• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.561-568
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• 2016

In this paper, the effect of absolute values on the behavior of functions f in the spaces $\mathcal{Q}_K$ is investigated. It is clear that $g{\in}\mathcal{Q}_K({\partial}{\mathbb{D}}){\Rightarrow}{\mid}g{\mid}{\in}\mathcal{Q}_K({\partial}{\mathbb{D}})$, but the converse is not always true. For f in the Hardy space $H^2$, we give a condition involving the modulus of the function only, such that the condition together with ${\mid}f{\mid}{\in}\mathcal{Q}_K({\partial}{\mathbb{D}})$ is equivalent to $f{\in}\mathcal{Q}_K$. As an application, a new criterion for inner-outer factorisation of $\mathcal{Q}_K$ spaces is given. These results are also new for $Q_p$ spaces.

• Communications of the Korean Mathematical Society
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• v.32 no.1
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• pp.55-64
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• 2017

For -2 < ${\alpha}$ < ${\infty}$ and 0 < p < ${\infty}$, the $\mathcal{Q}_K$-type space is the space of all analytic functions on the open unit disk ${\mathbb{D}}$ satisfying $$_{{\sup} \atop a{\in}{\mathbb{D}}}{\large \int_{\mathbb{D}}}{{\mid}f^{\prime}(z){\mid}}^p(1-{{\mid}z{\mid}^2})^{\alpha}K(g(z,a))dA(z)<{\infty}$$, where $g(z,a)=log\frac{1}{{\mid}{\sigma}_a(z){\mid}}$ is the Green's function on ${\mathbb{D}}$ and K : [0, ${\infty}$) [0, ${\infty}$), is a right-continuous and non-decreasing function. For 0 < s < ${\infty}$, the space $\mathcal{Q}_s$ consists of all analytic functions on ${\mathbb{D}}$ for which $$_{sup \atop a{\in}{\mathbb{D}}}{\large \int_{\mathbb{D}}}{{\mid}f^{\prime}(z){\mid}}^2(g(z,a))^sdA(z)<{\infty}$$. Boundedness and compactness of composition operators $C_{\varphi}$ acting on $\mathcal{Q}_K$-type spaces and $\mathcal{Q}_s$ spaces is characterized in terms of the norms of ${\varphi}^n$. Thus the author announces a solution to the problem raised by Wulan, Zheng and Zhou.

• Journal of the Chungcheong Mathematical Society
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• v.14 no.2
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• pp.37-46
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• 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 $L^1_a(\mathcal{B})$.

• Kyungpook Mathematical Journal
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• v.58 no.2
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• pp.271-289
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• 2018

Let 0 < q < ${\infty}$. In this study, we introduce the spaces ${\mathcal{BV}}_q$ and ${\mathcal{LS}}_q$ of q-bounded variation double sequences and q-summable double series as the domain of four-dimensional backward difference matrix ${\Delta}$ and summation matrix S in the space ${\mathcal{L}}_q$ of absolutely q-summable double sequences, respectively. Also, we determine their ${\alpha}$- and ${\beta}-duals$ and give the characterizations of some classes of four-dimensional matrix transformations in the case 0 < q ${\leq}$ 1.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.1
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• pp.131-138
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• 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 $\mathbb{C}^n$.

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

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

• East Asian mathematical journal
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• v.27 no.1
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• pp.83-89
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• 2011

We denote by $\mathcal{Q}(A)$ the set of all matrices with the same sign pattern as A. A matrix A has signed -space provided there exists a set S of sign patterns such that the set of sign patterns of vectors in the -space of e $\tilde{A}$ is S, for each e $\tilde{A}{\in}\mathcal{Q}(A)$. In this paper, we show that the number of sign patterns of elements in the row space of $\mathcal{S}^*$-matrix is $3^{m+1}-2^{m+2}+2$. Also the number of sign patterns of vectors in the -space of a totally L-matrix is obtained.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1071-1085
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• 2016

For $b{\in}L^1_{loc}({\mathbb{R}}^n)$, let ${\mathcal{I}}_{\alpha}$ be the bilinear fractional integral operator, and $[b,{\mathcal{I}}_{\alpha}]_i$ be the commutator of ${\mathcal{I}}_{\alpha}$ with pointwise multiplication b (i = 1, 2). This paper shows that if the commutator $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 is bounded from the product Morrey spaces $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to the Morrey space $L^{q,{\lambda}}({\mathbb{R}}^n)$ for some suitable indexes ${\lambda}$, ${\lambda}_1$, ${\lambda}_2$ and $p_1$, $p_2$, q, then $b{\in}BMO({\mathbb{R}}^n)$, as well as that the compactness of $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 from $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to $L^{q,{\lambda}}({\mathbb{R}}^n)$ implies that $b{\in}CMO({\mathbb{R}}^n)$ (the closure in $BMO({\mathbb{R}}^n)$of the space of $C^{\infty}({\mathbb{R}}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO({\mathbb{R}}^n)$ functions or $CMO({\mathbb{R}}^n)$ functions in essential ways.

• Honam Mathematical Journal
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• v.30 no.4
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• pp.659-670
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• 2008

We denote by $\mathcal{Q}$(A) the set of all real matrices with the same sign pattern as a real matrix A. A matrix A is an SN-matrix provided there exists a set S of sign pattern such that the set of sign patterns of vectors in the -space of $\tilde{A}$ is S, for each ${\tilde{A}}{\in}\mathcal{Q}(A)$. Some properties of SN-matrices arc investigated.