• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.19 no.2
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• pp.189-195
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• 2015

The goal of this note is to provide a detailed proof for local boundedness estimate near the boundary for weak solutions for second order elliptic equations with bounded measurable coefficients subject to Neumann boundary condition.

• East Asian mathematical journal
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• v.36 no.3
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• pp.377-387
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• 2020

In this paper, we provide detailed proof of the Sawyer type characterization of the two weight estimate for the dyadic paraproduct. Although the dyadic paraproduct is known to be a well localized operators and the testing conditions obtained from checking boundedness of the given localized operator on a collection of test functions are provided by many authors. The main purpose of this paper is to present the necessary and sufficient conditions on the weights to ensure boundedness of the dyadic paraproduct directly.

• 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 $m=(n+1)(\rho-1)$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1123-1148
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• 2016

The aim of this note is to provide detailed proofs for local estimates near the boundary for weak solutions of second order parabolic equations in divergence form with time-dependent measurable coefficients subject to Neumann boundary condition. The corresponding parabolic equations with Dirichlet boundary condition are also considered.

• Journal of the Korean Mathematical Society
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• v.58 no.2
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• pp.487-499
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• 2021

We prove a decay estimate for oscillatory integrals with Hölder amplitudes and polynomial phases. The estimate allows us to answer certain questions concerning the uniform boundedness of oscillatory singular integrals on various spaces.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.61-82
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• 2000

In this paper two so-called regularized Green's functions are introduced to derive the optimal maximum norm error estimates for the unknown function and the adjoint vector-valued function for mixed finite element methods of Laplacian operator. One contribution of the paper is a demonstration of how the boundedness of $L^1$-norm estimate for the second Green's function ${\lambda}_2$ and the optimal maximum norm error estimate for the adjoint vector-valued function are proved. These results are seemed to be to be new in the literature of the mixed finite element methods.

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1711-1719
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• 2015

We obtain a new criterion for the boundedness and compactness of the weighted composition operators ${\psi}C_{\varphi}$ from ${\ss}^{{\alpha}}$(0 < ${\alpha}$ < 1) to ${\ss}^{{\beta}}$ in terms of the sequence $\{{\psi}{\varphi}^n\}$. An estimate for the essential norm of ${\psi}C_{\varphi}$ is also given.

• East Asian mathematical journal
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• v.33 no.1
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• pp.103-113
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• 2017

We provide quantitative two weight estimates for the commutator of the Hilbert transform under certain conditions on a pair of weights (u, v) and b in $Carl_{u,v}$. In [10] and [11], Bloom's inequality is shown in a modern setting, and the boundedness of the commutators is provided by assuming both weights u, v are $A_2$ and $b{\in}BMO_{\rho}$. In the present paper we show that the condition on b can be replaced by $Carl_{u,v}$ by using the joint $A^d_2$ condition.

• East Asian mathematical journal
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• v.38 no.3
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• pp.339-346
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• 2022

In this paper, a comparison of two types of quantitative two weight conditions for the boundedness of the dyadic paraproduct and the commutator of the Hilbert transform is provided. In the case of the commutator [b, H], the conditions of the well-known Bloom's inequality [2] and the slightly different types of two weight inequality introduced in [1] are compared around the A₂-conditions on weights and the novel conditions on the function b.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1471-1493
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• 2022

The goal of this paper is to establish the boundedness of bilinear Calderón-Zygmund operator BT and its commutator [b₁, b₂, BT] which is generated by b₁, b₂ ∈ BMO(ℝⁿ) (or ${\dot{\Lambda}}_{\alpha}$(ℝⁿ)) and the BT on generalized variable exponent Morrey spaces 𝓛^p(·),𝜑(ℝⁿ). Under assumption that the functions 𝜑₁ and 𝜑₂ satisfy certain conditions, the authors proved that the BT is bounded from product of spaces 𝓛^{p₁(·),𝜑₁}(ℝⁿ)×𝓛^{p₂(·),𝜑₂}(ℝⁿ) into space 𝓛^p(·),𝜑(ℝⁿ). Furthermore, the boundedness of commutator [b₁, b₂, BT] on spaces L^p(·)(ℝⁿ) and on spaces 𝓛^p(·),𝜑(ℝⁿ) is also established.