Cartan-type estimates for potentials with Cauchy

V. Ya. Èiderman

Sbornik. Mathematics, Tome 198 (2007) no. 8, pp. 1175-1220

Résumé

Let $\nu$ be a (complex) Radon measure in $\mathbb C$ with compact support and finite variation and let $$ \mathscr C_*\nu(z)=\sup_{\varepsilon>0} \biggl|\int_{|\zeta-z|>\varepsilon}\frac{d\nu(\zeta)}{\zeta-z}\biggr| $$ be the maximal Cauchy integral. Estimates for the Hausdorff $h$-content of the set $\mathscr Z^*(\nu,P)=\bigl\{z\in\mathbb C:\mathscr C_*\nu(z)>P\bigr\}$ are obtained, where $h$ is a measuring function and $P$ is a fixed positive number. These estimates are shown to be sharp up to the values of the absolute constants involved. A similar problem is also considered for potentials with arbitrary real non-increasing kernels of positive measure in $\mathbb R^m$, $m\geqslant1$. As an application of the so-developed machinery, results on connections between the analytic capacity and the Hausdorff measure are obtained (for instance, an analogue of Frostman's theorem on classical capacities). Bibliography: 37 titles.