On multilinear fractional strong maximal operator associated with rectangles and multiple weights

In this paper, we introduce the multilinear fractional strong maximal operator MR, α and the corresponding multiple weights A(p→, q), R associated with rectangles. Under the dyadic reverse doubling condition, a necessary and sufficient condition for two-weight inequalities is given. As consequences, we first obtain a necessary and sufficient condition for one-weight inequalities. Then, we present a new proof for the weighted estimates of multilinear fractional integral operator and fractional maximal operator associated with cubes, which is quite different from and simpler than the proof that has been presented previously.