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On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance

2020; Mittag-Leffler Institute; Volume: 58; Issue: 2 Linguagem: Inglês

10.4310/arkiv.2020.v58.n2.a5

1871-2487

Christina Karafyllia,

Analytic and geometric function theory

Let ψ be a conformal map on D with ψ (0)=0 and let Fα ={z ∈D:|ψ (z)|=α} for α>0.Denote by H p (D) the classical Hardy space with exponent p>0 and by h (ψ) the Hardy number of ψ.Consider the limitswhere ω D (0, Fα) denotes the harmonic measure at 0 of Fα and d D (0, Fα) denotes the hyperbolic distance between 0 and Fα in D. We study a problem posed by P. Poggi-Corradini.What is the relation between L, μ and h (ψ)?Motivated by the result of Kim and Sugawa that h (ψ)=lim inf α→+∞ (log ω D (0, Fα) -1 log α), we show that h (ψ)=lim inf α→+∞ (d D (0, Fα)/log α).We also provide conditions for the existence of L and μ and for the equalities L=μ=h (ψ).Poggi-Corradini proved that ψ / ∈H μ (D) for a wide class of conformal maps ψ.We present an example of ψ such that ψ∈H μ (D).