On the Hardy number of comb domains

Abstract

Let H p (D) be the Hardy space of all holomorphic functions on the unit disk D with exponent p > 0. If D = C is a simply connected domain and f is the Riemann mapping from D onto D, then the Hardy number of D, introduced by Hansen, is the supremum of all p for which f ∈ H p (D). 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.

The range of Hardy number on comb domains

“…The domain C with x n = n and b n = 1 for every n ∈ Z has Hardy number equal to infinity. This has already been proved in [9].…”

“…There are only some ways to estimate it for certain types of domains. See, for example, [7,[9][10][11][12] and references therein.…”

“…Comb domains is an interesting class of simply connected domains and thus they have been studied from various points of view. See, for example, [3,9] and references therein.…”

A collection of results relating the geometry of plane domains and the exit time of planar Brownian motion

“…We now construct a second counterexample to the conjecture in which the domain and its complement both have finite p th moment of the exit time for any p > 0. To construct this second counterexample we must recall theory from hyperbolic geometry ([3], [15]).…”