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

摘要

In this paper, the multilinear fractional strong maximal operator$\mathcal{M}_{\mathcal{R},\alpha}$ associated with rectangles and correspondingmultiple weights $A_{(\vec{p},q),\mathcal{R}}$ are introduced. Under the dyadicreverse doubling condition, a necessary and sufficient condition for two-weightinequalities is given. As consequences, we first obtain a necessary andsufficient condition for one-weight inequalities. Then, we give a new proof forthe weighted estimates of multilinear fractional maximal operator$\mathcal{M}_\alpha$ associated with cubes and multilinear fractional integraloperator $\mathcal{I}_{\alpha}$, which is quite different and simple from theproof known before.