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 $0<\alpha <mn\mathrm{.}$ 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.