FOURIER DECAY OF FRACTAL MEASURES ON HYPERBOLOIDS

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 (mu) over cap More precisely, if H is a truncated hyperbolic paraboloid in R-d we study the optimal beta for which integral(H)vertical bar(mu) over cap (R xi)vertical bar 2 d sigma(xi)<= 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 H; if this number is as large as possible our estimates are sharp in all dimensions.