Angular limits of holomorphic functions which satisfy an integrability condition |
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Authors: | Stephen J Gardiner |
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Institution: | (1) Department of Mathematics, University College, Dublin 4, Ireland |
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Abstract: | Let ![beta](/content/p044238401g2400m/xxlarge946.gif) (–1,1), let 2/(1– ) p< , letp denote the Hölder conjugate ofp, and let be an open arc of the unit circle. It is shown that, iff is a holomorphic function on the unit disc such that: (i) (1–|z|) log+|f(z)| isL
p
-integrable on the sector {r :0![isin](/content/p044238401g2400m/xxlarge8712.gif) }, and (ii) the subset of wheref has an infinite asymptotic value has -finite (2–(1+ )p )-dimensional Hausdorff, measure, thenf has finite angular limits on a subset of of positive linear measure. In fact, a stronger conclusion will be established. |
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Keywords: | |
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