Real Analysis

Name: Real Analysis
Brand: Birkhäuser
Price: 85.59 EUR
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Emmanuele DiBenedetto(Author)

Birkhäuser (Publisher)

2nd Edition

Published on 17. September 2016

XXXII, 596 pages

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The second edition of this classic textbook presents a rigorous and self-contained introduction to real analysis with the goal of providing a solid foundation for future coursework and research in applied mathematics. Written in a clear and concise style, it covers all of the necessary subjects as well as those often absent from standard introductory texts. Each chapter features a "Problems and Complements" section that includes additional material that briefly expands on certain topics within the chapter and numerous exercises for practicing the key concepts.

The first eight chapters explore all of the basic topics for training in real analysis, beginning with a review of countable sets before moving on to detailed discussions of measure theory, Lebesgue integration, Banach spaces, functional analysis, and weakly differentiable functions. More topical applications are discussed in the remaining chapters, such as maximal functions, functions of bounded mean oscillation, rearrangements, potential theory, and the theory of Sobolev functions. This second edition has been completely revised and updated and contains a variety of new content and expanded coverage of key topics, such as new exercises on the calculus of distributions, a proof of the Riesz convolution, Steiner symmetrization, and embedding theorems for functions in Sobolev spaces.

Ideal for either classroom use or self-study, Real Analysis is an excellent textbook both for students discovering real analysis for the first time and for mathematicians and researchers looking for a useful resource for reference or review.

Praise for the First Edition:

"[This book] will be extremely useful as a text. There is certainly enough material for a year-long graduate course, but judicious selection would make it possible to use this most appealing book in a one-semester course for well-prepared students."

-Mathematical Reviews

Reviews / Votes

"The book is a valuable, comprehensive reference source on real analysis. The first eight chapters cover core material that is part of most courses taught on the subject, followed by a collection of special topics that stay within the framework of real analysis. In addition to the content, what makes the book especially useful as a reference source is its organization. . Summing Up: Recommended. Graduate students and faculty. This work should be used solely as a reference." (M. Bona, Choice, Vol. 54 (9), May, 2017)

"The reader can find many interesting details which serve to illuminate the diamonds of analysis. The list of references contains the main books and articles which form the modern real analysis. The book can be recommended as one of the main readings on real analysis for those who are interested in this subject and its numerous applications." (Sergei V. Rogosin, zbMATH 1353.26001, 2017)

