Minimal smoothness conditions for bilinear Fourier multipliers

Abstract

The problem of finding the differentiability conditions for bilinear Fourier multipliers that are as small as possible to ensure the boundedness of the corresponding operators from products of Hardy spaces $H^{p_1}\times H^{p_2}$ to $L^p$, $1/p_1+1/p_2=1/p$, is considered. The minimal conditions in terms of the product type Sobolev norms are given for the whole range $0<p_1,p_2\le \infty$.