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

Kuzman Adzievski. "Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ." Annales Polonici Mathematici 88.1 (2006): 59-82. <http://eudml.org/doc/280581>.

@article{KuzmanAdzievski2006,
abstract = {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 $lim_\{z→ζ \atop z∈_\{τ,c\}(ζ)\} V_f(z) = 0$ 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, $_\{τ,c\}(ζ)$ are the regions defined by $_\{τ,c\}(ζ) = \{z ∈ B:|1 - ⟨z,ζ⟩|^\{τ\} < c(1-|z|²)\}. $},
author = {Kuzman Adzievski},
journal = {Annales Polonici Mathematici},
keywords = {nonisotropic Hausdorff capacity; pluri-Green potentials},
language = {eng},
number = {1},
pages = {59-82},
title = {Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ},
url = {http://eudml.org/doc/280581},
volume = {88},
year = {2006},
}

TY - JOUR
AU - Kuzman Adzievski
TI - Non-isotropic Hausdorff capacity of exceptional sets for pluri-Green potentials in the unit ball of ℂⁿ
JO - Annales Polonici Mathematici
PY - 2006
VL - 88
IS - 1
SP - 59
EP - 82
AB - 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 $lim_{z→ζ \atop z∈_{τ,c}(ζ)} V_f(z) = 0$ 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, $_{τ,c}(ζ)$ are the regions defined by $_{τ,c}(ζ) = {z ∈ B:|1 - ⟨z,ζ⟩|^{τ} < c(1-|z|²)}. $
LA - eng
KW - nonisotropic Hausdorff capacity; pluri-Green potentials
UR - http://eudml.org/doc/280581
ER -