• Title, Summary, Keyword: Bergman metric

Search Result 15, Processing Time 0.036 seconds

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
    • /
    • v.37 no.1
    • /
    • pp.127-134
    • /
    • 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 $z_0$.

  • PDF

NOTES ON BERGMAN PROJECTION TYPE OPERATOR RELATED WITH BESOV SPACE

  • CHOI, KI SEONG
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.28 no.3
    • /
    • pp.473-482
    • /
    • 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.

LIPSCHITZ TYPE INEQUALITY IN WEIGHTED BLOCH SPACE Bq

  • Park, Ki-Seong
    • Journal of the Korean Mathematical Society
    • /
    • v.39 no.2
    • /
    • pp.277-287
    • /
    • 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
    • /
    • v.39 no.5
    • /
    • pp.783-800
    • /
    • 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].

MULTIPLIERS OF WEIGHTED BLOCH SPACES AND BESOV SPACES

  • Yang, Gye Tak;Choi, Ki Seong
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.22 no.4
    • /
    • pp.727-737
    • /
    • 2009
  • Let M(X) be the space of all pointwise multipliers of Banach space X. We will show that, for each $\alpha>1$, $M(\mathfrak{B}_\alpha)=M(\mathfrak{B}_{\alpha,0})=H^\infty{(B)}$. We will also show that, for each $0<{\alpha}<1$, $M(\mathfrak{B}_\alpha)$ and $M(\mathfrak{B}_{\alpha,0})$ are Banach algebras. It is established that certain inclusion relationships exist between the weighted Bloch spaces and holomorphic Besov spaces.

  • PDF

BOUNDED LINEAR FUNCTIONAL ON L1a(B) RELATED WITH $\mathcal{B}_q$q

  • Choi, Ki Seong
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.14 no.2
    • /
    • pp.37-46
    • /
    • 2001
  • In this paper, weighted Bloch spaces $\mathcal{B}_q$ are considered on the open unit ball in $\mathbb{C}^n$. In this paper, we will show that every Bloch function in $B_q$ induces a bounded linear functional on $L^1_a(\mathcal{B})$.

  • PDF

HIGHER ORDER ASYMPTOTIC BEHAVIOR OF CERTAIN KÄHLER METRICS AND UNIFORMIZATION FOR STRONGLY PSEUDOCONVEX DOMAINS

  • Joo, Jae-Cheon;Seo, Aeryeong
    • Journal of the Korean Mathematical Society
    • /
    • v.52 no.1
    • /
    • pp.113-124
    • /
    • 2015
  • We provide some relations between CR invariants of boundaries of strongly pseudoconvex domains and higher order asymptotic behavior of certain complete K$\ddot{a}$hler metrics of given domains. As a consequence, we prove a rigidity theorem of strongly pseudoconvex domains by asymptotic curvature behavior of metrics.

ON SOME MEASURE RELATED WITH POISSON INTEGRAL ON THE UNIT BALL

  • Yang, Gye Tak;Choi, Ki Seong
    • Journal of the Chungcheong Mathematical Society
    • /
    • v.22 no.1
    • /
    • pp.89-99
    • /
    • 2009
  • Let $\mu$ be a finite positive Borel measure on the unit ball $B{\subset}\mathbb{C}^n$ and $\nu$ be the Euclidean volume measure such that ${\nu}(B)=1$. For the unit sphere $S=\{z:{\mid}z{\mid}=1\}$, $\sigma$ is the rotation-invariant measure on S such that ${\sigma}(S)=1$. Let $\mathcal{P}[f]$ be the invariant Poisson integral of f. We will show that there is a constant M > 0 such that $\int_B{\mid}{\mathcal{P}}[f](z){\mid}^{p}d{\mu}(z){\leq}M\;{\int}_B{\mid}{\mathcal{P}}[f](z)^pd{\nu}(z)$ for all $f{\in}L^p({\sigma})$ if and only if ${\parallel}{\mu}{\parallel_r}\;=\;sup_{z{\in}B}\;\frac{\mu(E(z,r))}{\nu(E(z,r))}\;<\;\infty$.

  • PDF