# Focal points and sup-norms of eigenfunctions

### Christopher D. Sogge

The Johns Hopkins University, Baltimore, USA### Steve Zelditch

Northwestern University, Evanston, USA

## Abstract

If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(\lambda)$ saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point $x_0 \in M$ for the geodesic flow at which the associated Perron–Frobenius operator $U_{x_0}\colon L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M)$ has a nontrivial invariant $L^2$ function. The proof is based on an explicit Duistermaat–Guillemin–Safarov pre-trace formula and von Neumann's ergodic theorem.

## Cite this article

Christopher D. Sogge, Steve Zelditch, Focal points and sup-norms of eigenfunctions. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 971–994

DOI 10.4171/RMI/904