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arXiv · 2004.06553

Fourier decay of fractal measures on hyperboloids

Alex Barron·M. Burak Erdogan·Terence L. J. Harris
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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 $$\int_{\mathbb{H}} |\hat{\mu}(R\xi)|^2 \, d \sigma (\xi)\leq C(\alpha, \mu) R^{-\beta}$$ 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.

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Alex Barron, M. Burak Erdogan, Terence L. J. Harris. 2020-04-14. Fourier decay of fractal measures on hyperboloids. https://doi.org/10.1090/tran/8283

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