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 ℂⁿ,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<β<n, and somep>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 .