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

Abstract

Given a curve ${\vec{\hat{I}}}^3=(t^{\hat{I}{\pm }_1},t^{\hat{I}{\pm }_2},t^{\hat{I}{\pm }_3})$ with $\vec{\hat{I}}\pm =(\hat{I}{\pm }_1,\hat{I}{\pm }_2,\hat{I}{\pm }_3)\in {\mathbb{R}}_{+}^3$, we define the Carleson-Radon transform along ${\vec{\hat{I}}}^3$ by the formula $$C_{[\vec{\hat{I}}\pm ]}f(x,y):=\underset{a\in \mathbb{R}}{\sup }\left|p.v.{\int }_{\mathbb{R}}f(x-t^{\hat{I}{\pm }_1},y-t^{\hat{I}{\pm }_2})e^{ia{t}^{\hat{I}{\pm }_3}}\frac{dt}{t}\right|.$$ We show that in the \emph{non-resonant} case, that is, when the coordinates of $\vec{\hat{I}}\pm$ are pairwise disjoint, our operator $C_{[\vec{\hat{I}}\pm ]}$ 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.

37 pages, no figures