By Alexander Volberg
Singular quintessential operators play a primary position in glossy harmonic research. least difficult examples of singular kernels are given by way of Calderon-Zygmund kernels. Many vital 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 idea of Calderon-Zygmund operators may be generalized from Euclidean areas to areas of homogeneous variety. the aim of this booklet is to make the reader think that homogeneity (previously regarded as a cornerstone of the idea) isn't really wanted. This declare is illustrated by way of offering harmonic research difficulties well-known for his or her trouble. the 1st challenge treats semiadditivity of analytic and Lipschitz harmonic capacities. the amount offers the 1st self-contained and unified evidence of the semiadditivity of those capacities. The publication information Tolsa's resolution of Painleve's and Vitushkin's difficulties and explains why those are difficulties of the speculation of Calderon-Zygmund operators on nonhomogeneous areas. The exposition isn't dimension-specific, which permits the writer to regard Lipschitz harmonic ability and analytic capability whilst. the second one challenge thought of within the quantity is a two-weight estimate for the Hilbert rework. This challenge lately stumbled on very important functions in operator thought, the place it truly is in detail on the topic of spectral thought of small perturbations of unitary operators. The publication offers a method that may be priceless in overcoming quite undesirable degeneracies (i.e., exponential progress or decay) of underlying degree (volume) at the area the place the singular indispensable operator is taken into account. those events ensue, for instance, in boundary price 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 booklet is acceptable for graduate scholars and learn mathematicians drawn to harmonic research.
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Additional resources for Calderon-Zygmund Capacities and Operators on Nonhomogeneous Spaces (Cbms Regional Conference Series in Mathematics)
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