Non-isotropic hausdorff capacity of exceptional sets of invariant potentials

Abstract

In the paper we investigate tangential boundary limits of invariant Green potentials on the unit ballB in ℂ n ,n≥1. LetG(z, w) denote the Green function for the Laplace-Beltrami operator onB, and let λ denote the invariant measure onB. If μ is a non-negative measure, orf is a non-negative measurable function onB,G μ andG f denote the Green potential of μ andf respectively. For ξ∈S=δB, τ≥1, andc>0, let $$\mathcal{T}_{\tau ,c} (\zeta ) = \{ z \in B:\left| {1 - \left\langle {z,\xi } \right\rangle } \right|^\tau< c(1 - \left| z \right|^2 )\} $$ . The main result of the paper is as follows: Letf be a non-negative measurable function onB satisfying $$\int_B {(1 - \left| w \right|^2 )^\beta f^p (w)d\lambda (w)< \infty } $$ for some β, 0 n. Then for each τ, 1≤τ<n/β, there exists a setE t ⊂S withH βτ (E t )=0, such that $$\mathop {\lim }\limits_{\mathop {z \to \zeta }\limits_{z \in \mathcal{T}_{\tau ,c} (\zeta )} } G_f (z) = 0,forall\zeta \in S \sim E_\tau $$ In the above, for 0<α≤n,H α denotes the non-isotropic α-dimensional Hausdorff capacity onS. We also prove that if {a k } is a sequence inB satisfying Σ(1−|a k |2) β <∞ for some β, 0 <β<n, and μ=Σδ ak , where δ a denotes point mass measure ata, then the same conclusion holds for the potentialG μ .