Georgia Institute of Technology College of Sciences

We go over some relevant history and related problems to motivate the study of the Carleson-Radon operator and the difficulty exhibiting in the planar case. Our main result confirms that the planar Carleson-Radon operator along homogenous curve with general monomial \(t^\alpha\) term modulation admits full range \(L^p\) bound assuming the natural non-resonant condition. In the talk, I'll provide a brief overview of the three key ingredients of the LGC based proof:

1. A sparse-uniform dichotomy of the input function adapted to appropriate time-frequency foliation of the phase-space;
2. A joint structural analysis of the linearizing stopping-time function in the phase in relation to the Gabor coefficients of the input;
3. A level set analysis on the time-frequency correlation set.