A Weak-Type Inequality for Submartingales and Itô Processes
Tom 63 / 2015
Bulletin Polish Acad. Sci. Math. 63 (2015), 73-88
MSC: Primary 60G44; Secondary 60G42.
DOI: 10.4064/ba63-1-9
Streszczenie
Let $\alpha\in [0,1]$ be a fixed parameter. We show that for any nonnegative submartingale $X$ and any semimartingale $Y$ which is $\alpha$-subordinate to $X$, we have the sharp estimate $$ \|Y\|_{W}\leq \frac{2(\alpha+1)^2}{2\alpha+1}\|X\|_{L^\infty}.$$ Here $W$ is the weak-$L^\infty$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of $\alpha$-subordinate It\^o processes.
