Cartan-type estimates for potentials with Cauchy kernels and real-valued kernels

Let v be a (complex) Radon measure in C with compact support and finite variation and let L-*(z) = sup(epsilon > 0)|integral(|sigma-z|>epsilon)dv(sigma)/sigma-z| be the maximal Cauchy integral. Estimates for the Hausdorff h-content of the set T-*(v,P) = {z is an element of C : l(*)v(z) > P} 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 R-m, m >= 1. 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).