• 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 applied mathematics & informatics
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• v.39 no.3_4
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• pp.251-257
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• 2021

This article is devoted to introduce the notion of fuzzy cleanly covered fuzzy topological spaces; in addition two strong fuzzy separation axioms are studied. By means of fuzzy maximal open sets some properties of fuzzy cleanly covered fuzzy topological spaces are obtained and also by means of fuzzy maximal closed sets few identical results of a fuzzy topological spaces are investigated. Through fuzzy minimal open and fuzzy maximal closed sets, two strong fuzzy separation axioms are discussed.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.277-303
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• 2021

Under the rough kernels Ω belonging to the block spaces B0,qr (Sn-1) or the radial Grafakos-Stefanov kernels W����(Sn-1) for some r, �� > 1 and q ≤ 0, the boundedness and continuity were proved for two classes of rough maximal singular integrals and maximal operators associated to polynomial mappings on the Triebel-Lizorkin spaces and Besov spaces, complementing some recent boundedness and continuity results in [27, 28], in which the authors established the corresponding results under the conditions that the rough kernels belong to the function class L(log L)α(Sn-1) or the Grafakos-Stefanov class ����(Sn-1) for some α ∈ [0, 1] and �� ∈ (2, ∞).

• Journal of Applied and Pure Mathematics
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• v.4 no.1_2
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• pp.79-84
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• 2022

We have observed that there exist certain fuzzy topological spaces with no fuzzy minimal open sets. This observation motivates us to investigate fuzzy topological spaces with neither fuzzy minimal open sets nor fuzzy maximal open sets. We have observed if such fuzzy topological spaces exist and if it is connected are not fuzzy cut-point spaces. We also study and characterize certain properties of fuzzy mean open sets in fuzzy T₁-connected fuzzy topological spaces.

• Bulletin of the Korean Mathematical Society
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• v.35 no.3
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• pp.591-603
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• 1998

Taking into account the smoothness determined by sharp maximal functions, Sobolev type embedding results on Triebel-Lizorkin and Besov spaces are obtained.

• Journal of the Korean Mathematical Society
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• v.60 no.1
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• pp.33-69
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• 2023

Let α ∈ (0, ∞), p ∈ (0, ∞) and q(·) : ℝⁿ → [1, ∞) satisfy the globally log-Hölder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator M and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley g-function and the Littlewood-Paley $g^*_{\lambda}$-function.

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.383-391
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• 2020

We study the mapping property of multilinear fractional maximal operators in Lipschitz spaces. It should be pointed out that some of the techniques employed in the study of fractional integral operators do not apply to fractional maximal operators.

• Communications of the Korean Mathematical Society
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• v.32 no.1
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• pp.175-181
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• 2017

In this paper, we introduce a notion of cleanly covered topological spaces along with two strong separation axioms. Some properties of cleanly covered topological spaces are obtained in term of maximal open sets including some similar properties of a topological space in term of maximal closed sets. Two strong separation axioms are also investigated in terms of minimal open and maximal closed sets.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.43-46
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• 2018

It is concerned with the continuity of the Hardy-Little wood maximal function between the classical Lebesgue spaces or the Orlicz spaces. A new approach to the continuity of the Hardy-Littlewood maximal function is presented through the observation that the continuity is closely related to the existence of solutions for a certain type of first order ordinary differential equations. It is applied to verify the continuity of the Hardy-Littlewood maximal function from $L^p({\mathbb{R}}^n)$ to $L^q({\mathbb{R}}^n)$ for 1 ${\leq}$ q < p < ${\infty}$.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.305-320
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• 2013

Topological semilattices with path-connected intervals are special abstract convex spaces. In this paper, we obtain generalized KKM type theorems and their analytic formulations, maximal element theorems and collectively fixed point theorems on abstract convex spaces. We also apply them to topological semilattices with path-connected intervals, and obtain generalized forms of the results of Horvath and Ciscar, Luo, and Al-Homidan et al..