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Transference on certain multilinear multiplier operators

Dashan Fan, Shuichi Sato

2001
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Journal of the Australian Mathematical Society
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We study DeLeeuw type theorems for certain multilinear operators on the Lebesgue spaces and on the Hardy spaces. As applications, on the torus we obtain an analog of Lacey-Thiele's theorem on the bilinear Hilbert transform, as well as analogies of some recent theorems on multilinear singular integrals by Kenig-Stein and by Grafakos-Torres. 2000 Mathematics subject classification: primary 42B15,42B20, 42B25. T e (f u f 2 ,...,f m )(x) for all fj e y(R"),j = 1,2,... ,m, where f) is the Fourier
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... ) is the Fourier transform of fj , i e l " and (uj,x) is the inner product of uj andx. We denote T = T t if e = 1. The significance of studying such kind of multilinear operators can be illustrated by following two simple model cases. First, in the case m = 1, T is the classical multiplier which plays very important roles in harmonic analysis and in partial differential equations (see [S]). Secondly, the study of the case m > 1 is much more involved. This topic can be dated back by the pioneering work of Coifman-Meyer started from 70's [CM1, CM2, CM3, CM4], as well as some recent works by Lacey-Thiele, Kenig-Stein and many others [KeS, CG, GK, GT, GW, LT]. Readers can see these references for more details about the background and significance in this topic. Here we list a simple example by letting n -1, m = 2 and taking k(u\, u 2 ) =' k(u 2 -Hi) with X(t) = i sgn(/), where sgn(O is the sign function on R 1 . Then it is easy to check that is the bilinear Hilbert transform, which is related to a famous conjecture by Calderon in studying certain problems of Cauchy integrals. Very recently, Lacey and Thiele [LT, La] solved this conjecture by proving that \\T(f,g)\\ p < C | | / | | , | | s | | r provided \/p = \/q + \/r, 1 < q, r < oo and 2/3 < p < oo. Analogously, we can define multilinear operators on the torus. The n-torus T" can be identified with R"/A, where A is the unit lattice which is an additive group of points in R" having integer coordinates. Let T" m be the m-fold product space T" x T" x • • • x T". The multilinear operators t e , s > 0, on T nm associated with the function k are defined by (1.2) f e ( / , , . . . J~m)(x) for all C°°(J") functions fj(x) = ^2a kj exp(2ni(kj,x)), j = 1 , 2 , We denote T = T e if e = 1. As we mentioned before, in the case m -1, T becomes the ordinary multiplier operator. One of the well-known results in that case is a theorem by DeLeeuw [L] (see also [SW, page 260]) which says that if k(u) is a continuous function on R" and if p > 1, then T is bounded on U (OT) if and only if f e is uniformly bounded on U (T") for e > 0. This theorem was later extended to many different settings. Readers can see [K, KT, AC, F, T, KaS] for further details of these generalizations. The main purpose of this paper is to extend DeLeeuw's theorem to the case m > 2. Letting \/p = £"_, \/Pj , we will establish the following theorems.

doi:10.1017/s1446788700002263
fatcat:oyioura6lfdyhhklqnnmwyj2qm