• East Asian mathematical journal
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• 제31권1호
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• pp.55-63
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• 2015

Let $\mathbb{B}$ be the unit ball in $\mathbb{C}^n$. For a weight function ${\omega}$, we define the generalized growth space $A^{\omega}(\mathbb{B})$ by the space of holomorphic functions f on $\mathbb{B}$ such that $${\mid}f(z){\mid}{\leq}C{\omega}({\mid}{\rho}(z){\mid},\;z{\in}\mathbb{B}$$. Our main purpose in this note is to get the corona type decomposition in generalized growth spaces on $\mathbb{B}$.

• 대한수학회보
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• 제52권3호
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• pp.751-759
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• 2015

In this paper, we give new characterizations of the boundedness and compactness of composition operators $C_{\varphi}$ between Bloch type spaces in the unit ball $\mathbb{B}^n$, in terms of the power of the components of ${\varphi}$, where ${\varphi}$ is a holomorphic self-map of $\mathbb{B}^n$.

• Korean Journal of Mathematics
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• 제18권2호
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• pp.195-200
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• 2010

Let C be a closed convex set in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$. Assume that ${\Sigma}$ is an n-dimensional compact minimal submanifold outside C such that ${\Sigma}$ is orthogonal to ${\partial}C$ along ${\partial}{\Sigma}{\cap}{\partial}C$ and ${\partial}{\Sigma}$ lies on a geodesic sphere centered at a fixed point $p{\in}{\partial}{\Sigma}{\cap}{\partial}C$ and that r is the distance in ${\mathbb{S}}^m$ or ${\mathbb{H}}^m$ from p. We make use of a modified volume $M_p({\Sigma})$ of ${\Sigma}$ and obtain a sharp relative isoperimetric inequality $$\frac{1}{2}n^n{\omega}_nM_p({\Sigma})^{n-1}{\leq}Vol({\partial}{\Sigma}{\sim}{\partial}C)^n$$, where ${\omega}_n$ is the volume of a unit ball in ${\mathbb{R}}^n$ Equality holds if and only if ${\Sigma}$ is a totally geodesic half ball centered at p.

• 대한수학회지
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• 제42권1호
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• pp.111-127
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• 2005

Let B and S be the unit ball and the unit sphere in $\mathbb{C}^n$, respectively. Let ${\sigma}$ be the normalized Lebesgue measure on S. Define the Privalov spaces $N^P(B)\;(1\;<\;p\;<\;{\infty})$ by $$N^P(B)\;=\;\{\;f\;{\in}\;H(B) : \sup_{0 where H(B) is the space of all holomorphic functions in B. Let ${\varphi}$ be a holomorphic self-map of B. Let ${\mu}$ denote the pull-back measure ${\sigma}o({\varphi}^{\ast})^{-1}$. In this paper, we prove that the composition operator $C_{\varphi}$ is metrically bounded on $N^P$(B) if and only if ${\mu}(S(\zeta,\delta)){\le}C{\delta}^n$ for some constant C and $C_{\varphi}$ is metrically compact on $N^P(B)$ if and only if ${\mu}(S(\zeta,\delta))=o({\delta}^n)$ as ${\delta}\;{\downarrow}\;0$ uniformly in ${\zeta}\;\in\;S. Our results are an analogous results for Mac Cluer's Carleson-measure criterion for the boundedness or compactness of $C_{\varphi}$ on the Hardy spaces $H^P(B)$.

• 대한수학회보
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• 제47권6호
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• pp.1171-1180
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• 2010

In this paper we obtain some new characterizations for weighted Bergman spaces in the unit ball of $\mathbb{C}^n$.

• 대한수학회보
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• 제49권1호
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• pp.89-98
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• 2012

In this paper we obtain some new characterizations of Besov spaces on the unit ball of $\mathbb{C}^n$. These characterizations are also completely new even in the settings of the unit disk.

• 대한수학회지
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• 제50권1호
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• pp.189-202
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• 2013

On the unit ball of $\mathbb{C}^n$, the space of those holomorphic functions satisfying the mean Lipschitz condition $${\int}_0^1\;{\omega}_p(t,f)^q\frac{dt}{t^1+{\alpha}q}\;<\;{\infty}$$ is characterized by integral growth conditions of the tangential derivatives as well as the radial derivatives, where ${\omega}_p(t,f)$ denotes the $L^p$ modulus of continuity defined in terms of the unitary transformations of $\mathbb{C}^n$.

• 대한수학회논문집
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• 제24권2호
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• pp.187-195
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• 2009

We investigate the boundary values of the holomorphic mean Lipschitz function. In fact, we prove that the admissible limit exists at every boundary point of the unit ball for the holomorphic mean Lipschitz functions under some assumptions on the Lipschitz order. Moreover, we get embedding theorems of holomorphic mean Lipschitz spaces into Hardy spaces or into the Bloch space on the unit ball in $\mathbb{C}_n$.

• East Asian mathematical journal
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• 제34권5호
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• pp.577-585
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• 2018

For the purpose of characterizing subharmonic or ${\mathcal{M}}$-subharmonic Hardy classes in the unit ball of ${\mathbb{C}}^n$, we establish fundamental identities between integral means in terms of volume integrals and Green's functions.

• 대한수학회보
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• 제56권3호
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• pp.675-680
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• 2019

We, in this note, decompose the invariant Laplacian of the unit complex ball of ${\mathbb{C} }^n$ by the radial part and tangential part as ${\tilde{\Delta}}={\tilde{\Delta}}_{rad}+{\tilde{\Delta}}_{tan}$. We give several properties and interpretations involved with this decomposition.