Calderon-Zygmund capacities and operators on nonhomogeneous spaces
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A central part of modern harmonic analysis deals with "singular operators" of one type or another. Such operators are pervasive in the scientific landscape: they turn up in mathematical physics, probability, engineering, image processing, etc. The expression "singular operator" can take many different meanings, but whatever its precise meaning within a given context, the term almost always reflects the fact that only subtle tools can be applied to the investigation of such operators.
They are resistant to probing by "brute force" methods.
In particular, one needs to employ techniques that are sensitive to certain intrinsic cancellations in order to gain control over the singular behavior. So-called Calderon-Zygmund theory appeared in the early s as a first means of coping with singular operators of the type under scrutiny in this project. The PIs broaden this theory in two directions. First, they allow the number of degrees of freedom enjoyed by the operators to increase, thus exposing a whole new kind of cancellation.
Calderon-Zygmund Capacities and Operators on Nonhomogeneous Spaces cover image. CBMS Regional Conference Series in Mathematics. comparison o f analytic capacit y an d "positiv e analyti c capacity". . Calderon- Zygmund Capacities and Operators on Nonhomogeneous Spaces · Cover · Title .
Second, they make a rather bold move by disgarding one of the basic and, it was earlier thought, indispensible assumptions of the theory. In light of discoveries by the PIs, this assumption known technically as the homogeneity of the underlying space seems now to be completely superflous. The PIs' approach has already had several significant consequences: solutions to several long-standing problems, streamlined proofs for other important but difficult results, and the general feature of making complicated arguments much shorter and more lucid.
The book is suitable for graduate students and research mathematicians interested in harmonic analysis. You may print 0 more time s before then. Print Price 1: Electronic Media? Print Price 1 Label: List. The first problem treats semiadditivity of analytic and Lipschitz harmonic capacities. Overview Singular integral operators play the central part in modern harmonic analysis.
Nazarov, S. Treil, A.
Uniform Boundedness for Approximations of the Identity with Nondoubling Measures
Theory Adv. Hukovic, S.
- Calderon-Zygmund Capacities and Operators on Nonhomogeneous Spaces?
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Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces
Presidential and Honorary Awards. 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 '80s and early '90s, 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.
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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.
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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. 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.