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Hausdorff measure and Carleson thin sets


Author: Joel H. Shapiro
Journal: Proc. Amer. Math. Soc. 79 (1980), 67-71
MSC: Primary 28A05; Secondary 28A12, 30D55
DOI: https://doi.org/10.1090/S0002-9939-1980-0560586-0
MathSciNet review: 560586
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Abstract: A Carleson set is a closed subset of the unit circle T having measure zero, whose complement is a disjoint union of open subarcs $ ({I_n})$ with $ \Sigma \vert{I_n}\vert\log \vert{I_n}\vert > - \infty $. Suppose H is the Hausdorff measure on T induced by the determining function h, where $ h(t)/t$ is strictly decreasing. We show that $ H(E) = 0$ for every Carleson set E if and only if $ \smallint _0^1h{(t)^{ - 1}}dt = \infty $. Consequently the nonintegrability of $ {h^{ - 1}}$ is necessary and sufficient for every positive Borel measure $ \mu $ on T with modulus of continuity $ {\omega _\mu }(\delta ) = O(h(\delta ))$ to place zero mass on every Carleson set.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0560586-0
Keywords: Hausdorff measure, Carleson set, singular measure, modulus of continuity
Article copyright: © Copyright 1980 American Mathematical Society

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