• 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 $L^1_a(\mathcal{B})$.

• Korean Journal of Mathematics
• /
• v.30 no.3
• /
• pp.467-474
• /
• 2022

In this article, we consider the integral operator 𝓒^b,c, which is defined as follows: $${\mathcal{C}}^{b,c}(f)(z)={\displaystyle\smashmargin{2}{\int\nolimits_{0}}^z}{\frac{f(w)*F(1,1;c;w)}{w(1-w)^{b+1-c}}}dw,$$ where * denotes the Hadamard/ convolution product of power series, F(a, b; c; z) is the classical hypergeometric function with b, c > 0, b + 1 > c and f(0) = 0. We investigate the boundedness of the 𝓒^b,c operators on Bloch spaces.

• Korean Journal of Mathematics
• /
• v.27 no.3
• /
• pp.581-594
• /
• 2019

In this study a new generalization for operators of two parameters type of fractional in the unit disk is proposed. The fractional operators in this generalization are in the Srivastava-Owa sense. Concerning with the related applications, the generalized Gauss hypergeometric function is introduced. Further, some boundedness properties on Bloch space are also discussed.

• Korean Journal of Mathematics
• /
• v.26 no.1
• /
• pp.87-101
• /
• 2018

The purpose of this paper is to define a new class of hyperholomorphic functions spaces, which will be called $F^{\alpha}_{{\omega},G}$(p, q, s) type spaces. For this class, we characterize hyperholomorphic weighted ${\alpha}$-Bloch functions by functions belonging to $F^{\alpha}_{{\omega},G}$(p, q, s) spaces under some mild conditions. Moreover, we give some essential properties for the extended weighted little ${\alpha}$-Bloch spaces. Also, we give the characterization for the hyperholomorphic weighted Bloch space by the integral norms of $F^{\alpha}_{{\omega},G}$(p, q, s) spaces of hyperholomorphic functions. Finally, we will give the relation between the hyperholomorphic ${\mathcal{B}}^{\alpha}_{{\omega},0}$ type spaces and the hyperholomorphic valued-functions space $F^{\alpha}_{{\omega},G}$(p, q, s).

• Journal of the Chungcheong Mathematical Society
• /
• v.23 no.3
• /
• pp.523-534
• /
• 2010

In this paper, we consider the weighted Bloch spaces ${\mathcal{B}}_q$(q > 0) on the open unit ball in ${\mathbb{C}}^n$. We prove a certain integral representation theorem that is used to determine the degree of growth of the functions in the space ${\mathcal{B}}_q$ for q > 0. This means that for each q > 0, the Banach dual of $L_a^1$ is ${\mathcal{B}}_q$ and the Banach dual of ${\mathcal{B}}_{q,0}$ is $L_a^1$ for each $q{\geq}1$.

• Bulletin of the Korean Mathematical Society
• /
• v.61 no.4
• /
• pp.959-968
• /
• 2024

Suppose f belongs to the Bloch space with f(0) = 0. For 0 < r < 1 and 0 < p < ∞, we show that $$M_p(r,\,f)\,=\,({\frac{1}{2\pi}}{\int_{0}^{2\pi}}\,{\mid}f(re^{it}){\mid}^pdt)^{1/p}\,{\leq}\,({\frac{{\Gamma}(\frac{p}{2}+1)}{{\Gamma}(\frac{p}{2}+1-k)}})^{1/p}\,{\rho}{\mathcal{B}}(log\frac{1}{1-r^2})^{1/2},$$ where ρʙ(f) = sup_z∈ⅅ(1 - |z|²)|f'(z)| and k is the integer satisfying 0 < p - 2k ≤ 2. Moreover, we prove that for 0 < r < 1 and p > 1, $${\parallel}f_r{\parallel}_{B_q}\,{\leq}\,r\,{\rho}{\mathcal{B}}(f)(\frac{1}{(1-r^2)(q-1)})^{1/q},$$ where f_r(z) = f(rz) and ||·||ʙ_q is the Besov seminorm given by ║f║ʙ_q = (∫_𝔻 |f'(z)|^q(1-|z|²)^q-2dA(z)). These results improve previous results of Clunie and MacGregor.

• Journal of the Chungcheong Mathematical Society
• /
• v.13 no.1
• /
• pp.131-138
• /
• 2000

Let D be the open unit disk in the complex plane $\mathbb{C}$. For any q > 0, the operator $Q_q$ defined by $$Q_qf(z)=q\int_{D}\frac{f(\omega)}{(1-z{\bar{\omega}})^{1+q}}d{\omega},\;z{\in}D$$. maps $L^{\infty}(D)$ boundedly onto $B_q$ for each q > 0. In this paper, weighted Bloch spaces $\mathcal{B}_q$ (q > 0) are considered on the open unit ball in $\mathbb{C}^n$. In particular, we will investigate the possibility of extension of this operator to the Weighted Bloch spaces $\mathcal{B}_q$ in $\mathbb{C}^n$.

• Kyungpook Mathematical Journal
• /
• v.63 no.3
• /
• pp.413-424
• /
• 2023

The main purpose of this paper is to show that over a large class of bounded domains Ω ⊂ ℂ², for 1 < p < ∞, the Bergman projection 𝓟 is bounded from L^p(Ω, dV ) to the Bergman space A^p(Ω); from L^∞(Ω) to the holomorphic Bloch space BlHol(Ω); and from L¹(Ω, P(z, z)dV) to the holomorphic Besov space Besov(Ω), where P(ζ, z) is the Bergman kernel for Ω.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.1
• /
• pp.85-103
• /
• 2015

We prove some sharp extremal distance results for functions in various spaces of analytic functions on bounded strictly pseudoconvex domains with smooth boundary. Also, we obtain atomic decompositions in multifunctional Bloch and weighted Bergman spaces of analytic functions on strictly pseudoconvex domains with smooth boundary, which extend known results in the classical case of a single function.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.5
• /
• pp.1711-1719
• /
• 2015

We obtain a new criterion for the boundedness and compactness of the weighted composition operators ${\psi}C_{\varphi}$ from ${\ss}^{{\alpha}}$(0 < ${\alpha}$ < 1) to ${\ss}^{{\beta}}$ in terms of the sequence $\{{\psi}{\varphi}^n\}$. An estimate for the essential norm of ${\psi}C_{\varphi}$ is also given.