$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve

TL;DR

This paper establishes sharp conditions on the integrability of functions with Fourier support on fractal sets on the moment curve, showing that such functions must be zero under certain $L^p$ conditions and exploring implications for restriction estimates.

Contribution

The paper proves optimal $L^p$-integrability conditions for functions supported on fractal sets on the moment curve, extending previous results and including endpoint cases.

Findings

Functions with Fourier support on fractal sets on the moment curve are zero if in $L^p$ with specific $p$ ranges.

The $p$ range for which this holds is proven to be optimal using random Cantor sets.

Applications include demonstrating failure of certain restriction estimates and Wiener Tauberian Theorem.

Abstract

For $0 < \alpha \leq 1$, let $E$ be a compact subset of the $d$-dimensional moment curve in $\mathbb{R}^d$ such that $N(E,\varepsilon) \lesssim \varepsilon^{-\alpha}$ for $0 <\varepsilon <1$ where $N(E,\varepsilon)$ is the smallest number of $\varepsilon$-balls needed to cover $E$. We proved that if $f \in L^p(\mathbb{R}^d)$ with \begin{align*} 1 \leq p\leq p_\alpha:= \begin{cases} \frac{d^2+d+2\alpha}{2\alpha} & d \geq 3, \frac{4}{\alpha} &d =2, \end{cases} \end{align*} and $\widehat{f}$ is supported on the set $E$, then $f$ is identically zero. We also proved that the range of $p$ is optimal by considering random Cantor sets on the moment curve. We extended the result of Guo, Iosevich, Zhang and Zorin-Kranich, including the endpoint. We also considered applications of our results to the failure of the restriction estimates and Wiener Tauberian Theorem.