Revista Matemática Iberoamericana

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Volume 32, Issue 3, 2016, pp. 971–994
DOI: 10.4171/RMI/904

Published online: 2016-10-03

Focal points and sup-norms of eigenfunctions

Christopher D. Sogge[1] and Steve Zelditch[2]

(1) The Johns Hopkins University, Baltimore, USA
(2) Northwestern University, Evanston, USA

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.

Keywords: Eigenfunctions, $L^\infty$ bounds

Sogge Christopher, Zelditch Steve: Focal points and sup-norms of eigenfunctions. Rev. Mat. Iberoamericana 32 (2016), 971-994. doi: 10.4171/RMI/904