**CBMS Regional Conference Series in Mathematics**

Volume: 100;
2003;
167 pp;
Softcover

MSC: Primary 42;
Secondary 32; 31

Print ISBN: 978-0-8218-3252-3

Product Code: CBMS/100

List Price: $46.00

Individual Price: $36.80

**Electronic ISBN: 978-1-4704-2461-9
Product Code: CBMS/100.E**

List Price: $46.00

Individual Price: $36.80

#### Supplemental Materials

# Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces

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*Alexander Volberg*

A co-publication of the AMS and CBMS

Singular integral operators play a central role 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 1980's and early 1990's, 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 that can be helpful in overcoming rather bad
degeneracies (i.e., exponential growth or decay) of underlying measure (volume)
on the space where the singular integral operator is considered. These
situations occur, for example, in boundary value problems for elliptic PDE's in
domains with extremely singular boundaries. Another example involves harmonic
analysis on the boundaries of pseudoconvex domains that goes beyond the scope
of Carnot-Carathéodory spaces.

The book is suitable for graduate students and research mathematicians
interested in harmonic analysis.

#### Readership

Graduate students and research mathematicians interested in harmonic analysis.

#### Reviews & Endorsements

...this book will interest anyone who would like to learn these new beautiful techniques in harmonic analysis and apply them...

-- Hervé Pajot for Mathematical Reviews

#### Table of Contents

# Table of Contents

## Calderon-Zygmund Capacities and Operators on Nonhomogeneous Spaces

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- Chapter 1. Introduction 16 free
- Chapter 2. Preliminaries on Capacities 712 free
- Chapter 3. Localization of Newton and Riesz Potentials 1116
- Chapter 4. From Distribution to Measure. Carleson Property 2126
- Chapter 5. Potential Neighborhood that has Properties (3.13)–(3.14) 2530
- Chapter 6. The Tree of the Proof 5156
- Chapter 7. The First Reduction to Nonhomogeneous Tb Theorem 5560
- Chapter 8. The Second Reduction 6166
- Chapter 9. The Third Reduction 7176
- Chapter 10. The Fourth Reduction 7378
- Chapter 11. The Proof of Nonhomogeneous Cotlar's Lemma. Arbitrary Measure 8388
- Chapter 12. Starting the Proof of Nonhomogeneous Nonaccretive Tb Theorem 9398
- Chapter 13. Next Step in Theorem 10.6. Good and Bad Functions 101106
- 13.1. Good functions and bad functions again 101106
- 13.2. Reduction to estimates on good functions 102107
- 13.3. Splitting 〈T[sub([omitted]good, ψgood)〉 to three sums 103108
- 13.4. Three types of estimates of ∫ k(x, y)f(x)g(y) dμ,(x) dμ(y) 104109
- 13.5. Estimate of long range interaction sum σ[sub(2)] 106111
- 13.6. Short range interaction sum σ[sub(3)]. Nonhomogeneous paraproducts 109114

- Chapter 14. Estimate of the Diagonal Sum. Remainder in Theorem 3.3 121126
- Chapter 15. Two Weight Estimate for the Hilbert Transform. Preliminaries 127132
- Chapter 16. Necessity in the Main Theorem 133138
- Chapter 17. Two Weight Hilbert Transform. Towards the Main Theorem 135140
- Chapter 18. Long Range Interaction 139144
- Chapter 19. The Rest of the Long Range Interaction 143148
- Chapter 20. The Short Range Interaction 145150
- Chapter 21. Difficult Terms and Several Paraproducts 153158
- Chapter 22. Two-Weight Hilbert Transform and Maximal Operator 161166
- Bibliography 165170
- Back Cover Back Cover1176