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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Analytic capacity, Hölder conditions and $ \tau $-spikes


Author: Anthony G. O’Farrell
Journal: Trans. Amer. Math. Soc. 196 (1974), 415-424
MSC: Primary 30A98; Secondary 30A82, 46J15
DOI: https://doi.org/10.1090/S0002-9947-1974-0361116-0
MathSciNet review: 0361116
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Abstract: We consider the uniform algebra $ R(X)$, for compact $ X \subset {\text{C}}$, in relation to the condition $ {I_{p + \alpha }} = \Sigma _1^\infty {2^{(p + \alpha + 1)n}}\gamma ({A_n}(x)\backslash X) < + \infty $, where $ 0 \leq p \in {\mathbf{Z}},0 < \alpha < 1,\gamma $ is analytic capacity, and $ {A_n}(x)$ is the annulus $ \{ z \in {\text{C}}:{2^{ - n - 1}} < \vert z - x\vert < {2^{ - n}}\} $. We introduce the notion of $ \tau $-spike for $ \tau > 0$, and show that $ {I_{p + \alpha }} = + \infty $ implies x is a $ p + \alpha $-spike. If $ \mathop X\limits^ \circ $ satisfies a cone condition at x, and $ {I_{p + \alpha }} < + \infty $, we show that the pth derivatives of the functions in $ R(X)$ satisfy a uniform Hölder condition at x for nontangential approach. The structure of the set of non-$ \tau $-spikes is examined and the results are applied to rational approximation. A geometric question is settled.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0361116-0
Keywords: Uniform norm, analytic capacity, bounded point derivation, peak point, Gleason metric, potential
Article copyright: © Copyright 1974 American Mathematical Society