Bilinear Bochner-Riesz means for convex domains and Kakeya Maximal function

Abstract

In this paper we introduce bilinear Bochner-Riesz means associated with convex domains in the plane $\mathbb R^2$ and study their $L^p-$boundedness properties for a wide range of exponents. One of the important aspects of our proof involves the use of bilinear Kakeya maximal function in the context of bilinear Bochner-Riesz problem. This amounts to establish suitable $L^p-$ estimates for the later. We also point out some natural connections between bilinear Kakeya maximal function and Lacey's bilinear maximal function.