Predictable subordination, sharp martingale inequalities and applications

Abstract

We introduce a new method for obtaining sharp $L^p$ estimates for martingales under predictable analogues of differential subordination. By allowing suitable comparisons between continuous and jump variation adapted to the relevant range of $p$, we retain sharp constants in settings where pathwise differential subordination fails. The key idea is to study together the continuous and jump contributions arising from the appropriate Bellman functions. Although these contributions need not be nonpositive separately, we quantify their defects and show that they compensate at the predictable level. This compensation mechanism is inspired by the author's earlier work. As an application, we substantially improve the explicit dimension-free bounds for the Riesz vectors on the Hamming cube and on ${\mathbb{Z}}^n$.

19 pages