Boundedness of bilinear radial Fourier multipliers

Abstract

We show that a bilinear radial Fourier multiplier operator with symbol $\sigma$ is $L^2(\R^n)\times L^2(\R^n) \to L^1(\R^n)$ bounded, $n\in \mathbb N,$ if the function $\sigma$ satisfies the smoothness condition $\sigma(2^j\cdot)\Phi\in L^2_{1/2 +\epsilon}(\mathbb R^{2n})$ 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.