Transference and restriction of Fourier multipliers on Orlicz spaces

Let.. be a locally compact abelian group with Haar measure mG and Phi(1), Phi(2) be Young functions. A bounded measurable function.. on.. is called a Fourier (Phi(1), Phi(2))-multiplier if. Tm(f)(gamma) = integral a m(x)f(x)gamma(x)dm,(x), defined for functions in.....1(..<^>) such that..<^>...1(..), extends to a bounded operator from..F1 (..<^>) to..F2 (..<^>). We write. F1,F2 (..) for the space of (F1, F2)multipliers on.. and study some properties of this class. We give necessary and sufficient conditions for.. to be a (F1, F2)-multiplier on various groups such as R,.., Z, and... In particular, we prove that regulated (F1, F2)-multipliers defined on R coincide with (F1, F2)-multipliers defined on the real line with the discrete topology.., under certain assumptions involving the norm of the dilation operator acting on Orlicz spaces. Also, several transference and restriction results on multipliers acting on Z and.. are achieved.