We prove an analogue of Sogge's local L p estimates [So16] for L p norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-Gérard-Tzvetkov [BuGeTz07], Hu [Hu09], and Chen-Sogge [ChSo14]. The improvements are logarithmic on negatively curved manifolds (without boundary) and by o(1) for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get o(1) improvements on L ∞ estimates of Cauchy data away from a shrinking neighborhood of the corners, and as a result using the methods of [GhReSa13, JuZe16a, JuZe16b], we get that the number of nodal domains of 2-dimensional ergodic billiards tends to infinity as λ → ∞. These results work only for a full density subsequence of any given orthonormal basis of eigenfunctions.We also present an extension of the L p estimates of [BuGeTz07, Hu09, ChSo14] for the restrictions of Dirichlet and Neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary. This part does not assume ergodicity on the manifolds. 1 See [LuSa95, Yo16, HeRi16b, LeRu16] for parallel results in the arithmetic setting.