Doob’s maximal inequality and Burkholder–Davis–Gundy’s inequality on Musielak–Orlicz spaces

• Published in: Mathematische Zeitschrift 2023-03, Vol.303 (3), Article 53
• Main Authors: He, Lechen, Long, Long, Xie, Guangheng, Yang, Dachun
• Format: Article
• Language: English
• Subjects: Fourier analysis; Harmonic analysis; Martingales; Mathematics; Mathematics and Statistics; Orlicz space; Statistical analysis
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-023-03221-w

Given a probability space ( Ω , F , P ) and an unbounded Musielak–Orlicz function φ : Ω × [ 0 , ∞ ) → [ 0 , ∞ ] , the authors establish Doob’s maximal inequality and weak type Doob’s maximal inequality on the Musielak–Orlicz space L φ ( Ω ) . This result can be seen as a probabilistic counterpart of the analogue result in harmonic analysis proved by Hästö (J Funct Anal 269: 4038–4048, 2015). Using Doob’s maximal inequality and establishing the Davis decomposition for martingale Musielak–Orlicz Hardy spaces, the authors further prove Burkholder–Davis–Gundy’s inequality for the space L φ ( Ω ) . As applications, the authors obtain Doob’s maximal inequality and Burkholder–Davis–Gundy’s inequality for the Musielak–Orlicz function φ ( x , t ) with particular structure, including the variable Orlicz functions Φ ( t p ( x ) ) and [ Φ ( t ) ] p ( x ) , the perturbed variable exponent function t p ( x ) log ( e + t ) , and the variable double phase-growth t p ( x ) + a ( x ) t q ( x ) , which are completely new.