Fourier decay of fractal measures on hyperboloids

Alex Barron, M. Burak Erdogan, Terence L.J. Harris

Research output: Contribution to journalArticlepeer-review

Abstract

Let μ be an α-dimensional probability measure. We prove new upper and lower bounds on the decay rate of hyperbolic averages of the Fourier transform μ. More precisely, if H is a truncated hyperbolic paraboloid in Rd we study the optimal β for which / H |μ(Rξ)|2 dσ(ξ) ≤ C(α, μ)Rβ for all R > 1. Our estimates for β depend on the minimum between the number of positive and negative principal curvatures of H; if this number is as large as possible our estimates are sharp in all dimensions.

Original languageEnglish (US)
Pages (from-to)1041-1075
Number of pages35
JournalTransactions of the American Mathematical Society
Volume374
Issue number2
DOIs
StatePublished - Feb 2021

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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