Transference on certain multilinear multiplier operators
Journal of the Australian Mathematical Society
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
... ) 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.