Fourier decay of fractal measures on hyperboloids
| Autor: | M. Burak Erdogan, Alex Barron, Terence L. J. Harris |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Applied Mathematics
General Mathematics 010102 general mathematics 01 natural sciences Upper and lower bounds Combinatorics symbols.namesake Fourier transform Fractal Principal curvature Mathematics - Classical Analysis and ODEs symbols Classical Analysis and ODEs (math.CA) FOS: Mathematics 0101 mathematics 42B37 Probability measure Mathematics |
| Popis: | 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. Comment: Final version, to appear in Transactions of the AMS |
| Databáze: | OpenAIRE |
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