The Boundedness of Fractional Integral Operators in Local and Global Mixed Morrey-type Spaces

In this paper, we introduced the local and global mixed Morrey-type spaces, and some properties of these spaces are also studied. After that, the necessary conditions of the boundedness of fractional integral operators $I_{\alpha}$ are studied respectively in mixed-norm Lebesgue spaces and the local mixed Morrey-type spaces. Last but not least, the boundedness of $I_{\alpha}$ is obtained by Hardy operators' boundedness in weighted Lebesgue spaces. As corollaries, we acquire the boundedness of $I_{\alpha}$ in mixed Morrey spaces(\hyperlink{Corollary 5.6}{Corollary 5.6}) and mixed Lebesgue spaces(\hyperlink{Corollary 5.7}{Corollary 5.7}) respectively. Particularly, \hyperlink{Corollary 5.7}{Corollary 5.7} obtains a sufficient and necessary condition.