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

Christina Karafyllia Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York, U.S.A.

Complex Variables and Complex Analysis mathscidoc:2203.08004

Arkiv for Matematik, 58, (2), 307-331, 2020.11
Let ψ be a conformal map on D with ψ(0)=0 and let F_α={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 limits L:=lim_{α→+∞} (log ω_D(0,Fα)^{−1} / logα), μ:=lim_{α→+∞}(d_D(0,Fα) / logα), where ω_D(0,Fα) denotes the harmonic measure at 0 of F_α and d_D(0,Fα) denotes the hyperbolic distance between 0 and F_α 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} logα), we show that h(ψ)=lim inf_{α→+∞} (d_D(0,Fα) / logα). 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).
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@inproceedings{christina2020on,
  title={On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance},
  author={Christina Karafyllia},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220311100226753510945},
  booktitle={Arkiv for Matematik},
  volume={58},
  number={2},
  pages={307-331},
  year={2020},
}
Christina Karafyllia. On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance. 2020. Vol. 58. In Arkiv for Matematik. pp.307-331. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220311100226753510945.
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