Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds. I

In this paper, we investigate the Tomas-Stein restriction estimates on convex cocompact hyperbolic manifolds $\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{H}}^{n+1}$. Via the spectral measure of the Laplacian, we prove that the Tomas-Stein restriction estimate holds when the limit set has Hausdorff dimension ${\delta }_{\mathrm{\Gamma }}<n\mathrm{/}2$. This provides an example for which restriction estimate holds in the presence of hyperbolic geodesic trapping.