$L^p$ Fourier asymptotics, Hardy type inequality and fractal measures

TL;DR

This paper investigates the asymptotic behavior of Fourier transforms of functions with respect to fractal measures, establishing new $L^p$ bounds and a Hardy type inequality that generalizes previous one-dimensional results.

Contribution

It introduces novel $L^p$ asymptotic bounds for Fourier transforms on fractal measures and proves a Hardy type inequality in higher dimensions, extending earlier one-dimensional findings.

Findings

Established $L^p$ asymptotics for Fourier transforms of fractal measure functions.

Proved a Hardy type inequality relating function values and Fourier transform asymptotics.

Generalized one-dimensional Hardy inequalities to higher dimensions for fractal measures.

Abstract

Suppose $\mu$ is an $\alpha$-dimensional fractal measure for some $0<\alpha<n$. Inspired by the results proved by R. Strichartz in 1990, we discuss the $L^p$-asymptotics of the Fourier transform of $fd\mu$ by estimating bounds of $$\underset{L\rightarrow\infty}{\liminf}\ \frac{1}{L^k} \int_{|\xi|\leq L}\ |\widehat{fd\mu}(\xi)|^pd\xi,$$ for $f\in L^p(d\mu)$ and $2<p<2n/\alpha$. In a different direction, we prove a Hardy type inequality, that is, $$\int\frac{|f(x)|^p}{(\mu(E_x))^{2-p}}d\mu(x)\leq C\ \underset{L\rightarrow\infty}{\liminf} \frac{1}{L^{n-\alpha}} \int_{B_L(0)} |\widehat{fd\mu}(\xi)|^pd\xi$$ where $1\leq p\leq 2$ and $E_x=E\cap(-\infty,x_1]\times(-\infty,x_2]...(-\infty,x_n]$ for $x=(x_1,...x_n)\in\R^n$ generalizing the one dimensional results proved by Hudson and Leckband in 1992.