The Polynomial Carleson Operator correlates to the understanding of some parabolic PDE and serves as a basic model for operators with implicit higher-order modulation symmetry. Stein conjectured the full range bound of the operator, and Victor Lie proved exactly that in 2019 by rejuvenating Fefferman’s idea and introducing a framework for analysis on polynomial modulation. Later on, Pavel Zorin-Kranich generalized the result to a higher dimensional setting through the same philosophy and simplified some of the constructions in Lie’s arguments.

 

In this talk, aside from a brief history of the development, I will provide a heuristic explanation for Lie’s and Zorin-Kranich’s arguments, and, if time permits, we can discuss some interesting details.