Necessary Condition for Boundedness of Stein-Weiss Operator on Orlicz-Morrey Spaces - Al Hazmy, Sofihara; Masta, Al Azhary; Fatimah, Siti; Batkunde, Harmanus (Journal of the Indonesian Mathematical Society, 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.