  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
$$\int_{\mathbb{H}} |\hat{\mu}(R\xi)|^2 \, d \sigma (\xi)\leq C(\alpha, \mu)
R^{-\beta}$$ 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.
