On the Boundedness of Multilinear Fractional Strong Maximal Operator with multiple weights

In this paper, we investigated the boundedness of multilinear fractional strong maximal operator $\mathcal{M}_{\mathcal{R},\alpha}$ associated with rectangles or related to more general basis with multiple weights $A_{(\vec{p},q),\mathcal{R}}$. In the rectangles setting, we first gave an end-point estimate of $\mathcal{M}_{\mathcal{R},\alpha}$, which not only extended the famous linear result of Jessen, Marcinkiewicz and Zygmund, but also extended the multilinear result of Grafakos, Liu, P\'{e}rez and Torres ($\alpha=0$) to the case $0<\alpha<mn.$ Then, in one weight case, we gave several equivalent characterizations between $\mathcal{M}_{\mathcal{R},\alpha}$ and $A_{(\vec{p},q),\mathcal{R}}$, by applying a different approach from what we have used before. Moreover, a sufficient condition for the two weighted norm inequality of $\mathcal{M}_{\mathcal{R},\alpha}$ was presented and a version of vector-valued two weighted inequality for the strong maximal operator was established when $m=1$. In the general basis setting, we further studied the properties of the multiple weights $A_{(\vec{p},q),\mathcal{R}}$ conditions, including the equivalent characterizations and monotonic properties, which essentially extended one's previous understanding. Finally, a survey on multiple strong Muckenhoupt weights was given, which demonstrates the properties of multiple weights related to rectangles systematically.