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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
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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

$\displaystyle \int _{\mathbb{H}} \vert\widehat {\mu }(R\xi )\vert^2 \, d \sigma (\xi )\leq C(\alpha , \mu ) R^{-\beta }$    

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.

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Additional Information

Alex Barron
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: aabarron@illinois.edu

M. Burak Erdoğan
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: berdogan@illinois.edu

Terence L. J. Harris
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: terence2@illinois.edu

DOI: https://doi.org/10.1090/tran/8283
Received by editor(s): April 14, 2020
Received by editor(s) in revised form: April 29, 2020
Published electronically: December 3, 2020
Additional Notes: The second author was partially supported by the Simons collaboration grant, 634269.
Article copyright: © Copyright 2020 American Mathematical Society