Necessary Condition for Boundedness of Stein-Weiss Operator on Orlicz-Morrey Spaces - Al Hazmy, Sofihara; Masta, Al Azhary; Fatimah, Siti; Batkunde, Harmanus (IndoMS, 2025)
This study aims to find necessary conditions for the boundedness of Stein-Weiss Operator on Orlicz-Morrey spaces. It is well known that the Orlicz-Morrey space is the generalization of the Lebesque space. In particular, by considering power function as Young’s function and zero as Morrey Space index, the Orlicz-Morrey space is a Lebesgue space itself. Orlicz-Morrey space and several operators in the space have been studied intensively by several researchers. In this study, we find a necessary condition for the boundedness of the Stein-Weiss operator on Orlicz-Morrey spaces. The technique to achieve our purpose is by substituting dilation of a radial function on a ball into inequality of boundedness assumption, and some basic properties of Young’s function. Due to the properties of the dilation, the result will be presented as an inequality involving suitable parameters. As a result we get the necessary condition. As a discussion, we try to find several examples that satisfy the inequalities. The most important, the result shows that there is a significant factor to see the boundedness of the Stein-Weiss operator on Orlicz-Morrey spaces. Since Stein-Weiss operator is a generalization of fractional integral operator, the result shows that it generalize the boundedness of fractional integral on Orlicz space.",