On the growth and range of functions in Möbius invariant spaces

Donaire, Juan Jesús; Girela, Daniel; Vukotić, Dragan

doi:10.1007/s11854-010-0029-9

Cited by 9 publications

(7 citation statements)

References 27 publications

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“…We remark that Donaire, Girela, and Vukotić [17] have proved that K ∩ H ∞ ⊂ B 1 . Using this result, recalling that B 1 ⊂ B p ⊂ VMOA, and using also Twomey's result (2.9), the inclusion K ⊂ S , (2.10), (2.2), (1.2), (1.3), and Theorem 3, we obtain the following.…”

Section: Theorem 3 Let Be An Unbounded Convex Domain Inmentioning

confidence: 80%

“…We recall here that (2.8) shows that this is not true with S in the place of C. Now Donaire, Girela, and Vukotić have proved in [17] that there exists f ∈ S ∩H ∞ which does not belong to B p , if p < 2. Putting these results together we obtain our result.…”

Section: Theorem 2 There Exists a Bounded Univalent Functionmentioning

confidence: 95%

“…The univalent functions in Besov spaces have been extensively studied in [14,16,17], and [47]. It seems natural to look for possible relations between the space B log and Besov spaces.…”

Section: Logarithmic Bloch Spaces and Besov Spacesmentioning

confidence: 99%

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Univalent Functions, VMOA and Related Spaces

2010

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This paper is concerned mainly with the logarithmic Bloch space B log which consists of those functions f which are analytic in the unit disc D and satisfy sup |z|<1 (1−|z|) log 1 1−|z| |f (z)| < ∞, and the analytic Besov spaces B p , 1 ≤ p < ∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of:• A bounded univalent function in p>1 B p but not in the logarithmic Bloch space.• A bounded univalent function in B log but not in any of the Besov spaces B p with p < 2.We also prove that the situation changes for certain subclasses of univalent functions. Namely, we prove that the convex univalent functions in D which belong to any of the spaces B 0 , VMOA, B p (1 ≤ p < ∞), B log , or some other related spaces are the same, the bounded ones. Communicated by Michael Lacey. 666 P. Galanopoulos et al.We also consider the question of when the logarithm of the derivative, log g , of a univalent function g belongs to Besov spaces. We prove that no condition on the growth of the Schwarzian derivative Sg of g guarantees log g ∈ B p . On the other hand, we prove that the condition D (1 − |z| 2 ) 2p−2 |Sg(z)| p dA(z) < ∞ implies that log g ∈ B p and that this condition is sharp. We also study the question of finding geometric conditions on the image domain g(D) which imply that log g lies in B p . First, we observe that the condition of g(D) being a convex Jordan domain does not imply this. On the other hand, we extend results of Pommerenke and Warschawski, obtaining for every p ∈ (1, ∞), a sharp condition on the smoothness of a Jordan curve which implies that if g is a conformal mapping from D onto the inner domain of , then log g ∈ B p .

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Section: Theorem 3 Let Be An Unbounded Convex Domain Inmentioning

confidence: 80%

Section: Theorem 2 There Exists a Bounded Univalent Functionmentioning

confidence: 95%

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Univalent Functions, VMOA and Related Spaces

2010

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“…An important and well-studied case is the classical Dirichlet space B , usually denoted by D, of analytic functions whose image has a nite area, counting multiplicities. We mention [3,6,7,21,22,29,30] as references to nd a lot of information on Besov spaces.…”

Section: Introductionmentioning

confidence: 99%

Radial growth of the derivatives of analytic functions in Besov spaces

Domínguez

Girela

2020

Concrete Operators

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For 1 < p < ∞, the Besov space Bp consists of those functions f which are analytic in the unit disc 𝔻 = {z ∈ 𝔺 : |z| < 1} and satisfy ∫𝔻(1 − |z|2)p−2|f ′(z)|p dA(z) < ∞. The space B2 reduces to the classical Dirichlet space 𝒟. It is known that if f ∈ 𝒟then |f ′(reiθ)| = o[(1 − r)−1/2], for almost every ∈ [0, 2π]. Hallenbeck and Samotij proved that this result is sharp in a very strong sense. We obtain substitutes of the above results valid for the spaces Bp (1 < p < ∞) an we give also an application of our them to questions concerning multipliers between Besov spaces.

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“…The articles [8] and [44] are fundamental references for the theory of Möbius invariant spaces, spaces which have attracted attracted much attention in recent years (see, e.˙g., [3,16,17,30,47,57,58]).…”

Section: Introductionmentioning

confidence: 99%

Multipliers and integration operators between conformally invariant spaces

Girela¹,

Merchán²

2019

Preprint

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In this paper we are concerned with two classes of conformally invariant spaces of analytic functions in the unit disc D, the Besov spaces B p (1 ≤ p < ∞) and the Qs spaces (0 < s < ∞). Our main objective is to characterize for a given pair (X, Y ) of spaces in these classes, the space of pointwise multipliers M (X, Y ), as well as to study the related questions of obtaining characterizations of those g analytic in D such that the Volterra operator Tg or the companion operator Ig with symbol g is a bounded operator from X into Y .

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On the growth and range of functions in Möbius invariant spaces

Abstract: This paper is a continuation of our earlier work and focuses on the structural and geometric properties of functions in analytic Besov spaces, primarily on univalent functions in such spaces and their image domains. We improve several earlier results.

Cited by 9 publications

References 27 publications

Univalent Functions, VMOA and Related Spaces

Univalent Functions, VMOA and Related Spaces

Radial growth of the derivatives of analytic functions in Besov spaces

Multipliers and integration operators between conformally invariant spaces

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