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Content

Intro
Preface to the Second Edition
Preface to the First Edition
Acknowledgments
Contents
1 Preliminaries
1 Countable Sets
2 The Cantor Set
3 Cardinality
3.1 Some Examples
4 Cardinality of Some Infinite Cartesian Products
5 Orderings, the Maximal Principle, and the Axiom of Choice
6 Well Ordering
6.1 The First Uncountable
1c Countable Sets
2c The Cantor Set
2.1c A Generalized Cantor Set of Positive Measure
2.2c A Generalized Cantor Set of Measure Zero
2.3c Perfect Sets
3c Cardinality
2 Topologies and Metric Spaces
1 Topological Spaces
1.1 Hausdorff and Normal Spaces
2 Urysohn's Lemma
3 The Tietze Extension Theorem
4 Bases, Axioms of Countability and Product Topologies
4.1 Product Topologies
5 Compact Topological Spaces
5.1 Sequentially Compact Topological Spaces
6 Compact Subsets of mathbbRN
7 Continuous Functions on Countably Compact Spaces
8 Products of Compact Spaces
9 Vector Spaces
9.1 Convex Sets
9.2 Linear Maps and Isomorphisms
10 Topological Vector Spaces
10.1 Boundedness and Continuity
11 Linear Functionals
12 Finite Dimensional Topological Vector Spaces
12.1 Locally Compact Spaces
13 Metric Spaces
13.1 Separation and Axioms of Countability
13.2 Equivalent Metrics
13.3 Pseudo Metrics
14 Metric Vector Spaces
14.1 Maps Between Metric Spaces
15 Spaces of Continuous Functions
15.1 Spaces of Continuously Differentiable Functions
15.2 Spaces of Hölder and Lipschitz Continuous Functions
16 On the Structure of a Complete Metric Space
16.1 The Uniform Boundedness Principle
17 Compact and Totally Bounded Metric Spaces
17.1 Pre-Compact Subsets of X
1c Topological Spaces
1.12c Connected Spaces
1.19c Separation Properties of Topological Spaces
4c Bases, Axioms of Countability and Product Topologies
4.10c The Box Topology
5c Compact Topological Spaces
5.8c The Alexandrov One-Point Compactification of {X
mathcalU} ([3])
7c Continuous Functions on Countably Compact Spaces
7.1c Upper-Lower Semi-continuous Functions
7.2c Characterizing Lower-Semi Continuous Functions in mathbbRN
7.3c On the Weierstrass-Baire Theorem
7.4c On the Assumptions of Dini's Theorem
9c Vector Spaces
9.3c Hamel Bases
9.6c On the Dimension of a Vector Space
10c Topological Vector Spaces
13c Metric Spaces
13.10c The Hausdorff Distance of Sets
13.11c Countable Products of Metric Spaces
14c Metric Vector Spaces
15c Spaces of Continuous Functions
15.1c Spaces of Hölder and Lipschitz Continuous Functions
16c On the Structure of a Complete Metric Space
16.3c Completion of a Metric Space
16.4c Some Consequences of the Baire Category Theorem
17c Compact and Totally Bounded Metric Spaces
17.1c An Application of the Lebesgue Number Lemma
3 Measuring Sets
1 Partitioning Open Subsets of mathbbRN
2 Limits of Sets, Characteristic Functions, and s-Algebras
3 Measures
3.1 Finite, s-Finite, and Complete Measures
3.2 Some Examples
4 Outer Measures and Sequential Coverings
4.1 The Lebesgue Outer Measure in mathbbRN
4.2 The Lebesgue--Stieltjes Outer Measure [89, 154]
5 The Hausdorff Outer Measure in mathbbRN [71]
5.1 Metric Outer Measures
6 Constructing Measures from Outer Measures [26]
7 The Lebesgue--Stieltjes Measure on mathbbR
7.1 Borel Measures
8 The Hausdorff Measure on mathbbRN
9 Extending Measures from Semi-algebras to s-Algebras
