Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces ( CBMS Regional Conference Series in Mathematics )

Publication series ：CBMS Regional Conference Series in Mathematics

Author： Alexander Volberg  

Publisher： American Mathematical Society‎

Publication year： 2003

E-ISBN: 9781470424619

P-ISBN(Hardback):  9780821832523

Subject： O177 functional analysis

Keyword： 暂无分类

Language： ENG

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Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces

Description

Singular integral operators play the central part in modern harmonic analysis. Simplest examples of singular kernels are given by Calderón–Zygmund kernels. Many important properties of singular integrals have been thoroughly studied for Calderón–Zygmund operators. In the ’80s and early ’90s, Coifman, Weiss, and Christ noticed that the theory of Calderón–Zygmund operators can be generalized from Euclidean spaces to spaces of homogeneous type. The purpose of this book is to make the reader believe that homogeneity (previously considered as a cornerstone of the theory) is not needed. This claim is illustrated by presenting two harmonic analysis problems famous for their difficulty. The first problem treats semiadditivity of analytic and Lipschitz harmonic capacities. The volume presents the first self-contained and unified proof of the semiadditivity of these capacities. The book details Tolsa's solution of Painlevé's and Vitushkin's problems and explains why these are problems of the theory of Calderón–Zygmund operators on nonhomogeneous spaces. The exposition is not dimension-specific, which allows the author to treat Lipschitz harmonic capacity and analytic capacity at the same time. The second problem considered in the volume is a two-weight estimate for the Hilbert transform. This problem recently found important applications in operator theory, where it is intimately related to spectral theory of small perturbations of unitary operators. The book presents a technique

Chapter

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Title

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Contents

Chapter 1. Introduction

Chapter 2. Preliminaries on Capacities

Chapter 3. Localization of Newton and Riesz Potentials

3.1. Localization lemmas

3.2. A building block for the construction of special measures

3.3. Localization on special cubes

3.4. Modification of distribution S. Construction of auxiliary measures

3.5. Ahlfors balls

3.6. The principal estimate for auxiliary measures

Chapter 4. From Distribution to Measure. Carleson Property

Chapter 5. Potential Neighborhood that has Properties (3.13)–(3.14)

5.1. Capacities with Calderon-Zygmund (CZ) kernels

5.2. Variational capacity and extremal measures

5.3. L[sup(p)] theory of nonhomogeneous CZ operators. Measure of order m

5.4. Riesz and Cauchy kernels: γ[sub(+)] = γ[sub(op)]

5.5. Cauchy kernel and analytic capacity

Chapter 6. The Tree of the Proof

Chapter 7. The First Reduction to Nonhomogeneous Tb Theorem

Chapter 8. The Second Reduction

8.1. Suppressed kernels

8.2. From real-valued kernel to vector valued kernel

8.3. From one lattice to two lattices

8.4. Core suppression

Chapter 9. The Third Reduction

Chapter 10. The Fourth Reduction

10.1. μ, b, D, η decomposition

10.2. Good functions and bad functions

10.3. Estimates of nonhomogeneous Calderón-Zygmund operators on good functions

10.4. The reduction of Theorem 9.1 to estimates of nonhomogeneous Calderón-Zygmund operator, namely to Theorem 10.6

Chapter 11. The Proof of Nonhomogeneous Cotlar's Lemma. Arbitrary Measure

Chapter 12. Starting the Proof of Nonhomogeneous Nonaccretive Tb Theorem

12.1. Terminal and transit cubes

12.2. Projections Λ and Δ[sub(Q)]

Chapter 13. Next Step in Theorem 10.6. Good and Bad Functions

13.1. Good functions and bad functions again

13.2. Reduction to estimates on good functions

13.3. Splitting 〈T[sub([omitted]good, ψgood)〉 to three sums

13.4. Three types of estimates of ∫ k(x, y)f(x)g(y) dμ,(x) dμ(y)

13.5. Estimate of long range interaction sum σ[sub(2)]

13.6. Short range interaction sum σ[sub(3)]. Nonhomogeneous paraproducts

Chapter 14. Estimate of the Diagonal Sum. Remainder in Theorem 3.3

14.1. Estimate of ∑[sub(term)]

14.2. Estimate of ∑[sub(tr)]

Chapter 15. Two Weight Estimate for the Hilbert Transform. Preliminaries

Chapter 16. Necessity in the Main Theorem

Chapter 17. Two Weight Hilbert Transform. Towards the Main Theorem

17.1. Bad and good parts of f and g

17.2. Estimates on good functions

Chapter 18. Long Range Interaction

Chapter 19. The Rest of the Long Range Interaction

Chapter 20. The Short Range Interaction

20.1. The estimate of neighbor-terms

20.2. The estimate of stopping terms

20.3. The choice of stopping intervals

Chapter 21. Difficult Terms and Several Paraproducts

21.1. First paraproduct

21.2. Two more paraproducts

21.3. Second paraproduct: miraculous improvement of the Carleson property

Chapter 22. Two-Weight Hilbert Transform and Maximal Operator

22.1. Doubling

22.2. No doubling

Bibliography

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