Affine transformations and analytic capacities

Authors:
Thomas Dowling and Anthony G. O’Farrell

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

MSC:
Primary 30E10; Secondary 30H05, 46E15

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

MathSciNet review:
1273488

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[L]**Lars V. Ahlfors,*Bounded analytic functions*, Duke Math. J.**14**(1947), 1–11. MR**0021108****[T]**Thomas Bagby,*𝐿_{𝑝} approximation by analytic functions*, J. Approximation Theory**5**(1972), 401–404. Collection of articles dedicated to J. L. Walsh on his 75th birthday, IV. MR**0348116****[E]**Elias M. Stein,*Singular integrals and differentiability properties of functions*, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR**0290095****[L]**Lars Inge Hedberg,*Approximation in the mean by analytic functions*, Trans. Amer. Math. Soc.**163**(1972), 157–171. MR**0432886**, https://doi.org/10.1090/S0002-9947-1972-0432886-6**[S]**V. Hrushchev,*Simple proof of a theorem on removable singularities for analytic functions*, LOMI Notes, Leningrad, 1975.**[A]**Anthony G. O’Farrell,*Hausdorff content and rational approximation in fractional Lipschitz norms*, Trans. Amer. Math. Soc.**228**(1977), 187–206. MR**0432887**, https://doi.org/10.1090/S0002-9947-1977-0432887-2**[T]**Theodore W. Gamelin,*Uniform algebras*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1969. MR**0410387****[R]**Robert Kaufman,*Hausdorff measure, BMO, and analytic functions*, Pacific J. Math.**102**(1982), no. 2, 369–371. MR**686557**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30E10,
30H05,
46E15

Retrieve articles in all journals with MSC: 30E10, 30H05, 46E15

Additional Information

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

Keywords:
Analytic capacity,
symmetric concrete space

Article copyright:
© Copyright 1995
American Mathematical Society