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 $({I_n})$ with $\Sigma |{I_n}|\log |{I_n}| > - \infty$. Suppose H is the Hausdorff measure on T induced by the determining function h, where $h(t)/t$ 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.