Localized Lp-estimates of eigenfunctions: A note on an article of Hezari and Rivière

We use a straightforward variation on a recent argument of Hezari and Rivière [8] to obtain localized LpLp-estimates for all exponents larger than or equal to the critical exponent pc=2(n+1)n−1. We are able to do this directly by just using the LpLp-bounds for spectral projection operators from our much earlier work [12]. The localized bounds we obtain here imply, for instance, that, for a density one sequence of eigenvalues on a manifold whose geodesic flow is ergodic, all of the LpLp, 2