• 대한수학회보
• /
• 제60권6호
• /
• pp.1439-1451
• /
• 2023

This paper provides a constructive proof of the weak factorizations of the classical Hardy space H¹(ℝⁿ) in terms of multilinear fractional integral operator on the variable Lebesgue spaces, which the result is new even in the linear case. As a direct application, we obtain a new proof of the characterization of BMO(ℝⁿ) via the boundedness of commutators of the multilinear fractional integral operator on the variable Lebesgue spaces.

• 한국수학사학회지
• /
• 제21권3호
• /
• pp.67-96
• /
• 2008

19세기에 푸리에와 디리클레가 한 개의 식으로 표현되지 않을 수도 있는 "임의의" 함수를 삼각급수로 표현하는 것과 관련하여 연속함수의 적분을 다루었던 코시의 적분보다 더 일반적인 적분의 필요성을 제기하여 리만적분론으로 이끌었다. 한동안 리만적분이 가장 일반적인 적분으로 간주되었고, 이 적분론이 집중적으로 다루어진 결과 리만적분의 약점들이 보였으나, 적어도 초기에는 이것들이 리만적분에 대한 비판으로 보이지 않았다. 그러나 죠르단이 1892년에 용량개념을 소개하며 리만적분론을 측도론적 배경에서 다루었고, 이로부터 몇 년 후에 보렐이 죠르단의 용량론을 측도론으로 발전시킨 후에 르베그가 이 둘의 이론을 합쳐서 지금 "르베그적분"으로 알고 있는 적분의 새 개념을 얻게 되었다.

• Journal of applied mathematics & informatics
• /
• 제30권1_2호
• /
• pp.211-217
• /
• 2012

In this paper, we give the weighted Lebesgue-Radon-Nikodym theorem with respect to $p$-adic invariant integral on $\mathbb{Z}_p$.

• 충청수학회지
• /
• 제27권3호
• /
• pp.489-494
• /
• 2014

In this paper, we define the extension $f^*:[a,b]{\rightarrow}\mathbb{R}$ of a function $f:[a,b]_{\mathbb{T}}{\rightarrow}\mathbb{R}$ for a time scale $\mathbb{T}$ and investigate the properties of the Lebesgue delta integral of f on $[a,b]_{\mathbb{T}}$ by using the function $f^*$.

• 한국지능시스템학회논문지
• /
• 제17권4호
• /
• pp.455-459
• /
• 2007

Interval-valued fuzzy sets were suggested for the first time by Gorzafczany(1983) and Turksen(1986). Based on this, Zeng and Li(2006) introduced concepts of similarity measure and entropy on interval-valued fuzzy sets which are different from Bustince and Burillo(1996). In this paper, by using Choquet integral with respect to a fuzzy measure, we introduce distance measure and similarity measure defined by Choquet integral on interval-valued fuzzy sets and discuss some properties of them. Choquet integral is a generalization concept of Lebesgue inetgral, because the two definitions of Choquet integral and Lebesgue integral are equal if a fuzzy measure is a classical measure.

• 대한수학회보
• /
• 제33권3호
• /
• pp.411-417
• /
• 1996

Some generalizations of the Riemann integral have been studied for real-valued functions. One of these generalizations leads to an integral, often called the McShane integral, that is equivalent to the Lebesgue integral.

• 한국수학교육학회지시리즈A:수학교육
• /
• 제19권1호
• /
• pp.23-30
• /
• 1980

• Kyungpook Mathematical Journal
• /
• 제52권3호
• /
• pp.299-306
• /
• 2012

We will give a new proof of the Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on $Z_p$, using Mahler expansion of continuous functions, studied by the authors in 2012. In the special case, q = 1, we can derive the same result as in Kim, 2012, Kim et al, 2011.

• International Journal of Fuzzy Logic and Intelligent Systems
• /
• 제7권1호
• /
• pp.66-70
• /
• 2007

Note that Choquet integral is a generalized concept of Lebesgue integral, because two definitions of Choquet integral and Lebesgue integral are equal if a fuzzy measure is a classical measure. In this paper, we consider interval-valued Choquet integrals with respect to fuzzy measures(see [4,5,6,7]). Using these Choquet integrals, we define a mappings on the classes of Choquet integrable functions and give an example of a mapping defined by interval-valued Choquet integrals. And we will investigate some relations between m-convex mappings ${\phi}$ on the class of Choquet integrable functions and m-convex mappings $T_{\phi}$, defined by the class of closed set-valued Choquet integrals with respect to fuzzy measures.

• Korean Journal of Mathematics
• /
• 제14권2호
• /
• pp.137-160
• /
• 2006

We describe the weight functions for which Hardy's inequality of nonincreasing functions is satisfied. Further we characterize the pairs of weight functions $(w,v)$ for which the Laplace transform $\mathcal{L}f(x)={\int}^{\infty}_0e^{-xy}f(y)dy$, with monotone function $f$, is bounded from the weighted Lebesgue space $L^p(w)$ to $L^q(v)$.