Real Analysis Exchange

A Fractal Plancherel Theorem

Ursula M. Molter and Leandro Zuberman

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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 $h(x)=x^\alpha$.

Article information

Real Anal. Exchange Volume 34, Number 1 (2008), 69-86.

First available in Project Euclid: 19 May 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 28A80: Fractals [See also 37Fxx]

Hausdorff Measures Fourier Transform Dimension Plancherel


Molter, Ursula M.; Zuberman, Leandro. A Fractal Plancherel Theorem. Real Anal. Exchange 34 (2008), no. 1, 69--86.

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