The Lᵖ restriction bounds for Neumann data on surface

Let _λ\ be a sequence of L²-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold $(M,g)$. We seek to get an Lᵖ restriction bounds of the Neumann data λ⁻¹ ∂_ν uλ\,_γ along a unit geodesic γ. Using the T-T^* argument one can transfer the problem to an estimate of the norm of a Fourier integral operator and show that such bound is O(λ^-1p+32). The Van De Corput theorem (Lemma 2.1) plays the crucial role in our proof. Moreover, this upper bound is shown to be optimal.