Weighted estimates for the multilinear maximal function

TL;DR

This paper develops weighted estimates for the multilinear maximal function, extending classical theorems like Carleson embedding, Sawyer's two weight theorem, and the B_p theorem into the multilinear setting, providing new bounds and inequalities.

Contribution

It introduces a multilinear Carleson embedding theorem, a multilinear Sawyer's two weight theorem, and extends the B_p theorem to the multilinear context, along with mixed weighted bounds.

Findings

Established a multilinear Carleson embedding theorem.

Proved a multilinear analogue of Sawyer's two weight theorem.

Derived a mixed A_{\vec{P}}-W_{\vec{P}}^{\infty} bound for the multilinear maximal function.

Abstract

A formulation of the Carleson embedding theorem in the multilinear setting is proved which allows to obtain a multilinear analogue of Sawyer's two weight theorem for the multisublinear maximal function \mathcal{M} introduced in Lerner et al. A multilinear version of the B_p theorem from Hyt\"onen and P\'erez is also obtained and a mixed A_{\vec{P}}-W_{\vec{P}}^{\infty} bound for \mathcal{M} is proved as well.