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1 Statement of Main Results Concerning the Divergence Theorem |
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1 | (128) |
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1.1 The De Giorgi--Federer Version of the Divergence Theorem |
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1 | (16) |
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1.2 The Case When the Divergence Is Absolutely Integrable |
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17 | (12) |
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1.3 The Case Without Decay and When the Divergence Is a Measure |
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29 | (8) |
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1.4 The Divergence Theorem for Singular Vector Fields Without Decay |
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37 | (10) |
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1.5 Non-doubling Surface Measures and Maximally Singular Vector Fields |
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47 | (5) |
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1.6 Divergence Formulas Without Lower Ahlfors Regularity |
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52 | (5) |
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1.7 Integration by Parts in Open Sets with Ahlfors Regular Boundaries |
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57 | (8) |
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1.8 Higher-Order Integration by Parts |
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65 | (5) |
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1.9 The Divergence Theorem with Weak Boundary Traces |
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70 | (6) |
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1.10 The Divergence Theorem Involving an Averaged Nontangential Maximal Operator |
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76 | (3) |
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1.11 The Manifold Setting and a Sharp Version of Stokes' Formula |
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79 | (17) |
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1.12 Integrating by Parts on Boundaries of Ahlfors Regular Domains on Manifolds |
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96 | (33) |
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2 Examples, Counterexamples, and Additional Perspectives |
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129 | (114) |
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2.1 Failure of Hypotheses on the Nontangential Boundary Trace |
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131 | (15) |
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2.2 Failure of Hypotheses on Behavior at Infinity |
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146 | (11) |
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2.3 Failure of Hypotheses on the Nontangential Maximal Function |
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157 | (14) |
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2.4 Failure of Hypotheses of Geometric Measure Theoretic Nature |
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171 | (3) |
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2.5 Failure of Hypotheses on the Nature of the Divergence of the Vector Field |
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174 | (3) |
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2.6 Relationship with Classical Results in the One-Dimensional Setting |
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177 | (8) |
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2.7 Examples and Counterexamples Pertaining to Weak Traces |
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185 | (5) |
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2.8 Other Versions of the Gauss-Green Formula |
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190 | (53) |
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3 Measure Theoretical and Topological Rudiments |
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243 | (48) |
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3.1 Sigma-Algebras, Measures, Lebesgue Spaces |
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245 | (5) |
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3.2 The Topology on the Space of Measurable Functions |
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250 | (2) |
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252 | (4) |
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3.4 Borel-Regular Measure and Outer Measures |
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256 | (11) |
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267 | (7) |
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274 | (5) |
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3.7 Density Results for Lebesgue Spaces |
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279 | (4) |
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3.8 The Support of a Measure |
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283 | (4) |
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3.9 The Riesz Representation Theorem |
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287 | (4) |
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4 Selected Topics in Distribution Theory |
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291 | (56) |
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4.1 Distribution Theory on Arbitrary Sets |
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291 | (11) |
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302 | (8) |
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4.3 The Product Rule for Weak Derivatives |
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310 | (7) |
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4.4 Pointwise Divergence Versus Distributional Divergence |
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317 | (3) |
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4.5 Removability of Singularities for Distributional Derivatives |
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320 | (9) |
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4.6 The Algebraic Dual of the Space of Smooth and Bounded Functions |
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329 | (4) |
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4.7 The Contribution at Infinity of a Vector Field |
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333 | (14) |
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5 Sets of Locally Finite Perimeter and Other Categories of Euclidean Sets |
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347 | (148) |
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5.1 Thick Sets and Corkscrew Conditions |
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347 | (4) |
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5.2 The Geometric Measure Theoretic Boundary |
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351 | (4) |
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5.3 Area/Coarea Formulas, and Countable Rectifiability |
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355 | (12) |
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5.4 Approximate Tangent Planes |
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367 | (1) |
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5.5 Functions of Bounded Variation |
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368 | (2) |
