Transference and restriction of Fourier multipliers on Orlicz spaces

Let G be a locally compact abelian group with Haar measure mG$m_G$ and Φ1,Φ2$\Phi _1,\,\Phi _2$ be Young functions. A bounded measurable function m on G is called a Fourier (Φ1,Φ2)$(\Phi _1,\,\Phi _2)$‐multiplier if Tm(f)(γ)=∫Gm(x)f̂(x)γ(x)dmG(x),$$\begin{equation*}\hskip7pc T_m (f)(\gamma )= \int _{G} m(x) \hat{f}(x) \gamma (x) dm_G(x),\hskip-7pc \end{equation*}$$defined for functions in f∈L1(Ĝ)$f\in L^1(\hat{G})$ such that f̂∈L1(G)$\hat{f}\in L^1(G)$, extends to a bounded operator from LΦ1(Ĝ)$L^{\Phi _1}(\hat{G})$ to LΦ2(Ĝ)$L^{\Phi _2}(\hat{G})$. We write MΦ1,Φ2(G)$\mathcal {M}_{\Phi _1,\Phi _2}(G)$ for the space of (Φ1,Φ2)$(\Phi _1,\Phi _2)$‐multipliers on G and study some properties of this class. We give necessary and sufficient conditions for m to be a (Φ1,Φ2)$(\Phi _1,\,\Phi _2)$‐multiplier on various groups such as R,D,Z$\mathbb {R},\, {\bf D},\, \mathbb {Z}$, and T$\mathbb {T}$. In particular, we prove that regulated (Φ1,Φ2)$(\Phi _1,\,\Phi _2)$‐multipliers defined on R$\mathbb {R}$ coincide with (Φ1,Φ2)$(\Phi _1,\,\Phi _2)$‐multipliers defined on the real line with the discrete topology D, 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$\mathbb {Z}$ and T$\mathbb {T}$ are achieved.