A $B_p$ condition for the strong maximal function

TL;DR

This paper introduces a new $B_p$ condition for the strong maximal function, characterizes its boundedness, and explores two-weight inequalities, extending results to multilinear and other maximal operators.

Contribution

It defines a natural $B_p$ condition for the rectangle case and provides a sufficient condition for two-weight inequalities, including multilinear and other maximal functions.

Findings

Characterization of boundedness via $B_p$ condition

Sufficient conditions for two-weight inequalities

Extension to multilinear and other maximal operators

Abstract

A strong version of the Orlicz maximal operator is introduced and a natural $B_p$ condition for the rectangle case is defined to characterize its boundedness. This fact let us to describe a sufficient condition for the two weight inequalities of the strong maximal function in terms of power and logarithmic bumps. Results for the multilinear version of this operator and for others multi(sub)linear maximal functions associated with bases of open sets are also studied.