Softcover ISBN:  9780821832523 
Product Code:  CBMS/100 
List Price:  $49.00 
Individual Price:  $39.20 
Electronic ISBN:  9781470424619 
Product Code:  CBMS/100.E 
List Price:  $46.00 
Individual Price:  $36.80 

Book DetailsCBMS Regional Conference Series in MathematicsVolume: 100; 2003; 167 ppMSC: Primary 42; Secondary 32; 31;
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 selfcontained 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 dimensionspecific, 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 twoweight 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 CarnotCarathéodory spaces.
The book is suitable for graduate students and research mathematicians interested in harmonic analysis.ReadershipGraduate students and research mathematicians interested in harmonic analysis.

Table of Contents

Chapters

Chapter 1. Introduction

Chapter 2. Preliminaries on capacities

Chapter 3. Localization of Newton and Riesz potentials

Chapter 4. From distribution to measure. Carleson property

Chapter 5. Potential neighborhood that has properties (3.13)–(3.14)

Chapter 6. The tree of the proof

Chapter 7. The first reduction to nonhomogeneous $Tb$ theorem

Chapter 8. The second reduction

Chapter 9. The third reduction

Chapter 10. The fourth reduction

Chapter 11. The proof of nonhomogeneous Cotlar’s lemma. Arbitrary measure

Chapter 12. Starting the proof of nonhomogeneous nonaccretive $Tb$ theorem

Chapter 13. Next step in theorem 10.6. Good and bad functions

Chapter 14. Estimate of the diagonal sum. Remainder in theorem 3.3

Chapter 15. Twoweight estimate for the Hilbert transform. Preliminaries

Chapter 16. Necessity in the main theorem

Chapter 17. Twoweight Hilbert transform. Towards the main theorem

Chapter 18. Long range interaction

Chapter 19. The rest of the long range interaction

Chapter 20. The short range interaction

Chapter 21. Difficult terms and several paraproducts

Chapter 22. Twoweight Hilbert transform and maximal operator


Additional Material

Reviews

...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


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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 selfcontained 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 dimensionspecific, 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 twoweight 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 CarnotCarathéodory spaces.
The book is suitable for graduate students and research mathematicians interested in harmonic analysis.
Graduate students and research mathematicians interested in harmonic analysis.

Chapters

Chapter 1. Introduction

Chapter 2. Preliminaries on capacities

Chapter 3. Localization of Newton and Riesz potentials

Chapter 4. From distribution to measure. Carleson property

Chapter 5. Potential neighborhood that has properties (3.13)–(3.14)

Chapter 6. The tree of the proof

Chapter 7. The first reduction to nonhomogeneous $Tb$ theorem

Chapter 8. The second reduction

Chapter 9. The third reduction

Chapter 10. The fourth reduction

Chapter 11. The proof of nonhomogeneous Cotlar’s lemma. Arbitrary measure

Chapter 12. Starting the proof of nonhomogeneous nonaccretive $Tb$ theorem

Chapter 13. Next step in theorem 10.6. Good and bad functions

Chapter 14. Estimate of the diagonal sum. Remainder in theorem 3.3

Chapter 15. Twoweight estimate for the Hilbert transform. Preliminaries

Chapter 16. Necessity in the main theorem

Chapter 17. Twoweight Hilbert transform. Towards the main theorem

Chapter 18. Long range interaction

Chapter 19. The rest of the long range interaction

Chapter 20. The short range interaction

Chapter 21. Difficult terms and several paraproducts

Chapter 22. Twoweight Hilbert transform and maximal operator

...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