On Multilinear Fourier Multipliers of Limited Smoothness

Abstract

Abstract. In this paper, we prove certain L 2 -estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we extend the result of Calderón and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed.

Minimal regularity conditions for the end-point estimate of bilinear Calderón-Zygmund operators

“…We refer to [9] for the precise technical details. Similar results on the product of Hardy H p were very recently obtained by Grafakos-Miyachi-Tomita [8], but, as far as we know and unlike the linear case, there are no results of this type that give the end-point estimate…”

Weighted version of Carleson measure and multilinear Fourier multiplier

“…Taking g 2 L 2 .! 1 /, and using (3.2), we have as in the estimate for m in this case in [6], we have as in the estimate for m in [6],…”

“…In this paper, we first study this endpoint case. We will show that we can remove the restriction p j < 1 and get endpoint estimates to some degree as the following theorem, which is a weighted version of the results in [6]:…”

“…Grafakos and Si [7] gave a similar result for the case p Ä 1 by using the L r -based Sobolev spaces (1 < r Ä 2). In [6], Grafakos, Miyachi and Tomita proved that if m 2 L 1 .R nN / satisfies sup k2Z km k k W .s 1 ;:::;s N / .R nN / < 1 with s 1 ; : : : ; s N > n 2 ;…”

Estimates for the commutator of bilinear Fourier multiplier

“…This lemma can be proved by repeating the argument used in the proof of Theorem 6.1 in [7]. We omit the details for brevity.…”