关键词: martingales, Doob's inequality, Orlicz-Lorentz-Karamata spaces, convex functions

Abstract: Let $1\leq q\leq \infty$, $b$ be a slowly varying function and let $ \Phi: [0,\infty ) \longrightarrow [0,\infty ) $ be an increasing convex function with $\Phi(0)=0$ and $\lim\limits_{r \rightarrow \infty}\Phi(r)=\infty$. In this paper, we present a new class of Doob's maximal inequality on Orlicz-Lorentz-Karamata spaces $L_{\Phi,q,b}$. The results are new, even for the Lorentz-Karamata spaces with $\Phi(t)=t^p$, the Orlicz-Lorentz spaces with $b\equiv1$, and weak Orlicz-Karamata spaces with $q=\infty$ in the framework of $L_{\Phi,q,b}$. Moreover, we obtain some even stronger qualitative results that can remove the $\vartriangle_2$-condition of Liu, Hou and Wang (Sci China Math, 2010, 53(4): 905--916).

Key words: martingales, Doob's inequality, Orlicz-Lorentz-Karamata spaces, convex functions