• Bulletin of the Korean Mathematical Society
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• v.21 no.2
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• pp.99-106
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• 1984

J. Yeh has recently introduced the concept of conditional Wiener integrals which are meant specifically the conditional expectation E$^{w}$ (Z vertical bar X) of a real or complex valued Wiener integrable functional Z conditioned by the Wiener measurable functional X on the Wiener measure space (A precise definition of the conditional Wiener integral and a brief discussion of the Wiener measure space are given in Section 2). In [3] and [4] he derived some inversion formulae for conditional Wiener integrals and evaluated some conditional Wiener integrals E$^{w}$ (Z vertical bar X) conditioned by X(x)=x(t) for a fixed t>0 and x in Wiener space. Thus E$^{w}$ (Z vertical bar X) is a real or complex valued function on R$^{1}$. In this paper we shall be concerned with the random vector X given by X(x) = (x(s$_{1}$),..,x(s$_{n}$ )) for every x in Wiener space where 0=s$_{0}$ $_{1}$<..$_{n}$ =t. In Section 3 we will evaluate some conditional Wiener integrals E$^{w}$ (Z vertical bar X) which are real or complex valued functions on the n-dimensional Euclidean space R$^{n}$ . Thus we extend Yeh's results [4] for the random variable X given by X(x)=x(t) to the random vector X given by X(x)=(x(s$_{1}$).., x(s$_{n}$ )).

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.1-10
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• 2007

The purpose of this paper is to define generalized twisted q-Bernoulli numbers by using p-adic q-integrals. Furthermore, we construct a q-analogue of the p-adic generalized twisted L-functions which interpolate generalized twisted q-Bernoulli numbers. This is the generalization of Kim's h-extension of p-adic q-L-function which was constructed in [5] and is a partial answer for the open question which was remained in [3].

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 1998.06a
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• pp.380-386
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• 1998

In this paper we introduce the concept of fuzzy integrals for set-valued mappings, which is an extension of fuzzy integrals for single-valued functions defined by Sugeno. And we give some properties including convergence theorems on fuzzy integrals for set-valued mappings.

• Proceedings of the Korean Institute of Intelligent Systems Conference
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• 2005.11a
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• pp.129-132
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• 2005

In this paper, we consider comonotonically additive compact set-valued functionals instead of interval-valued functionals and study some characterizations of them. And we also investigate some relation between compact set-valued functionals and compact set-valued Choquet integrals.

• Kyungpook Mathematical Journal
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• v.46 no.3
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• pp.377-387
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• 2006

In this paper, we study the $L^p$ mapping properties of singular integral operators related to homogeneous mappings on product spaces with kernels which belong to $L(log^+\;L)^2$. Our results extend as well as improve some known results on singular integrals.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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• pp.789-805
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• 2005

In this paper we consider analogous of Gronwall-type inequalities involving iterated integrals in the inequality (1.2) for functions when the function u in the right-hand side of the in­equality (1.2) is replaced by the function $u^P$ for some p. These inequalities are effective tools in the study of a system of an integral equation. We also provide some integral inequalities involving iterated integrals.

• Journal of the Korean Mathematical Society
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• v.39 no.1
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• pp.61-75
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• 2002

Under the cancellation property and the Lipschitz condition on kernels, we prove that the Marcinkiewicz integrals defined on a homogeneous group H are bounded from $H^1$(H) to $L^1$(H), from $L_{c}$ $^{\infty}$(H) to BMO (H), and from $L^{p}$ (H) to $L^{p}$ (H) for 1 < p < $\infty$ assuming the $L^{q}$ -boundedness for some q > 1.for some q > 1.

• Journal of the Korean Mathematical Society
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• v.38 no.2
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• pp.469-483
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• 2001

The configuration space and phase space oscillatory path integrals are computed in the case of the stochastic Schrodinger equation for the harmonic oscillator with a stochastic term of the form (K$\psi$(sub)t)(x) o dW(sub)t, where K is either the position operator or the momentum operator, and W(sub)t is the Wiener process. In this way formulae are derived for the stochastic analogues of the Mehler kernel.

• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.771-788
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• 2015

In this note we consider the parametric Marcinkiewicz integrals with mixed homogeneity along polynomials compound curves. Under the rather weakened size conditions on the integral kernels both on the unit sphere and in the radial direction, the $L^p$ bounds of such operators are given by an extrapolation argument. Some previous results are greatly extended and improved.

• Communications of the Korean Mathematical Society
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• v.13 no.2
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• pp.307-316
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• 1998

In this paper Denjoy*-Dunford, Denjoy*-Pettis, Denjoy*-McShane and Denjoy*-Bochner integrals of functions which map an interval [a,b] into a Banach space X are defined. And we give the relations among the integrals.