On the boundedness of multilinear fractional strong maximal operators with multiple weights

We investigate the boundedness of multilinear fractional strong maximal operator M_R,α associated with rectangles or related to more general basis with multiple weights A_{(P, Q), R}. In the rectangular setting, we first give an end-point estimate of M_R,α which not only extends the famous linear result of Jessen, Marcinkiewicz and Zygmund, but also extends the multilinear result of Grafakos, Liu, Perez and Torres (α = 0)to the case 0<α < mn. Then, in the one weight case, we give several equivalent characterizations between M_Rα and A_{(P, Q), R}. Ep Based on the Carleson embedding theorem regarding dyadic rectangles, we obtain a multilinear Fefferman-Stein type inequality, which is new even in the linear case. We present a sufficient condition for the two weighted norm inequality of M_Rα and establish a version of the vector-valued two weighted inequality for the strong maximal operator when m = 1. In the general basis setting, we study the properties of the multiple weight A_{(P, Q), R}. conditions, including the equivalent characterizations and monotonic properties, which essentially extends previous understanding. Finally, a survey on multiple strong Muckenhoupt weights is given, which demonstrates the properties of multiple weights related to rectangles systematically.