Transference and Restriction of Bilinear Fourier Multipliers on Orlicz Spaces

Abstract

AbstractLet G be a locally compact abelian group with Haar measure $$m_G$$ m G and let $$\Phi _i$$ Φ i , $$i=1,2,3$$ i = 1 , 2 , 3 , be Young functions. A bounded measurable function m on $$G\times G$$ G × G is a $$(\Phi _1,\Phi _2;\Phi _3)$$ ( Φ 1 , Φ 2 ; Φ 3 ) -bilinear multiplier if there exists $$C>0$$ C > 0 such that the bilinear map $$\begin{aligned} B_m (f,g)(\gamma )= \int _{G}\int _{G} m(x,y) {{\hat{f}}}(x) {{\hat{g}}}(y) \gamma (x+y) dm_G(x)dm_G(y), \end{aligned}$$ B m ( f , g ) ( γ ) = ∫ G ∫ G m ( x , y ) f ^ ( x ) g ^ ( y ) γ ( x + y ) d m G ( x ) d m G ( y ) , satisfies $$N_{\Phi _3}(B_m(f,g))\le CN_{\Phi _1}(f)N_{\Phi _2}(g)$$ N Φ 3 ( B m ( f , g ) ) ≤ C N Φ 1 ( f ) N Φ 2 ( g ) for functions in $$f,g\in L^1({{\hat{G}}})$$ f , g ∈ L 1 ( G ^ ) such that $${{\hat{f}}},{{\hat{g}}}\in L^1(G)$$ f ^ , g ^ ∈ L 1 ( G ) . We denote by $${\mathcal {B}}{\mathcal {M}}_{(\Phi _1,\Phi _2;\Phi _3)}(G)$$ B M ( Φ 1 , Φ 2 ; Φ 3 ) ( G ) the space of all bilinear multipliers on $$G\times G$$ G × G and study some properties of this class. We consider $$(\Phi _1,\Phi _2;\Phi _3)$$ ( Φ 1 , Φ 2 ; Φ 3 ) -bilinear multipliers on various groups such as $${\mathbb {R}}\times {\mathbb {R}},\, \textbf{D}\times \textbf{D},\, {\mathbb {Z}}\times {\mathbb {Z}}$$ R × R , D × D , Z × Z and $${\mathbb {T}}\times {\mathbb {T}}$$ T × T . In particular we prove, under certain assumptions involving the norm of the dilation operator on the Orlicz spaces, that regulated bilinear multipliers in $${\mathcal {B}}{\mathcal {M}}_{(\Phi _1,\Phi _2;\Phi _3)}({\mathbb {R}})$$ B M ( Φ 1 , Φ 2 ; Φ 3 ) ( R ) coincide with $${\mathcal {B}}{\mathcal {M}}_{(\Phi _1,\Phi _2;\Phi _3)}(\textbf{D})$$ B M ( Φ 1 , Φ 2 ; Φ 3 ) ( D ) with where $$\textbf{D}$$ D stands for the real line with the discrete topology. Moreover, we investigate several transference and restriction results on multipliers acting on $${\mathbb {Z}}\times {\mathbb {Z}}$$ Z × Z and $${\mathbb {T}}\times {\mathbb {T}}$$ T × T .