9.1 On the Lebesgue--Stieltjes and Hausdorff Measures
10 Necessary and Sufficient Conditions for Measurability
11 More on Extensions from Semi-algebras to s-Algebras
12 The Lebesgue Measure of Sets in mathbbRN
12.1 A Necessary and Sufficient Condition of Measurability
13 Vitali's Nonmeasurable Set [168]
14 Borel Sets, Measurable Sets, and Incomplete Measures
14.1 A Continuous Increasing Function f:[0,1]to[0,1]
14.2 On the Preimage of a Measurable Set
14.3 Proof of Propositions 14.1 and 14.2
15 Borel Measures
16 Borel, Regular, and Radon Measures
16.1 Regular Borel Measures
16.2 Radon Measures
17 Vitali Coverings
18 The Besicovitch Covering Theorem
19 Proof of Proposition 18.1
20 The Besicovitch Measure-Theoretical Covering Theorem
1c Partitioning Open Subsets of mathbbRN
2c Limits of Sets, Characteristic Functions and s-Algebras
3c Measures
3.1c Completion of a Measure Space
4c Outer Measures
5c The Hausdorff Outer Measure in mathbbRN
5.1c The Hausdorff Dimension of a Set EsubsetmathbbRN
5.2c The Hausdorff Dimension of the Cantor Set is ln 2/ln 3
8c The Hausdorff Measure in mathbbRN
8.1c Hausdorff Outer Measure of the Lipschitz Image of a Set
8.2c Hausdorff Dimension of Graphs of Lipschitz Functions
9c Extending Measures from Semi-algebras to s-Algebras
9.1c Inner and Outer Continuity of ? on Some Algebra mathcalQ
10c More on Extensions from Semi-algebras to s-Algebras
10.1c Self-extensions of Measures
10.2c Nonunique Extensions of Measures ? on Semi-algebras
12c The Lebesgue Measure of Sets in mathbbRN
12.1c Inner Measure and Measurability
12.2c The Peano--Jordan Measure of Bounded Sets in mathbbRN
12.3c Lipschitz Functions and Measurability
13c Vitali's Nonmeasurable Set
14c Borel Sets, Measurable Sets and Incomplete Measures
16c Borel, Regular and Radon Measures
16.1c Regular Borel Measures
16.2c Regular Outer Measures
17c Vitali Coverings
17.1c Pointwise and Measure-Theoretical Vitali Coverings
18c The Besicovitch Covering Theorem
18.1c The Besicovitch Theorem for Unbounded E
18.2c The Besicovitch Measure-Theoretical Inner Covering of Open Sets EsubsetmathbbRN
18.3c A Simpler Form of the Besicovitch Theorem
18.4c Another Besicovitch-Type Covering
4 The Lebesgue Integral
1 Measurable Functions
2 The Egorov--Severini Theorem [39, 145]
2.1 The Egorov--Severini Theorem in mathbbRN
3 Approximating Measurable Functions by Simple Functions
4 Convergence in Measure (Riesz [125], Fisher [46])
5 Quasicontinuous Functions and Lusin's Theorem
6 Integral of Simple Functions ([87])
7 The Lebesgue Integral of Nonnegative Functions
8 Fatou's Lemma and the Monotone Convergence Theorem
9 More on the Lebesgue Integral
10 Convergence Theorems
11 Absolute Continuity of the Integral
12 Product of Measures
13 On the Structure of (mathcalAtimesmathcalB)
14 The Theorem of Fubini--Tonelli
14.1 The Tonelli Version of the Fubini Theorem
15 Some Applications of the Fubini--Tonelli Theorem
15.1 Integrals in Terms of Distribution Functions
15.2 Convolution Integrals
15.3 The Marcinkiewicz Integral ([101, 102])
16 Signed Measures and the Hahn Decomposition
17 The Radon-Nikodým Theorem
17.1 Sublevel Sets of a Measurable Function
17.2 Proof of the Radon-Nikodým Theorem
18 Decomposing Measures
18.1 The Jordan Decomposition
18.2 The Lebesgue Decomposition
