Bilinear multipliers and transference

(defined for Schwarzt test functions f and g in ) extends to a bounded bilinear operator from Lp1 (R)×Lp2 (R) into Lp3 (R). The theory of these multipliers has been tremendously developed after the results proved by Lacey and Thiele (see [16, 18, 17]) which establish that m(ξ,ν) = sign(ξ +αν) is a (p1, p2)-multiplier for each triple (p1, p2, p3) such that 1 < p1, p2 ≤∞, p3 > 2/3, and each α∈R \ {0,1}. The study of such multipliers was started by Coifman and Meyer (see [3, 4, 19]) for smooth symbols and new results for nonsmooth symbols, extending the ones given by the bilinear Hilbert transform, have been achieved by Gilbert and Nahmod (see [8, 9, 10]) and also by Muscalu et al. (see [20]). We refer the reader also to [7, 12, 11, 15] for new results on bilinear multipliers and related topics. In a recent paper (see [7]), Fan and Sato have shown certain de Leeuw-type theorems for transferring multilinear operators on Lebesgue and Hardy spaces fromRn to Tn. Here we will consider bilinear multipliers on Lebesgue spaces Lp(R) and get a characterization which allows us to transfer not only to the bilinear multipliers on T but also on Z. Our approach will follow closely the ideas in the original paper by de Leeuw (see [6]) and will