Transference and restriction of Fourier multipliers on Orlicz spaces
Abstract: Let G be a locally compact abelian group with Haar measure and be Young functions. A bounded measurable function m on G is called a Fourier ‐multiplier if defined for functions in such that , extends to a bounded operator from to . We write for the space of ‐multipliers on G and study some properties of this class. We give necessary and sufficient conditions for m to be a ‐multiplier on various groups such as , and . In particular, we prove that regulated ‐multipliers defined on coincide with ‐multiplier… Show more
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Cited by 4 publications
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Transference and Restriction of Bilinear Fourier Multipliers on Orlicz Spaces
Let 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 .
Transference and Restriction of Bilinear Fourier Multipliers on Orlicz Spaces
Let 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 .
Fourier Type Operators on Orlicz Spaces and the Role of Orlicz Lebesgue Exponents
We deduce continuity and (global) wave-front properties of classes of Fourier multipliers, pseudo-differential, and Fourier integral operators when acting on Orlicz spaces, or more generally, on Orlicz–Sobolev type spaces. In particular, we extend Hörmander’s improvement of Mihlin’s Fourier multiplier theorem to the framework of Orlicz spaces. We also show how Young functions $$\Phi $$ Φ of the Orlicz spaces are linked to properties of certain Lebesgue exponents $$p_\Phi $$ p Φ and $$q_\Phi $$ q Φ emerged from $$\Phi $$ Φ .
Linear and bilinear Fourier multipliers on Orlicz modulation spaces
Let $$\Phi _i, \Psi _i$$ Φ i , Ψ i be Young functions, $$\omega _i$$ ω i be weights and $$M^{\Phi _i,\Psi _i}_{\omega _i}(\mathbb {R} ^{d})$$ M ω i Φ i , Ψ i ( R d ) be the corresponding Orlicz modulation spaces for $$i=1,2,3$$ i = 1 , 2 , 3 . We consider linear (respect. bilinear) multipliers on $$\mathbb {R} ^{d}$$ R d , that is bounded measurable functions $$m(\xi )$$ m ( ξ ) (respect. $$m(\xi ,\eta )$$ m ( ξ , η ) ) on $$\mathbb {R} ^{d}$$ R d (respect. $$\mathbb {R} ^{2d}$$ R 2 d ) such that $$\begin{aligned} T_m(f)(x)=\int _{\mathbb {R} ^{d}}{\hat{f}}(\xi ) m(\xi )e^{2\pi i \langle \xi , x\rangle }d\xi \end{aligned}$$ T m ( f ) ( x ) = ∫ R d f ^ ( ξ ) m ( ξ ) e 2 π i ⟨ ξ , x ⟩ d ξ (respect. $$\begin{aligned} B_m(f_1,f_2)(x)=\int _{\mathbb {R} ^{d}}\int _{\mathbb {R} ^{d}} \hat{f_1}(\xi ) \hat{f_2}(\eta )m(\xi ,\eta )e^{2\pi i \langle \xi +\eta , x\rangle }d\xi d\eta \end{aligned}$$ B m ( f 1 , f 2 ) ( x ) = ∫ R d ∫ R d f 1 ^ ( ξ ) f 2 ^ ( η ) m ( ξ , η ) e 2 π i ⟨ ξ + η , x ⟩ d ξ d η define a bounded linear (respect. bilinear) operator from $$M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})$$ M ω 1 Φ 1 , Ψ 1 ( R d ) to $$M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})$$ M ω 2 Φ 2 , Ψ 2 ( R d ) (respect. $$M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})\times M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})$$ M ω 1 Φ 1 , Ψ 1 ( R d ) × M ω 2 Φ 2 , Ψ 2 ( R d ) to $$M^{\Phi _3,\Psi _3}_{\omega _3}(\mathbb {R} ^{d})$$ M ω 3 Φ 3 , Ψ 3 ( R d ) ). In this paper we study some properties of these spaces and give methods to generate linear and bilinear multipliers between Orlicz modulation spaces.
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