A remark on John – Nirenberg theorem for martingales

Abstract

This paper is mainly devoted to establishing an extension of the John – Nirenberg theorem for martingales, more precisely, let $1 < p < \infty$ and $0 < q < \infty$. If the stochastic basis $(\scr {F_n})_n\geq 0$ is regular, then $BMO_{p,q} = BMO_1$ with the equivalent norms. Our method is to use a new atomic decomposition construction of the martingale Hardy space.