Martingale Transforms and Compensated Bellman Estimates for Dunkl Riesz Transforms

Abstract

We develop a martingale-transform framework for Dunkl harmonic analysis and apply it to prove $L^p$ estimates for the Dunkl Riesz transforms using the martingale decomposition of the Dunkl process. A fundamental difference from the classical Brownian setting is that the martingale representing the Dunkl Riesz transform of a function is not, in general, differentially subordinated to the Poisson martingale of the function itself. Our main argument bypasses this obstruction by applying Burkholder's Bellman function directly. The possible positive defect of the continuous Itô drift is compensated by the negative contribution produced by the reflection jumps. This yields $L^p$ estimates for single Dunkl Riesz transforms for $1<p<\infty$, and vector-valued estimates for $p\geq2$ with a spectral dependence on the root system. For $G$-invariant functions, differential subordination can be recovered and the resulting estimates are independent of the root system.