• Title, Summary, Keyword: holomorphic boundedness

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EXTENDED CESÀRO OPERATORS BETWEEN α-BLOCH SPACES AND QK SPACES

  • Wang, Shunlai;Zhang, Taizhong
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.567-578
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    • 2017
  • Many scholars studied the boundedness of $Ces{\grave{a}}ro$ operators between $Q_K$ spaces and Bloch spaces of holomorphic functions in the unit disc in the complex plane, however, they did not describe the compactness. Let 0 < ${\alpha}$ < $+{\infty}$, K(r) be right continuous nondecreasing functions on (0, $+{\infty}$) and satisfy $${\displaystyle\smashmargin{2}{\int\nolimits_0}^{\frac{1}{e}}}K({\log}{\frac{1}{r}})rdr<+{\infty}$$. Suppose g is a holomorphic function in the unit disk. In this paper, some sufficient and necessary conditions for the extended $Ces{\grave{a}}ro$ operators $T_g$ between ${\alpha}$-Bloch spaces and $Q_K$ spaces in the unit disc to be bounded and compact are obtained.

WEIGHTED COMPOSITION OPERATORS ON BERS-TYPE SPACES OF LOO-KENG HUA DOMAINS

  • Jiang, Zhi-jie;Li, Zuo-an
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.583-595
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    • 2020
  • Let HEI, HEII, HEIII and HEIV be the first, second, third and fourth type Loo-Keng Hua domain respectively, �� a holomorphic self-map of HEI, HEII, HEIII, or HEIV and u ∈ H(��) the space of all holomorphic functions on �� ∈ {HEI, HEII, HEIII, HEIV}. In this paper, motivated by the well known Hua's matrix inequality, first some inequalities for the points in the Bers-type spaces of the Loo-Keng Hua domains are obtained, and then the boundedness and compactness of the weighted composition operators W��,u : f ↦ u · f ◦ �� on Bers-type spaces of these domains are characterized.

GENERALIZED COMPOSITION OPERATORS FROM GENERALIZED WEIGHTED BERGMAN SPACES TO BLOCH TYPE SPACES

  • Zhu, Xiangling
    • Journal of the Korean Mathematical Society
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    • v.46 no.6
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    • pp.1219-1232
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    • 2009
  • Let H(B) denote the space of all holomorphic functions on the unit ball B of $\mathbb{C}^n$. Let $\varphi$ = (${\varphi}_1,{\ldots}{\varphi}_n$) be a holomorphic self-map of B and $g{\in}2$(B) with g(0) = 0. In this paper we study the boundedness and compactness of the generalized composition operator $C_{\varphi}^gf(z)=\int_{0}^{1}{\mathfrak{R}}f(\varphi(tz))g(tz){\frac{dt}{t}}$ from generalized weighted Bergman spaces into Bloch type spaces.

A BOUNDED KOHN NIRENBERG DOMAIN

  • Calamai, Simone
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.5
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    • pp.1339-1345
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    • 2014
  • Building on the famous domain of Kohn and Nirenberg we give an example of a domain which shares the important features of the Kohn Nirenberg domain, but which can also be shown to be ${\phi}$-bounded As an application, we remark that this example has compact automorphism group.

NEW CHARACTERIZATIONS OF COMPOSITION OPERATORS BETWEEN BLOCH TYPE SPACES IN THE UNIT BALL

  • Fang, Zhong-Shan;Zhou, Ze-Hua
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.3
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    • pp.751-759
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    • 2015
  • In this paper, we give new characterizations of the boundedness and compactness of composition operators $C_{\varphi}$ between Bloch type spaces in the unit ball $\mathbb{B}^n$, in terms of the power of the components of ${\varphi}$, where ${\varphi}$ is a holomorphic self-map of $\mathbb{B}^n$.

COMPOSITION OPERATORS ON THE PRIVALOV SPACES OF THE UNIT BALL OF ℂn

  • UEKI SEI-ICHIRO
    • Journal of the Korean Mathematical Society
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    • v.42 no.1
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    • pp.111-127
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    • 2005
  • Let B and S be the unit ball and the unit sphere in $\mathbb{C}^n$, respectively. Let ${\sigma}$ be the normalized Lebesgue measure on S. Define the Privalov spaces $N^P(B)\;(1\;<\;p\;<\;{\infty})$ by $$N^P(B)\;=\;\{\;f\;{\in}\;H(B) : \sup_{0 where H(B) is the space of all holomorphic functions in B. Let ${\varphi}$ be a holomorphic self-map of B. Let ${\mu}$ denote the pull-back measure ${\sigma}o({\varphi}^{\ast})^{-1}$. In this paper, we prove that the composition operator $C_{\varphi}$ is metrically bounded on $N^P$(B) if and only if ${\mu}(S(\zeta,\delta)){\le}C{\delta}^n$ for some constant C and $C_{\varphi}$ is metrically compact on $N^P(B)$ if and only if ${\mu}(S(\zeta,\delta))=o({\delta}^n)$ as ${\delta}\;{\downarrow}\;0$ uniformly in ${\zeta}\;\in\;S. Our results are an analogous results for Mac Cluer's Carleson-measure criterion for the boundedness or compactness of $C_{\varphi}$ on the Hardy spaces $H^P(B)$.

CESÀRO OPERATORS IN THE BERGMAN SPACES WITH EXPONENTIAL WEIGHT ON THE UNIT BALL

  • Cho, Hong Rae;Park, Inyoung
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.705-714
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    • 2017
  • Let $A^2_{{\alpha},{\beta}}(\mathbb{B}_n)$ denote the space of holomorphic functions that are $L^2$ with respect to a weight of form ${\omega}_{{\alpha},{\beta}}(z)=(1-{\mid}z{\mid}^{\alpha}e^{-{\frac{\beta}{1-{\mid}z{\mid}}}}$, where ${\alpha}{\in}\mathbb{R}$ and ${\beta}$ > 0 on the unit ball $\mathbb{B}_n$. We obtain some results for the boundedness and compactness of $Ces{\grave{a}}ro$ operator on $A^2_{{\alpha},{\beta}(\mathbb{B}_n)$.

ON A POSITIVE SUBHARMONIC BERGMAN FUNCTION

  • Kim, Jung-Ok;Kwon, Ern-Gun
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.623-632
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    • 2010
  • A holomorphic function F defined on the unit disc belongs to $A^{p,{\alpha}}$ (0 < p < $\infty$, 1 < ${\alpha}$ < $\infty$) if $\int\limits_U|F(z)|^p \frac{1}{1-|z|}(1+log)\frac{1}{1-|z|})^{-\alpha}$ dxdy < $\infty$. For boundedness of the composition operator defined by $C_{fg}=g{\circ}f$ mapping Blochs into $A^{p,{\alpha}$ the following (1) is a sufficient condition while (2) is a necessary condition. (1) $\int\limits_o^1\frac{1}{1-r}(1+log\frac{1}{1-r})^{-\alpha}M_p(r,\lambda{\circ}f)^p\;dr$ < $\infty$ (2) $\int\limits_o^1\frac{1}{1-r}(1+log\frac{1}{1-r})^{-\alpha+p}(1-r)^pM_p(r,f^#)^p\;dr$ < $\infty$.