Atomic subspaces of $${L_1}$$ L 1 -martingale spaces

Abstract

We introduce and study some new martingale spaces \({\mathcal{H}_r}\) between martingale Hardy spaces \({H_1}\) and \({L_1}\) for \({0 < r \leqq \infty}\). We investigate the dual spaces of \({\mathcal{H}_r}\) for \({1 < r \leqq \infty}\) and some embeddings for \({0 < r \leqq \infty}\). As applications, we obtain the atomic decomposition of \({L_1}\)-martingales, and measure the distance from \({L_1}\)-martingale space to the Hardy space \({H_1}\). We also obtain some similar results for two-parameter martingales when the \({\sigma}\)-algebra filtration \({\mathcal{F} = ({\mathcal{F}}_{n,m}, \, (n, \, m) \in {\mathbb{N}}^{2})}\) is generated by finitely many atoms.