Communications in Analysis and Geometry

Volume 24 (2016)

Number 4

Boundedness of Laplacian eigenfunctions on manifolds of infinite volume

Pages: 753 – 768

DOI: https://dx.doi.org/10.4310/CAG.2016.v24.n4.a3

Authors

Leonardo P. Bonorino (Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil)

Patrícia K. Klaser (Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil)

Miriam Telichevesky (Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil)

Abstract

In a Hadamard manifold $M$, it is proved that if $u$ is a $\lambda$-eigenfunction of the Laplacian that belongs to $L^p(M)$ for some $p \geq 2$, then $u$ is bounded and $\lVert u \rVert L^{\infty} \leq C \lVert u \rVert L^p$, where $C$ depends only on $p, \lambda$ and the dimension of $M$. This result is obtained in the more general context of a Riemannian manifold endowed with an isoperimetric function $H$ satisfying some integrability condition. In this case, the constant $C$ depends on $p$, $\lambda$ and $H$.

2010 Mathematics Subject Classification

58J05

Published 3 November 2016