On the Hardy number of comb domains

TL;DR

This paper investigates the Hardy number of comb domains, characterizing when it is infinite using quasi-hyperbolic distance, and links this to Brownian motion exit times.

Contribution

It provides a necessary and sufficient condition for the Hardy number of comb domains to be infinite, connecting geometric properties with probabilistic behavior.

Findings

Hardy number characterized by quasi-hyperbolic distance

Condition for infinite Hardy number derived

Relation between domain geometry and Brownian motion exit times

Abstract

Let ${H^p}\left( \mathbb{D} \right)$ be the Hardy space of all holomorphic functions on the unit disk $\mathbb{D}$ with exponent $p>0$. If $D\ne \mathbb{C}$ is a simply connected domain and $f$ is the Riemann mapping from $\mathbb{D}$ onto $D$, then the Hardy number of $D$, introduced by Hansen, is the supremum of all $p$ for which $f \in {H^p}\left( \mathbb{D} \right)$. Comb domains are a well-studied class of simply connected domains that, in general, have the form of the entire plane minus an infinite number of vertical rays. In this paper we study the Hardy number of a class of comb domains with the aid of the quasi-hyperbolic distance and we establish a necessary and sufficient condition for the Hardy number of these domains to be equal to infinity. Applying this condition, we derive several results that show how the mutual distances and the distribution of the rays affect the finiteness of the Hardy number. By a result of Burkholder our condition is also necessary and sufficient for all moments of the exit time of Brownian motion from comb domains to be infinite.