The $L^p$ restriction bounds for Neumann data on surface

Let $\{u_{\lambda }\}$ be a sequence of $L^2$-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold $(M,g)$. We seek to get an $L^p$ restriction bounds of the Neumann data ${\lambda }^{-1}{\mathrm{\partial }}_{\nu }{u}_{\lambda }{\text{vline}}_{\gamma }$ along a unit geodesic $\gamma$. Using the $T$-$T^{\ast }$ argument one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is $O({\lambda }^{-\frac{1}{p}+\frac{3}{2}})$. The Van De Corput theorem (Lemma 2.1) plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.