A Fractal Plancherel Theorem

Abstract

A measure $\mu$ on $\R^n$ is called locally and uniformly $h$-dimensional if $\mu(B_r(x))\leq h(r)$ for all $x\in\R^n$ and for all $0<r<1$, where $h$ is a real valued function. If $f\in L^2(\mu)$ and $\Fmu f$ denotes its Fourier transform with respect to $\mu$, it is not true (in general) that $\Fmu f \in L^2$ (e.g. \cite{Str90}). 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 \ref {cotasupr}). By imposing certain restrictions on the measure $\mu$, we can also obtain a lower bound for this ratio (Theorem \ref{cotainf}). These results generalize the ones obtained by Strichartz in \cite {Str90} where he considered the particular case in which $h(x)=x^\alpha$.