Weak Orlicz–Hardy martingale spaces

Jiao, Yong; Wu, Liang; Peng, Lihua

doi:10.1142/s0129167x15500627

Cited by 23 publications

(24 citation statements)

References 28 publications

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“…Since then, atomic decompositions were established for many other martingale spaces, see [9] for weak martingale Hardy spaces, [10,12] for martingale Hardy-Lorentz spaces, [17] for Orlicz-Hardy martingale spaces, [11] for weak Orlicz-Hardy martingale spaces, [24] for weak Orlicz-Lorentz martingale spaces, [8] for Lorentz-Karamata martingale spaces, and so on.…”

Section: Introductionmentioning

confidence: 99%

New weak martingale Hardy spaces of Musielak–Orlicz type

Yang¹

2017

Ann. Acad. Sci. Fenn. Math.

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Abstract. The main purpose of this paper is to introduce the weak Musielak-Orlicz martingale spaces and establish several weak atomic decompositions for them. With the help of weak atomic decompositions, a sufficient condition for sublinear operators defined on weak MusielakOrlicz martingale spaces to be bounded is given. Using the sufficient condition, a series of martingale inequalities are obtained. These results are generalizations of the previous results of weak martingale Hardy spaces and weak martingale Orlicz-Hardy spaces.

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Section: Introductionmentioning

confidence: 99%

New weak martingale Hardy spaces of Musielak–Orlicz type

Yang¹

2017

Ann. Acad. Sci. Fenn. Math.

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“…All the results in [12] need the assumptions that Φ ∈ G ℓ for some ℓ ∈ (0, 1] and q Φ −1 ∈ (0, ∞), here Φ −1 denotes the inverse function of Φ. Observe that, when ϕ(x, t) := Φ(t) for any x ∈ Ω and t ∈ (0, ∞), ϕ satisfies Assumption 1.A if and only if Φ ∈ G ℓ for some ℓ ∈ (0, 1].…”

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confidence: 95%

“…On another hand, Jiao et al [12] studied weak martingale Orlicz-Hardy spaces under the following assumption. For any ℓ ∈ (0, ∞), let G ℓ be the set of all Orlicz functions Φ satisfying that Φ is of lower type ℓ and of upper type 1 (see, for example, [12,19]). Let Φ be a concave function and Φ ′ its derivative function.…”

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confidence: 99%

Atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applications

Xie¹,

Yang²

2019

Banach J. Math. Anal.

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Let (Ω, F , P) be a probability space and ϕ : Ω × [0, ∞) → [0, ∞) be a Musielak-Orlicz function. In this article, the authors establish the atomic characterizations of weak martingale Musielak-Orlicz Hardy spaces WH s ϕ (Ω), WH M ϕ (Ω), WH S ϕ (Ω), WP ϕ (Ω) and WQ ϕ (Ω). Using these atomic characterizations, the authors then obtain the boundedness of sublinear operators from weak martingale Musielak-Orlicz Hardy spaces to weak Musielak-Orlicz spaces, and some martingale inequalities which further clarify the relationships among WH s ϕ (Ω), WH M ϕ (Ω), WH S ϕ (Ω), WP ϕ (Ω) and WQ ϕ (Ω). All these results improve and generalize the corresponding results on weak martingale Orlicz-Hardy spaces. Moreover, the authors also improve all the known results on weak martingale Musielak-Orlicz Hardy spaces. In particular, both the boundedness of sublinear operators and the martingale inequalities, for the weak weighted martingale Hardy spaces as well as for the weak weighted martingale Orlicz-Hardy spaces, are new. IntroductionAs is well known, the classical weak Hardy spaces naturally appear when studying the boundedness of operators in critical cases. Indeed, Fefferman and Soria [6] originally introduced the weak Hardy space W H 1 (R n ) and proved in [6, Theorem 5] that some Calderón-Zygmund operators are bounded from W H 1 (R n ) to weak Lebesgue spaces W L 1 (R n ). It should also point out that Fefferman et al. [5] proved that the weak Hardy spaces are the intermediate spaces of Hardy spaces in the real interpolation method. Recently, various martingale Hardy spaces were investigated; see, for example, Weisz [26, 24, 23], Ho [7, 8], Nakai et al. [19, 20, 21], Sadasue [22] and Jiao et al. [11, 27] for various different martingale Hardy spaces and their applications. Moreover, the theory of weak martingale Hardy spaces has also been developed rapidly. The weak Hardy spaces consisting of Vilenkin martingales were originally studied by Weisz [25] and then fully generalized by Hou and Ren [9]. Inspired by these, Jiao et al. [13, 12] and Liu et al. [17, 16] investigated the weak martingale Orlicz-Hardy spaces associated with concave functions. Zhou et al. [31] introduced the weak martingale 2010 Mathematics Subject Classification. Primary 60G42; Secondary 60G46, 42B25, 42B35.

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“…In this section, we shall construct some new atomic decomposition theorems. We start with the following definition of weak atoms, see for example . Definition A measurable function a is said to be a w‐1‐ atom (or w‐2‐ atom , w‐3‐ atom , resp.)…”

Section: Modular Atomic Decompositionsmentioning

confidence: 99%

“…In this section, we shall construct some new atomic decomposition theorems. We start with the following definition of weak atoms, see for example [14,16,32].…”

Section: Modular Atomic Decompositionsmentioning

confidence: 99%