• 대한수학회지
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• 제58권2호
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• pp.383-400
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• 2021

The aim of this paper is to mainly establish the sufficient and necessary conditions for the boundedness of the commutator ����Ω,b which is generated by the parameter Marcinkiwicz integral ����Ω and the Lipschitz function b on generalized Orlicz-Morrey space L��,��(Rd) in the sense of the Adams type result (or Spanne type result). Moreover, the necessary conditions for the parameter Marcinkiewizcz integral ����Ω on the L��,��(Rd), and the commutator [b,����Ω] generated by the ����Ω and the space BMO on the L��,��(Rd), are also obtained, respectively.

• 대한수학회보
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• 제60권2호
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• pp.541-560
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• 2023

Let T be an m-linear Calderón-Zygmund operator. $T_{{\vec{b}S}}$ is the generalized commutator of T with a class of measurable functions {b_i}^∞_i=1. In this paper, we will give some new estimates for $T_{{\vec{b}S}}$ when {b_i}^∞_i=1 belongs to Orlicz-type space and Lipschitz space, respectively.

• 대한수학회보
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• 제59권6호
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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.

• 대한수학회논문집
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• 제15권3호
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• pp.511-519
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• 2000

If G is a strict generalized orthomodular lattice and H={I｜I=[0, $\chi$, $\chi$$\in$G}, then H is prime ideal of the Janowitz's hull J(G) of G. If f is the janowitz's embedding, then the set of all commutatiors of f(G) equals the set of all commutators of the Janowitz's hull J(G) of G. Let L be an OML. Then L J(G) for a strict GOML G if and only if ther exists a proper nonprincipal prime ideal G in L.

• 대한수학회지
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• 제61권2호
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• pp.207-226
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• 2024

In this paper, the weighted L^p boundedness of multilinear commutators and multilinear iterated commutators generated by the multilinear singular integral operators with generalized kernels and BMO functions is established, where the weight is multiple weight. Our results are generalizations of the corresponding results for multilinear singular integral operators with standard kernels and Dini kernels under certain conditions.

• 대한수학회논문집
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• 제34권1호
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• pp.83-93
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• 2019

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R and f($x_1,{\ldots},x_n$) a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all the evaluations of f($x_1,{\ldots},x_n$) on R. If d is a nonzero derivation of R and G a nonzero generalized derivation of R such that $$d(G(u)u){\in}Z(R)$$ for all $u{\in}f(R)$, then $f(x_1,{\ldots},x_n)^2$ is central-valued on R and there exists $b{\in}U$ such that G(x) = bx for all $x{\in}R$ with $d(b){\in}C$. As an application of this result, we investigate the commutator $[F(u)u,G(v)v]{\in}Z(R)$ for all $u,v{\in}f(R)$, where F and G are two nonzero generalized derivations of R.

• Korean Journal of Mathematics
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• 제5권2호
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• pp.177-183
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• 1997

Let ${\sigma}(X,x_0,G)$ be the fundamental group of a transformation group (X,G). Let $R({\varphi},{\psi})$) be the generalized Reidemeister number for an endomorphism $({\varphi},{\psi}):(X,G){\rightarrow}(X,G)$. In this paper, our main results are as follows ; we prove some sufficient conditions for $R({\varphi},{\psi})$ to be the cardinality of $Coker(1-({\varphi},{\psi})_{\bar{\sigma}})$, where 1 is the identity isomorphism and $({\varphi},{\psi})_{\bar{\sigma}}$ is the endomorphism of ${\bar{\sigma}}(X,x_0,G)$, the quotient group of ${\sigma}(X,x_0,G)$ by the commutator subgroup $C({\sigma}(X,x_0,G))$, induced by (${\varphi},{\psi}$). In particular, we prove $R({\varphi},{\psi})={\mid}Coker(1-({\varphi},{\psi})_{\bar{\sigma}}){\mid}$, provided that (${\varphi},{\psi}$) is eventually commutative.

• Kyungpook Mathematical Journal
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• 제55권1호
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• pp.63-71
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• 2015

Let H be a separable infinite dimensional complex Hilbert space, and let $\mathcal{L}(H)$ denote the algebra of all bounded linear operators on H into itself. Let $A,B{\in}\mathcal{L}(H)$ we define the generalized derivation ${\delta}_{A,B}:\mathcal{L}(H){\mapsto}\mathcal{L}(H)$ by ${\delta}_{A,B}(X)=AX-XB$, we note ${\delta}_{A,A}={\delta}_A$. If the inequality ${\parallel}T-(AX-XA){\parallel}{\geq}{\parallel}T{\parallel}$ holds for all $X{\in}\mathcal{L}(H)$ and for all $T{\in}ker{\delta}_A$, then we say that the range of ${\delta}_A$ is orthogonal to the kernel of ${\delta}_A$ in the sense of Birkhoff. The operator $A{\in}\mathcal{L}(H)$ is said to be finite [22] if ${\parallel}I-(AX-XA){\parallel}{\geq}1(*)$ for all $X{\in}\mathcal{L}(H)$, where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.