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Harmonic and Geometric Analysis [Mīkstie vāki]

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  • Formāts: Paperback / softback, 170 pages, height x width: 240x168 mm, weight: 380 g, 12 Illustrations, color; 7 Illustrations, black and white; IX, 170 p. 19 illus., 12 illus. in color., 1 Paperback / softback
  • Sērija : Advanced Courses in Mathematics - CRM Barcelona
  • Izdošanas datums: 22-May-2015
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3034804075
  • ISBN-13: 9783034804073
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  • Mīkstie vāki
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Harmonic and Geometric AnalysisPalielināt
  • Formāts: Paperback / softback, 170 pages, height x width: 240x168 mm, weight: 380 g, 12 Illustrations, color; 7 Illustrations, black and white; IX, 170 p. 19 illus., 12 illus. in color., 1 Paperback / softback
  • Sērija : Advanced Courses in Mathematics - CRM Barcelona
  • Izdošanas datums: 22-May-2015
  • Izdevniecība: Birkhauser Verlag AG
  • ISBN-10: 3034804075
  • ISBN-13: 9783034804073
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783034804073.html
Keywords:
This book contains an expanded version of lectures delivered by the authors at the CRM in Spring of 2009. It contains four series of lectures. The first one is an application of harmonic analysis and the Heisenberg group to understand human vision. The second and third series of lectures cover some of the main topics on linear and multilinear harmonic analysis. The last one is a clear introduction to a deep result of De Giorgi, Moser and Nash on regularity of elliptic partial differential equations in divergence form.

This volume contains two surveys of new results on linear and multilinear analysis. It offers an insightful presentation of the De Giorgi–Moser–Nash result and contains elegant applications of harmonic analysis to human vision.
Foreword ix
1 Models of the Visual Cortex in Lie Groups
Giovanna Citti
Alessandro Sarti
1.1 Introduction
1(2)
1.2 Perceptual completion phenomena
3(3)
1.2.1 Gestalt rules and association fields
3(1)
1.2.2 The phenomenological model of elastica
4(2)
1.3 Functional architecture of the visual cortex
6(2)
1.3.1 The retinotopic structure
7(1)
1.3.2 The hypercolumnar structure
8(1)
1.3.3 The neural circuitry
8(1)
1.4 The visual cortex modeled as a Lie group
8(14)
1.4.1 A first model in the Heisenberg group
8(2)
1.4.2 A sub-Riemannian model in the rototranslation group
10(5)
1.4.3 Hormander vector fields and sub-Riemannian structures
15(2)
1.4.4 Connectivity property
17(3)
1.4.5 Control distance
20(1)
1.4.6 Riemannian approximation of the metric
21(1)
1.4.7 Geodesies and elastica
21(1)
1.5 Activity propagation and differential operators in Lie groups
22(6)
1.5.1 Integral curves, association fields, and the experiment of Bosking
22(1)
1.5.2 Differential calculus in a sub-Riemannian setting
22(3)
1.5.3 Sub-Riemannian differential operators
25(3)
1.6 Regular surfaces in a sub-Riemannian setting
28(7)
1.6.1 Maximum selectivity and lifting images to regular surfaces
28(1)
1.6.2 Definition of a regular surface
29(1)
1.6.3 Implicit function theorem
30(3)
1.6.4 Non-regular and non-linear vector fields
33(2)
1.7 Completion and minimal surfaces
35(22)
1.7.1 A completion process
35(1)
1.7.2 Minimal surfaces in the Heisenberg group
35(2)
1.7.3 Uniform regularity for the Riemannian approximating minimal graph
37(9)
1.7.4 Regularity of the viscosity minimal surface
46(1)
1.7.5 Foliation of minimal surfaces and completion result
46(2)
Bibliography
48(9)
2 Multilinear Calderon--Zygmund Singular Integrals
Loukas Grafakos
2.1 Introduction
57(3)
2.2 Bilinear Calderon--Zygmund operators
60(5)
2.3 Endpoint estimates and interpolation for bilinear Calderon--Zygmund operators
65(4)
2.4 The bilinear T1 theorem
69(4)
2.5 Orthogonality properties for bilinear multiplier operators
73(6)
2.6 The bilinear Hilbert transform and the method of rotations
79(4)
2.7 Counterexample for the higher-dimensional bilinear ball multiplier
83(8)
Bibliography
88(3)
3 Singular Integrals and Weights
Carlos Perez
Summary
91(1)
3.1 Introduction
91(11)
3.2 Three applications of the Besicovitch covering lemma to the maximal function
102(3)
3.3 Two applications of Rubio de Francia's algorithm: Optimal factorization and extrapolation
105(5)
3.3.1 The sharp factorization theorem
105(2)
3.3.2 The sharp extrapolation theorem
107(3)
3.4 Three more applications of Rubio de Francia's algorithm
110(5)
3.4.1 Building A1 weights from duality
110(1)
3.4.2 Improving inequalities with weights
111(4)
3.5 The sharp reverse Holder property of A1 weights
115(2)
3.6 Main lemma and proof of the linear growth theorem
117(1)
3.7 Proof of the logarithmic growth theorem
118(1)
3.8 Properties of Ap weights
119(3)
3.9 Improvements in terms of mixed A1-A∞ constants
122(2)
3.10 Quadratic estimates for commutators
124(6)
3.10.1 A preliminary result: a sharp connection between the John-Nirenberg theorem and the A2 class
124(2)
3.10.2 Results in the Ap context
126(2)
3.10.3 Examples
128(1)
3.10.4 The A1 case
129(1)
3.11 Rearrangement type estimates
130(4)
3.12 Proof of Theorem 3.42
134(2)
3.13 The exponential decay lemma
136(9)
Bibliography
139(6)
4 De Giorgi--Nash--Moser Theory
Xiao Zhong
4.1 Introduction
145(6)
4.1.1 Equations
145(1)
4.1.2 Motivation: a variational problem
146(5)
4.2 Sobolev spaces
151(4)
4.2.1 A brief introduction to Sobolev spaces
151(3)
4.2.2 Definition of weak solutions
154(1)
4.3 Moser's iteration
155(6)
4.3.1 Harnack's inequality
155(1)
4.3.2 Weak Harnack's inequality: sup
155(4)
4.3.3 Weak Harnack's inequality: inf
159(2)
4.4 De Giorgi's method
161(5)
4.4.1 De Giorgi's class of functions
161(1)
4.4.2 Boundedness of functions in DG(Ω, γ)
161(2)
4.4.3 Holder continuity of functions in DG(Ω, γ)
163(3)
4.5 Further discussions
166
4.5.1 Degenerate elliptic equations
166(3)
Bibliography
169