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# ðŸ“™ Calderon-Zygmund capacities and operators on nonhomogeneous spaces by Alexander Volberg â€” free download

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Singular integral operators play a central role in modern harmonic analysis. Simplest examples of singular kernels are given by Calderon-Zygmund kernels. Many important properties of singular integrals have been thoroughly studied for Calderon-Zygmund operators. In the 1980's and early 1990's, Coifman, Weiss, and Christ noticed that the theory of Calderon-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 Painleve's and Vitushkin's problems and explains why these are problems of the theory of Calderon-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-Caratheodory spaces. The book is suitable for graduate students and research mathematicians interested in harmonic analysis.

## About book:

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- Series:
**CBMS regional conference series in mathematics 100** - Author:
**Alexander Volberg** - Year:
**2003** - Publisher:
**American Mathematical Society** - Language:
**English** - ISBN:
**0821832522**

- DPI:
**300** - File size:
**1 121 156** - Format:
**djvu**

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Singular integral operators play a central role in modern harmonic analysis. Simplest examples of singular kernels are given by Calderon-Zygmund kernels. Many important properties of singular integrals have been thoroughly studied for Calderon-Zygmund ope...

This volume of the "Mathematics Studies" series presents work done on composition operators during the last 25 years. Composition operators form a simple but interesting class of operators having interactions with different branches of mathematics and mat...

This volume of the "Mathematics Studies" series presents work done on composition operators during the last 25 years. Composition operators form a simple but interesting class of operators having interactions with different branches of mathematics and mat...

The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, SchrÃ¶dinger operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-...

The theory of the numerical range of a linear operator on an arbitrary normed space had its beginnings around 1960, and during the 1970s the subject has developed and expanded rapidly. This book presents a self-contained exposition of the subject as a who...

The theory of the numerical range of a linear operator on an arbitrary normed space had its beginnings around 1960, and during the 1970s the subject has developed and expanded rapidly. This book presents a self-contained exposition of the subject as a who...

Narrow operators are those operators defined on function spaces which are "small" at signs, i.e. at {-1,0,1}-valued functions. Numerous works and research papers exist on these, but no coherent monograph yet to place them in context. This book gives compr...

The theory of the numerical range of a linear operator on an arbitrary normed space had its beginnings around 1960, and during the 1970s the subject has developed and expanded rapidly. This book presents a self-contained exposition of the subject as a who...

This English edition is almost identical to the German original Lineare Operatoren in Hilbertriiumen, published by B. G. Teubner, Stuttgart in 1976. A few proofs have been simplified, some additional exercises have been included, and a small number of new...