鞅变换与补偿Bellman估计在Dunkl Riesz变换中的应用
Martingale Transforms and Compensated Bellman Estimates for Dunkl Riesz Transforms
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中文总结 AI 辅助
本文提出鞅变换框架，通过补偿Bellman函数证明Dunkl Riesz变换的$L^p$估计，并给出向量值估计及$G$-不变函数情形下的根系无关结果。
中文摘要 AI 辅助
我们为Dunkl调和分析发展了一个鞅变换框架，并利用Dunkl过程的鞅分解将其应用于证明Dunkl Riesz变换的$L^p$估计。与经典布朗设定相比，一个根本区别在于：表示函数Dunkl Riesz变换的鞅，通常并不微分从属于该函数本身的Poisson鞅。我们的主要论证通过直接应用Burkholder的Bellman函数绕过了这一障碍。连续Itô漂移可能产生的正缺陷被反射跳跃产生的负贡献所补偿。这为$1<p<\infty$时的单个Dunkl Riesz变换提供了$L^p$估计，并为$p\geq2$时提供了具有根系谱依赖的向量值估计。对于$G$-不变函数，微分从属可以被恢复，且所得估计与根系无关。
英文摘要
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.
发表机构
- Washington University in St. Louis（圣路易斯华盛顿大学）
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