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

Abstract

We use a straightforward variation on a recent argument of Hezari and Rivière [8] to obtain localized Lp-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 Lp-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 Lp, 2<p≤∞, bounds of the corresponding eigenfunctions are relatively small compared to the general ones in [12], which are saturated on round spheres. The connection with quantum ergodicity was established for exponents 2<p<pc in the recent results of the author [13] and Blair and the author [3]; however, the article of Hezari and Rivière [8] was the first one to make this connection (in the case of negatively curved manifolds) for the critical exponent, pc. As is well known, and we indicate here, bounds for the critical exponent, pc, imply ones for all the other exponents 2<p≤∞. The localized estimates involve L2-norms over small geodesic balls Br of radius r, and we shall go over what happens for these in certain model cases on the sphere and on manifolds of nonpositive curvature. We shall also state a problem of determining when one can improve on the trivial O(r12) estimates for these L2(Br) bounds. If r=λ−1, one can improve on the trivial estimates if one has improved Lpc(M) bounds just by using Hölder's inequality; however, obtaining improved bounds for r≫λ−1 seems to be subtle.