Cauchy-Szegö integrals for systems of harmonic functions
Let be a closed polar subset of a domain in . We give a complete description of the pluripolar hull of the graph of a holomorphic function defined on . To achieve this, we prove for pluriharmonic measure certain semi-continuity properties and a localization principle.
Let be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that is either 0 or 1; the first case occurs if and only if is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also obtained.
It is well known that certain transformations decrease the capacity of a condenser. We prove equality statements for the condenser capacity inequalities under symmetrization and polarization without connectivity restrictions on the condenser and without regularity assumptions on the boundary of the condenser.