The Hörmander multiplier theorem for multilinear operators

Abstract

Abstract. In this paper, we provide a version of the Mihlin-Hörmander multiplier theorem for multilinear operators in the case where the target space is L p for p ≤ 1. This extends a recent result of Tomita [15] who proved an analogous result for p > 1.

Weighted estimates for multilinear Fourier multipliers

“…As a consequence, T m is bounded from L p 1 (R n ) × · · · × L p N (R n ) to L p (R n ) for all 1 < p 1 , · · · , p N , p < ∞ satisfying 1/p 1 + · · · + 1/p N = 1/p with s = ⌊N n/2⌋ + 1 in (1.1), where ⌊N n/2⌋ is the integer part of N n/2. Grafakos and Si [11] gave similar results for the case p ≤ 1 by using L r -based Sobolev spaces, 1 < r ≤ 2.…”

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

“…However, to obtain other values of 1/2 < r ≤ 1, which is also natural in the bilinear case, it appears that one needs to impose higher regularity. Grafakos-Si [9] showed after Tomita's work that one can push p, q to 1 + (i.e. r to 1/2 + /2 ) if…”

“…Essentially, one may say that 2n derivatives in L 1 may be required to get the full range of exponents. 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…”

Minimal smoothness conditions for bilinear Fourier multipliers

“…These fundamental results were given by Coifman-Meyer [3,4], Kenig-Stein [13], and Grafakos-Torres [12]. In the last decade, the research on bilinear (multilinear) multipliers of limited smoothness has been developed by several authors; here we mention Tomita [20], Grafakos-Si [11], Grafakos-Miyachi-Tomita [9], Miyachi-Tomita [15], and Park [17].…”