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)R−β 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.
ASJC Scopus subject areas
- Applied Mathematics