### A Fractal Plancherel Theorem

Ursula M. Molter and Leandro Zuberman
Source: Real Anal. Exchange Volume 34, Number 1 (2008), 69-86.

#### Abstract

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

First Page:
Primary Subjects: 42B10
Secondary Subjects: 28A80
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Permanent link to this document: http://projecteuclid.org/euclid.rae/1242738921
Zentralblatt MATH identifier: 05578215
Mathematical Reviews number (MathSciNet): MR2527123

### References

S. Agmon & L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math., 30 (1976), 1–38.
Mathematical Reviews (MathSciNet): MR466902
Zentralblatt MATH: 0335.35013
Digital Object Identifier: doi:10.1007/BF02786703
John J. Benedetto & Joseph D. Lakey, The definition of the Fourier transform for weighted inequalities, J. Funct. Anal., 120(2) (1994), 403–439.
Mathematical Reviews (MathSciNet): MR1266315
Zentralblatt MATH: 0804.46040
Digital Object Identifier: doi:10.1006/jfan.1994.1037
C. Cabrelli, F. Mendivil, U. Molter, & R. Shonkwiler, On the $h$-Hausdorff measure of Cantor sets, Pacific J. Math., 217(1) (2004), 29–43.
Mathematical Reviews (MathSciNet): MR2105765
Zentralblatt MATH: 1063.28006
Digital Object Identifier: doi:10.2140/pjm.2004.217.45
K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, Cambridge, 1985.
Mathematical Reviews (MathSciNet): MR867284
K. J. Falconer, Techniques in Fractal Geometry, John Wiley & Sons, New York, 1997.
Mathematical Reviews (MathSciNet): MR1449135
Zentralblatt MATH: 0869.28003
K-S. Lau, Fractal measures and mean $p$-variations, J. Funct. Anal., 108(2) (1992), 427–457.
Mathematical Reviews (MathSciNet): MR1176682
Zentralblatt MATH: 0767.28007
Digital Object Identifier: doi:10.1016/0022-1236(92)90031-D
K. S. Lau & Y. Wang, Characterizations of $\mathcal L^p$-solutions for the two scale dilation equations, Preprint, 1997.
P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, Cambridge, 1995.
Mathematical Reviews (MathSciNet): MR1333890
C. A. Rogers, Hausdorff Measures, Cambridge University Press, Cambridge, UK, $2^{nd}$ Edition, 1998.
Mathematical Reviews (MathSciNet): MR1692618
R. Strichartz, Fourier asymptotics of fractal measures, J. Funct. Anal., 89 (1990), 154–187.
Mathematical Reviews (MathSciNet): MR1040961
Zentralblatt MATH: 0693.28005
Digital Object Identifier: doi:10.1016/0022-1236(90)90009-A
R. Strichartz, Self-similar measures and their Fourier transforms I, Indiana Univ. Math. J., 39(3) (1990), 797–817.
Mathematical Reviews (MathSciNet): MR1078738
Zentralblatt MATH: 0695.28003
Digital Object Identifier: doi:10.1512/iumj.1990.39.39038
R. Strichartz, Self-similar measures and their Fourier transforms II, Trans. Amer. Math. Soc., (1993).
Mathematical Reviews (MathSciNet): MR1081941
Zentralblatt MATH: 0765.28007
Digital Object Identifier: doi:10.2307/2154350
R. Strichartz, Self-similarity in harmonic analysis, J. Fourier Anal. Appl., 1(1) (1994), 1–37.
Mathematical Reviews (MathSciNet): MR1307067
Zentralblatt MATH: 0851.42001
Digital Object Identifier: doi:10.1007/s00041-001-4001-z