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

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