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

Abstract

Let be a (complex) Radon measure in with compact support and finite variation and let be the maximal Cauchy integral. Estimates for the Hausdorff -content of the set are obtained, where is a measuring function and 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 , . 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.