Weak Musielak-Orlicz Hardy spaces and applications

Let φ:Rn×[0,∞)→[0,∞) satisfy that φ(x,·), for any given x∈Rn, is an Orlicz function and φ(·,t) is a Muckenhoupt A∞(Rn) weight uniformly in t∈(0,∞). In this article, the authors introduce the weak Musielak–Orlicz Hardy space WHφ(Rn) via the grand maximal function and then obtain its vertical or its n...

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Veröffentlicht in:Mathematische Nachrichten 2016-04, Vol.289 (5-6), p.634-677
Hauptverfasser: Liang, Yiyu, Yang, Dachun, Jiang, Renjin
Format: Artikel
Sprache:eng
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Zusammenfassung:Let φ:Rn×[0,∞)→[0,∞) satisfy that φ(x,·), for any given x∈Rn, is an Orlicz function and φ(·,t) is a Muckenhoupt A∞(Rn) weight uniformly in t∈(0,∞). In this article, the authors introduce the weak Musielak–Orlicz Hardy space WHφ(Rn) via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of WHφ(Rn), respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or gλ*‐function. All these characterizations for weighted weak Hardy spaces WHwp(Rn) (namely, φ(x,t):=w(x)tp for all t∈[0,∞) and x∈Rn with p∈(0,1] and w∈A∞(Rn)) are new and part of these characterizations even for weak Hardy spaces WHp(Rn) (namely, φ(x,t):=tp for all t∈[0,∞) and x∈Rn with p∈(0,1]) are also new. As an application, the boundedness of Calderón–Zygmund operators from Hφ(Rn) to WHφ(Rn) in the critical case is presented.
ISSN:0025-584X
1522-2616
DOI:10.1002/mana.201500152