Hardy spaces and unbounded quasidisks

Kim, Yong Chan; Sugawa, Toshiyuki

doi:10.5186/aasfm.2011.3618

Cited by 12 publications

(13 citation statements)

References 15 publications

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“…By Lemma 3.2 in [15] and Theorem 1.1 in [14], an immediate consequence of Theorem 1.3 is that if f is a conformal mapping on D, then…”

Section: Introductionmentioning

confidence: 99%

Geometric characterizations for conformal mappings in weighted Bergman spaces

Karafyllia¹,

Karamanlis²

2021

Preprint

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We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of conformal mappings in Hardy or weighted Bergman spaces by studying Euclidean areas. Applying these results, we prove several consequences for such mappings that extend known results for Hardy spaces to weighted Bergman spaces. Moreover, we introduce a number which is the analogue of the Hardy number for weighted Bergman spaces. We derive various expressions for this number and hence we establish new results for the Hardy number and the relation between Hardy and weighted Bergman spaces.

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“…By Lemma 3.2 in [15] and Theorem 1.1 in [14], an immediate consequence of Theorem 1.3 is that if f is a conformal mapping on D, then…”

Section: Introductionmentioning

confidence: 99%

Geometric characterizations for conformal mappings in weighted Bergman spaces

Karafyllia¹,

Karamanlis²

2021

Preprint

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“…Poggi-Corradini [20] studied the range domains D of univalent Koenigs functions (see also [21]) and found that the number h (D) can be described in terms of the essential norm of the associated composition operators. Finally, based on Essén' s main lemma [7], Kim and Sugawa [15] proved that…”

Section: Introductionmentioning

confidence: 99%

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

Karafyllia

2020

Arkiv För Matematik

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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).

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“…A classical problem in geometric function theory is to find geometric conditions for a holomorphic function on the unit disk to belong in Hardy spaces (see e.g. [1], [11], [12], [14], [17], [19] and [21]). In this paper we study this problem in the case of conformal mappings from the unit disk onto a comb domain.…”

Section: Introductionmentioning

confidence: 99%

“…He also proved [19] for a certain class of functions, which give a geometric model for the self-mappings of D, that the Hardy number is equal to infinity if and only if the image region does not contain a twisted sector. Furthermore, in [9] and [17] Essén, and Kim and Sugawa, respectively, gave a description of the Hardy number of a plane domain in terms of harmonic measure. In [15] the current author gave a formula for the Hardy number of a simply connected domain in terms of hyperbolic distance.…”

Section: Introductionmentioning

confidence: 99%

On the Hardy number of comb domains

Karafyllia¹

2021

Preprint

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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.

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Hardy spaces and unbounded quasidisks

Abstract: Abstract. We study the maximal number 0 ≤ h ≤ +∞ for a given plane domain Ω such that f ∈ H p whenever p < h and f is analytic in the unit disk with values in Ω. One of our main contributions is an estimate of h for unbounded K-quasidisks.

Cited by 12 publications

References 15 publications

Geometric characterizations for conformal mappings in weighted Bergman spaces

Geometric characterizations for conformal mappings in weighted Bergman spaces

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

On the Hardy number of comb domains

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