Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces
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Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces Xianjie YAN,
Dachun YANG,
Wen YUAN
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
c Higher Education Press 2020
Abstract Let X be a ball quasi-Banach function space satisfying some mild additional assumptions and HX (Rn ) the associated Hardy-type space. In this article, we first establish the finite atomic characterization of HX (Rn ). As an application, we prove that the dual space of HX (Rn ) is the Campanato space associated with X. For any given α ∈ (0, 1] and s ∈ Z+ , using the atomic and the Littlewood–Paley function characterizations of HX (Rn ), we also establish its s-order intrinsic square function characterizations, respectively, in terms of the intrinsic Lusin-area function Sα,s , the intrinsic g-function gα,s , and the ∗ intrinsic gλ∗ -function gλ,α,s , where λ coincides with the best known range. Keywords Ball quasi-Banach function space, Hardy space, finite atomic characterization, Campanato space, intrinsic square function MSC 42B25, 42B30, 42B35, 46E30 1
Introduction
As a family of function spaces, quasi-Banach function spaces include many known function spaces, for example, Lebesgue spaces, Lorentz spaces, variable Lebesgue spaces, and Orlicz spaces. For more details on quasi-Banach function spaces, we refer the reader to [1,4]. However, weighted Lebesgue spaces, Morrey spaces, mixed-norm Lebesgue spaces, Orlicz-slice spaces, and Musielak– Orlicz spaces are not necessarily quasi-Banach function spaces (see, for instance, [22,23,28,36] for more details and examples). Motivated by this, Sawano et al. [22] first introduced the ball quasi-Banach function spaces X, which extend quasi-Banach function spaces further so that the spaces we mentioned above are included. Sawano et al. [22] also introduced the Hardy-type spaces HX (Rn ), Received May 20, 2020; accepted June 25, 2020 Corresponding author: Wen YUAN, E-mail: [email protected]
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associated with X, and established their various maximal function characterizations by assuming that the Hardy–Littlewood maximal function is bounded on the p-convexification of X, and several other characterizations, respectively, in terms of atoms, molecules, and Lusin-area functions by further assuming the Fefferman–Stein vector-valued maximal inequality on X and the boundedness on the associated space of the powered Hardy–Littlewood maximal operator. Moreover, the local Hardy-type space hX (Rn ) and the Hardy-type space HX,L (Rn ), associated with an operator L, in this setting were also studied in [22]. Later, Wang et al. [26] further established the characterizations of HX (Rn ) and hX (Rn ), respectively, in terms of the Littlewood–Paley g-functions and gλ∗ -functions, and obtained the boundedness of Calder´ on– Zygmund operators and pseudo-differential operators, respectively, on HX (Rn ) and hX (Rn ). Furtherm
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