Multilinear Fourier multipliers with minimal Sobolev regularity, I
Volume 144 / 2016
Colloquium Mathematicum 144 (2016), 1-30
MSC: 42B15, 42B30.
DOI: 10.4064/cm6771-10-2015
Published online: 4 February 2016
Abstract
We find optimal conditions on $m$-linear Fourier multipliers that give rise to bounded operators from products of Hardy spaces $H^{p_k}$, $0 \lt p_k\le 1$, to Lebesgue spaces $L^p$. These conditions are expressed in terms of $L^2$-based Sobolev spaces with sharp indices within the classes of multipliers we consider. Our results extend those obtained in the linear case ($m=1 $) by Calderón and Torchinsky (1977) and in the bilinear case ($m=2$) by Miyachi and Tomita (2013). We also prove a coordinate-type Hörmander integral condition which we use to obtain certain endpoint cases.
