Bilinear multipliers on weighted Orlicz spaces,Georgian Mathematical Journal

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Bilinear multipliers on weighted Orlicz spaces
Georgian Mathematical Journal ( IF 0.7 ) Pub Date : 2023-11-19 , DOI: 10.1515/gmj-2023-2099
Rüya Üster ₁

Affiliation

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.

更新日期：2023-11-19

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