scholarly journals LIPSCHITZ SPACES AND BOUNDED MEAN OSCILLATION OF HARMONIC MAPPINGS

2013 ◽  
Vol 88 (1) ◽  
pp. 143-157 ◽  
Author(s):  
SH. CHEN ◽  
S. PONNUSAMY ◽  
M. VUORINEN ◽  
X. WANG
Keyword(s):  
Banach Space ◽  
Bloch Space ◽  
Bounded Mean Oscillation ◽  
Harmonic Mappings ◽  
Lipschitz Spaces ◽  
Sharp Estimates ◽  
Mean Oscillation ◽  
Type Spaces ◽  
Equivalent Modulus ◽  
Planar Harmonic Mappings

AbstractWe first study the bounded mean oscillation of planar harmonic mappings. Then we establish a relationship between Lipschitz-type spaces and equivalent modulus of real harmonic mappings. Finally, we obtain sharp estimates on the Lipschitz number of planar harmonic mappings in terms of the bounded mean oscillation norm, which shows that the harmonic Bloch space is isomorphic to$BM{O}_{2} $as a Banach space.

scholarly journals Characterizations of Bloch-Type Spaces of Harmonic Mappings

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Munirah Aljuaid ◽  
Flavia Colonna
Keyword(s):  
Banach Space ◽  
Analytic Functions ◽  
Bloch Space ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Space Of Analytic Functions ◽  
Type Spaces ◽  
Growth Space ◽  
Derivatives Of

We study the Banach space BHα (α>0) of the harmonic mappings h on the open unit disk D satisfying the condition supz∈D⁡(1-z2)α(hzz+hz¯z)<∞, where hz and hz¯ denote the first complex partial derivatives of h. We show that several properties that are valid for the space of analytic functions known as the α-Bloch space extend to BHα. In particular, we prove that for α>0 the mappings in BHα can be characterized in terms of a Lipschitz condition relative to the metric defined by dH,α(z,w)=sup⁡{hz-hw:h∈BHα,hBHα≤1}. When α>1, the harmonic α-Bloch space can be viewed as the harmonic growth space of order α-1, while for 0<α<1, BHα is the space of harmonic mappings that are Lipschitz of order 1-α.

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 Composition operators on some Möbius invariant Banach spaces

2000 ◽  
Vol 62 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Shamil Makhmutov ◽  
Maria Tjani
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Disk ◽  
Composition Operator ◽  
Composition Operators ◽  
Bloch Space ◽  
Holomorphic Functions ◽  
Mean Oscillation

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

scholarly journals On M-ideals and o-O type spaces

2017 ◽  
Vol 121 (1) ◽  
pp. 151 ◽  
Author(s):  
Karl-Mikael Perfekt
Keyword(s):  
Banach Spaces ◽  
Analytic Functions ◽  
Function Spaces ◽  
Weighted Spaces ◽  
Bounded Mean Oscillation ◽  
Mean Oscillation ◽  
Type Spaces ◽  
Invariant Spaces ◽  
Holder Spaces ◽  
Spaces Of Analytic Functions

We consider pairs of Banach spaces $(M_0, M)$ such that $M_0$ is defined in terms of a little-$o$ condition, and $M$ is defined by the corresponding big-$O$ condition. The construction is general and pairs include function spaces of vanishing and bounded mean oscillation, vanishing weighted and weighted spaces of functions or their derivatives, Möbius invariant spaces of analytic functions, Lipschitz-Hölder spaces, etc. It has previously been shown that the bidual $M_0^{**}$ of $M_0$ is isometrically isomorphic with $M$. The main result of this paper is that $M_0$ is an M-ideal in $M$. This has several useful consequences: $M_0$ has Pełczýnskis properties (u) and (V), $M_0$ is proximinal in $M$, and $M_0^*$ is a strongly unique predual of $M$, while $M_0$ itself never is a strongly unique predual.

