Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces

Abstract

Let X be a ball quasi-Banach function space satisfying some mild additional assumptions and H_X({ℝ}^n) the associated Hardy-type space. In this article, we first establish the finite atomic characterization of H_X({ℝ}^n). As an application, we prove that the dual space of H_X({ℝ}^n) is the Campanato space associated with X. For any given \alpha \in (\mathrm{0},\mathrm{1}] and s\in {\mathbb{Z}}_{+}, using the atomic and the Littlewood–Paley function characterizations of H_X({ℝ}^n),we also establish its s-order intrinsic square function characterizations, respectively, in terms of the intrinsic Lusin-area function S_{\alpha ,s},the intrinsic g-function g_{\alpha ,s},and the intrinsic g_{\lambda }^{\ast }-function g_{\lambda ,\alpha ,s}^{\ast }, where λ coincides with the best known range.

Key words： Ball quasi-Banach function space; Hardy space; finite atomic characterization; Campanato space; intrinsic square function

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