Commutators of Bilinear Pseudo-differential Operators on Local Hardy Spaces with Variable Exponents
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Commutators of Bilinear Pseudo-differential Operators on Local Hardy Spaces with Variable Exponents Guanghui Lu1 Received: 11 January 2019 / Accepted: 9 November 2019 © Sociedade Brasileira de Matemática 2019
Abstract The aim of this paper is to establish the boundedness of the commutator [b1 , b2 , Tσ ] generated by the bilinear pseudo-differential operator Tσ with smooth symbols and b1 , b2 ∈ BMO(Rn ) on product of local Hardy spaces with variable exponents. By applying the refined atomic decomposition result, the authors prove that the bilinear pseudo-differential operator Tσ is bounded from the Lebesgue space L p (Rn ) into h p1 (·) (Rn ) × h p2 (·) (Rn ). Moreover, the boundedness of the commutator [b1 , b2 , Tσ ] on product of local Hardy spaces with variable exponents is also obtained. Keywords Bilinear pseudo-differential operator · Commutator · BMO function · Local Hardy spaces with variable exponent Mathematics Subject Classification 42B20 · 42B35 · 47G30
1 Introduction m with δ, ρ ≥ 0 and m ∈ R, Hörmander (1967) introduced the class of symbols Sρ,δ n composed of a smooth function σ (x, ξ ) defined on R × Rn , such that for all multiindices α and β,
|Dξα Dxβ σ (x, ξ )| ≤ Cα,β (1 + |ξ |)m−ρ|α|+δ|α| ,
(1.1)
where Cα,β > 0 is independent of x and ξ . Respectively, the linear pseudo-differential operator Tσ associated with the symbol σ is defined by
The research is supported by the Doctoral Scientific Research Foundation of Northwest Normal University (No. 0002020203).
B 1
Guanghui Lu [email protected] College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, People’s Republic of China
123
G. Lu
Tσ f (x) =
Rn
σ (x, ξ )e2πi x·ξ f (x)dξ,
where f is a Schwartz function and f represents the Fourier transform of f . Since then, many authors have paid much attention to study the boundedness properties of the pseudo-differential operators with various symbols and the corresponding commutators on different function spaces, for example, see (Hörmander 1967; Alvarez and Hounie 1990; Chanillo and Torchinsky 1986; Taylor 1991; Hwang and Lee 1994; Tang 2012; Hu and Zhou 2018; Lin 2008) and their references therein. On the other hand, the theory of pseudo-differential operators has played an important role in harmonic analysis and PDE (see Hörmander 1967; Taylor 1991; Bényi and Torres 2003). In 2003, Bényi and Torres (2004) and Herbert and Naibo (2014) gave out the definition of the bilinear pseudo-differential operators with the bilinear Hörmander classes and got the bounded properties of the bilinear pseudo-differential operators. In addition, the bilinear theory exhibits the similarities and differences with its linear counterpart. Therefore, some authors have spent much more time on researching the properties of the bilinear pseudo-differential operators on different kind of the function spaces and theirs applications (see Bényi et al. 2010; Xiao et al. 2012; Grafakos et al. 2002; Gilbert and Nahmod 1999; Michalowski et al. 2014). Regarding as an important space in
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