Bilinear multipliers on weighted Orlicz spaces

Abstract

Let Φ i {\Phi_{i}} be Young functions and ω i {\omega_{i}} be weights on ℝ d {\mathbb{R}^{d}} , i = 1 , 2 , 3 {i=1,2,3} . A locally integrable function m ⁢ ( ξ , η ) {m(\xi,\eta)} on ℝ d × ℝ d {\mathbb{R}^{d}\times\mathbb{R}^{d}} is said to be a bilinear multiplier on ℝ d {\mathbb{R}^{d}} of type ( Φ 1 , ω 1 ; Φ 2 , ω 2 ; Φ 3 , ω 3 ) {(\Phi_{1},\omega_{1};\Phi_{2},\omega_{2};\Phi_{3},\omega_{3})} if 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 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 defines a bounded bilinear operator from L ω 1 Φ 1 ⁢ ( ℝ d ) × L ω 2 Φ 2 ⁢ ( ℝ d ) {L^{\Phi_{1}}_{\omega_{1}}(\mathbb{R}^{d})\times L^{\Phi_{2}}_{\omega_{2}}(% \mathbb{R}^{d})} to L ω 3 Φ 3 ⁢ ( ℝ d ) {L^{\Phi_{3}}_{\omega_{3}}(\mathbb{R}^{d})} . We deduce some properties of this class of operators. Moreover, we give the methods to generate bilinear multipliers between weighted Orlicz spaces.