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

Kuzman Adzievski
2006 Annales Polonici Mathematici  
We study questions related to exceptional sets of pluri-Green potentials V µ in the unit ball B of C n in terms of non-isotropic Hausdorff capacity. For suitable measures µ on the ball B, the pluri-Green potentials V µ are defined by where for a fixed z ∈ B, φ z denotes the holomorphic automorphism of B satisfying 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
more » ... main result of this paper is as follows: Let f be a non-negative measurable function on B satisfying 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 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 )}.
doi:10.4064/ap88-1-5 fatcat:exvg7zvlmzghje7nduq5xdjauy