A fractal Plancherel theorem. (English) Zbl 1174.42009

Molter, Ursula M.; Zuberman, Leandro

Real Anal. Exch. 34(2008-2009), No. 1, 69-85 (2009).

The authors focus on the behavior around 0 of Fourier transforms of functions \(f \in L^2(\mu)\), where \(\mu\) is a measure on \(\mathbb{R}^n\) enjoying the following property (which implies a uniform local behavior of \(\mu\)): for all \(x\), for all \(0<r<1\), \(\mu(B(x,r)) \leq h(r)\). Here \(h\) is a gauge function, satisfying the doubling property.
For any \(f \in L^2(\mu)\), it is proved the the norm of the Fourier transform of \(f\) on balls of the form \(B(x,r)\) behaves like \(h(1/r)r^n\), when \(r\) tends to zero (in the sense that it is bounded by above and by below modulo some uniform constants by \(h(1/r)r^n\)). More precisely, the upper bound is general, and the lower bound is obtained for measures \(\mu\) which are restrictions of the \(\mathcal{H}^h\)-Hausdorff measure on \(h\)-dimensional quasi-regular sets.
The proof is based on tools coming from classical integration theory and Fourier analysis. The paper extends previous results by R. Strichartz [J. Funct. Anal. 89, No. 1, 154–187 (1990; Zbl 0693.28005)].

Reviewer: Stéphane Seuret (Creteil)

MSC:

42B10	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
28A80	Fractals

Keywords:

Hausdorff measures; Fourier transform; dimension; Plancherel

Citations:

Zbl 0693.28005