Transference and restriction of Fourier multipliers on Orlicz spaces

Let G be a locally compact abelian group with Haar measure m $m_G$ and Φ 1 , 2 $\Phi _1,\,\Phi _2$ Young functions. A bounded measurable function on is called Fourier ( ) $(\Phi _2)$ -multiplier if T f γ = ∫ x ̂ d $$\begin{equation*}\hskip7pc T_m (f)(\gamma )= \int _{G} m(x) \hat{f}(x) \gamma (x) dm_G(x),\hskip-7pc \end{equation*}$$ defined for functions in ∈ L $f\in L^1(\hat{G})$ such that $\hat{f}\in L^1(G)$ extends to operator from $L^{\Phi _1}(\hat{G})$ _2}(\hat{G})$ . We write M $\mathcal {M}_{\Phi _1,\Phi _2}(G)$ the space of -multipliers study some properties this class. give necessary sufficient conditions various groups as R D Z $\mathbb {R},\, {\bf D},\, \mathbb {Z}$ {T}$ In particular, we prove regulated {R}$ coincide real line discrete topology D, under certain assumptions involving norm dilation acting Orlicz spaces. Also, several transference restriction results multipliers are achieved.