Abstract
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–Nirenberg 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 a theory of martingale spaces and hence the obtained results have a wide generality.
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