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

TL;DR

This paper introduces a multilinear fractional strong maximal operator related to rectangles and multiple weights, establishing two-weight inequalities and providing new, simplified proofs for related weighted estimates.

Contribution

It defines a new operator associated with rectangles, characterizes two-weight inequalities under reverse doubling conditions, and offers simplified proofs for existing weighted estimates.

Findings

Necessary and sufficient conditions for two-weight inequalities.

Necessary and sufficient conditions for one-weight inequalities.

Simplified proofs for weighted estimates of multilinear fractional operators.

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.