On the boundedness of multilinear Fourier multipliers on Hardy spaces

TL;DR

This paper establishes boundedness results for multilinear Fourier multipliers of Hörmander type on Hardy spaces, extending previous estimates and improving known results in limiting cases.

Contribution

It proves boundedness of multilinear Fourier multipliers of Hörmander type on Hardy spaces for a range of exponents, extending and refining prior multilinear estimates.

Findings

Extended trilinear estimates to general multilinear cases.

Proved boundedness under suitable cancellation conditions.

Improved previous boundedness results in limiting scenarios.

Abstract

In this paper, we study multilinear Fourier multiplier operators on Hardy spaces. In particular, we prove that the multilinear Fourier multiplier operator of H\"ormander type is bounded from $H^{p_1} \times \cdots \times H^{p_m}$ to $H^p$ for $0<p_1,\dots,p_m\le 1$ with $1/p_1 + \cdots 1/p_m = 1/p$, under suitable cancellation conditions. As a result, we extend the trilinear estimates of the authors(arXiv:2107.00225) to general multilinear ones and improve the boundedness result of the authors, Lee, Heo, Hong, Park, and Yang(Math. Ann. 381 : 499-555, 2021) in limiting situations.