Unbounded H\"{o}lder domains

TL;DR

This paper introduces and characterizes unbounded H"{o}lder domains, extending classical bounded H"{o}lder domains, and explores their relation to Hardy spaces with applications to Hardy number bounds.

Contribution

It defines unbounded H"{o}lder domains, provides analytic and geometric characterizations, and establishes their significance in Hardy space theory.

Findings

Unbounded H"{o}lder domains are characterized using spherical and hyperbolic metrics.

A relation between unbounded and bounded H"{o}lder domains is established.

A sharp bound for the Hardy number of unbounded H"{o}lder domains is proved.

Abstract

Motivated by the classical bounded H\"{o}lder domains, we introduce the notion of an unbounded simply connected H\"{o}lder domain. We prove analytic and geometric characterizations of those domains with the aid of the spherical metric and the hyperbolic metric. We also study the relation of our definition to the definition of classical bounded H\"{o}lder domains. It turns out that unbounded H\"older domains form a natural class of domains for the study of the Hardy number (which determines the Hardy spaces to which the corresponding Riemann mapping belongs to). As an application of our characterizations, we prove a sharp bound for the Hardy number of an unbounded H\"{o}lder domain.