Transference and Restriction of Bilinear Fourier Multipliers on Orlicz Spaces

Let G be a locally compact abelian group with Haar measure mG and let Phi(i, i) = 1,2,3, be Young functions. A bounded measurable function m on GxG is a (Phi(1), Phi(2); Phi(3))-bilinear multiplier if there exists C>0 B-m(f, g)(gamma) = integral(G)integral(G) m(x, y)(f) over cap (x)(g) over cap (y)gamma(x+y)dmG(x)dmG(y), satisfies N Phi(3)(B-m(f, g)) <= CN Phi 1(f)N-Phi 2(g) for functions in f,g is an element of L-1((G) over cap) such that (f) over cap,(g) over cap is an element of L-1(G). We denote by BM(Phi(1),Phi(2); Phi(3))(G) the space of all bilinear multipliers on GxG and study some properties of this class. We consider (Phi(1),Phi(2);Phi(3))-bilinear multipliers on various groups such as R x R, D x D, Z x Z and T x T. In particular we prove, under certain assumptions involving the norm of the dilation operator on the Orlicz spaces, that regulated bilinear multipliers in BM(Phi(1), Phi(2); Phi(3))(R) coincide with BM(Phi 1,Phi 2; Phi 3)(D) with where D stands for the real line with the discrete topology. Moreover, we investigate several transference and restriction results on multipliers acting on Z x Z and T x T.