inequality and isoperimetric inequality
2.8 Alternate Proofs of the Fundamental Inequalities
Hardy-Littlewood-Wiener maximal theorem
Sobolev's inequality for Riesz potentials
2.9 Limiting Cases of the Sobolev Inequality
The case kp = n by infinite series
The best constant in the case kp = n
An L∞-bound in the limiting case
2.10 Lorentz Spaces, A Slight Improvement
Young's inequality in the context of Lorentz spaces
Sobolev's inequality in Lorentz spaces
The limiting case
Exercises
Historical Notes
3 Pointwise Behavior of Sobolev Functions
3.1 Limits of Integral Averages of Sobolev Functions
Limiting values of integral averages except for capacity null set
3.2 Densities of Measures
3.3 Lebesgue Points for Sobolev Functions
Existence of Lebesgue points except for capacity null set
Approximate continuity
Fine continuity everywhere except for capacity null set
3.4 LP-Derivatives for Sobolev Functions
Existence of Taylor expansions LP
3.5 Properties of LP-Derivatives
The spaces Tk, tk, Tk,p,tk,p
The implication of a function being in Tk,p at all points of a closed set
3.6 An LP-Version of the Whitney Extension Theorem
Existence of a Coo function comparable to the distance function to a closed set
The Whitney extension theorem for functions in Tk,p and tk,p
3.7 An Observation on Different
