Characterization of two-weighted inequalities for multilinear fractional maximal operator

In this paper, we restudied the two-weight problem of multilinear fractional maximal operator MαMα. First, we gave a characterization of two-weight inequalities for MαMα related to a multilinear analogue of Sawyer’s two-weight condition S(p→,q), which essentially improved and extended some known results before. This was done mainly by using the technique of the well known atomic decomposition of tent space. Then we obtained the strong boundedness of MαMα associated with multiple weight A(p→,q) class, which removed the power bump condition assumed in the known results before. Finally, a new two-weight B(p→,q) condition was introduced and the two-weight inequality of MαMα with B(p→,q) condition was established.