O'Farrell, Anthony G.
Capacities in function theory.
Potential theory : proceedings of the International Conference on Potential Theory, Nagoya (Japan) August 30-September 4, 1990.
Walter de Gruyter, Berlin, New York, pp. 93-105.
Ever since the famous thesis of Frostman, capacities have been important in many areas of function theory. In this talk I shall be concerned only with one–variable
function theory on arbitrary open subsets of the complex plane, C. It is important to stress that the open sets need not be connected. I will discuss the use of analytic
capacities in connection with problems of removable singularities, holomorphic approximation, and boundary smoothness. A brief reference to the applications is in order. The connections between analytic (and harmonic) functions and the physics of perfect fluids, electrostatics,
magnetostatics, classical gravitation and heat are well–known. Much of what I shall say about analytic functions (solutions of the d–bar equation) applies also to solutions
of other elliptic equations, and so there are other applications, for instance to elasticity (connected to the bi-Laplacian and the d-bar-squared operator). It should
also be noted that students of capacitary problems were among the first to discover and examine many of the weird–looking sets formerly regarded as pathological by
most people, but now known as fractals and accepted as natural objects of study for many applications.
||Capacities; Function theory; Banach–Grothendieck theorem; Cauchy transform; Beurling transform.
||Faculty of Science and Engineering > Mathematics and Statistics
Prof. Anthony O'Farrell
||19 Jan 2010 16:06
||Walter de Gruyter
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