Bilinear Bochner-Riesz Square Function and Applications

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 [22]. Another application concerns the L-p-estimates for bilinear fractional Schrodinger multipliers. Finally, we improve upon a result of Grafakos, He and Honzik [17] in the context of bilinear radial multipliers and provide a dimension-free sufficient condition on the bilinear multipliers for L-2 x L-2 -> L-1-boundedness of the associated maximal function. The generalised bilinear spherical maximal function is a particular example of such maximal functions.