The multilinear strong maximal function

TL;DR

This paper introduces a multivariable strong maximal function, establishes sharp distributional estimates, characterizes its boundedness on weighted Lebesgue spaces, and explores related multilinear maximal functions and interpolation results.

Contribution

It presents a new multivariable strong maximal function, derives sharp distributional estimates, and characterizes boundedness on weighted Lebesgue spaces, extending classical results to a multivariable setting.

Findings

Sharp distributional estimates for the multivariable strong maximal function

Characterization of boundedness on weighted Lebesgue spaces with multiple weights

Interpolation results between distributional estimates

Abstract

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Certain bilinear interpolation results between distributional estimates, such as that obtained for the multivariable strong maximal function, are also proved.