Fourier decay of fractal measures on hyperboloids

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 R^d 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.