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

Abstract: In this paper, the multilinear fractional strong maximal operator $\mathcal{M}{\mathcal{R},\alpha}$ associated with rectangles and corresponding multiple weights $A{(\vec{p},q),\mathcal{R}}$ are introduced. 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 give a new proof for the weighted estimates of multilinear fractional maximal operator $\mathcal{M}\alpha$ associated with cubes and multilinear fractional integral operator $\mathcal{I}{\alpha}$, which is quite different and simple from the proof known before.