Bilinear Bochner-Riesz Means on the Complex Sphere

Abstract

In this paper, we establish the boundedness of the bilinear Bochner-Riesz means ${\mathcal{B}}_R^{\alpha }$ on the complex sphere $\mathbb{S}$. More precisely, we prove that ${\mathcal{B}}_R^{\alpha }$ is bounded from $L^{p_1}(\mathbb{S})\times L^{p_2}(\mathbb{S})\to L^p(\mathbb{S})$ where $1/p_1+1/p_2=1/p$ and $1\le p_1,p_2\le \infty$, for an admissible range of exponents, with the required smoothness parameter $\alpha$ described in terms of the topological dimension of $\mathbb{S}$. To facilitate our proof, we establish several analytic estimates, including restriction-type estimates, weighted Plancherel estimates with large power of weights and bilinear weighted Plancherel estimates, which are derived from the ground up in our setting and may be considered of independent interest.

56 pages