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Download Calderon-Zygmund capacities and operators on nonhomogeneous by Alexander Volberg PDF

By Alexander Volberg

Singular imperative operators play a principal function in sleek harmonic research. easiest examples of singular kernels are given via Calderon-Zygmund kernels. Many very important homes of singular integrals were completely studied for Calderon-Zygmund operators. within the 1980's and early 1990's, Coifman, Weiss, and Christ spotted that the speculation of Calderon-Zygmund operators might be generalized from Euclidean areas to areas of homogeneous variety. the aim of this ebook is to make the reader think that homogeneity (previously regarded as a cornerstone of the speculation) isn't really wanted. This declare is illustrated via offering harmonic research difficulties recognized for his or her hassle. the 1st challenge treats semiadditivity of analytic and Lipschitz harmonic capacities. the quantity provides the 1st self-contained and unified evidence of the semiadditivity of those capacities. The publication info Tolsa's answer of Painleve's and Vitushkin's difficulties and explains why those are difficulties of the idea of Calderon-Zygmund operators on nonhomogeneous areas. The exposition isn't dimension-specific, which permits the writer to regard Lipschitz harmonic means and analytic skill whilst. the second one challenge thought of within the quantity is a two-weight estimate for the Hilbert remodel. This challenge lately came upon very important functions in operator idea, the place it's in detail concerning spectral thought of small perturbations of unitary operators. The e-book offers a method that may be invaluable in overcoming particularly undesirable degeneracies (i.e., exponential development or decay) of underlying degree (volume) at the house the place the singular critical operator is taken into account. those events take place, for instance, in boundary worth difficulties for elliptic PDE's in domain names with super singular limitations. one other instance includes harmonic research at the limitations of pseudoconvex domain names that is going past the scope of Carnot-Caratheodory areas. The e-book is appropriate for graduate scholars and examine mathematicians drawn to harmonic research.

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