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Fourier asymptotics of fractal measures. (English) Zbl 0693.28005

Let \(\mu\) be a Borel measure in \({\mathbb{R}}^ n\). The author studies relations between the local behavior of \(\mu\) and the asymptotic behavior in average at infinity of the Fourier transforms \((f d\mu){\hat{\;}}\) for \(f\in L^ 2(\mu).\) He shows that if \(0\leq \alpha \leq n\) and \(\mu B_ r(x)\leq cr^{\alpha}\) for \(x\in {\mathbb{R}}^ n,\quad 0<r\leq 1,\) then for \(r\geq 1\) \[ (1)\quad r^{\alpha -n}\int_{B_ r(y)}| (f d\mu){\hat{\;}}(\xi)|^ 2d\xi \leq c\int | f|^ 2d\mu. \] The opposite inequality holds under some additional hypothesis, for example if \(\mu B(x,r)\geq br^{\alpha}\) for x in the support of \(\mu\) and \(0<r\leq 1.\) In certain regular cases (where \(\alpha\) has to be an integer) the left hand side of (1) tends to \(c\int | f|^ 2d\mu\) as \(r\to \infty.\) These results can be considered as intermediate cases between the well-known theorems of Plancherel \((\alpha =n)\) and Wiener \((\alpha =0),\) and they generalize earlier results on Fourier transforms on smooth surfaces.
Reviewer: P.Mattila

MSC:

28A75 Length, area, volume, other geometric measure theory
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