• Title, Summary, Keyword: integral operator

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Sequential operator-valued function space integral as an $L({L_p},{L_p'})$ theory

  • Ryu, K.S.
    • Journal of the Korean Mathematical Society
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    • v.31 no.3
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    • pp.375-391
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    • 1994
  • In 1968k Cameron and Storvick introduced the analytic and the sequential operator-valued function space integral [2]. Since then, the theo교 of the analytic operator-valued function space integral has been investigated by many mathematicians - Cameron, Storvick, Johnson, Skoug, Lapidus, Chang and author etc. But there are not that many papers related to the theory of the sequential operator-valued function space integral. In this paper, we establish the existence of the sequential operator-valued function space integral as an operator from $L_p$ to $L_p'(1 and investigated the integral equation related to this integral.

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ANALYTIC OPERATOR-VALUED FUNCTION SPACE INTEGRAL REPRESENTED AS THE BOCHNER INTEGRAL:AN$L(L_2)$ THEORY

  • Chang, Kun-Soo;Park, Ki-Seong;Ryu, Kun-Sik
    • Communications of the Korean Mathematical Society
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    • v.9 no.3
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    • pp.599-606
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    • 1994
  • In [1], Cameron and Storvick introduced the analytic operator-valued function space integral. Johnson and Lapidus proved that this integral can be expressed in terms of an integral of operator-valued functions [6]. In this paper, we find some operator-valued Bochner integrable functions and prove that the analytic operator-valued function space integral of a certain function is represented as the Bochner integral of operator-valued functions on some conditions.

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A CLASS OF THE OPERATOR-VALUED FEYNMAN INTEGRAL

  • Ahn, Byung-Moo
    • Journal of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.569-579
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    • 1997
  • We investigate the existence of the operator-valued Feynman integral when a Wiener functional is given by a Fourier transform of complex Borel measure.

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Existence theorems of an operator-valued feynman integral as an $L(L_1,C_0)$ theory

  • Ahn, Jae-Moon;Chang, Kun-Soo;Kim, Jeong-Gyoo;Ko, Jung-Won;Ryu, Kun-Sik
    • Bulletin of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.317-334
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    • 1997
  • The existence of an operator-valued function space integral as an operator on $L_p(R) (1 \leq p \leq 2)$ was established for certain functionals which involved the Labesgue measure [1,2,6,7]. Johnson and Lapidus showed the existence of the integral as an operator on $L_2(R)$ for certain functionals which involved any Borel measures [5]. J. S. Chang and Johnson proved the existence of the integral as an operator from L_1(R)$ to $C_0(R)$ for certain functionals involving some Borel measures [3]. K. S. Chang and K. S. Ryu showed the existence of the integral as an operator from $L_p(R) to L_p'(R)$ for certain functionals involving some Borel measures [4].

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CERTAIN FRACTIONAL INTEGRAL INEQUALITIES INVOLVING HYPERGEOMETRIC OPERATORS

  • Choi, Junesang;Agarwal, Praveen
    • East Asian mathematical journal
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    • v.30 no.3
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    • pp.283-291
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    • 2014
  • A remarkably large number of inequalities involving the fractional integral operators have been investigated in the literature by many authors. Very recently, Baleanu et al. [2] gave certain interesting fractional integral inequalities involving the Gauss hypergeometric functions. Using the same fractional integral operator, in this paper, we present some (presumably) new fractional integral inequalities whose special cases are shown to yield corresponding inequalities associated with Saigo, Erd$\acute{e}$lyi-Kober and Riemann-Liouville type fractional integral operators. Relevant connections of the results presented here with those earlier ones are also pointed out.

QUANTITATIVE WEIGHTED BOUNDS FOR THE VECTOR-VALUED SINGULAR INTEGRAL OPERATORS WITH NONSMOOTH KERNELS

  • Hu, Guoen
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1791-1809
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    • 2018
  • Let T be the singular integral operator with nonsmooth kernel which was introduced by Duong and McIntosh, and $T_q(q{\in}(1,{\infty}))$ be the vector-valued operator defined by $T_qf(x)=({\sum}_{k=1}^{\infty}{\mid}T\;f_k(x){\mid}^q)^{1/q}$. In this paper, by proving certain weak type endpoint estimate of L log L type for the grand maximal operator of T, the author establishes some quantitative weighted bounds for $T_q$ and the corresponding vector-valued maximal singular integral operator.

CERTAIN NEW PATHWAY TYPE FRACTIONAL INTEGRAL INEQUALITIES

  • Choi, Junesang;Agarwal, Praveen
    • Honam Mathematical Journal
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    • v.36 no.2
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    • pp.455-465
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    • 2014
  • In recent years, diverse inequalities involving a variety of fractional integral operators have been developed by many authors. In this sequel, here, we aim at establishing certain new inequalities involving pathway type fractional integral operator by following the same lines, recently, used by Choi and Agarwal [7]. Relevant connections of the results presented here with those earlier ones are also pointed out.