Calderón-Zygmund capacities and operators on nonhomogeneous spaces
著者
書誌事項
Calderón-Zygmund capacities and operators on nonhomogeneous spaces
（Regional conference series in mathematics, no. 100）
Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society with support from the National Science Foundation, c2003
大学図書館所蔵 件 / 全39件
-
該当する所蔵館はありません
- すべての絞り込み条件を解除する
注記
"CBMS Regional Research Conference Nonhomogeneous Harmonic Analysis, Weights, and Applications to Problems in Complex Analysis and Operator Theory, University of North Carolina, May 13-17, 2002"--T.p. verso
Bibliography: p. 165-167
内容説明・目次
内容説明
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.
目次
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.
「Nielsen BookData」 より