A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions
Volume 59 / 2011
Bulletin Polish Acad. Sci. Math. 59 (2011), 261-274
MSC: Primary 60G44; Secondary 31B05.
DOI: 10.4064/ba59-3-7
Abstract
Let $X$ be a submartingale starting from $0$, and $Y$ be a semimartingale which is orthogonal and strongly differentially subordinate to $X$. The paper contains the proof of the sharp estimate $$ \mathbb{P}\left(\sup_{t\geq 0}|Y_t|\geq 1\right)\leq 3.375\ldots \|X\|_1. $$ As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.
