Multilinear Fourier Multipliers with Minimal Sobolev Regularity, I

TL;DR

This paper establishes optimal Sobolev space conditions for multilinear Fourier multipliers to be bounded from products of Hardy spaces to Lebesgue spaces, extending previous linear and bilinear results.

Contribution

It provides necessary and sufficient Sobolev regularity conditions for boundedness of multilinear Fourier multipliers, generalizing earlier linear and bilinear cases.

Findings

Derived optimal Sobolev space conditions for boundedness

Extended linear and bilinear multiplier results to multilinear case

Introduced a coordinate-type Hörmander integral condition

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 $) by Calder\'on and Torchinsky [http://www.sciencedirect.com/science/article/pii/S0001870877800169] and in the bilinear case ($m=2$) by Miyachi and Tomita [http://www.ems-ph.org/journals/show_abstract.php?issn=0213-2230&vol=29&iss=2&rank=4]. We also prove a coordinate-type H\"ormander integral condition which we use to obtain certain extreme cases.