Speaker: Lingxiao Zhang (University of Connecticut)

 

Title: Hilbert transform along flat curves on the Heisenberg group

 

 

Abstract:  The Stein-Street condition characterizes the \(L^p\) boundedness of multi-parameter singular Radon transforms in the real-analytic setting, establishing a foundational principle: geometric structure precisely governs operator behavior. Beyond the real-analytic category, however, this condition is no longer decisive, and new phenomena arise. To capture these phenomena, we study Hilbert transforms along flat curves in the Heisenberg group \(ℍ^1\), thereby resolving a 30-year-old problem in work of Carbery, Wainger, and Wright. This is the first non-Euclidean theorem of the same strength and structural form as the Euclidean result of Carbery, Christ, Vance, Wainger, and Watson. Our work demonstrates that geometric structure continues to govern operator behavior beyond the real-analytic setting by developing a framework that unifies Euclidean and non-Euclidean theories. The construction of operator-adapted metrics and an associated Calderón-Zygmund theory forms one key component, alongside further structural innovations. The framework adapts to different settings and is expected to apply to related operators and other Carnot groups.

 

 

For more details, please visit https://webhome.auburn.edu/~lzc0090/SASA/20260422_Zhang.html.