Littlewood-Paley Characterizations of Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces
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Complex Analysis and Operator Theory
Littlewood-Paley Characterizations of Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces In Memory of Professor Carlos Berenstein Der-Chen Chang1,2 · Songbai Wang3 · Dachun Yang4
· Yangyang Zhang4
Received: 9 November 2019 / Accepted: 11 March 2020 © Springer Nature Switzerland AG 2020
Abstract Let X be a ball quasi-Banach function space on Rn . In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X and is bounded on the associated space, the authors establish various Littlewood–Paley function characterizations of the Hardy space H X (Rn ) associated with X , under some weak assumptions on the Littlewood– Paley functions. To this end, the authors also establish a useful estimate on the change of angles in tent spaces associated with X . All these results have wide p applications. Particularly, when X := Mr (Rn ) (the Morrey space), X := L p (Rn ) (the mixed-norm Lebesgue space), X := L p(·) (Rn ) (the variable Lebesgue space), p r ) (Rn ) (the Orlicz-slice X := L ω (Rn ) (the weighted Lebesgue space) and X := (E t space), the Littlewood–Paley function characterizations of H X (Rn ) obtained in this article improve the existing results via weakening the assumptions on the Littlewood– Paley functions and widening the range of λ in the Littlewood–Paley gλ∗ -function characterization of H X (Rn ). Keywords Ball quasi-Banach function space · Hardy space · Littlewood–Paley function · Extrapolation · Tent space · Maximal function Mathematics Subject Classification Primary 42B25; Secondary 42B30 · 42B35 · 46E30
Communicated by Irene Sabadini. This article is part of the topical collection “In honor of CA Berenstein” edited by Irene Sabadini and D. C. Struppa. This project is supported by the National Natural Science Foundation of China (Grant Nos. 11971058, 11761131002 and 11671185). Der-Chen Chang is partially supported by an NSF Grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University. Extended author information available on the last page of the article 0123456789().: V,-vol
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D.-C. Chang et al.
1 Introduction The real-variable theory of the classical Hardy space H p (Rn ) with p ∈ (0, 1] was originally initiated by Stein and Weiss [63] and further developed by Fefferman and Stein [24]. It is well known that the classical Hardy space H p (Rn ) with p ∈ (0, 1] plays a key role in harmonic analysis, partial differential equations and other analysis subjects. In particular, when p ∈ (0, 1], H p (Rn ) is a good substitute of the Lebesgue space L p (Rn ) in the study on the boundedness of Calderón–Zygmund operators. In recent decades, in order to meet the requirements arising in the study on the boundedness of operators, partial differential equations and some other analysis subjects, various variants of Hardy spaces have been introduced and their real-variable theories have been well developed; these variants of Hardy spaces were built on
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