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Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ

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Adzievski K.

Annales Polonici Mathematici

2006

tom 88

nr 1

59-82

We study questions related to exceptional sets of pluri-Green potentials $V_{μ}$ in the unit ball B of ℂⁿ in terms of non-isotropic Hausdorff capacity. For suitable measures μ on the ball B, the pluri-Green potentials $V_{μ}$ are defined by $V_{μ}(z) = ∫_B log(1/|ϕ_z(w)|)dμ(w)$, where for a fixed z ∈ B, $ϕ_z$ denotes the holomorphic automorphism of B satisfying $ϕ_z(0) = z$, $ϕ_z(z) = 0$ and $(ϕ_z ∘ ϕ_z)(w) = w$ for every w ∈ B. If dμ(w) = f(w)dλ(w), where f is a non-negative measurable function of B, and λ is the measure on B, invariant under all holomorphic automorphisms of B, then $V_{μ}$ is denoted by $V_f$. The main result of this paper is as follows: Let f be a non-negative measurable function on B satisfying $∫_B (1-|z|²) f^p(z) dλ(z) < ∞$ for some p with 1 < p < n/(n-1) and some α with 0 < α < n + p - np. Then for each τ with 1 ≤ τ ≤ n/α, there exists a set $E_{τ} ⊆ S$ with $H_{ατ}(E_{τ}) = 0$ such that $lim_{z→ζ \atop z∈𝓣_{τ,c}(ζ)} V_f(z) = 0$ for all points $ζ ∈ S∖E_{τ}$. In the above, for α > 0, $H_{α}$ denotes the non-isotropic Hausdorff capacity on S, and for ζ ∈ S = ∂B, τ ≥ 1, and c > 0, $𝓣_{τ,c}(ζ)$ are the regions defined by $𝓣_{τ,c}(ζ) = {z ∈ B:|1 - ⟨z,ζ⟩|^{τ} < c(1-|z|²)}. $

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1 Annales Polonici Mathematici

1 Adzievski K.

1 2006

Wyniki wyszukiwania

Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