Abstract
In 2011, Dekel et al. developed highly geometric Hardy spaces Hp(Θ), for the full range 0<p≤1, which were constructed by continuous multi-level ellipsoid covers Θ of Rn with high anisotropy in the sense that the ellipsoids can rapidly change shape from point to point and from level to level. In this article, when the ellipsoids in Θ rapidly change shape from level to level, the authors further obtain some real-variable characterizations of Hp(Θ) in terms of the radial, the non-tangential, and the tangential maximal functions, which generalize the known results on the anisotropic Hardy spaces of Bownik.
Highlights
As a generalization of the classical isotropic Hardy spaces Hp(Rn) [1], anisotropic Hardy spaces HAp (Rn) were introduced and investigated by Bownik [2] in 2003
We present several lemmas that are isotropic extensions in the setting of variable anisotropy, and we show the proof for the main result
Preliminary and Some Basic Propositions we first recall the notion of continuous ellipsoid covers Θ and we introduce the pointwise continuity for Θ
Summary
As a generalization of the classical isotropic Hardy spaces Hp(Rn) [1], anisotropic Hardy spaces HAp (Rn) were introduced and investigated by Bownik [2] in 2003. By Remark 1, we know that, for every continuous ellipsoid cover Θ, there exists an equivalent pointwise continuous ellipsoid cover Ξ This implies that their corresponding (quasi-)norms ρΘ(·, ·) and ρΞ(·, ·) are equivalent, and the corresponding Hardy spaces Hp(Θ) = Hp(Ξ)(0 < p ≤ 1) with equivalent (quasi-)norms (see ([9], Theorem 5.8)). N, N with equivalent (quasi-)norms, where Hqp, l(Θ) denotes the atomic Hardy space with pointwise variable anisotropy; see ([9], Definition 4.2). By Definition 3, we obtain that, for any N ≥ Np and N ≥ (a4N + 1)/a6, Hp (Θ) ⊆ Hp (Θ)
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