$L^p$-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 Strichartz (J Funct Anal 89:154–187, 1990), we discuss the $L^p$-asymptotics of the Fourier transform of $fd\mu$ by estimating bounds of

$$\begin{aligned} \underset{L\rightarrow \infty }{\liminf }\ \frac{1}{L^k} \int _{|\xi |\le L}\ |\widehat{fd\mu }(\xi )|^pd\xi , \end{aligned}$$

for $f\in L^p(d\mu )$ and $2<p<2n/\alpha$. In a different direction, we prove a Hardy type inequality, that is,

$$\begin{aligned} \int \frac{|f(x)|^p}{(\mu (E_x))^{2-p}}d\mu (x)\le C\ \underset{L\rightarrow \infty }{\liminf }\frac{1}{L^{n-\alpha }} \int _{B_L(0)}|\widehat{fd\mu }(\xi )|^pd\xi \end{aligned}$$

where $1\le p\le 2$ and $E_x=E\cap (-\infty ,x_1]\times (-\infty ,x_2]\ldots (-\infty ,x_n]$ for $x=(x_1,\ldots x_n)\in {\mathbb R}^n$ generalizing the one dimensional results by Hudson and Leckband (J Funct Anal 108:133–160, 1992).