18.3 A General Version of the Radon-Nikodým Theorem
1c Measurable Functions
1.1c Sublevel Sets
2c The Egorov--Severini Theorem
3c Approximating Measurable Functions by Simple Functions
4c Convergence in Measure
7c The Lebesgue Integral of Nonnegative Measurable Functions
7.1c Comparing the Lebesgue Integral with the Peano-Jordan Integral
7.2c On the Definition of the Lebesgue Integral
9c More on the Lebesgue Integral
10c Convergence Theorems
10.1c Another Version of Dominated Convergence
11c Absolute Continuity of the Integral
12c Product of Measures
12.1c Product of a Finite Sequence of Measure Spaces
13c On the Structure of (mathcalAtimesmathcalB)
13.1c Sections and Their Measure
14c The Theorem of Fubini--Tonelli
15c Some Applications of the Fubini--Tonelli Theorem
15.1c Integral of a Function as the ``Area Under the Graph''
15.2c Distribution Functions
17c The Radon-Nikodým Theorem
18c A Proof of the Radon-Nikodým Theorem When Both µ and ? Are s-Finite
5 Topics on Measurable Functions of Real Variables
1 Functions of Bounded Variation ([78])
2 Dini Derivatives ([37])
3 Differentiating Functions of Bounded Variation
4 Differentiating Series of Monotone Functions
5 Absolutely Continuous Functions ([91, 169])
6 Density of a Measurable Set
7 Derivatives of Integrals
8 Differentiating Radon Measures
9 Existence and Measurability of Dµ?
9.1 Proof of Proposition 9.2
10 Representing Dµ?
10.1 Representing Dµ? for ?llµ
10.2 Representing Dµ? for ?perpµ
11 The Lebesgue-Besicovitch Differentiation Theorem
11.1 Points of Density
11.2 Lebesgue Points of an Integrable Function
12 Regular Families
13 Convex Functions
14 The Jensen's Inequality
15 Extending Continuous Functions
15.1 The Concave Modulus of Continuity of f
16 The Weierstrass Approximation Theorem
17 The Stone-Weierstrass Theorem
18 Proof of the Stone-Weierstrass Theorem
18.1 Proof of Stone's Theorem
19 The Ascoli-Arzelà Theorem
19.1 Pre-compact Subsets of C(barE)
1c Functions of Bounded Variations
1.1c The Function of The Jumps
1.2c The Space BV[a,b]
2c Dini Derivatives
2.1c A Continuous, Nowhere Differentiable Function ([167])
2.2c An Application of the Baire Category Theorem
4c Differentiating Series of Monotone Functions
5c Absolutely Continuous Functions
5.1c The Cantor Ternary Function ([23])
5.2c A Continuous Strictly Monotone Function with a.e. Zero Derivative
5.3c Absolute Continuity of the Distribution Function of a Measurable Function
7c Derivatives of Integrals
7.1c Characterizing BV[a,b] Functions
7.2c Functions of Bounded Variation in N Dimensions [55]
13c Convex Functions
13.8c Convex Functions in mathbbRN
13.14c The Legendre Transform ([92])
13.15c Finiteness and Coercivity
13.16c The Young's Inequality
14c Jensen's Inequality
14.1c The Inequality of the Geometric and Arithmetic Mean
14.2c Integrals and Their Reciprocals
15c Extending Continuous Functions
16c The Weierstrass Approximation Theorem
17c The Stone-Weierstrass Theorem
19c A General Version of the Ascoli-Arzelà Theorem
6 The Lp Spaces
1 Functions in Lp(E) and Their Norm
2 The Hölder and Minkowski Inequalities
3 More on the Spaces Lp and Their Norm
3.1 Characterizing the Norm "026B30D f"026B30D p for 1lep&infty
3.2 The Norm "026B30D cdot"026B30D infty for E of Finite Measure
3.3 The Continuous Version of the Minkowski Inequality
