• Title/Summary/Keyword: Bergman metric

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THE BERGMAN METRIC AND RELATED BLOCH SPACES ON THE EXPONENTIALLY WEIGHTED BERGMAN SPACE

  • Byun, Jisoo;Cho, Hong Rae;Lee, Han-Wool
    • East Asian mathematical journal
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    • v.37 no.1
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    • pp.19-32
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    • 2021
  • We estimate the Bergman metric of the exponentially weighted Bergman space and prove many different geometric characterizations for related Bloch spaces. In particular, we prove that the Bergman metric of the exponentially weighted Bergman space is comparable to some Poincaré type metric.

LOWER HOUNDS ON THE HOLOMORPHIC SECTIONAL CURVATURE OF THE BERGMAN METRIC ON LOCALLY CONVEX DOMAINS IN $C^{n}$

  • Cho, Sang-Hyun;Lim, Jong-Chun
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.1
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    • pp.127-134
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    • 2000
  • Let $\Omega$ be a bounded pseudoconvex domain in$C^{n}$ with smooth defining function r and let$z_0\; {\in}\; b{\Omega}$ be a point of finite type. We also assume that $\Omega$ is convex in a neighborhood of $z_0$. Then we prove that all the holomorphic sectional curvatures of the Bergman metric of $\Omega$ are bounded below by a negative constant near $z_0$.

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NOTES ON BERGMAN PROJECTION TYPE OPERATOR RELATED WITH BESOV SPACE

  • CHOI, KI SEONG
    • Journal of the Chungcheong Mathematical Society
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    • v.28 no.3
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    • pp.473-482
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    • 2015
  • Let Qf be the maximal derivative of f with respect to the Bergman metric $b_B$. In this paper, we will find conditions such that $(1-{\parallel}z{\parallel})^s(Qf)^p(z)$ is bounded on B. We will also find conditions such that Bergman projection type operator $P_r$ is bounded operator from $L^p(B,d{\mu}_r)$ to the holomorphic Besov p-space Bs $B^s_p(B)$ with weight s.

THE TOEPLITZ OPERATOR INDUCED BY AN R-LATTICE

  • Kang, Si Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.25 no.3
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    • pp.491-499
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    • 2012
  • The hyperbolic metric is invariant under the action of M$\ddot{o}$bius maps and unbounded. For 0 < $r$ < 1, there is an r-lattice in the Bergman metric. Using this r-lattice, we get the measure ${\mu}_r$ and the Toeplitz operator $T^{\alpha}_{\mu}_r$ and we prove that $T^{\alpha}_{\mu}_r$ is bounded and $T^{\alpha}_{\mu}_r$ is compact under some condition.

LIPSCHITZ TYPE INEQUALITY IN WEIGHTED BLOCH SPACE Bq

  • Park, Ki-Seong
    • Journal of the Korean Mathematical Society
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    • v.39 no.2
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    • pp.277-287
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    • 2002
  • Let B be the open unit ball with center 0 in the complex space $C^n$. For each q>0, B$_{q}$ consists of holomorphic functions f : B longrightarrow C which satisfy sup z $\in$ B $(1-\parallel z \parallel^2)^q\parallel\nabla f(z)\parallel < \infty$ In this paper, we will show that functions in weighted Bloch spaces $B_{q}$ (0 < q < 1) satifies the following Lipschitz type result for Bergman metric $\beta$: |f(z)-f($\omega$)|< $C\beta$(z, $\omega$) for some constant C.

A CHARACTERIZATION OF WEIGHTED BERGMAN-PRIVALOV SPACES ON THE UNIT BALL OF Cn

  • Matsugu, Yasuo;Miyazawa, Jun;Ueki, Sei-Ichiro
    • Journal of the Korean Mathematical Society
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    • v.39 no.5
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    • pp.783-800
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    • 2002
  • Let B denote the unit ball in $C^n$, and ν the normalized Lebesgue measure on B. For $\alpha$ > -1, define $dv_\alpha$(z) = $c_\alpha$$(1-\midz\mid^2)^{\alpha}$dν(z), z $\in$ B. Here $c_\alpha$ is a positive constant such that $v_\alpha$(B) = 1. Let H(B) denote the space of all holomorphic functions in B. For $p\geq1$, define the Bergman-Privalov space $(AN)^{p}(v_\alpha)$ by $(AN)^{p}(v_\alpha)$ = ${f\inH(B)$ : $\int_B{log(1+\midf\mid)}^pdv_\alpha\;<\;\infty}$ In this paper we prove that a function $f\inH(B)$ is in $(AN)^{p}$$(v_\alpha)$ if and only if $(1+\midf\mid)^{-2}{log(1+\midf\mid)}^{p-2}\mid\nablaf\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case 1<p<$\infty$, or $(1+\midf\mid)^{-2}\midf\mid^{-1}\mid{\nabla}f\mid^2\;\epsilon\;L^1(v_\alpha)$ in the case p = 1, where $nabla$f is the gradient of f with respect to the Bergman metric on B. This is an analogous result to the characterization of the Hardy spaces by M. Stoll [18] and that of the Bergman spaces by C. Ouyang-W. Yang-R. Zhao [13].

LIPSCHITZ REGULARITY OF M-HARMONIC FUNCTIONS

  • Youssfi, E.H.
    • Journal of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.959-971
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    • 1997
  • In the paper we introduce Hausdorff measures which are suitable or the study of Lipschitz regularity of M-harmonic function in the unit ball B in $C^n$. For an M-harmonic function h which satisfies certain integrability conditions, we show that there is an open set $\Omega$, whose Hausdorff content is arbitrarily small, such that h is Lipschitz smooth on $B \backslash \Omega$.

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