Distortion of locally biholomorphic Bloch mappings on bounded symmetric domains

Chu, Cho-Ho; Hamada, Hidetaka; Honda, Tatsuhiro; Kohr, Gabriela

doi:10.1016/j.jmaa.2016.04.035

Cited by 16 publications

(3 citation statements)

References 25 publications

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“…We recall a constant

c ({double-struckB}_{X})

defined in [11]. Let h 0 be the Bergman metric on

B_{X}

at 0 and let

c ({double-struckB}_{X}) = \frac{1}{2} \underset{x, y \in {double-struckB}_{X}}{trueprefixsup} | h_{0} false(x, y false) | .

It is proved in [5] that

\frac{dim X + r}{2} \leq c ({double-struckB}_{X}) \leq dim X,

where r is the rank of X . Theorem Let G be a balanced domain in a complex Banach space E .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A rigidity theorem at the boundary for holomorphic mappings with values in finite dimensional bounded symmetric domains

Hamada

Kohr

2021

Mathematische Nachrichten

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Let 𝔹 𝑋 be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB * -triple 𝑋. In this paper, we obtain a rigidity theorem at the boundary for holomorphic mappings from a balanced domain 𝐺 in a complex Banach space 𝐸 into 𝔹 𝑋 . We also obtain a rigidity theorem at the boundary for holomorphic self-mappings of 𝔹 𝑋 . Our results give generalizations of the recent results obtained on the Euclidean unit ball or the unit polydisc in ℂ 𝑛 .

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“…We recall a constant

c ({double-struckB}_{X})

defined in [11]. Let h 0 be the Bergman metric on

B_{X}

at 0 and let

c ({double-struckB}_{X}) = \frac{1}{2} \underset{x, y \in {double-struckB}_{X}}{trueprefixsup} | h_{0} false(x, y false) | .

It is proved in [5] that

\frac{dim X + r}{2} \leq c ({double-struckB}_{X}) \leq dim X,

where r is the rank of X . Theorem Let G be a balanced domain in a complex Banach space E .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

A rigidity theorem at the boundary for holomorphic mappings with values in finite dimensional bounded symmetric domains

Hamada

Kohr

2021

Mathematische Nachrichten

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“…Note that the author [10] (cf. [5], [13]) gave distortion theorems for other Bloch type mappings on bounded symmetric domains in C n . As in [10], [13], as a corollary of the distortion theorem, we obtain the lower estimate for the radius of the largest schlicht ball in the image of f centered at f (0) for α-Bloch mappings f on B X (Theorem 3.2).…”

Section: Introductionmentioning

confidence: 99%

Distortion theorems, Lipschitz continuity and their applications for Bloch type mappings on bounded symmetric domains in C^n

Hamada¹

2019

Ann. Acad. Sci. Fenn. Math.

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Let B X be a bounded symmetric domain realized as the unit ball of an ndimensional JB *-triple X = (C n , • X). In this paper, we give a new definition of Bloch type mappings on B X and give distortion theorems for Bloch type mappings on B X. When B X is the Euclidean unit ball in C n , this new definition coincides with that given by Chen and Kalaj or by the author. As a corollary of the distortion theorem, we obtain the lower estimate for the radius of the largest schlicht ball in the image of f centered at f (0) for α-Bloch mappings f on B X. Next, as another corollary of the distortion theorem, we show the Lipschitz continuity of (det B(z, z)) 1/2n | det Df (z)| 1/n for Bloch type mappings f on B X with respect to the Kobayashi metric, where B(z, z) is the Bergman operator on X, and use it to give a sufficient condition for the composition operator C ϕ to be bounded from below on the Bloch type space on B X , where ϕ is a holomorphic self mapping of B X. In the case B X = B n , we also give a necessary condition for C ϕ to be bounded from below which is a converse to the above result. Finally, as another application of the Lipschitz continuity, we obtain a result related to the interpolating sequences for the Bloch type space on B X .

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“…Timoney gave several equivalent definitions for Bloch functions on a finite dimensional bounded homogeneous domain. Chu, Hamada, Honda and Kohr generalized the Bloch space to a bounded symmetric domain realized as the open unit ball of a JB*‐triple X . We remark that all four types of classical Cartan domains are the open unit balls of JB*‐triples, and the same holds for any finite product of these Cartan domains (, see also ).…”

Section: Introductionmentioning

confidence: 99%

Operators from the Hardy space to the α‐Bloch space

Honda

2019

Mathematische Nachrichten

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Let BX be a bounded symmetric domain realized as the open unit ball of a finite dimensional JB*‐triple X. In this paper, we characterize the bounded weighted composition operators from the Hardy space H∞false(BXfalse) into the α‐Bloch space scriptBαfalse(BXfalse) on BX. Also, we show the multiplication operator from H∞false(BXfalse) into scriptBαfalse(BXfalse) is bounded. Finally, we show that there exist no isometric composition operators.

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Distortion of locally biholomorphic Bloch mappings on bounded symmetric domains

Abstract: We generalize Bonk's distortion theorem on the unit disc in the complex plane to locally biholomorphic mappings on finite dimensional bounded symmetric domains. As an application, we obtain a lower bound for the Bloch constant for various classes of locally biholomorphic Bloch mappings.

Cited by 16 publications

References 25 publications

A rigidity theorem at the boundary for holomorphic mappings with values in finite dimensional bounded symmetric domains

A rigidity theorem at the boundary for holomorphic mappings with values in finite dimensional bounded symmetric domains

Distortion theorems, Lipschitz continuity and their applications for Bloch type mappings on bounded symmetric domains in C^n

Operators from the Hardy space to the α‐Bloch space

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