scholarly journals Representations and interpolations of harmonic Bergman functions on half-spaces

1998 ◽  
Vol 151 ◽  
pp. 51-89 ◽  
Author(s):  
Boo Rim Choe ◽  
Heungsu Yi
Keyword(s):  
Uniqueness Theorem ◽  
Half Space ◽  
Bloch Space ◽  
Distance Estimate ◽  
Bergman Metric ◽  
Representation Theorems ◽  
Bloch Spaces ◽  
Little Bloch Space ◽  
Metric Balls

Abstract.On the setting of the half-space of the euclidean n-space, we prove representation theorems and interpolation theorems for harmonic Bergman functions in a constructive way. We also consider the harmonic (little) Bloch spaces as limiting spaces. Our results show that well-known phenomena for holomorphic cases continue to hold. Our proofs of representation theorems also yield a uniqueness theorem for harmonic Bergman functions. As an application of interpolation theorems, we give a distance estimate to the harmonic little Bloch space. In the course of the proofs, pseudohyperbolic balls are used as substitutes for Bergman metric balls in the holomorphic case.

scholarly journals Linear Combinations of Composition Operators on the Bloch Spaces

2011 ◽  
Vol 63 (4) ◽  
pp. 862-877 ◽  
Author(s):  
Takuya Hosokawa ◽  
Pekka J. Nieminen ◽  
Shûichi Ohno
Keyword(s):  
Composition Operators ◽  
Bloch Space ◽  
Linear Combinations ◽  
Bloch Spaces ◽  
Little Bloch Space

Abstract We characterize the compactness of linear combinations of analytic composition operators on the Bloch space. We also study their boundedness and compactness on the little Bloch space.

scholarly journals Norm and Essential Norm of Composition Followed by Differentiation from Logarithmic Bloch Spaces toHμ∞

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Shanli Ye
Keyword(s):  
Lower Bound ◽  
Unit Disk ◽  
Weighted Space ◽  
Bloch Space ◽  
Essential Norm ◽  
Bloch Spaces

In this note we express the norm of composition followed by differentiationDCφfrom the logarithmic Bloch and the little logarithmic Bloch spaces to the weighted spaceHμ∞on the unit disk and give an upper and a lower bound for the essential norm of this operator from the logarithmic Bloch space toHμ∞.

On Analytic Functions of Bergman BMO in the Ball

1999 ◽  
Vol 42 (1) ◽  
pp. 97-103 ◽  
Author(s):  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Composition Operator ◽  
Analytic Functions ◽  
Bloch Space ◽  
Bounded Mean Oscillation ◽  
Open Unit ◽  
Bergman Metric ◽  
Open Unit Ball ◽  
Mean Oscillation ◽  
Volume Measure

AbstractLet B = Bn be the open unit ball of Cn with volume measure v, U = B1 and B be the Bloch space on , 1 ≤ α < 1, is defined as the set of holomorphic f : B → C for whichif 0 < α < 1 and , the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic f : B → U for which the composition operator defined by , is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric.

scholarly journals The n-th derivative characterisation of Möbius invariant Dirichlet space

1998 ◽  
Vol 58 (1) ◽  
pp. 43-56 ◽  
Author(s):  
Rauno Aulaskari ◽  
Maria Nowak ◽  
Ruhan Zhao
Keyword(s):  
Carleson Measure ◽  
Bloch Space ◽  
Dirichlet Space ◽  
Function Spaces ◽  
Multiplier Space ◽  
Little Bloch Space ◽  
Special Parameter

In this paper we give the n-th derivative criterion for functions belonging to recently defined function spaces Qp and Qp, 0. For a special parameter value p = 1 this criterion is applied to BMOA and VMOA, and for p > 1 it is applied to the Bloch space and the little Bloch space . Further, a Carleson measure characterisation is given to Qp, and in the last section the multiplier space from Hq into Qp is considered.

COMPOSITION OPERATORS ON THE LITTLE BLOCH SPACE IN POLYDISCS

2005 ◽  
Vol 25 (4) ◽  
pp. 629-638
Author(s):  
Zehua Zhou ◽  
Min Zhu ◽  
Jihuai Shi
Keyword(s):  
Composition Operators ◽  
Bloch Space ◽  
Little Bloch Space

scholarly journals Nevanlinna-type characterizations for the Bloch space and related spaces

1990 ◽  
Vol 33 (1) ◽  
pp. 123-141 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Bloch Space ◽  
Bloch Function ◽  
Value Distribution ◽  
Nevanlinna Characteristic ◽  
Little Bloch Space

We give a characterisation of the Bloch space in terms of an area version of the Nevanlinna characteristic, analogous to Baernstein's description of the space BMOA in terms of the usual Nevanlinna characteristic. We prove analogous results for the little Bloch space and the space VMOA, and give value distribution characterizations for all these spaces. Finally we give valence conditions on a Bloch or little Bloch function for containment in BMOA or VMOA.

scholarly journals The Bloch space and Besov spaces of analytic functions

1996 ◽  
Vol 54 (2) ◽  
pp. 211-219 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Besov Spaces ◽  
Analytic Functions ◽  
Elementary Proof ◽  
Bloch Space ◽  
Little Bloch Space ◽  
Spaces Of Analytic Functions

We shall give an elementary proof of a characterisation for the Bloch space due to Holland and Walsh, and obtain analogous characterisations for the little Bloch space and Besov spaces of analytic functions on the unit disk in the complex plane.

scholarly journals Logarithmic Bloch space and its predual

2016 ◽  
Vol 100 (114) ◽  
pp. 1-16 ◽  
Author(s):  
Miroslav Pavlovic
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Bloch Space ◽  
Positive Coefficients ◽  
Little Bloch Space

We consider the space B1log?, of analytic functions on the unit disk D, defined by the requirement ?D|f?(z)|?(|z|) dA(z) < ?, where ?(r) = log?(1/(1?r)) and show that it is a predual of the ?log?-Bloch? space and the dual of the corresponding little Bloch space. We prove that a function f(z)=??n=0 an zn with an ? 0 is in B1 log? iff ??n=0 log?(n+2)/(n+1) < ? and apply this to obtain a criterion for membership of the Libera transform of a function with positive coefficients in B1 log?. Some properties of the Cesaro and the Libera operator are considered as well.

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