Noncommutative continuous differentially subordinate martingales and second-order Riesz transforms

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发表日期：

2025年4月16日

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Yong Jiao,
Adam Osękowski,
Lian Wu

摘要

We introduce the notion of differential subordination in the context of noncommutative continuous-time martingales and show that for $2<p<\infty $ $2 < p < \infty$ this domination enforces the corresponding $L^p$ $L^{p}$ bound between the two processes. As applications, we obtain best-order $L^p$ $L^{p}$ inequalities for the second-order Riesz transforms on a class of group von Neumann algebras equipped with a conditionally negative length function. The proof relies on a novel stochastic representation of these second-order operators and reveals an interesting interplay between the cylindrical geometric Brownian motion, the crossed product and the cocycle law.