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Radial variation of functions in Dirichlet-type spaces

Part of: Series expansions

Published online by Cambridge University Press:  26 February 2010

J. B. Twomey
J. B. Twomey
Affiliation:
Department of Mathematics, University College Cork, Ireland.
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Extract

Let W denote a positive, increasing and continuous function on [1, ∞]. We write to denote the Dirichlettype space of functions f that are holomorphic in the unit disc and for which

Where If W(x) = x for all x, then is the classicial Dirichlet space for which Note also that for every so, by Fatu's theoreum, every function in . ha finite radial(and angular) limits a.e. on the boundary of U. The question of the existence a.e. on ∂U of certain tangential limits for functions in has been considered in [6,11], but we shall be concerned here with the radial variation

i.e., the length of the image of the ray from 0 to eiθ under the mapping w = f(z), and, in particular, with the size of the set of values of θ for which Lf(θ) can be infinite when

MSC classification

Secondary: 30B30: Boundary behavior of power series, over-convergence
Type
Research Article
Information
Mathematika , Volume 44 , Issue 2 , December 1997 , pp. 267 - 277
Copyright
Copyright © University College London 1997

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