A Weak-Type Inequality for Submartingales and Itô Processes

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.