Affine transformations and analytic capacities

Dowling, Thomas; O’Farrell, Anthony G.

doi:10.1090/S0002-9947-1995-1273488-7

Affine transformations and analytic capacities
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by Thomas Dowling and Anthony G. O’Farrell

Trans. Amer. Math. Soc. 347 (1995), 2643-2655

DOI: https://doi.org/10.1090/S0002-9947-1995-1273488-7

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Abstract:

Analytic capacities are set functions defined on the plane which may be used in the study of removable singularities, boundary smoothness and approximation of analytic functions belonging to some function space. The symmetric concrete Banach spaces form a class of function spaces that includes most spaces usually studied. The Beurling transform is a certain singular integral operator that has proved useful in analytic function theory. It is shown that the analytic capacity associated to each Beurling-invariant symmetric concrete Banach space behaves reasonably under affine transformation of the plane. It is not known how general analytic capacities behave under affine maps.

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