Fourier decay of fractal measures on hyperboloids

Let \(\mu\) be an \(\alpha\)-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform \(\widehat{\mu}\). More precisely, if \(\mathbb{H}\) is a truncated hyperbolic paraboloid in \(\mathbb{R}^d\) we study the optimal \(\be...

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Bibliographic Details
Published in:arXiv.org 2020-08
Main Authors: Barron, Alex, Erdogan, M Burak, Harris, Terence L J
Format: Article
Language:English
Subjects:
Decay rate
Fourier transforms
Lower bounds
Parabolic bodies
Online Access:Get full text
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Summary:Let \(\mu\) be an \(\alpha\)-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform \(\widehat{\mu}\). More precisely, if \(\mathbb{H}\) is a truncated hyperbolic paraboloid in \(\mathbb{R}^d\) we study the optimal \(\beta\) for which $$\int_{\mathbb{H}} |\hat{\mu}(R\xi)|^2 \, d \sigma (\xi)\leq C(\alpha, \mu) R^{-\beta}$$ for all \(R > 1\). Our estimates for \(\beta\) depend on the minimum between the number of positive and negative principal curvatures of \(\mathbb{H}\); if this number is as large as possible our estimates are sharp in all dimensions.
ISSN:2331-8422
DOI:10.48550/arxiv.2004.06553