New weak martingale Hardy spaces of Musielak–Orlicz type

Abstract

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.

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

“…All these results improve and generalize the corresponding results on weak martingale Orlicz-Hardy spaces (see [12]). Moreover, we also improve all the results on weak martingale Musielak-Orlicz Hardy spaces in [29]. 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.…”

“…However, we establish the qatomic characterizations for any q ∈ (max{p + ϕ , 1}, ∞] in this article, where p + ϕ is the uniformly upper type index of ϕ. Moreover, in [12] for weak martingale Orlicz-Hardy spaces and [29] for weak martingale Musielak-Orlicz Hardy spaces, the results of atomic characterizations need the index p + ϕ = 1. Differently from [12,29], we allow p + ϕ ∈ (0, ∞) in Theorems 3.1, 3.2 and 3.5 below.…”

“…Indeed, for any given p ∈ (1, ∞), if ϕ(x, t) := t p for any x ∈ Ω and t ∈ (0, ∞), then ϕ is of uniformly lower type p and also of uniformly upper type p. However, in this case, ϕ is not of uniformly upper type 1. Thus, under Assumption 1.A, all the results in [29] can not cover the corresponding results on weak Lebesgue spaces W L p (Ω) with any given p ∈ (1, ∞) in [25,9].…”

“…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]. The first motivation of this article is to weaken Assumption 1.A of [29] and to remove the unnecessary assumption q Φ −1 ∈ (0, ∞) of [12]. Indeed, instead of Assumption 1.A, in this article, we always make the following assumption.…”

“…Observe that (1.3) is also quite restrictive. Indeed, using (1.3) with E = Ω, we find that, for any x ∈ Ω and t ∈ (0, ∞), Observe that all these assumptions for the boundedness of sublinear operators used in [9,25,12,29] ensure that T is bounded from some martingale Hardy spaces to some Lebesgue spaces, which, together with the fact that Musielak-Orlicz functions unify Orlicz functions and weights, motivates us to introduce the following assumption. Assumption 1.2.…”