Bilinear Bochner-Riesz square function and applications

TL;DR

This paper introduces a bilinear Stein's square function related to Bochner-Riesz means, explores its $L^p$ boundedness, and applies it to bilinear multipliers, maximal functions, and fractional Schr"odinger multipliers, extending previous results.

Contribution

It develops a new bilinear Stein's square function, establishes its $L^p$ bounds, and applies it to various bilinear multiplier problems, including maximal functions and fractional Schr"odinger operators.

Findings

Established $L^p$ boundedness of the bilinear Stein's square function.

Derived $L^p$ estimates for bilinear fractional Schr"odinger multipliers.

Provided a dimension-free condition for $L^2 imes L^2 o L^1$ boundedness of bilinear maximal functions.

Abstract

In this paper we introduce Stein's square function associated with bilinear Bochner-Riesz means and investigate its $L^p$ boundedness properties. Further, we discuss several applications of the square function in the context of bilinear multipliers. In particular, we obtain results for maximal function associated with generalised bilinear Bochner-Riesz means. This extends the results proved in~\cite{JS}. Another application concerns the $L^p$ estimates for bilinear fractional Schr\"{o}dinger multipliers. Finally, we improve upon a result of Grafakos, He and Honzik~\cite{GHH} in the context of bilinear radial multipliers and provide a dimension free sufficient condition on the bilinear multipliers for $L^2\times L^2\rightarrow L^1$ boundedness of the associated maximal function. The generalised bilinear spherical maximal function is a particular example of such maximal functions.