Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces

Volberg, Alexander

doi:10.1090/cbms/100

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Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces

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Alexander Volberg, Michigan State University, East Lansing, MI

Publication: CBMS Regional Conference Series in Mathematics
Publication Year: 2003; Volume 100
ISBNs: 978-0-8218-3252-3 (print); 978-1-4704-2461-9 (online)
DOI: https://doi.org/10.1090/cbms/100
MathSciNet review: MR2019058
MSC: Primary 42B20; Secondary 31A15, 31C05, 32A55, 47G10

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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 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.

Readership

Graduate students and research mathematicians interested in harmonic analysis.

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. 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
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. Two-weight Hilbert transform and maximal operator

Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces

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