Weak Musielak–Orlicz Hardy spaces and applications

Abstract

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∞(double-struckRn) weight uniformly in t∈(0,∞). In this article, the authors introduce the weak Musielak–Orlicz Hardy space WHφ(double-struckRn) 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(double-struckRn) (namely, φ(x,t):=w(x)tp for 0.33em 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(double-struckRn) (namely, φ(x,t):=tp for 0.33em 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(double-struckRn) in the critical case is presented.

Variable Hardy–Lorentz spaces

“…For , j and , set and where ϕ is as in , and . It follows from [, Theorem 4.5] that converges in . Denote this limitation by Following a similar argument used in the proof of [, Theorem 6.1], we find that for any and j , supp…”

New weak martingale Hardy spaces of Musielak–Orlicz type

“…Recently, Liang, Yang and Jiang [15] introduced the weak Musielak-Orlicz Hardy spaces and the results in [15] generalized the real-variable theory of weak Hardy spaces and weighted weak Hardy spaces. Inspired by [15], we introduce the martingale version of weak Musielak-Orlicz Hardy spaces.…”

“…Inspired by [15], we introduce the martingale version of weak Musielak-Orlicz Hardy spaces. The results obtained in this paper will generalize the previous results of weak martingale Hardy spaces and weak martingale Orlicz-Hardy spaces.…”

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation

“…Using the some ideas from Liu [54], we prove that the sum of the sequence of molecules indeed converges in the sense of distribution [see (3.27) below]. Then, applying some ideas used in the proof of [67,Theorem 3.21], we prove that, under some even weaker assumptions on the Lusin area function, the distribution which the sum of the aforementioned sequence of molecules converges to equals f in the sense of distribution, and hence seal the aforementioned gap existed in the proof of [52,Theorem 4.5]. To obtain the Littlewood-Paley g-function characterization of W H X (R n ), we first establish a discrete Calderón reproducing formula under some weak assumptions (see Lemma 3.19(i) below), and then we apply an approach initiated by Ullrich [71] and further developed by Liang et al [51], which provides one way to control the W X norm of the Lusin area function by the corresponding norm of the Littlewood-Paley g-function, via a key technical lemma (see Lemma 3.21 below) and an auxiliary Peetre type square function g a, * ( f ).…”

“…In lines 14 and 16 of [52, p. 662], when they used Calderón reproducing formula to decompose a distribution f into a sequence of atoms, Liang et al did not prove that the sum of the sequence of atoms converges to f in the sense of distribution. To seal this gap, we employ a different method from [52]. Via borrowing some ideas from [67], we first introduce the weak tent space associated to X and establish its atomic characterization (see Theorem 3.7 below), which is a key tool to decompose a distribution f into a sequence of molecules.…”