4 Lp(E) for 1lepleinfty as Normed Spaces of Equivalence Classes
4.1 Lp(E) for 1lePleinfty as a Metric Topological Vector Space
5 Convergence in Lp(E) and Completeness
6 Separating Lp(E) by Simple Functions
7 Weak Convergence in Lp(E)
7.1 Counterexample
8 Weak Lower Semi-continuity of the Norm in Lp(E)
9 Weak Convergence and Norm Convergence
9.1 Proof of Proposition 9.1 for pge2
9.2 Proof of Proposition 9.1 for 1&p&2
10 Linear Functionals in Lp(E)
11 The Riesz Representation Theorem
11.1 Proof of Theorem 11.1: The Case of {X,mathcalA,µ} Finite
11.2 Proof of Theorem 11.1: The Case of {X,mathcalA,µ} s-Finite
11.3 Proof of Theorem 11.1: The Case 1&p&infty
12 The Hanner and Clarkson Inequalities
12.1 Proof of Hanner's Inequalities
12.2 Proof of Clarkson's Inequalities
13 Uniform Convexity of Lp(E) for 1&p&infty
14 The Riesz Representation Theorem By Uniform Convexity
14.1 Proof of Theorem 14.1. The Case 1&p&infty
14.2 The Case p=1 and E of Finite Measure
14.3 The Case p=1 and {X,mathcalA,µ} s-Finite
15 If EsubsetmathbbRN and pin[1,infty), then Lp(E) Is Separable
15.1 Linfty(E) Is Not Separable
16 Selecting Weakly Convergent Subsequences
17 Continuity of the Translation in Lp(E) for 1lep&infty
17.1 Continuity of the Convolution
18 Approximating Functions in Lp(E) with Functions in Cinfty(E)
19 Characterizing Pre-compact Sets in Lp(E)
1c Functions in Lp(E) and Their Norm
1.1c The Spaces Lp for 0&p&1
1.2c The Spaces Lq for q&0
1.3c The Spaces ellp for 1lePleinfty
2c The Inequalities of Hölder and Minkowski
2.1c Variants of the Hölder and Minkowski Inequalities
2.2c Some Auxiliary Inequalities
2.3c An Application to Convolution Integrals
2.4c The Reverse Hölder and Minkowski Inequalities
2.5c Lp(E)-Norms and Their Reciprocals
3c More on the Spaces Lp and Their Norm
3.4c A Metric Topology for Lp(E) when 0&p&1
3.5c Open Convex Subsets of Lp(E) for 0&p&1
5c Convergence in Lp(E) and Completeness
5.1c The Measure Space {X,mathcalA,µ} and the Metric Space {mathcalA
d}
6c Separating Lp(E) by Simple Functions
7c Weak Convergence in Lp(E)
7.3c Comparing the Various Notions of Convergence
7.5c Weak Convergence in ellp
9c Weak Convergence and Norm Convergence
9.1c Proof of Lemmas 9.1 and 9.2
11c The Riesz Representation Theorem
11.1c Weakly Cauchy Sequences in Lp(X) for 1& pleinfty
11.2c Weakly Cauchy Sequences in Lp(X) for p=1
11.3c The Riesz Representation Theorem in ellp
14c The Riesz Representation Theorem By Uniform Convexity
14.1c Bounded Linear Functional in Lp(E) for 0&p&1
14.2c An Alternate Proof of Proposition 14.1c
15c If EsubsetmathbbRN and pin[1,infty), then Lp(E) Is Separable
18c Approximating Functions in Lp(E) with Functions in Cinfty(E)
18.1c Caloric Extensions of Functions in Lp(mathbbRN)
18.2c Harmonic Extensions of Functions in Lp(mathbbRN)
18.3c Characterizing Hölder Continuous Functions
19c Characterizing Pre-compact Sets in Lp(E)
19.1c The Helly's Selection Principle
20c The Vitali-Saks-Hahn Theorem [59, 138, 170]
21c Uniformly Integrable Sequences of Functions
22c Relating Weak and Strong Convergence and Convergence in Measure
23c An Independent Proof of Corollary 22.1c
7 Banach Spaces
1 Normed Spaces
1.1 Semi-norms and Quotients
2 Finite and Infinite Dimensional Normed Spaces
2.1 A Counterexample
2.2 The Riesz Lemma
2.3 Finite Dimensional Spaces
