ON THE BOUNDEDNESS OF MULTILINEAR FRACTIONAL STRONG MAXIMAL OPERATORS WITH MULTIPLE WEIGHTS

We investigate the boundedness of multilinear fractional strong maximal operator M-R,M-alpha associated with rectangles or related to more general basis with multiple weights A(((p) over right arrowp,q)),(R). In the rectangular setting, we first give an end-point estimate of M-R,M-alpha, 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 (alpha = 0) to the case 0 < alpha < mn. Then, in the one weight case, we give several equivalent characterizations between M-R,M-alpha and A(((p) over right arrowp,q)),(R). 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-R,M-alpha 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) over right arrowp,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.