Boundedness of Laplacian eigenfunctions on manifolds of infinite volume

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 $p\ge 2$, then $u$ is bounded and $\Vert u{\Vert }_{\infty }\le C\Vert u{\Vert }_p,$ 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 $H.$

arXiv comment13 pages; change in the title; change in the abstract; improvement in the introduction; added some details in the proof of Theorem 2.2

Cited by in corpus (1)

• Bounded $\lambda$-harmonic functions in domains of ${\mathbb{H}}^n$ with asymptotic boundary with fractional dimension