A Fractal Plancherel Theorem

Abstract

A measure $\mu$ on $\mathbb{R}^n$ is called locally and uniformly $h$-dimensional if $\mu(B_r(x))\leq h(r)$ for all $x \in \mathbb{R}^n$ and for all $0 <r < 1 $, where $h$ is a real valued function. If $f\in L^2(\mu)$ and $\mathcal{F}_\mu f$ denotes its Fourier transform with respect to $\mu$, it is not true (in general) that s. However in this paper we prove that, under certain hypothesis on $h$, for any $f\in L^2(\mu)$ the $L^2$-norm of its Fourier transform restricted to a ball of radius $r$ has the same order of growth as $r^n h(r^{-1})$ when $r\to\infty$. Moreover we prove that the ratio between these quantities is bounded by the $L^2(\mu)$-norm of $f$ (Theorem 3.2). By imposing certain restrictions on the measure $\mu$, we can also obtain a lower bound for this ratio (Theorem 4.3). These results generalize the ones obtained by Strichartz in [10] where he considered the particular case in which $h(x)=x^\alpha$.