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

MathSciNet review:
560586

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Abstract | References | Similar Articles | Additional Information

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 with . Suppose *H* is the Hausdorff measure on *T* induced by the determining function *h*, where is strictly decreasing. We show that for every Carleson set *E* if and only if . Consequently the nonintegrability of is necessary and sufficient for every positive Borel measure on *T* with modulus of continuity to place zero mass on every Carleson set.

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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