Localized $L^p$-estimates of eigenfunctions: A note on an article of Hezari and Rivi\`ere

We use a straightforward variation on a recent argument of Hezari and Rivi\`ere~\cite{HR} to obtain localized $L^p$-estimates for all exponents larger than or equal to the critical exponent $p_c=\tfrac{2(n+1)}{n-1}$. We are able to this directly by just using the $L^{p}$-bounds for spectral projection operators from our much earlier work \cite{Seig}. 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 $L^p$, $2<p\le \infty$, bounds of the corresponding eigenfunctions are relatively small compared to the general ones in \cite{Seig}, which are saturated on round spheres. The connection with quantum ergodicity was established for exponents $2<p<p_c$ in the recent results of the author \cite{SK} and Blair and the author \cite{BS2}; however, the article of Hezari and Rivi\`ere~\cite{HR} was the first one to make this connection (in the case of negatively curved manifolds) for the critical exponent, $p_c$. As is well known, and we indicate here, bounds for the critical exponent, $p_c$, imply ones for all of the other exponents $2<p\le \infty$. The localized estimates involve $L^2$-norms over small geodesic balls $B_r$ 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 as to when one can improve on the trivial $O(r^{\frac12})$ estimates for these $L^2(B_r)$ bounds. If $r=\lambda^{-1}$, one can improve on the trivial estimates if one has improved $L^{p_c}(M)$ bounds just by using H\"older's inequality; however, obtaining improved bounds for $r\gg \lambda^{-1}$ seems to be subtle.