3 Linear Maps and Functionals
4 Examples of Maps and Functionals
4.1 Functionals
4.2 Linear Functionals on C(barE)
4.3 Linear Functionals on Ca(barE) for Some ain(0,1)
5 Kernels of Maps and Functionals
6 Equibounded Families of Linear Maps
6.1 Another Proof of Proposition 6.1
7 Contraction Mappings
7.1 Applications to Some Fredholm Integral Equations
8 The Open Mapping Theorem
8.1 Some Applications
8.2 The Closed Graph Theorem
9 The Hahn--Banach Theorem
10 Some Consequences of the Hahn--Banach Theorem
10.1 Tangent Planes
11 Separating Convex Subsets of a Hausdorff, Topological Vector Space {X
mathcalU}
11.1 Separation in Locally Convex, Hausdorff, Topological Vector Spaces {X
mathcalU}
12 Weak Topologies
12.1 Weak Boundedness
12.2 Weakly and Strongly Closed Convex Sets
13 Reflexive Banach Spaces
14 Weak Compactness
14.1 Weak Sequential Compactness
15 The Weak* Topology of X*
16 The Alaoglu Theorem
17 Hilbert Spaces
17.1 The Schwarz Inequality
17.2 The Parallelogram Identity
18 Orthogonal Sets, Representations and Functionals
18.1 Bounded Linear Functionals on H
19 Orthonormal Systems
19.1 The Bessel Inequality
19.2 Separable Hilbert Spaces
20 Complete Orthonormal Systems
20.1 Equivalent Notions of Complete Systems
20.2 Maximal and Complete Orthonormal Systems
20.3 The Gram--Schmidt Orthonormalization Process ([142])
20.4 On the Dimension of a Separable Hilbert Space
1c Normed Spaces
1.1c Semi-Norms and Quotients
2c Finite and Infinite Dimensional Normed Spaces
3c Linear Maps and Functionals
6c Equibounded Families of Linear Maps
8c The Open Mapping Theorem
9c The Hahn--Banach Theorem
9.1c The Complex Hahn--Banach Theorem
9.2c Linear Functionals in Linfty(E)
11c Separating Convex Subsets of X
11.1c A Counterexample of Tukey [164]
11.2c A Counterexample of Goffman and Pedrick [56]
11.3c Extreme Points of a Convex Set
11.4c A General Version of the Krein--Milman Theorem
12c Weak Topologies
12.1c Infinite Dimensional Normed Spaces
12.2c About Corollary 12.5
12.3c Weak Closure and Weak Sequential Closure
14c Weak Compactness
14.1c Linear Functionals on Subspaces of C(barE)
14.2c Weak Compactness and Boundedness
15c The Weak* Topology of X*
15.1c Total Sets of X
15.2c Metrization Properties of Weak* Compact Subsets of X*
16c The Alaoglu Theorem
16.1c The Weak* Topology of X**
16.2c Characterizing Reflexive Banach Spaces
16.3c Metrization Properties of the Weak Topology of the Closed Unit Ball of a Banach Space
16.4c Separating Closed Sets in a Reflexive Banach Space
17c Hilbert Spaces
17.1c On the Parallelogram Identity
18c Orthogonal Sets, Representations and Functionals
19c Orthonormal Systems
8 Spaces of Continuous Functions, Distributions, and Weak Derivatives
1 Bounded Linear Functionals on Co(mathbbRN)
1.1 Positive Linear Functionals on Co(mathbbRN)
1.2 The Riesz Representation Theorem
2 Partition of Unity
3 Proof of Theorem 1.1. Constructing µ
4 An Auxiliary Positive Linear Functional on Co(mathbbRN)+
4.1 Measuring Compact Sets by T+
5 Representing T+ on Co(mathbbRN)+ as in (1.1) for a Unique µmathcalB
6 Proof of Theorem 1.1. Representing T on Co(mathbbRN) as in (1.3) for a Unique µ-Measurable w
7 A Topology for Coinfty(E) for an Open Set EsubsetmathbbRN
8 A Metric Topology for Coinfty(E)
8.1 Equivalence of These Topologies
