Trilinear Fourier multipliers on Hardy spaces

Abstract

In this paper, we obtain the H p 1 × H p 2 × H p 3 → H p boundedness for trilinear Fourier multiplier operators, which is a trilinear analogue of the multiplier theorem of Calderón and Torchinsky [4]. Our result improves the trilinear estimate in [22] by additionally assuming an appropriate vanishing moment condition, which is natural in the boundedness into the Hardy space H p for 0 < p ≤ 1.

On the boundedness of multilinear Fourier multipliers on Hardy spaces

“…, pm ≤ 1 with 1/p1 + • • • 1/pm = 1/p, under suitable cancellation conditions. As a result, we extend the trilinear estimates in [17] to general multilinear ones and improve the boundedness result in [18] in limiting situations.…”

“…, p m ≤ 1. As mentioned in [17], the condition (1.13) is very natural and necessary in the boundedness into the Hardy space H p for 0 < p ≤ 1, in view of (1.8). For a multilinear extension of the above boundedness result, we now consider a bounded function σ on (R n ) m having the property…”