• Bulletin of the Korean Mathematical Society
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• v.52 no.1
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• pp.223-228
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• 2015

In this paper, we study the eigenvalue problem of Schr$\ddot{o}$dinger-type operator on bounded domains in strictly pseudoconvex CR manifolds and obtain some universal inequalities for lower order eigenvalues. Moreover, we will give some generalized Reilly-type inequalities of the first nonzero eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex CR manifold without boundary.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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• pp.259-263
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• 1995

The following classical Oka-Weil approximation theorem on pseudoconvex domains in $C^n$ is well-known. Suppose that $M \subseteq C^n$ is pseudoconvex and that K is a compact subset of M with K = K, where K is the usual holomorphic hull of K in M. Then any function holomorphic in a neighborhood of K can be approximated uniformly on K by functions holomorphic on M (see [5], [6]).

• Honam Mathematical Journal
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• v.25 no.1
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• pp.83-91
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• 2003

We prove a unique continuation theorem for $C^{\infty}$ functions in pseudoconvex domains in ${\mathbb{C}}^{n}$. More specifically, we show that if ${\Omega}$ is a pseudoconvex domain in ${\mathbb{C}}^n$, if f is in $C^{\infty}({\Omega})$ such that for all multi-indexes ${\alpha},{\beta}$ with ${\mid}{\beta}{\mid}{\geq}1$ and for any positive integer k, there exists a positive constant $C_{{\alpha},{\beta},{\kappa}}$ such that $$|{\frac{{\partial}^{{\mid}{\alpha}{\mid}+{\mid}{\beta}{\mid}}f}{{\partial}z^{\alpha}{\partial}{\bar{z}}^{\beta}}{\mid}{\leq}C_{{\alpha},{\beta},{\kappa}}{\mid}f{\mid}^{\kapp}}\;in\;{\Omega}$$, and if there exists $z_0{\in}{\Omega}$ such that f vanishes to infinite order at $z_0$, then f is identically zero. We also have a sharp result for the case of strongly pseudoconvex domains.

• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.993-1002
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• 2017

Let ${\Omega}$ be a bounded, uniformly totally pseudoconvex domain in ${\mathbb{C}}^2$ with the smooth boundary b${\Omega}$. Assuming that ${\Omega}$ satisfies the negative ${\bar{\partial}}$ property. Let M be a positive, finite area divisor of ${\Omega}$. In this paper, we will prove that: if ${\Omega}$ admits a maximal type F and the ${\check{C}}eck$ cohomology class of the second order vanishes in ${\Omega}$, there is a bounded holomorphic function in ${\Omega}$ such that its zero set is M. The proof is based on the method given by Shaw [27].

• East Asian mathematical journal
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• v.19 no.1
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• pp.49-61
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• 2003

We study the asymptotic boundary behavior of the Bergman kernel function on the diagonal, the Bergman metric and the holomorphic sectional curvatures of the Bergman metric in bounded strongly pseudoconvex domains.

• East Asian mathematical journal
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• v.31 no.3
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• pp.363-370
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• 2015

The automorphism group of domains is main stream of classification problem coming from E. Cartan's work. In this paper, I introduce classical technique of computations of automorphism group of domains and recent development of automorphism group. Moreover, I suggest new research problems in computations of automorphism group.

• Communications of the Korean Mathematical Society
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• v.10 no.1
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• pp.85-92
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• 1995

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.425-437
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• 2002

Let $\Omega$ be a smoothly bounded pseudoconvex domain in $C^{n}$ and let $z^{0}$ $\in$b$\Omega$ a point of finite type. We also assume that the Levi form of b$\Omega$ is comparable in a neighborhood of $z^{0}$ . Then we get precise estimates of the Bergman kernel function, $K_{\Omega}$(z, w), and its derivatives in a neighborhood of $z^{0}$ . .

• Communications of the Korean Mathematical Society
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• v.10 no.2
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• pp.349-355
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• 1995

Let $D \subset C^n$ be a smoothly bounded pseudoconvex domain and let ${\bar{D}_r}_r$ be a family of smooth perturbations of $\bar{D}$ such that $\bar{D} \subset \bar{D}_r$. Let $K_D(z, w)$ be the Bergman kernel function on $D \times D$. Then $lim_{r \to 0} K_{D_r}(z, w) = K_D(z, w)$ locally uniformally on $D \times D$.

• Journal of the Korean Mathematical Society
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• v.55 no.4
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• pp.897-921
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• 2018

Given bounded pseudoconvex domains in 2-dimensional complex Euclidean space, we derive analytical and geometric conditions which guarantee the Diederich-$Forn{\ae}ss$ index is 1. The analytical condition is independent of strongly pseudoconvex points and extends $Forn{\ae}ss$-Herbig's theorem in 2007. The geometric condition reveals the index reflects topological properties of boundary. The proof uses an idea including differential equations and geometric analysis to find the optimal defining function. We also give a precise domain of which the Diederich-$Forn{\ae}ss$ index is 1. The index of this domain can not be verified by formerly known theorems.