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5.6 Sets of Locally Finite Perimeter |
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370 | (38) |
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5.7 Sets of Finite Perimeter |
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408 | (12) |
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420 | (11) |
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431 | (25) |
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5.10 Uniformly Rectifiable Sets |
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456 | (16) |
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5.11 Nontangentially Accessible Domains |
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472 | (23) |
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6 Tools from Harmonic Analysis |
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495 | (74) |
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6.1 The Regularized Distance Function and Whitney's Extension Theorem |
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495 | (3) |
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6.2 Short Foray into Lorentz Spaces |
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498 | (14) |
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6.3 The Fractional Hardy--Littlewood Maximal Operator in a Non-Metric Setting |
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512 | (8) |
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6.4 Clifford Algebra Fundamentals |
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520 | (25) |
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6.5 Subaveraging Functions, Reverse Holder Estimates, and Interior Estimates |
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545 | (8) |
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6.6 The Solid Maximal Function and Maximal Lebesgue Spaces |
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553 | (16) |
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7 Quasi-Metric Spaces and Spaces of Homogeneous Type |
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569 | (102) |
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7.1 Quasi-Metric Spaces and a Sharp Metrization Result |
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569 | (4) |
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7.2 Estimating Integrals Involving the Quasi-Distance |
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573 | (5) |
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7.3 Holder Spaces on Quasi-Metric Spaces |
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578 | (4) |
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7.4 Functions of Bounded Mean Oscillations on Spaces of Homogeneous Type |
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582 | (28) |
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7.5 Whitney Decompositions on Geometrically Doubling Quasi-Metric Spaces |
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610 | (11) |
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7.6 The Hardy--Littlewood Maximal Operator on Spaces of Homogeneous Type |
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621 | (17) |
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7.7 Muckenhoupt Weights on Spaces of Homogeneous Type |
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638 | (25) |
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7.8 The Fractional Integration Theorem |
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663 | (8) |
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8 Open Sets with Locally Finite Surface Measures and Boundary Behavior |
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671 | (166) |
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8.1 Nontangential Approach Regions in Arbitrary Open Sets |
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671 | (9) |
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8.2 The Definition and Basic Properties of the Nontangential Maximal Operator |
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680 | (7) |
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8.3 Elementary Estimates Involving the Nontangential Maximal Operator |
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687 | (10) |
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8.4 Size Estimates for the Nontangential Maximal Operator Involving a Doubling Measure |
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697 | (27) |
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8.5 Maximal Operators: Tangential Versus Nontangential |
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724 | (7) |
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8.6 Off-Diagonal Carleson Measure Estimates of Reverse Holder Type |
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731 | (24) |
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8.7 Estimates for Marcinkiewicz Type Integrals and Applications |
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755 | (19) |
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8.8 The Nontangentially Accessible Boundary |
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774 | (12) |
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8.9 The Nontangential Boundary Trace Operator |
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786 | (38) |
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8.10 The Averaged Nontangential Maximal Operator |
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824 | (13) |
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9 Proofs of Main Results Pertaining to Divergence Theorem |
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837 | (66) |
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9.1 Proofs of Theorems 1.2.1 and 1.3.1 and Corollaries 1.2.2, 1.2.4, and 1.3.2 |
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837 | (18) |
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9.2 Proof of Theorem 1.4.1 and Corollaries 1.4.2--1.4.4 |
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855 | (4) |
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9.3 Proofs of Theorem 1.5.1 and Corollary 1.5.2 |
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859 | (7) |
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9.4 Proofs of Theorem 1.6.1 and Corollaries 1.6.2--1.6.6 |
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866 | (4) |
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9.5 Proofs of Theorems 1.7.1, 1.7.2, and 1.7.6 |
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870 | (5) |
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9.6 Proofs of Theorems 1.8.2, 1.8.3, and 1.8.5 |
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875 | (10) |
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9.7 Proofs of Theorems 1.9.1--1.9.4 |
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885 | (4) |
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9.8 Proof of Theorem 1.10.1 |
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889 | (4) |
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9.9 Proofs of Theorems 1.11.3, 1.11.6, and 1.11.8--1.11.11 |
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893 | (10) |
| References |
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903 | (12) |
| Subject Index |
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915 | (6) |
| Symbol Index |
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921 | |
Lisainfo e-raamatute kohta