The Range of Hardy Numbers for Comb Domains

Abstract

Let $D\ne \mathbb {C}$ be a simply connected domain and f be a Riemann mapping from $\mathbb {D}$ onto D. The Hardy number of D is the supremum of all p for which f belongs in the Hardy space ${H^p}\left( \mathbb {D} \right)$. A comb domain is a domain whose complement is the union of an infinite number of vertical rays symmetric with respect to the real axis. In this paper we prove that, for $p>0$, there is a comb domain with Hardy number equal to p if and only if $p\in [1,+\infty ]$. It is known that the Hardy number is related to the moments of the exit time of Brownian motion from the domain. In fact, Burkholder proved that the Hardy number of a simply connected domain is twice the supremum of all $p>0$ for which the p-th moment of the exit time of Brownian motion is finite. Therefore, our result implies that given $p < q$ there exists a comb domain with finite p-th moment but infinite q-th moment if and only if $q\ge 1/2$. This answers a question posed by Boudabra and Markowsky.