Trilinear Fourier multipliers on Hardy spaces

TL;DR

This paper establishes boundedness results for trilinear Fourier multipliers on Hardy spaces, extending classical multiplier theorems and improving recent estimates by incorporating vanishing moment conditions.

Contribution

It introduces new boundedness criteria for trilinear Fourier multipliers on Hardy spaces, generalizing and strengthening previous results with additional vanishing moment assumptions.

Findings

Proves $H^{p_1} imes H^{p_2} imes H^{p_3} o H^p$ boundedness.

Extends Calderón-Torchinsky multiplier theorem to a trilinear setting.

Improves recent estimates by including vanishing moment conditions.

Abstract

In this paper, we obtain the $H^{p_1}\times H^{p_2}\times H^{p_3}\to H^p$ boundedness for trilinear Fourier multiplier operators, which is a trilinear analogue of the multiplier theorem of Calder\'on and Torchinsky (Adv. Math. 24 : 101-171, 1977). Our result improves the trilinear estimate in the very recent work of the authors, Lee, Heo, Hong, Park, and Yang (Math. Ann., to appear ) by additionally assuming an appropriate vanishing moment condition, which is natural in the boundedness into the Hardy space $H^p$ for $0<p\le 1$.