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

Abstract

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.

Convergence of Vilenkin–Fourier series in variable Hardy spaces

“…Similar results for the anisotropic Hardy spaces 𝐻 𝑝(⋅) (ℝ) and 𝐻 𝑝(⋅),𝑞 (ℝ) can be found in Liu et al [23,24]. Martingale Musielak-Orlicz Hardy spaces were investigated in Xie et al [44][45][46]. Very recently, these results were generalized for martingale Hardy spaces with variable exponent in Jiao et al [18].…”

“…Martingale Musielak–Orlicz Hardy spaces were investigated in Xie et al. [44–46]. Very recently, these results were generalized for martingale Hardy spaces with variable exponent in Jiao et al.…”

Dual spaces for weak martingale Hardy spaces associated with rearrangement-invariant spaces

“…Remark 2.9. We know from Remark 2.3 that the class of weak martingale Hardy-type spaces associated with quasi-Banach function lattices that we investigate here includes classical weak martingale Hardy spaces [49,25], weak martingale variable Hardy spaces [24], and weak martingale Musielak-Orlicz Hardy spaces [52]. Therefore, we deal with the martingale theory for weak Hardy-type spaces universally.…”

Characterization of the boundedness of generalized fractional integral and maximal operators on Orlicz-Morrey and weak Orlicz-Morrey spaces

“…For other kinds of Orlicz-Morrey spaces, see [4,5,8,11,55], etc. For the study related to weak Orlicz and weak Morrey spaces, see [7,10,15,16,19,20,29,30,64], etc.…”