Bilinear pseudo-differential operators on product of weighted spaces
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Bilinear pseudo-differential operators on product of weighted spaces Guanghui Lu1 · Jiang Zhou2 Received: 29 May 2020 / Revised: 14 August 2020 / Accepted: 2 September 2020 © Springer Nature Switzerland AG 2020
Abstract σ be a bilinear pseudo-differential operator and ω = (ω1 , ω2 ) ∈ A P , where Let T P = ( p1 , p2 ), pi ∈ [1, ∞) with i = 1, 2, and A P denotes the multiple weight σ is bounded from the product of weighted class. In this paper, we will prove that T p1 p2 p n n Lebesgue spaces L ω1 (R ) × L ω2 (R ) into the L νω (Rn ), where 1p = p11 + p12 p 2 pi and νω (x) = i=1 [ωi (x)] , and bounded from the weighted Morrey spaces p ,κ,ρ p ,κ,ρ p,κ,ρ L ω11 (Rn ) × L ω22 (Rn ) into the L νω (Rn ). Furthermore, the endpoint estimate σ on the weighted Hardy space Hω1 (Rn ) is also obtained. for T Keywords Bilinear pseudo-differential operator · Multiple weight · Weighted Morrey space · Weighted Hardy space Mathematics Subject Classification 42B35 · 47B37
1 Introduction As we all know, although the weighted inequalities aries naturally in Fourier analysis, their use is best justified by the many different kinds of applications, see [1]. In 1979, p García-Cuerva in [2] gave out the definition of weighted Hardy spaces Hω . Komori and Shirai in 2009 first introduced the weighted Morrey spaces and established the boundedness of the Hardy–Littlewood maximal operator and the Calderón–Zygmund
B
Guanghui Lu [email protected] Jiang Zhou [email protected]
1
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, People’s Republic of China
2
College of Mathematics and System Sciences, Xinjiang University, Ürümqi 830046, Xinjiang, People’s Republic of China
G. Lu, J. Zhou
operators on these spaces (see [3]). Especially, when ω is regarded as the constant 1, the above weighted Hardy space and the weighted Morrey space just go back to the classical Hardy space and the Morrey space introduced by [4,5], respectively. Since then, many authors have paid much attention to the properties and boundedness of the different kinds of operators on Hardy space and Morrey space, for example, the readers can see [6–11] and their references therein. In 1967, Hörmander [12] first introduced the definition of the Hörmander class of pseudo-differential operators. That is, for any f ∈ S, the pseudo-differential operator Tσ is defined by Tσ f (x) :=
Rn
σ (x, ξ ) f (ξ )ei x·ξ dξ,
(1.1)
m is smooth function for where f represents the Fourier transform of f , and σ ∈ Sρ,δ n n (x, ξ ) ∈ R × R and β
|∂xα ∂ξ σ (x, ξ )| ≤ Cα,β (1 + |ξ |)m−ρ|β|+δ|α| , m ∈ R and ρ, δ ∈ [0, 1], here the constants Cα,β is independent of x and ξ . What’s more, such pseudodifferential operators generalize the notion of differential operators with variable coefficients. Thus, many articles focus on the boundedness of the pseudo-differential operators on various of spaces. For example, in 1972, Calderón and Vaillancourt obtained the boundedness of the pseudo-differential operators on the Lebesgue space L p (Rn )
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