### Abstract

Let (Ω,F,P) be a probability space and φ:Ω×[0,∞)→[0,∞) be a Musielak–Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak–Orlicz space L^{φ}(Ω). Using this and extrapolation method, the authors then establish a Fefferman–Stein vector-valued Doob maximal inequality on L^{φ}(Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for L^{φ}(Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak–Orlicz Hardy spaces H_{φ}
^{s}(Ω), P_{φ}(Ω), Q_{φ}(Ω), H_{φ}
^{S}(Ω) and H_{φ}
^{M}(Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak–Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on H_{φ}
^{S}(Ω) and H_{φ}
^{M}(Ω), the authors obtain the Burkholder–Davis–Gundy inequality associated with Musielak–Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from H_{φ}[0,1) to L^{φ}[0,1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

Original language | English |
---|---|

Pages (from-to) | 143-192 |

Number of pages | 50 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 182 |

DOIs | |

Publication status | Published - May 1 2019 |

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### Keywords

- Atom
- Burkholder–Davis–Gundy inequality
- Doob maximal operator
- Fejér operator
- Martingale Musielak–Orlicz Hardy space
- Musielak–Orlicz space
- Probability space
- Quadratic variation
- Weight

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Nonlinear Analysis, Theory, Methods and Applications*,

*182*, 143-192. https://doi.org/10.1016/j.na.2018.12.011