Hausdorff measure and Carleson thin sets

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 ) ({I_n}) with Σ | I n | log ⁡ | I n | > − ∞ \Sigma |{I_n}|\log |{I_n}| > - \infty . Suppose H is the Hausdorff measure on T induced by the determining function h, where h ( t ) / t h(t)/t is strictly decreasing. We show that H ( E ) = 0 H(E) = 0 for every Carleson set E if and only if ∫ 0 1 h ( t ) − 1 d t = ∞ \smallint _0^1h{(t)^{ - 1}}dt = \infty . Consequently the nonintegrability of h − 1 {h^{ - 1}} is necessary and sufficient for every positive Borel measure μ \mu on T with modulus of continuity ω μ ( δ ) = O ( h ( δ ) ) {\omega _\mu }(\delta ) = O(h(\delta )) to place zero mass on every Carleson set.