8.2 D(E) Is Not Complete
9 A Topology for Coinfty(K) for a Compact Set KsubsetE
9.1 mathcalD(K) Is Complete
9.2 Relating the Topology of D(E) to the Topology of mathcalD(K)
10 The Schwartz Topology of mathcalD(E)
11 mathcalD(E) Is Complete
11.1 Cauchy Sequences in mathcalD(E) and Completeness
11.2 The Topology of mathcalD(E) Is Not Metrizable
12 Continuous Maps and Functionals
12.1 Distributions in E
12.2 Continuous Linear Maps T:mathcalD(E)tomathcalD(E)
13 Distributional Derivatives
14 Fundamental Solutions
14.1 The Fundamental Solution of the Wave Operator in mathbbR2
14.2 The Fundamental Solution of the Laplace Operator
15 Weak Derivatives and Main Properties
16 Domains and Their Boundaries
16.1 E of Class C1
16.2 Positive Geometric Density and E Piecewise Smooth
16.3 The Segment Property
16.4 The Cone Property
16.5 On the Various Properties of E
17 More on Smooth Approximations
18 Extensions into mathbbRN
19 The Chain Rule
20 Steklov Averagings
20.1 Characterizing W1,p(E) for 1&p&infty
20.2 Remarks on W1,infty(E)
21 The Rademacher's Theorem
1c Bounded Linear Functionals on Co(mathbbRN
mathbbRm)
2c Convergence of Measures
3c Calculus with DistributionsMost of the problems in Sections 3c-6c, were provided by U. Gianazza and V. Vespri.
4c Limits in mathcalD
5c Algebraic Equations in mathcalD
6c Differential Equations in mathcalD
7c Miscellaneous Problems
9 Topics on Integrable Functions of Real Variables
1 A Vitali-Type Covering
2 The Maximal Function (Hardy--Littlewood [69] and Wiener [175])
3 Strong Lp Estimates for the Maximal Function
3.1 Estimates of Weak and Strong Type
4 The Calderón--Zygmund Decomposition Theorem [20]
5 Functions of Bounded Mean Oscillation
5.1 Some Consequences of the John--Nirenberg Theorem
6 Proof of the John--Nirenberg Theorem 5.1
7 The Sharp Maximal Function
8 Proof of the Fefferman--Stein Theorem
9 The Marcinkiewicz Interpolation Theorem
9.1 Quasi-linear Maps and Interpolation
10 Proof of the Marcinkiewicz Theorem
11 Rearranging the Values of a Function
12 Some Integral Inequalities for Rearrangements
12.1 Contracting Properties of Symmetric Rearrangements
12.2 Testing for Measurable Sets E Such that E=E* a.e. in mathbbRN
13 The Riesz Rearrangement Inequality
13.1 Reduction to Characteristic Functions of Bounded Sets
14 Proof of (13.1) for N=1
14.1 Reduction to Finite Union of Intervals
14.2 Proof of (13.1) for N=1. The Case T+SleR
14.3 Proof of (13.1) for N=1. The Case S+T&R
14.4 Proof of the Lemma 14.1
15 The Hardy's Inequality
16 The Hardy--Littlewood--Sobolev Inequality for N=1
16.1 Some Reductions
17 Proof of Theorem 16.1
18 The Hardy--Littlewood--Sobolev Inequality for Nge1
18.1 Proof of Theorem 18.1
19 Potential Estimates
20 Lp Estimates of Riesz Potentials
20.1 Motivating Lp Estimates of Riesz Potentials as Embeddings
21 Lp Estimates of Riesz Potentials for p=1 and p&N
22 The Limiting Case p=N
23 Steiner Symmetrization of a Set EsubsetmathbbRN
24 Some Consequences of Steiner's Symmetrization
24.1 Symmetrizing a Set About the Origin
24.2 The Isodiametric Inequality
24.3 Steiner Rearrangement of a Function
25 Proof of the Riesz Rearrangement Inequality for N=2
25.1 The Limit of {Fn}
25.2 The Set F* Is the Disc F*
26 Proof of the Riesz Rearrangement Inequality for N&2
11c Rearranging the Values of a Function
