New variable martingale Hardy spaces

Abstract

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

New fractional maximal operators in the theory of martingale Hardy and Lebesgue spaces with variable exponents

“…Some years later, we have systematically studied and developed this theory in [16]. The applications of variable martingale Hardy spaces to Fourier analysis were investigated in [16] while its applications to stochastic integrals in [18]. Martingale Musielak-Orlicz Hardy spaces were considered in Xie et al [47][48][49].…”

Variable Hardy–Lorentz martingale spaces Hp(·),q(·)$\mathcal {H}_{p(\cdot ),q(\cdot )}$

“…Remark 4.4. In [25,Theorem 5.4], the authors obtained a similar result for martingale Hardy spaces associated with  𝑝(⋅) (Ω) under the condition that 𝑝(⋅) ∈  0 ([0, 1]) is locally log-Hölder continuous. As mentioned in Section 2.1, the condition we impose on 𝑝(⋅) and 𝑞(⋅) is weaker.…”

“…Then, 𝜏 𝑘 is nondecreasing (see [25,Theorem 3.6]). Also, it is easy to see that {𝜈 𝑘 = 𝑗} ⊂ {𝜏 𝑘 ≤ 𝑗 − 1}, which implies that 𝜏 𝑘 < 𝜈 𝑘 on the set {𝜈 𝑘 < ∞}.…”

“…It is also noted that, Jiao et al. [25] very recently introduced the variable martingale Lebesgue spaces $$\mathcal {L}_{p(\cdot )}(\Omega )$$ defined by rearrangement functions, and established the dual theory and martingale inequalities for corresponding martingale Hardy spaces.…”

“…[16] and Jiao et al. [25], our main concern of this paper is the variable Hardy–Lorentz martingale spaces $$\mathcal {H}_{p(\cdot ),q(\cdot )}(\Omega )$$ associated with the variable Lorentz spaces $$\mathcal {L}_{p(\cdot ),q(\cdot )}(\Omega )$$; see the definitions in Sections 2.3 and 2.4, respectively. More precisely, we first establish the atomic decomposition theorems in martingale Hardy–Lorentz space $$\mathcal {H}_{p(\cdot ),q(\cdot )}(\Omega )$$ (see Section 3).…”