On a Carleson-Radon Transform (the non-resonant setting)

TL;DR

This paper proves $L^p$ boundedness of a Carleson-Radon transform along specific curves in the non-resonant case, using advanced time-frequency analysis techniques.

Contribution

It establishes the boundedness of the Carleson-Radon transform for non-resonant exponents, employing the Rank I LGC-methodology with novel time-frequency analysis tools.

Findings

Proved $L^p$ boundedness for $1<p< $ in the non-resonant case.

Developed a partition of the time-frequency plane to linearize the phase.

Introduced a sparse-uniform dichotomy analysis of Gabor coefficients.

Abstract

Given a curve $\vec{\gamma}=(t^{\alpha_1}, t^{\alpha_2}, t^{\alpha_3})$ with $\vec{\alpha}=(\alpha_1,\alpha_2,\alpha_3)\in \mathbb{R}_{+}^3$, we define the Carleson-Radon transform along $\vec{\gamma}$ by the formula $$ C_{[\vec{\alpha}]}f(x,y):=\sup_{a\in \mathbb{R}}\left|p.v.\,\int_{\mathbb{R}} f (x-t^{\alpha_1},y-t^{\alpha_2})\,e^{i\,a\,t^{\alpha_3}}\,\frac{dt}{t}\right|\,.$$ We show that in the \emph{non-resonant} case, that is, when the coordinates of $\vec{\alpha}$ are pairwise disjoint, our operator $ C_{[\vec{\alpha}]}$ is $L^p$ bounded for any $1<p<\infty$. Our proof relies on the (Rank I) LGC-methodology introduced in arXiv:1902.03807 and employs three key elements: 1) a partition of the time-frequency plane with a linearizing effect on both the argument of the input function and on the phase of the kernel; 2) a sparse-uniform dichotomy analysis of the Gabor coefficients associated with the input/output function; 3) a level set analysis of the time-frequency correlation set.