The Hörmander multiplier theorem for n-linear operators

Trilinear Fourier multipliers on Hardy spaces

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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.

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“…The inequality (2.6) follows from the repeated use of the inequality in one dimensional setting that appears in [31, Chapter II, §5.10], and we omit the detailed proof here. Refer to [22,Appendix] for the argument.…”

“…The proof of the lemma is essentially same as that of [22,Lemma 2.4], so it is omitted here. and select q 1 , q 2 > 2 such that…”

“…, p m ≤ ∞, in the recent paper of the authors, Lee, Heo, Hong, Park, and Yang [22], we provide a multilinear multiplier theorem with standard Sobolev space conditions. Theorem D ( [22]). Let 0 < p 1 , • • • , p m ≤ ∞ and 0 < p < ∞ with 1/p = 1/p 1 + • • • + 1/p m .…”

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.…”