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

By Christopher D. Sogge. 2009

69 citations recorded by the source corpus.

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 $\lambda$. The condition that ensures this is that their $L^2$ norms over $O(\lambda^{-1/2})$ neighborhoods of arbitrary unit geodesics are small when $\lambda$ 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 H\"ormander's proof of the Carleson-Sj\"olin 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 $\lambda$ 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f90a7589510d078f1a70b1e14843487f380c870e

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Concerning Toponogov’s theorem and logarithmic improvement of estimates of eigenfunctions (2015, methodology): “Although not explicitly stated, the inequalities proved in [21] yield”
Improvement of eigenfunction estimates on manifolds of nonpositive curvature (2012, background): “Bourgain [2] and Sogge [13] study the relationship between ||u||L4(M) and ||u||L2(γ) where γ is a geodesic.”