• Bulletin of the Korean Mathematical Society
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• v.41 no.1
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• pp.153-166
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• 2004

We prove the Radon-Nikodym theorem for a nonabsolute integral on measure spaces endowed with metric topologies and hence provide a descriptive definition of the integral.

• East Asian mathematical journal
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• v.24 no.2
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• pp.201-212
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• 2008

We characterize operators between Banach spaces sending unconditionally weakly p-summable sequences into sequences that lie in the range of a vector measure of bounded variation. Further, we describe operators between Banach spaces taking unconditionally weakly p-summable sequences into sequences that lie in the range of a vector measure.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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• pp.541-555
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• 1996

The Fourier transform plays a central role in the theory of distribution on Euclidean spaces. Although Lebesgue measure does not exist in infinite dimensional spaces, the Fourier transform can be introduced in the space $(S)^*$ of generalized white noise functionals. This has been done in the series of paper by H.-H. Kuo [1, 2, 3], [4] and [5]. The Fourier transform $F$ has many properties similar to the finite dimensional case; e.g., the Fourier transform carries coordinate differentiation into multiplication and vice versa. It plays an essential role in the theory of differential equations in infinite dimensional spaces.

• Korean Journal of Mathematics
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• v.10 no.2
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• pp.103-116
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• 2002

We characterize operators between Banach spaces sending compact sets into sets that lie in the range of a vector measure. Further, we describe operators between Banach spaces taking compact sets into sets that lie in the range of a vector measure of bounded variation.

• Bulletin of the Korean Mathematical Society
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• v.38 no.1
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• pp.113-120
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• 2001

Using the notions of admissibility, local convexity and measure of noncompactness, we give new fixed point theorems for quasicompact or condensing multivalued mappings with domains that are not necessarily convex subsets of an arbitrary topological vector space.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.623-636
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• 2023

In this paper, we prove some vanishing theorems under the assumptions of weighted BiRic curvature or m-Bakry-Émery-Ricci curvature bounded from below.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.541-548
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• 2008

For a completely bounded linear maps between operator spaces, we introduce numbers which measure the degree of injectivity and subjectivity. The number measuring the injectivity is an operator space analogue of the minimum modulus of a linear map in normed spaces.

• 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 $H^P(B)$.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.749-763
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• 2022

Let C[0, T] denote an analogue of Weiner space, the space of real-valued continuous on [0, T]. In this paper, we investigate the translation of time interval [0, T] defining the analogue of Winer space C[0, T]. As applications of the result, we derive various relationships between the analogue of Wiener space and its product spaces. Finally, we express the analogue of Wiener measures on C[0, T] as the analogue of Wiener measures on C[0, s] and C[s, T] with 0 < s < T.

• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1569-1575
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• 2019

In this paper we calculate three geometric constants, namely the von Neumann-Jordan constant, the James constant, and the Dunkl-Williams constant, for Morrey spaces and discrete Morrey spaces. These constants measure uniformly nonsquareness of the associated spaces. We obtain that the three constants are the same as those for $L^1$ and $L^{\infty}$ spaces.