Dual spaces for martingale Musielak-Orlicz Lorentz Hardy spaces

In this article, under the mild assumption that the Doob maximal operator is bounded on Musielak-Orlicz spaces, the authors first establish the atomic characterization of martingale Musielak-Orlicz Lorentz Hardy spaces. Using atomic characterizations, the authors then clarify the relations among five martingale Musielak-Orlicz Lorentz Hardy spaces and construct the generalized martingale BMO type spaces which prove to be the dual spaces of martingale Musielak- Orlicz Lorentz Hardy spaces. As applications, the authors further investigate John-Nirenb erg inequalities by the dual method. A novelty of this article is to apply the assumption on the boundedness of the Doob maximal operator to develop