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

"...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

• Introduction
• Preliminaries on capacities
• Localization of Newton and Riesz potentials
• From distribution to measure. Carleson property
• Potential neighborhood that has properties (3.13)-(3.14)
• The tree of the proof
• The first reduction to nonhomogeneous $Tb$ theorem
• The second reduction
• The third reduction
• The fourth reduction
• The proof of nonhomogeneous Cotlar's lemma. Arbitrary measure
• Starting the proof of nonhomogeneous nonaccretive $Tb$ theorem
• Next step in theorem 10.6. Good and bad functions
• Estimate of the diagonal sum. Remainder in theorem 3.3
• Two-weight estimate for the Hilbert transform. Preliminaries
• Necessity in the main theorem
• Two-weight Hilbert transform. Towards the main theorem
• Long range interaction
• The rest of the long range interaction
• The short range interaction
• Difficult terms and several paraproducts
• Two-weight Hilbert transform and maximal operator
• Bibliography