12c Some Integral Inequalities for Rearrangements
20c Lp Estimates of Riesz Potentials
21c Lp Estimates of Riesz Potentials for p=1 and p&N
22c The Limiting Case p=Na
23c Some Consequences of Steiner's Symmetrization
23.1c Applications of the Isodiametric Inequality
10 Embeddings of W1,p(E) into Lq(E)
1 Multiplicative Embeddings of W1,po(E)
1.1 Proof of Theorem 1.1
2 Proof of Theorem 1.1 for N=1
3 Proof of Theorem 1.1 for 1lep&N
4 Proof of Theorem 1.1 for 1lep&N Concluded
5 Proof of Theorem 1.1 for pgeN&1
5.1 Estimate of I1(x,R)
5.2 Estimate of I2(x,R)
6 Proof of Theorem 1.1 for pgeN&1 Concluded
7 On the Limiting Case p=N
8 Embeddings of W1,p(E)
9 Proof of Theorem 8.1
10 Poincaré Inequalities
10.1 The Poincaré Inequality
10.2 Multiplicative Poincaré Inequalities
10.3 Extensions of (u-uE) for Convex E
11 Level Sets Inequalities
12 Morrey Spaces [110]
12.1 Embeddings for Functions in the Morrey Spaces
13 Limiting Embedding of W1,N(E)
14 Compact Embeddings
15 Fractional Sobolev Spaces in mathbbRN
16 Traces
17 Traces and Fractional Sobolev Spaces
18 Traces on E of Functions in W1,p(E)
18.1 Traces and Fractional Sobolev Spaces
19 Multiplicative Embeddings of W1,p(E)
20 Proof of Theorem 19.1. A Special Case
21 Constructing a Map Between E and Q. Part I
21.1 Case 1. Sn intersects B?
21.2 Case 2. Sn does not intersect B?
22 Constructing a Map Between E and Q. Part II
23 Proof of Theorem 19.1 Concluded
24 The Spaces W1,pp*(E)
1c Multiplicative Embeddings of W1,po(E)
8c Embeddings of W1,p(E)
8.1c Differentiability of Functions in W1,p(E) for p&N
14c Compact Embeddings
17c Traces and Fractional Sobolev Spaces
17.1c Characterizing Functions in W1-1p,p(mathbbRN) as Traces
18c Traces on E of Functions in W1,p(E)
18.1c Traces on a Sphere
11 Topics on Weakly Differentiable Functions
1 Sard's Lemma [140]
2 The Co-area Formula for Smooth Functions
3 The Isoperimetric Inequality for Bounded Sets E with Smooth Boundary E
3.1 Embeddings of W1,po(E) Versus the Isoperimetric Inequality
4 The p-Capacity of a Compact Set KsubsetmathbbRN, for 1lep&N
4.1 Enlarging the Class of Competing Functions
5 A Characterization of the p-Capacity of a Compact Set KsubsetmathbbRN, for 1lep&N
6 Lower Estimates of cp(K) for 1lep&N
6.1 A Simpler Proof of Lemma 6.1 with a Coarser Constant
6.2 p-Capacity of a Closed Ball barB?subsetmathbbRN, for 1lep&N
6.3 cp(barB?)= cp(B?)
7 The Norm "026B30D Du"026B30D p, for 1leP&N, in Terms of the p-Capacity ?
7.1 Some Auxiliary Estimates for 1&p&N
7.2 Proof of Theorem 7.1
8 Relating Gagliardo Embeddings, Capacities, and the Isoperimetric Inequality
9 Relating mathcalHN-p(K) to cp(K) for 1&p&N
9.1 An Auxiliary Proposition
9.2 Proof of Theorem 9.1
10 Relating cp(K) to mathcalHN-p+e(K) for 1lep&N
11 The p-Capacity of a Set EsubsetmathbbRN for 1lep&N
12 Limits of Sets and Their Outer p-Capacities
13 Capacitable Sets
14 Capacities Revisited and p-Capacitability of Borel Sets
14.1 The Borel Sets in mathbbRN Are p-Capacitable
14.2 Generating Measures by p-Capacities
15 Precise Representatives of Functions in L1loc(mathbbRN)
16 Estimating the p-Capacity of [u&t] for t&0
17 Precise Representatives of Functions in W1,ploc(mathbbRN)
17.1 Quasi-Continuous Representatives of Functions uinW1,ploc(mathbbRN)
References
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