Fourier decay of fractal measures on hyperboloids

Barron, Alex; Erdoğan, M.; Harris, Terence

doi:10.1090/tran/8283

Fourier decay of fractal measures on hyperboloids

Authors: Alex Barron, M. Burak Erdoğan and Terence L. J. Harris
Journal: Trans. Amer. Math. Soc. 374 (2021), 1041-1075
MSC (2020): Primary 42B37
DOI: https://doi.org/10.1090/tran/8283
Published electronically: December 3, 2020
MathSciNet review: 4196386
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Abstract: 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 \begin{equation*} \int _{\mathbb {H}} |\widehat {\mu }(R\xi )|^2 d \sigma (\xi )\leq C(\alpha , \mu ) R^{-\beta } \end{equation*} 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.

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