This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations.
Volume I establishes a sharp version of the Divergence Theorem (aka Fundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundary trace is only assumed to exist in a nontangential pointwise sense.
Author(s): Dorina Mitrea, Irina Mitrea, Marius Mitrea
Series: Developments in Mathematics, 72
Publisher: Springer
Year: 2022
Language: English
Pages: 939
City: Cham
Prefacing the Full Series
Acknowledgements
Description of Volume I
Contents
Compendium of Notation Used in Volume I
1 Statement of Main Results Concerning the Divergence Theorem
1.1 The De Giorgi–Federer Version of the Divergence Theorem
1.2 The Case When the Divergence Is Absolutely Integrable
1.3 The Case Without Decay and When the Divergence Is a Measure
1.4 The Divergence Theorem for Singular Vector Fields Without Decay
1.5 Non-doubling Surface Measures and Maximally Singular Vector Fields
1.6 Divergence Formulas Without Lower Ahlfors Regularity
1.7 Integration by Parts in Open Sets with Ahlfors Regular Boundaries
1.8 Higher-Order Integration by Parts
1.9 The Divergence Theorem with Weak Boundary Traces
1.10 The Divergence Theorem Involving an Averaged Nontangential Maximal Operator
1.11 The Manifold Setting and a Sharp Version of Stokes' Formula
1.12 Integrating by Parts on Boundaries of Ahlfors Regular Domains on Manifolds
2 Examples, Counterexamples, and Additional Perspectives
2.1 Failure of Hypotheses on the Nontangential Boundary Trace
2.2 Failure of Hypotheses on Behavior at Infinity
2.3 Failure of Hypotheses on the Nontangential Maximal Function
2.4 Failure of Hypotheses of Geometric Measure Theoretic Nature
2.5 Failure of Hypotheses on the Nature of the Divergence of the Vector Field
2.6 Relationship with Classical Results in the One-Dimensional Setting
2.7 Examples and Counterexamples Pertaining to Weak Traces
2.8 Other Versions of the Gauss–Green Formula
3 Measure Theoretical and Topological Rudiments
3.1 Sigma-Algebras, Measures, Lebesgue Spaces
3.2 The Topology on the Space of Measurable Functions
3.3 Outer Measures
3.4 Borel-Regular Measure and Outer Measures
3.5 Radon Measures
3.6 Separable Measures
3.7 Density Results for Lebesgue Spaces
3.8 The Support of a Measure
3.9 The Riesz Representation Theorem
4 Selected Topics in Distribution Theory
4.1 Distribution Theory on Arbitrary Sets
4.2 The Bullet Product
4.3 The Product Rule for Weak Derivatives
4.4 Pointwise Divergence Versus Distributional Divergence
4.5 Removability of Singularities for Distributional Derivatives
4.6 The Algebraic Dual of the Space of Smooth and Bounded Functions
4.7 The Contribution at Infinity of a Vector Field
5 Sets of Locally Finite Perimeter and Other Categories of Euclidean Sets
5.1 Thick Sets and Corkscrew Conditions
5.2 The Geometric Measure Theoretic Boundary
5.3 Area/Coarea Formulas, and Countable Rectifiability
5.4 Approximate Tangent Planes
5.5 Functions of Bounded Variation
5.6 Sets of Locally Finite Perimeter
5.7 Sets of Finite Perimeter
5.8 Planar Curves
5.9 Ahlfors Regular Sets
5.10 Uniformly Rectifiable Sets
5.11 Nontangentially Accessible Domains
6 Tools from Harmonic Analysis
6.1 The Regularized Distance Function and Whitney's Extension Theorem
6.2 Short Foray into Lorentz Spaces
6.3 The Fractional Hardy–Littlewood Maximal Operator in a Non-Metric Setting
6.4 Clifford Algebra Fundamentals
6.5 Subaveraging Functions, Reverse Hölder Estimates, and Interior Estimates
6.6 The Solid Maximal Function and Maximal Lebesgue Spaces
7 Quasi-Metric Spaces and Spaces of Homogeneous Type
7.1 Quasi-Metric Spaces and a Sharp Metrization Result
7.2 Estimating Integrals Involving the Quasi-Distance
7.3 Hölder Spaces on Quasi-Metric Spaces
7.4 Functions of Bounded Mean Oscillations on Spaces of Homogeneous Type
7.5 Whitney Decompositions on Geometrically Doubling Quasi-Metric Spaces
7.6 The Hardy–Littlewood Maximal Operator on Spaces of Homogeneous Type
7.7 Muckenhoupt Weights on Spaces of Homogeneous Type
7.8 The Fractional Integration Theorem
8 Open Sets with Locally Finite Surface Measures and Boundary Behavior
8.1 Nontangential Approach Regions in Arbitrary Open Sets
8.2 The Definition and Basic Properties of the Nontangential Maximal Operator
8.3 Elementary Estimates Involving the Nontangential Maximal Operator
8.4 Size Estimates for the Nontangential Maximal Operator Involving a Doubling Measure
8.5 Maximal Operators: Tangential Versus Nontangential
8.6 Off-Diagonal Carleson Measure Estimates of Reverse Hölder Type
8.7 Estimates for Marcinkiewicz Type Integrals and Applications
8.8 The Nontangentially Accessible Boundary
8.9 The Nontangential Boundary Trace Operator
8.10 The Averaged Nontangential Maximal Operator
9 Proofs of Main Results Pertaining to Divergence Theorem
9.1 Proofs of Theorems 1.2.1 and 1.3.1 and Corollaries 1.2.2, 1.2.4, and 1.3.2
9.2 Proof of Theorem 1.4.1 and Corollaries 1.4.2–1.4.4
9.3 Proofs of Theorem 1.5.1 and Corollary 1.5.2
9.4 Proofs of Theorem 1.6.1 and Corollaries 1.6.2–1.6.6
9.5 Proofs of Theorems 1.7.1, 1.7.2, and 1.7.6
9.6 Proofs of Theorems 1.8.2, 1.8.3, and 1.8.5
9.7 Proofs of Theorems 1.9.1–1.9.4
9.8 Proof of Theorem 1.10.1
9.9 Proofs of Theorems 1.11.3, 1.11.6, and 1.11.8–1.11.11
Appendix References
Appendix Subject Index
Index
Appendix Symbol Index
Symbol Index
