%0 Journal Article
%T $L^p$ Fourier asymptotics, Hardy type inequality and fractal measures
%A K. S. Senthil Raani
%J Mathematics
%D 2015
%I arXiv
%X Suppose $\mathcal{P}^{\alpha}$ is an $\alpha$-dimensional packing measure for some $0<\alpha<n$ and $\mu$ is a restriction of $\mathcal{P}^{\alpha}$ to a set $E$ with $\mathcal{P}^{\alpha}(E)<\infty$. Inspired by the results proved by 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 $x=(x_1,...x_n)\in\mathbb{R}^n$ generalizing the one dimensional results proved by Hudson and Leckband in 1992.
%U http://arxiv.org/abs/1509.04848v1