Generalized grand Lorentz martingale spaces

Abstract

In this article, we define a new class of function spaces $L_{p,q),\theta }$ which unify and generalize the Iwaniec–Sbordone spaces with $q=p$, grand Lorentz spaces $\theta =1$ and classical Lorentz spaces $\theta =0$. Based on the new space, we introduce a kind of Hardy-type space, $H_{p,q),\theta }^s$, via the martingale operators and then develop a theory of these martingale Hardy spaces. More specifically, the atomic decompositions of $H_{p,q),\theta }^s$ $(0<p\le 1$, $1<q<\infty )$ are established. As applications, the dual theorems for the new framework are provided. We also obtain a new John–Nirenberg-type inequality by the duality. The results extend the very recent work of Jiao et al. (2017) to the case of generalized grand Lorentz spaces. Our results are new, even for the grand Lorentz spaces.