NONISOTROPIC HAUSDORFF CAPACITY OF EXCEPTIONAL SETS OF INVARIANT POTENTIALS

In the paper we investigate tangential boundary limits of invariant Green potentials on the unit ball B in C-n, n greater than or equal to 1. Let G(z, w) denote the Green function for the Laplace-Beltrami operator on B, and let lambda denote the invariant measure on B. If mu is a non-negative measure, or f is a non-negative measurable function on B, G(mu) and G(f) denote the Green potential of mu and f respectively. For zeta is an element of S = partial derivative B, tau greater than or equal to 1, and c > 0, let T-tau,T-c(zeta) = {Z is an element of B: \1 - < z, xi\(tau) < c (1 - \z\(2))}. The main result of the paper is as follows: Let f be a non-negative measurable function on B satisfying integral(B) (1 - \w\(2))(beta)f(p)(w) d lambda(w) < infinity for some beta, 0 < beta < n, and some p > n. Then for each tau, 1 less than or equal to tau < n/beta, there exists a set E(tau) subset of S with H-beta tau(E(tau)) = 0, such that [GRAPHICS] In the above, for 0 < alpha less than or equal to n, H-alpha denotes the non-isotropic alpha-dimensional Hausdorff capacity on S. We also prove that if {a(k)} is a sequence in B satisfying Sigma(1 - \a(k)\(2))(beta) < infinity for some beta, 0 < beta < n, and mu = Sigma delta(ak), where delta(a) denotes point mass measure at a, then the same conclusion holds for the potential G(mu).