目錄
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 Differentiation
3.8 Rademacher's Theorem in the IP-Context
A function in Tk,p everywhere implies it is in tk,p almost everywhere
3.9 The Implications of Pointwise Differentiability
Comparison of LP-derivatives and distributional derivatives
If u ? tk,p(x) for every x,and if the
LP-derivatives are in Lp,then u ? Wk,p
3.10 A Lusin-Type Approximation for Sobolev Functions
Integral averages of Sobolev functions are uniformly close to their limits on the complement of sets of small capacity
Existence of smooth functions that agree with Sobolev functions on the complement of sets of small capacity
3.11 The Main Approximation
Existence of smooth functions that agree with Sobolev functions on the complement of sets of small capacity and are close in norm
Exercises
Historical Notes
4 Poincar? Inequalities-A Unified Approach
4.1 Inequalities in a General Setting
An abstract version of the Poincar? inequality
4.2 Applications to Sobolev Spaces
An interpolation inequality
4.3 The Dual of Wm,p(Ω)
The representation of (Wom,P(Ω))*
4.4 Some Measures in (Wom,P(Ω))*
Poincar? inequalities derived from the abstract version by identifying Lebesgue and Hausdorff measure with elements in (Wom,P(Ω))*
The trace of Sobolev functions on the boundary of Lipschitz domains
Poincar? inequalities involving the trace of a Sobolev function
4.5 Poincar? Inequalities
Inequalities involving the capacity of the set on which a function vanishes
4.6 Another Version of Poincar?'s Inequality
An inequality involving dependence on the set on which the function vanishes, not merely on its capacity
4.7 More Measures in(Wom,P(Ω))*
Sobolev's inequality for Riesz potentials involving measures other than Lebesgue measure
Characterization of measures in (Wm,p(Rn))*
4.8 Other Inequalities Involving Measures in (Wk,P)*
Inequalities involving the restriction of Hausdorff measure to lower dimensional manifolds
4.9 The Case p =1
Inequalities involving the L1-norm of the gradient
Exercises
Historical Notes
5 Functions of Bounded Variation
5.1 Definitions
Definition of BV functions
The total variation measure ||Du||
5.2 Elementary Properties of BV Functions
Lower semicontinuity of the total variation measure
A condition ensuring continuity of the total variation measure
Regularization of BV Functions
5.3 Regularization does not increase the BV norm
Approximation of BV functions by smooth functions
Compactness in L1 of the unit ball in BV
5.4 Sets of Finite Perimeter
Definition of sets of finite perimeter
The perimeter of domains with smooth boundaries
Isoperimetric and relative isoperimetric inequality for sets of finite perimeter
5.5 The Generalized Exterior Normal
A preliminary version of the Gauss-Green theorem
Density results at points of the reduced boundary
5.6 Tangential Properties of the Reduced Boundary and the Measure-Theoretic Normal
Blow-up at a point of the reduced boundary
The measure-theoretic normal
The reduced boundary is contained in the measure-theoretic boundary
A lower bound for the density of ||DXE||
Hausdorff measure restricted to the reduced boundary is bounded above by ||DXE||
5.7 Rectifiability of the Reduced Boundary
Countably (n-1)-rectifiable sets
Countable (n-1)-rectifiability of the measure-theoretic boundary
5.8 The Gauss-Green Theorem
The equivalence of the restriction of Hausdorff measure to the measure-theoretic boundary and ||DXE||
The Gauss-Green theorem for sets of finite perimeter
5.9 Pointwise Behavior of BV Functions
Upper and lower approximate limits
The Boxing inequality
The set of approximate jump discontinuities
5.10 The Trace of a BV Function
The bounded extension of BV functions
Trace of a BV function defined in terms of the upper and lower approximate limits of theextended function
The integrability of the trace over the measure-theoretic boundary
5.11 Sobolev-Type Inequalities for BV Functions
Inequalities involving elements in (BV(Ω))*
5.12 Inequalities Involving Capacity
Characterization of measure in(BV(Ω))*
Poincar? inequality for BV functions
5.13 Generalizations to the Case p>1
5.14 Trace Defined in Terms of Integral Averages
Exercises
Historical Notes
Bibliography
List of Symbols
Index
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