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A CHARACTERIZATION OF WEIGHTED BERGMAN-PRIVALOV SPACES ON THE UNIT BALL OF C<sup>n</sup>
Matsugu, Yasuo,Miyazawa, Jun,Ueki, Sei-Ichiro Korean Mathematical Society 2002 대한수학회지 Vol.39 No.5
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].
THE BERGMAN METRIC AND RELATED BLOCH SPACES ON THE EXPONENTIALLY WEIGHTED BERGMAN SPACE
Byun, Jisoo,Cho, Hong Rae,Lee, Han-Wool The Youngnam Mathematical Society 2021 East Asian mathematical journal Vol.37 No.1
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.
THE RADIAL DERIVATIVES ON WEIGHTED BERGMAN SPACES
Kang, Si-Ho,Kim, Ja-Young Korean Mathematical Society 2003 대한수학회논문집 Vol.18 No.2
We consider weighted Bergman spaces and radial derivatives on the spaces. We also prove that for each element f in B$\^$p, r/ there is a unique f in B$\^$p, r/ such that f is the radial derivative of f and for each f$\in$B$\^$r/(i), f is the radial derivative of some element of B$\^$r/(i) if and only if, lim f(tz)= 0 for all z$\in$H.
WEIGHTED HARMONIC BERGMAN FUNCTIONS ON HALF-SPACES
Koo, HYUNGWOON,NAM, KYESOOK,YI, HEUNGSU Korean Mathematical Society 2005 대한수학회지 Vol.42 No.5
On the setting of the upper half-space H of the Euclidean n-space, we show the boundedness of weighted Bergman projection for 1 < p < $\infty$ and nonorthogonal projections for 1 $\leq$ p < $\infty$ . Using these results, we show that Bergman norm is equiva lent to the normal derivative norms on weighted harmonic Bergman spaces. Finally, we find the dual of b$\_{$^{1}$.
PRODUCT-TYPE OPERATORS FROM WEIGHTED BERGMAN-ORLICZ SPACES TO WEIGHTED ZYGMUND SPACES
Zhi-Jie Jiang 대한수학회 2015 대한수학회보 Vol.52 No.4
Let D = {z ∈ C : |z| < 1} be the open unit disk in the complex plane C, an analytic self-map of D and ψ an analytic function in D. Let D be the differentiation operator and W ,ψ the weighted composition operator. The boundedness and compactness of the product-type operator W ,ψ D from the weighted Bergman-Orlicz space to the weighted Zygmund space on D are characterized.
PRODUCT-TYPE OPERATORS FROM WEIGHTED BERGMAN-ORLICZ SPACES TO WEIGHTED ZYGMUND SPACES
JIANG, ZHI-JIE Korean Mathematical Society 2015 대한수학회보 Vol.52 No.4
Let ${\mathbb{D}}=\{z{\in}{\mathbb{C}}:{\mid}z{\mid}<1\}$ be the open unit disk in the complex plane $\mathbb{C}$, ${\varphi}$ an analytic self-map of $\mathbb{D}$ and ${\psi}$ an analytic function in $\mathbb{D}$. Let D be the differentiation operator and $W_{{\varphi},{\psi}}$ the weighted composition operator. The boundedness and compactness of the product-type operator $W_{{\varphi},{\psi}}D$ from the weighted Bergman-Orlicz space to the weighted Zygmund space on $\mathbb{D}$ are characterized.
WEIGHTED COMPOSITION OPERATORS FROM BERGMAN SPACES INTO WEIGHTED BLOCH SPACES
LI SONGXIAO Korean Mathematical Society 2005 대한수학회논문집 Vol.20 No.1
In this paper we study bounded and compact weighted composition operator, induced by a fixed analytic function and an analytic self-map of the open unit disk, from Bergman space into weighted Bloch space. As a corollary, obtain the characterization of composition operator from Bergman space into weighted Bloch space.
Generalized composition operators from generalized weighted Bergman spaces to Bloch type spaces
Xiangling Zhu 대한수학회 2009 대한수학회지 Vol.46 No.6
Let H(B) denote the space of all holomorphic functions on the unit ball B of C^n. Let φ=(φ_1,…,φ_n) be a holomorphic self-map of B and g ∈ H(B) with g(0)=0. In this paper we study the boundedness and compactness of the generalized composition operator <수식> from generalized weighted Bergman spaces into Bloch type spaces. Let H(B) denote the space of all holomorphic functions on the unit ball B of C^n. Let φ=(φ_1,…,φ_n) be a holomorphic self-map of B and g ∈ H(B) with g(0)=0. In this paper we study the boundedness and compactness of the generalized composition operator <수식> from generalized weighted Bergman spaces into Bloch type spaces.
Yunus E. Zeytuncu 대한수학회 2015 대한수학회보 Vol.52 No.3
Two well known facts from elementary number theory are proven by using Bergman spaces.
NOTES ON THE SPACE OF DIRICHLET TYPE AND WEIGHTED BESOV SPACE
Choi, Ki Seong Chungcheong Mathematical Society 2013 충청수학회지 Vol.26 No.2
For 0 < $p$ < ${\infty}$, ${\alpha}$ > -1 and 0 < $r$ < 1, we show that if $f$ is in the space of Dirichlet type $\mathfrak{D}^p_{p-1}$, then ${\int}_{1}^{0}M_{p}^{p}(r,f^{\prime})(1-r)^{p-1}rdr$ < ${\infty}$ and ${\int}_{1}^{0}M_{(2+{\alpha})p}^{(2+{\alpha})p}(r,f^{\prime})(1-r)^{(2+{\alpha})p+{\alpha}}rdr$ < ${\infty}$ where $M_p(r,f)=\[\frac{1}{2{\pi}}{\int}_{0}^{2{\pi}}{\mid}f(re^{it}){\mid}^pdt\]^{1/p}$. For 1 < $p$ < $q$ < ${\infty}$ and ${\alpha}+1$ < $p$, we show that if there exists some positive constant $c$ such that ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathfrak{D}^p_{\alpha}}$ for all $f{\in}\mathfrak{D}^p_{\alpha}$, then ${\parallel}f{\parallel}_{L^{q(d{\mu})}}{\leq}c{\parallel}f{\parallel}_{\mathcal{B}_p(q)}$ where $\mathcal{B}_p(q)$ is the weighted Besov space. We also find the condition of measure ${\mu}$ such that ${\sup}_{a{\in}D}{\int}_D(k_a(z)(1-{\mid}a{\mid}^2)^{(p-a-1)})^{q/p}d{\mu}(z)$ < ${\infty}$.