A fractal Plancherel theorem
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 \textit{R. Strichartz} [J. Funct. Anal. 89, No. 1, 154--187 (1990; Zbl 0693.28005)].
scientific article; zbMATH DE number 5578215