• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1071-1085
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• 2016

For $b{\in}L^1_{loc}({\mathbb{R}}^n)$, let ${\mathcal{I}}_{\alpha}$ be the bilinear fractional integral operator, and $[b,{\mathcal{I}}_{\alpha}]_i$ be the commutator of ${\mathcal{I}}_{\alpha}$ with pointwise multiplication b (i = 1, 2). This paper shows that if the commutator $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 is bounded from the product Morrey spaces $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to the Morrey space $L^{q,{\lambda}}({\mathbb{R}}^n)$ for some suitable indexes ${\lambda}$, ${\lambda}_1$, ${\lambda}_2$ and $p_1$, $p_2$, q, then $b{\in}BMO({\mathbb{R}}^n)$, as well as that the compactness of $[b,{\mathcal{I}}_{\alpha}]_i$ for i = 1 or 2 from $L^{p_1,{\lambda}_1}({\mathbb{R}}^n){\times}L^{p_2,{\lambda}_2}({\mathbb{R}}^n)$ to $L^{q,{\lambda}}({\mathbb{R}}^n)$ implies that $b{\in}CMO({\mathbb{R}}^n)$ (the closure in $BMO({\mathbb{R}}^n)$of the space of $C^{\infty}({\mathbb{R}}^n)$ functions with compact support). These results together with some previous ones give a new characterization of $BMO({\mathbb{R}}^n)$ functions or $CMO({\mathbb{R}}^n)$ functions in essential ways.

• Bulletin of the Korean Mathematical Society
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• v.52 no.3
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• pp.851-864
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• 2015

In this paper, we study the generalized Littlewood-Paley operators. It is shown that the generalized g-function, Lusin area function and $g^*_{\lambda}$-function on any BMO function are either infinite everywhere, or finite almost everywhere, respectively; and in the latter case, such operators are bounded from BMO($\mathbb{R}^n$) to BLO($\mathbb{R}^n$), which improve and generalize some previous results.

• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1117-1127
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• 2019

Let ${\mathcal{L}}_2=(-{\Delta})^2+V^2$ be the $Schr{\ddot{o}}dinger$ type operator, where nonnegative potential V belongs to the reverse $H{\ddot{o}}lder$ class $RH_s$, s > n/2. In this paper, we consider the operator $T_{{\alpha},{\beta}}=V^{2{\alpha}}{\mathcal{L}}^{-{\beta}}_2$ and its conjugate $T^*_{{\alpha},{\beta}}$, where $0<{\alpha}{\leq}{\beta}{\leq}1$. We establish the $(L^p,\;L^q)$-boundedness of operator $T_{{\alpha},{\beta}}$ and $T^*_{{\alpha},{\beta}}$, respectively, we also show that $T_{{\alpha},{\beta}}$ is bounded from Hardy type space $H^1_{L_2}({\mathbb{R}}^n)$ into $L^{p_2}({\mathbb{R}}^n)$ and $T^*_{{\alpha},{\beta}}$ is bounded from $L^{p_1}({\mathbb{R}}^n)$ into BMO type space $BMO_{{\mathcal{L}}1}({\mathbb{R}}^n)$, where $p_1={\frac{n}{4({\beta}-{\alpha})}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})}}$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.1
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• pp.235-251
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• 2021

We consider the Schrödinger type operator ��k = (-∆)k+Vk on ℝn(n ≥ 2k + 1), where k = 1, 2 and the nonnegative potential V belongs to the reverse Hölder class RHs with n/2 < s < n. In this paper, we establish the (Lp, Lq)-boundedness of the higher order Riesz transform T��,�� = V2��∇2��-��2 (0 ≤ �� ≤ 1/2 < �� ≤ 1, �� - �� ≥ 1/2) and its adjoint operator T∗��,�� respectively. We show that T��,�� is bounded from Hardy type space $H^1_{\mathcal{L}_2}({\mathbb{R}}_n)$ into Lp2 (ℝn) and T∗��,�� is bounded from ��p1 (ℝn) into BMO type space $BMO_{\mathcal{L}_1}$ (ℝn) when �� - �� > 1/2, where $p_1={\frac{n}{4({\beta}-{\alpha})-2}}$, $p_2={\frac{n}{n-4({\beta}-{\alpha})+2}}$. Moreover, we prove that T��,�� is bounded from $BMO_{\mathcal{L}_1}({\mathbb{R}}_n)$ to itself when �� - �� = 1/2.

• Journal of the Korean Mathematical Society
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• v.42 no.5
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• pp.1087-1100
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• 2005

In this note we give the mapping properties of the Marcinkiewicz integral !-to. at some end spaces. More precisely, we first prove that !-to. is a bounded operator from H$^{1,($\mathbb{R) to H$^{1, ($\mathbb{R). As a corollary of the results above, we obtain again the weak type (1,1) boundedness of $\mu$$_{, but the condition assumed on n is weaker than Stein's condition. Finally, we show that !-to. is bounded from BMO($\mathbb{R) to BMO($\mathbb{R). The results in this note are the extensions of the results obtained by Lee and Rim recently.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.4
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• pp.635-641
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• 2014

The existence and the regularity of the particle trajectory flow X(x, t) along a velocity field u on $\mathbb{R}^n$ are discussed under the BMO-blow-up condition: $${\int}_{0}^{T}{\parallel}{\omega}({\tau}){\parallel}_{BMO}d{\tau}&lt;{\infty}$$ of the vorticity ${\omega}{\equiv}{\nabla}{\times}u$. A comment on our result related with the mystery of turbulence is presented.

• East Asian mathematical journal
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• v.29 no.1
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• pp.91-99
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• 2013

In this paper, we prove that the BMO norm and the Garsia norm are equivalent on a bounded domain in $\mathbb{R}^N$. Also, we investigate the equivalent relation between the Lipschitz norm and the Garsia-type norm for harmonic functions.

• 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.46 no.3
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• pp.577-588
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• 2009

In this paper, the author studies the k-th commutators of oscillatory singular integral operators with a BMO function and phases more general than polynomials. For 1 < p < $\infty$, the $L^p$-boundedness of such operators are obtained provided their kernels belong to the spaces $L(log+L)^{k+1}(S^{n-1})$. The results of the corresponding maximal operators are also established.

• East Asian mathematical journal
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• v.28 no.1
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• pp.73-79
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• 2012

It is well-known that the BMO norm is equivalent to the Garsia norm. In this paper, we characterize mean-Lipschitz spaces by using Garsia-type norms on the upper half space $\mathbb{R}_+^{n+1}$.