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By Xavier Tolsa

This ebook reviews a number of the groundbreaking advances which have been made concerning analytic ability and its courting to rectifiability within the decade 1995–2005. The Cauchy rework performs a primary function during this sector and is as a result one of many major topics coated. one other very important subject, that may be of self sustaining curiosity for plenty of analysts, is the so-called non-homogeneous Calderón-Zygmund idea, the improvement of which has been principally encouraged through the issues bobbing up in reference to analytic capability. The Painlevé challenge, which was once first posed round 1900, is composed to find an outline of the detachable singularities for bounded analytic features in metric and geometric phrases. Analytic potential is a key software within the examine of this challenge. within the Sixties Vitushkin conjectured that the detachable units that have finite size coincide with these that are basically unrectifiable. in addition, as a result of functions to the idea of uniform rational approximation, he posed the query to whether analytic ability is semiadditive. This paintings provides complete proofs of Vitushkin’s conjecture and of the semiadditivity of analytic skill, either one of which remained open difficulties till very lately. different comparable questions also are mentioned, comparable to the connection among rectifiability and the life of imperative values for the Cauchy transforms and different singular integrals. The ebook is basically self-contained and will be obtainable for graduate scholars in research, in addition to a important source for researchers.

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Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory: 307 (Progress in Mathematics) by Xavier Tolsa

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