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

Abstract

Given a curve $\vecγ=(t^{α_1}, t^{α_2}, t^{α_3})$ with $\vecα=(α_1,α_2,α_3)\in \mathbb{R}_{+}^3$, we define the Carleson-Radon transform along $\vecγ$ by the formula $$ C_{[\vecα]}f(x,y):=\sup_{a\in \mathbb{R}}\left|p.v.\,\int_{\mathbb{R}} f (x-t^{α_1},y-t^{α_2})\,e^{i\,a\,t^{α_3}}\,\frac{dt}{t}\right|\,.$$ We show that in the \emph{non-resonant} case, that is, when the coordinates of $\vecα$ are pairwise disjoint, our operator $ C_{[\vecα]}$ 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.

Theorems & Definitions (22)

• Remark 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof