$L^p$ Fourier asymptotics, Hardy type inequality and 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 $$\underset{L\to \infty }{\mathrm{liminf}}\frac{1}{L^k}{\int }_{|\xi |\le L}|\widehat{fd\mu }(\xi ){|}^{p}d\xi ,$$ for $f\in L^p(d\mu )$ and $2<p<2n/\alpha$. In a different direction, we prove a Hardy type inequality, that is, $$\int \frac{|f(x){|}^p}{(\mu (E_x){)}^{2-p}}d\mu (x)\le C\underset{L\to \infty }{\mathrm{liminf}}\frac{1}{L^{n-\alpha }}{\int }_{B_L(0)}|\widehat{fd\mu }(\xi ){|}^{p}d\xi$$ where $1\le p\le 2$ and $E_x=E\cap (-\infty ,x_1]\times (-\infty ,x_2]...(-\infty ,x_n]$ for $x=(x_1,...x_n)\in {\mathbb{R}}^n$ generalizing the one dimensional results proved by Hudson and Leckband in 1992.