On Removability Properties of $$\psi $$ ψ -Uniform Domains in Banach Spaces

Abstract

Suppose that E denotes a real Banach space with the dimension at least 2. The main aim of this paper is to show that a domain D in E is a ψ-uniform domain if and only if D\P is a ψ 1 -uniform domain, and D is a uniform domain if and only if D\P also is a uniform domain, whenever P is a closed countable subset of D satisfying a quasihyperbolic separation condition. This condition requires that the quasihyperbolic distance (w.r.t. D) between each pair of distinct points in P has a lower bound greater than or equal to 1 2 .

On the Removability Property of φ‐Natural Domains in Real Banach Spaces

“…e motivation of this study stems from the discussions in [1], where the authors studied the removability of uniform domains and ψ-uniform domains in Banach spaces. e following are the main results in [1].…”

“…A domain G⊊E is a c 1 -uniform domain if and only if G 0 G\ is a c 2 -uniform domain, where the uniformity coe cients c 1 and c 2 depend on each other. Theorem 2 ([see [1],…”

“…To investigate the relationships between conformal invariants and quasi hyperbolic metrics, in [14], Vuorinen introduced the so-called ψ-uniform domains, which are a generalization of uniform domains. See [1,[15][16][17][18][19] for the development of ψ-uniform domains in recent years.…”

“…Next, we construct a curve β in G 0 to connect z 1 and z 2 . Since B c ≠ ∅, there is an element in B c , denoted by B 1 , such that, (1)…”

Constructing uniform spaces

“…A special case of Theorem C, with κ = 1/2 and A ⊂ X a countable subset of a Banach space 2 uniform domain X, was proved in [HVW17]. Theorems B and C were established in the Euclidean setting in [Her89]; see also [Her87].…”