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 establishing 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.