Abstract
Abstract Let p(·) : ℝ n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝ n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces Hp (·)(Θ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces Hp (Θ) on ℝ n with pointwise variable anisotropy of Dekel et al. [16] and variable anisotropic Hardy spaces of Liu et al. [24]. As an application, we establish the boundedness of variable anisotropic singular integral operators from Hp (·)(Θ) to Lp (·)(ℝ n ) in general and from Hp (·)(Θ) to itself under the moment condition, which generalizes the previous work of Bownik et al. [6] on Hp (Θ).
Highlights
The main purpose of this article is to introduce and to investigate the variable anisotropic Hardy spaces Hp(·)(Θ) with variable exponents
We introduce highly geometric Hardy spaces Hp(·)(Θ) via the radial grand maximal function and obtain its atomic decomposition, which generalizes that of Hardy spaces Hp(Θ) on Rn with pointwise variable anisotropy of Dekel et al [16] and variable anisotropic Hardy spaces of Liu et al [24]
We establish the boundedness of variable anisotropic singular integral operators from Hp(·)(Θ) to Lp(·)(Rn) in general and from Hp(·)(Θ) to itself under the moment condition, which generalizes the previous work of Bownik et al [6] on Hp(Θ)
Summary
The main purpose of this article is to introduce and to investigate the variable anisotropic Hardy spaces Hp(·)(Θ) with variable exponents. Inspired by Dekel et al [16] and Liu et al [24], the rst goal is to further introduce the variable anisotropic Hardy space Hp(·)(Θ) with variable exponent de ned via the radial grand maximal function and obtain its atomic decomposition. For this purpose, we rst introduce the variable anisotropic atomic Hardy space Hqp,(·l)(Θ) (see De nition 4.2 below) and prove. Let p(·) ∈ Clog(Rn) and q ∈ (max{p+, }, ∞] with p+ as in (2.4), and let Θ be a continuous ellipsoid cover and (p(·), q, l) an admissible triple as in De nition 4.1, Hqp,(·l)(Θ) = Hp(·)(Θ) with equivalent quasi-norms.
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