• Journal of the Chungcheong Mathematical Society
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• v.8 no.1
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• pp.29-36
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• 1995

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1391-1408
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• 2023

Let M and M^# be Hardy-Littlewood maximal operator and sharp maximal operator, respectively. In this article, we present necessary and sufficient conditions for the boundedness properties for commutator operators [M, b] and [M^#, b] in a general context of Banach function spaces when b belongs to BMO(?ⁿ) spaces. Some applications of the results on weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces and Musielak-Orlicz spaces are also given.

• Journal of the Korean Mathematical Society
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• v.35 no.3
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• pp.713-725
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• 1998

Let X and Y be Ito processes with dX$_{s}$ = $\phi$$_{s}$dB$_{s}$ ＋ $\psi$$_{s}$ds and dY$_{s}$ = (equation omitted)dB$_{s}$ ＋ ξ$_{s}$ds. Burkholder obtained a sharp bound on the distribution of the maximal function of Y under the assumption that │Y$_{0}$│$\leq$│X$_{0}$│,│ζ│$\leq$│$\phi$│, │ξ│$\leq$│$\psi$│ and that X is a nonnegative local submartingale. In this paper we consider a wider class of Ito processes, replace the assumption │ξ│$\leq$│$\psi$│ by a more general one │ξ│$\leq$$\alpha$ │$\psi$│ , where a $\geq$ 0 is a constant, and get a weak-type inequality between X and the maximal function of Y. This inequality, being sharp for all a $\geq$ 0, extends the work by Burkholder.der.urkholder.der.

• East Asian mathematical journal
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• v.6 no.2
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• pp.133-144
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• 1990

In 1982, N. Miller [5] showed a weighted $L^p$ boundedness theorem for pseudo differential operators with symbols $S^0_{1.0}$. In this paper, we shall prove the pointwise estimates, in terms of the Fefferman, Stein sharp function and Hardy Littlewood maximal function, for pseudo differential operators with reduced symbols and show a weighted $L^p$-boundedness for pseudo differential operators with symbol in $S^m_{\rho,\delta}$, 0{$\leq}{\delta}{\leq}{\rho}{\leq}1$, ${\delta}{\neq}1$, ${\rho}{\neq}0$ and $m=(n+1)(\rho-1)$.

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.615-623
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• 1995

Consider a function m defined in $R^n$ and the associated convolution operator T given by the Fourier transform $\hat{T} = m\hat{f}$.

• Korean Journal of Exercise Nutrition
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• v.25 no.1
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• pp.42-55
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• 2021

[Purpose] This study investigated the effects of marine phytoplankton supplementation (Oceanix®, Tetraselmis chuii) on 1) maximal isometric strength and immune function in healthy humans following a oneweek high-intensity resistance-training program and 2) the proinflammatory cytokine response to exercise in a rat model. [Methods] In the human trial, 22 healthy male and female participants were randomly divided into marine phytoplankton and placebo groups. Following baseline testing, participants underwent a 14-day supplement loading phase before completing five consecutive days of intense resistance training. In the rat model, rats were randomly divided into four groups (n=7 per condition): (i) control, (ii) exercise, (iii) exercise + marine phytoplankton (2.55 mg/kg/day), or (iv) exercise + marine phytoplankton (5.1 mg/kg/day). Rats in the exercising groups performed treadmill exercise 5 days per week for 6 weeks. [Results] In the human model, marine phytoplankton prevented significant declines in the isometric peak rate of force development compared to placebo. Additionally, salivary immunoglobulin A concentration was significantly lower following the resistance training protocol in the placebo group but not in the marine phytoplankton group. Marine phytoplankton in exercising rats decreased intramuscular levels and serum concentrations of tumor necrosis factor-alpha (TNF-α) and interleukin-1 beta (IL-1β) and intramuscular concentrations of malondialdehyde. [Conclusion] Marine phytoplankton prevented decrements in indices of functional exercise recovery and immune function. Mechanistically, these outcomes could be prompted by modulating the oxidative stress and proinflammatory cytokine response to exercise.