Boundedness of bilinear radial Fourier multipliers

Abstract

We show that a bilinear radial Fourier multiplier operator with symbol $\sigma$ is $L^2({\mathbb{R}}^n)\times L^2({\mathbb{R}}^n)\to L^1({\mathbb{R}}^n)$ bounded, $n\in \mathbb{N},$ if the function $\sigma$ satisfies the smoothness condition $\sigma (2^j\cdot )\Phi \in L_{1/2+\epsilon }^2({\mathbb{R}}^{2n})$ for some $\epsilon >0$ and every $j\in \mathbb{Z},$ where $\Phi$ is a smooth cutoff function adapted to the annulus $|x|\in [1/4,4]$. This condition is dimension free. We also apply similar reasoning to provide alternative proof of the initial result concerning multilinear Bochner-Riesz operator and prove an estimate for generalized bilinear Bochner-Riesz operator.