• Bulletin of the Korean Mathematical Society
• /
• v.50 no.1
• /
• pp.285-303
• /
• 2013

The asymptotic expansions of the Bergman kernels on the diagonals near the boundary points of exponentially-flat infinite type for pseudoconvex tube domain in $\mathbb{C}^2$ are obtained.

• Bulletin of the Korean Mathematical Society
• /
• v.57 no.5
• /
• pp.1241-1249
• /
• 2020

Let Ω be a domain in ℂⁿ. Suppose that ∂Ω is smooth pseudoconvex of D'Angelo finite type near a boundary point ξ₀ ∈ ∂Ω and the Levi form has corank at most 1 at ξ₀. Our goal is to show that if the squeezing function s_Ω(𝜂_j) tends to 1 or the Fridman invariant h_Ω(𝜂_j) tends to 0 for some sequence {𝜂_j} ⊂ Ω converging to ξ₀, then this point must be strongly pseudoconvex.

• Journal of the Korean Mathematical Society
• /
• v.53 no.6
• /
• pp.1211-1223
• /
• 2016

Let M be an n-dimensional $K{\ddot{a}}hler$ manifold with positive holomorphic bisectional curvature and let ${\Omega}{\Subset}M$ be a pseudoconvex domain of order $n-q$, $1{\leq}q{\leq}n$, with $C^2$ smooth boundary. Then, we study the (weighted) $\bar{\partial}$-equation with support conditions in ${\Omega}$ and the closed range property of ${\bar{\partial}}$ on ${\Omega}$. Applications to the ${\bar{\partial}}$-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number ${\ell}_0$ > 0 such that the ${\bar{\partial}}$-Neumann problem and the Bergman projection are regular in the Sobolev space $W^{\ell}({\Omega})$ for ${\ell}$ < ${\ell}_0$.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.3
• /
• pp.781-794
• /
• 2021

Let Ω be a smoothly bounded pseudoconvex domain in ℂⁿ and assume that the (n - 2)-eigenvalues of the Levi-form are comparable in a neighborhood of z₀ ∈ bΩ. Also, assume that there is a smooth 1-dimensional analytic variety V whose order of contact with bΩ at z₀ is equal to 𝜂 and 𝚫_n-2(z₀) < ∞. We show that the maximal gain in Hölder regularity for solutions of the ${\bar{\partial}}$-equation is at most ${\frac{1}{\eta}}$.

• Communications of the Korean Mathematical Society
• /
• v.33 no.2
• /
• pp.409-421
• /
• 2018

Let X be a complex manifold of dimension n $n{\geqslant}2$ and let ${\Omega}{\Subset}X$ be a weakly q-convex domain with smooth boundary. Assume that E is a holomorphic line bundle over X and $E^{{\otimes}m}$ is the m-times tensor product of E for positive integer m. If there exists a strongly plurisubharmonic function on a neighborhood of $b{\Omega}$, then we solve the ${\bar{\partial}}$-problem with support condition in ${\Omega}$ for forms of type (r, s), $s{\geqslant}q$ with values in $E^{{\otimes}m}$. Moreover, the solvability of the ${\bar{\partial}}_b$-problem on boundaries of weakly q-convex domains with smooth boundary in $K{\ddot{a}}hler$ manifolds are given. Furthermore, we shall establish an extension theorem for the ${\bar{\partial}}_b$-closed forms.

• Bulletin of the Korean Mathematical Society
• /
• v.37 no.1
• /
• pp.127-134
• /
• 2000

Let $\Omega$ be a bounded pseudoconvex domain in$C^{n}$ with smooth defining function r and let$z_0\; {\in}\; b{\Omega}$ be a point of finite type. We also assume that $\Omega$ is convex in a neighborhood of $z_0$. Then we prove that all the holomorphic sectional curvatures of the Bergman metric of $\Omega$ are bounded below by a negative constant near $z_0$.