• Journal of the Korean Mathematical Society
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• v.42 no.1
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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 $H^P(B)$.

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.623-669
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• 2018

In Donaldson-Thomas theory, moduli spaces are locally the critical locus of a holomorphic function defined on a complex manifold. In this paper, we develop a theory of critical virtual manifolds which are the gluing of critical loci of holomorphic functions. We show that a critical virtual manifold X admits a natural semi-perfect obstruction theory and a virtual fundamental class $[X]^{vir}$ whose degree $DT(X)=deg[X]^{vir}$ is the Euler characteristic ${\chi}_{\nu}$(X) weighted by the Behrend function ${\nu}$. We prove that when the critical virtual manifold is orientable, the local perverse sheaves of vanishing cycles glue to a perverse sheaf P whose hypercohomology has Euler characteristic equal to the Donaldson-Thomas type invariant DT(X). In the companion paper, we proved that a moduli space X of simple sheaves on a Calabi-Yau 3-fold Y is a critical virtual manifold whose perverse sheaf categorifies the Donaldson-Thomas invariant of Y and also gives us a mathematical theory of Gopakumar-Vafa invariants.

• Journal of the Korean Mathematical Society
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• v.39 no.2
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• pp.277-287
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• 2002

Let B be the open unit ball with center 0 in the complex space $C^n$. For each q>0, B$_{q}$ consists of holomorphic functions f : B longrightarrow C which satisfy sup z $\in$ B $(1-\parallel z \parallel^2)^q\parallel\nabla f(z)\parallel < \infty$ In this paper, we will show that functions in weighted Bloch spaces $B_{q}$ (0 < q < 1) satifies the following Lipschitz type result for Bergman metric $\beta$: ｜f(z)-f($\omega$)｜< $C\beta$(z, $\omega$) for some constant C.

• 제어로봇시스템학회:학술대회논문집
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• 2004.08a
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• pp.1847-1852
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• 2004

This article provides the connection between feedback stabilization and interpolation conditions for n-D linear systems (n > 1). In addition to internal stability, if one demands performance as a design goal, then there results an n-D matrix Nevanlinna-Pick interpolation problem. Application of recent work on Nevanlinna-Pick interpolation on the polydisk yields a solution of the problem for the 2-D case. The same analysis applies in the n-D case (n > 2), but leads to solutions which are contractive in a norm (the "Schur-Agler norm") somewhat stronger than the $H^{\infty}$ norm. This is an analogous version of the connection between the standard $H^{\infty}$ control problem and an interpolation problem of Nevanlinna-Pick type in the classical 1-D linear time-invariant systems.

• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.569-608
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• 1994

The subject matter of this survey has to do with holomorphic maps from a compact Riemann surface to projective space, which are also called algebrac curves; the theory we survey lies at the crossroads of function theory, projective geometry, and commutative algebra (although we should mention that the present survey de-emphasizes the algebraic aspect). Algebraic curves have been vigorously and continuously investigated since the time of Riemann. The reasons for the preoccupation with algebraic curves amongst mathematicians perhaps have to do with-other than the usual usual reason, namely, the herd mentality prompting us to follow the leads of a few great pioneering methematicians in the field-the fact that algebraic curves possess a certain simple unity together with a rich and complex structure. From a differential-topological standpoint algebraic curves are quite simple as they are neatly parameterized by a single discrete invariant, the genus. Even the possible complex structures of a fixed genus curve afford a fairly complete description. Yet there are a multitude of diverse perspectives (algebraic, function theoretic, and geometric) often coalescing to yield a spectacular result.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.705-714
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• 2017

Let $A^2_{{\alpha},{\beta}}(\mathbb{B}_n)$ denote the space of holomorphic functions that are $L^2$ with respect to a weight of form ${\omega}_{{\alpha},{\beta}}(z)=(1-{\mid}z{\mid}^{\alpha}e^{-{\frac{\beta}{1-{\mid}z{\mid}}}}$, where ${\alpha}{\in}\mathbb{R}$ and ${\beta}$ > 0 on the unit ball $\mathbb{B}_n$. We obtain some results for the boundedness and compactness of $Ces{\grave{a}}ro$ operator on $A^2_{{\alpha},{\beta}(\mathbb{B}_n)$.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1001-1015
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• 2010

Let M be a $C^{\infty}$ real hypersurface in $\mathbb{C}^{n+1}$, $n\;{\geq}\;1$, locally given as the zero locus of a $C^{\infty}$ real valued function r that is defined on a neighborhood of the reference point $P\;{\in}\;M$. For each k = 1,..., n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n - k at P. The problem is to find an integral manifold of the real 1-form $i{\partial}r$ on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.

• East Asian mathematical journal
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• v.31 no.1
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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 $\mathbb{B}$.

• Communications of the Korean Mathematical Society
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• v.20 no.2
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• pp.339-350
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• 2005

We obtain the following two inequalities on a strongly pseudoconvex domain $\Omega\;in\;\mathbb{C}^n\;:\;for\;f\;{\in}\;O(\Omega)$ $$\int_{0}^{{\delta}0}t^{a{\mid}a{\mid}+b}M_p^a(t, D^{a}f)dt\lesssim\int_{0}^{{\delta}0}t^{b}M_p^a(t,\;f)dt\;\int_{O}^{{\delta}O}t_{b}M_p^a(t,\;f)dt\lesssim\sum_{j=0}^{m}\int_{O}^{{\delta}O}t^{am+b}M_{p}^{a}\(t,\;\aleph^{i}f\)dt$$. In [9], Shi proved these results for the unit ball in $\mathbb{C}^n$. These are generalizations of some classical results of Hardy and Littlewood.

• Journal of the Korean Mathematical Society
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• v.41 no.1
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• pp.39-50
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• 2004

In [25] Taskinen shows that if $\{E_n\}_n\;and\;\{F_n\}_n$ are two sequences of Frechet spaces such that ($E_m,\;F_n$) has the BB-property for all m and n then (${\Pi}_m\;E_m,\;{\Pi}_n\;F_n$) also has the ΒΒ-property. Here we investigate when this result extends to (i) arbitrary products of Frechet spaces, (ii) countable products of DFN spaces, (iii) countable direct sums of Frechet nuclear spaces. We also look at topologies properties of ($H(U),\;\tau$) for U balanced open in a product of Frechet spaces and $\tau\;=\;{\tau}_o,\;{\tau}_{\omega}\;or\;{\tau}_{\delta}$.