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 $\widehat{\mu }$. More precisely, if $\mathbb{H}$ is a truncated hyperbolic paraboloid in ${\mathbb{R}}^d$ we study the optimal $\beta$ for which $${\int }_{\mathbb{H}}\mathrm{\mid }\widehat{\mu }(R\xi ){\mathrm{\mid }}^{2}d\sigma (\xi )\le C(\alpha ,\mu )R^{-\beta }$$ 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.