Kakeya-Nikodym averages and $L^p$-norms of eigenfunctions

We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $λ$. The condition that ensures this is that their $L^2$ norms over $O(λ^{-1/2})$ neighborhoods of arbitrary unit geodesics are small when $λ$ is large (which is not the case for the highest weight spherical harmonics on $S^2$ for instance). The proof exploits Gauss' lemma and the fact that the bilinear oscillatory integrals in Hörmander's proof of the Carleson-Sjölin theorem become better and better behaved away from the diagonal. Our results are related to a recent work of Bourgain who showed that $L^2$ averages over geodesics of eigenfunctions are small compared to a natural power of the eigenvalue $λ$ provided that the $L^4(M)$ norms are similarly small. Our results imply that QUE cannot hold on a compact boundaryless Riemannian manifold $(M,g)$ of dimension two if $L^p$-norms are saturated for a given $2<p<6$. We also show that eigenfunctions cannot have a maximal rate of $L^2$-mass concentrating along unit portions of geodesics that are not smoothly closed.

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