In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak-type and strong-type inequalities on Doob’s maximal operator, and we get a -atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John–Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with a probabilistic characterization of to replace the so-called log-Hölder continuity condition in .
"Martingale Hardy spaces with variable exponents." Banach J. Math. Anal. 10 (4) 750 - 770, October 2016. https://doi.org/10.1215/17358787-3649326