CHARACTERIZATIONS OF BMO AND LIPSCHITZ SPACES IN TERMS OF WEIGHTS AND THEIR APPLICATIONS

2019 ◽  
Vol 107 (3) ◽  
pp. 381-391
Author(s):  
DINGHUAI WANG ◽  
JIANG ZHOU ◽  
ZHIDONG TENG
Keyword(s):  
Integrable Function ◽  
Lipschitz Space ◽  
Bounded Mean Oscillation ◽  
Lebesgue Spaces ◽  
Lipschitz Spaces ◽  
Mean Oscillation ◽  
Weighted Lebesgue Spaces ◽  
Locally Integrable Function ◽  
Image Position

Let $0<\unicode[STIX]{x1D6FC}<n,1\leq p<q<\infty$ with $1/p-1/q=\unicode[STIX]{x1D6FC}/n$, $\unicode[STIX]{x1D714}\in A_{p,q}$, $\unicode[STIX]{x1D708}\in A_{\infty }$ and let $f$ be a locally integrable function. In this paper, it is proved that $f$ is in bounded mean oscillation $\mathit{BMO}$ space if and only if $$\begin{eqnarray}\sup _{B}\frac{|B|^{\unicode[STIX]{x1D6FC}/n}}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty ,\end{eqnarray}$$ where $\unicode[STIX]{x1D714}^{p}(B)=\int _{B}\unicode[STIX]{x1D714}(x)^{p}\,dx$ and $f_{\unicode[STIX]{x1D708},B}=(1/\unicode[STIX]{x1D708}(B))\int _{B}f(y)\unicode[STIX]{x1D708}(y)\,dy$. We also show that $f$ belongs to Lipschitz space $Lip_{\unicode[STIX]{x1D6FC}}$ if and only if $$\begin{eqnarray}\sup _{B}\frac{1}{\unicode[STIX]{x1D714}^{p}(B)^{1/p}}\bigg(\int _{B}|f(x)-f_{\unicode[STIX]{x1D708},B}|^{q}\unicode[STIX]{x1D714}(x)^{q}\,dx\bigg)^{1/q}<\infty .\end{eqnarray}$$ As applications, we characterize these spaces by the boundedness of commutators of some operators on weighted Lebesgue spaces.

scholarly journals Lengths, areas and Lipschitz-type spaces of planar harmonic mappings

2015 ◽  
Vol 115 ◽  
pp. 62-70 ◽  
Author(s):  
Shaolin Chen ◽  
Saminathan Ponnusamy ◽  
Antti Rasila
Keyword(s):  
Harmonic Mappings ◽  
Type Spaces ◽  
Planar Harmonic Mappings

scholarly journals BMO and the John-Nirenberg Inequality on Measure Spaces

2020 ◽  
Vol 8 (1) ◽  
pp. 335-362
Author(s):  
Galia Dafni ◽  
Ryan Gibara ◽  
Andrew Lavigne
Keyword(s):  
Banach Space ◽  
Measure Space ◽  
Sufficient Conditions ◽  
General Setting ◽  
Finite Measure ◽  
Bounded Mean Oscillation ◽  
Muckenhoupt Weights ◽  
Mean Oscillation ◽  
Nirenberg Inequality ◽  
Measure Spaces

Abstract We study the space BMO𝒢 (𝕏) in the general setting of a measure space 𝕏 with a fixed collection 𝒢 of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in 𝒢. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space 𝕏 for BMO𝒢 (𝕏) to be a Banach space modulo constants. We also develop the notion of a Denjoy family 𝒢, which guarantees that functions in BMO𝒢 (𝕏) satisfy the John-Nirenberg inequality on the elements of 𝒢.

scholarly journals COEFFICIENT ESTIMATES, LANDAU’S THEOREM AND LIPSCHITZ-TYPE SPACES ON PLANAR HARMONIC MAPPINGS

2013 ◽  
Vol 96 (2) ◽  
pp. 198-215 ◽  
Author(s):  
SHAOLIN CHEN ◽  
SAMINATHAN PONNUSAMY ◽  
ANTTI RASILA
Keyword(s):  
Harmonic Mappings ◽  
Coefficient Estimates ◽  
Type Spaces ◽  
Univalent Harmonic Mappings ◽  
Planar Harmonic Mappings

AbstractIn this paper, we investigate the properties of locally univalent and multivalent planar harmonic mappings. First, we discuss coefficient estimates and Landau’s theorem for some classes of locally univalent harmonic mappings, and then we study some Lipschitz-type spaces for locally univalent and multivalent harmonic mappings.

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