Boundedness of Laplacian eigenfunctions on manifolds of infinite volume

Leonardo Bonorino, Patrícia Klaser, Miriam Telichevesky

Published 2014-03-21, updated 2014-08-07Version 2

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 $p \ge 2$, then $u$ is bounded and $\|u\|_{\infty} \le C \|u\|_p,$ where $C$ depends only on $p$, $\lambda$ and on the dimension of $M$. This result is obtained in the more general context of a complete Riemannian manifold endowed with an isoperimetric function $H$ satisfying some integrability condition. In this case, the constant $C$ depends on $p,\lambda$ and $H.$