Maximal operators associated with bilinear multipliers of limited decay

TL;DR

This paper extends classical results to bilinear maximal operators with limited decay, establishing new boundedness results and improving dimension restrictions for bilinear spherical maximal operators.

Contribution

It introduces new boundedness results for bilinear maximal functions and reduces the dimension requirement for bilinear spherical maximal operators.

Findings

Boundedness of bilinear maximal Bochner-Riesz operator established.

Bilinear spherical maximal operator boundedness improved from dimension 8 to 4.

Results generalize Rubio de Francia's classical theorems to bilinear settings.

Abstract

Results analogous to those proved by Rubio de Francia are obtained for a class of maximal functions formed by dilations of bilinear multiplier operators of limited decay. We focus our attention to $L^2\times L^2\to L^1$ estimates. We discuss two applications: the boundedness of the bilinear maximal Bochner-Riesz operator and of the bilinear spherical maximal operator. For the latter we improve the known results by reducing the dimension restriction from $n\ge 8$ to $n\ge 4$.