Geometric harmonic analysis, I : a sharp divergence theorem with nontangential pointwise traces / / Dorina Mitrea, Irina Mitrea, and Marius Mitrea

• Autore: Mitrea Dorina
• Pubblicazione: Cham, Switzerland : , : Springer, , [2022]; ©2022
• Descrizione fisica: 1 online resource (940 pages)
• Disciplina: 620.00151
• Soggetto topico: Divergence theorem; Functional analysis; Anàlisi funcional
• Soggetto genere / forma: Llibres electrònics
• Persona (resp. second.): MitreaIrina; MitreaMarius
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: Geometric harmonic analysis, I
• ISBN: 3-031-05950-6
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910629292003321
• Lo trovi qui: Univ. Federico II
• Serie: Developments in Mathematics