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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis > Complex analysis
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Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderon-Zygmund Theory (Paperback, Softcover reprint of the original 1st ed. 2014)
Loot Price: R4,107
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Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderon-Zygmund Theory (Paperback, Softcover reprint of the original 1st ed. 2014)
Series: Progress in Mathematics, 307
Expected to ship within 10 - 15 working days
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This book studies some of the groundbreaking advances that have
been made regarding analytic capacity and its relationship to
rectifiability in the decade 1995-2005. The Cauchy transform plays
a fundamental role in this area and is accordingly one of the main
subjects covered. Another important topic, which may be of
independent interest for many analysts, is the so-called
non-homogeneous Calderon-Zygmund theory, the development of which
has been largely motivated by the problems arising in connection
with analytic capacity. The Painleve problem, which was first posed
around 1900, consists in finding a description of the removable
singularities for bounded analytic functions in metric and
geometric terms. Analytic capacity is a key tool in the study of
this problem. In the 1960s Vitushkin conjectured that the removable
sets which have finite length coincide with those which are purely
unrectifiable. Moreover, because of the applications to the theory
of uniform rational approximation, he posed the question as to
whether analytic capacity is semiadditive. This work presents full
proofs of Vitushkin's conjecture and of the semiadditivity of
analytic capacity, both of which remained open problems until very
recently. Other related questions are also discussed, such as the
relationship between rectifiability and the existence of principal
values for the Cauchy transforms and other singular integrals. The
book is largely self-contained and should be accessible for
graduate students in analysis, as well as a valuable resource for
researchers.
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