Muutke küpsiste eelistusi

Geometric Harmonic Analysis I: A Sharp Divergence Theorem with Nontangential Pointwise Traces 2022 ed. [Kõva köide]

, ,
  • Formaat: Hardback, 924 pages, kõrgus x laius: 235x155 mm, kaal: 1586 g, 20 Illustrations, color; 24 Illustrations, black and white; XXVIII, 924 p. 44 illus., 20 illus. in color., 1 Hardback
  • Sari: Developments in Mathematics 72
  • Ilmumisaeg: 05-Nov-2022
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3031059492
  • ISBN-13: 9783031059490
Teised raamatud teemal:
  • Kõva köide
  • Hind: 159,88 €*
  • * hind on lõplik, st. muud allahindlused enam ei rakendu
  • Tavahind: 188,09 €
  • Säästad 15%
  • Raamatu kohalejõudmiseks kirjastusest kulub orienteeruvalt 2-4 nädalat
  • Kogus:
  • Lisa ostukorvi
  • Tasuta tarne
  • Tellimisaeg 2-4 nädalat
  • Lisa soovinimekirja
Geometric Harmonic Analysis I: A Sharp Divergence Theorem with Nontangential Pointwise Traces 2022 ed.Suurem pilt
  • Formaat: Hardback, 924 pages, kõrgus x laius: 235x155 mm, kaal: 1586 g, 20 Illustrations, color; 24 Illustrations, black and white; XXVIII, 924 p. 44 illus., 20 illus. in color., 1 Hardback
  • Sari: Developments in Mathematics 72
  • Ilmumisaeg: 05-Nov-2022
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3031059492
  • ISBN-13: 9783031059490
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783031059490.html
Märksõnad:
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.

Arvustused

The theory is developed in a consistent manner, and the motivation behind the results and tools is made clear to the reader. All of the main results and also a vast majority of the auxiliary results come with full and carefully written proofs, making the book highly self-contained. Thus this work can be a useful and enjoyable reference. (Juha Lehrbäck, Mathematical Reviews, August, 2024)

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