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This article is cited in 1 scientific paper (total in 1 paper)
Derivatives at the boundary for analytic Lipschitz functions
A. G. O'Farrell Department of Mathematics and Statistics, National University of Ireland, Maynooth, Co. Kildare, Ireland
Abstract:
We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $\alpha$, with $0<\alpha<1$, in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We show that whenever such a bounded point derivation exists at a boundary point $b$, it may be evaluated by taking a limit of classical difference quotients, approaching from a set having full area density at $b$.
Bibliography: 13 titles.
Keywords:
analytic function, boundary, Lipschitz condition, point derivation, difference quotient, capacity, Hausdorff content.
Received: 31.10.2015 and 12.05.2016
Citation:
A. G. O'Farrell, “Derivatives at the boundary for analytic Lipschitz functions”, Mat. Sb., 207:10 (2016), 119–140; Sb. Math., 207:10 (2016), 1471–1490
Linking options:
https://www.mathnet.ru/eng/sm8629https://doi.org/10.1070/SM8629 https://www.mathnet.ru/eng/sm/v207/i10/p119
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Abstract page: | 386 | Russian version PDF: | 163 | English version PDF: | 3 | References: | 47 | First page: | 37 |
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