Multilinear Fourier multipliers\cr with minimal Sobolev regularity, I

Abstract

We find optimal conditions on $m$-linear Fourier multipliers to give rise to bounded operators from a product of Hardy spaces $H^{p_j}$, $0<p_j\le 1$, to Lebesgue spaces $L^p$. The conditions we obtain are necessary and sufficient for boundedness and are expressed in terms of $L^2$-based Sobolev spaces. Our results extend those obtained in the linear case ($m=1 $) by Calder\'on and Torchinsky [this http URL] and in the bilinear case ($m=2$) by Miyachi and Tomita [this http URL&vol=29&iss=2&rank=4]. We also prove a coordinate-type H\"ormander integral condition which we use to obtain certain extreme cases.