Boundedness of Laplacian eigenfunctions on manifolds of infinite volume (Q503441)

Let \((M,g)\) be a Hadamard manifold of dimension \(m\). Let \(\Delta\) be the scalar Laplacian. Suppose that \(f\in C^\infty(M)\) satisfies \(\Delta f=\lambda f\) and that \(f\in L^p(M)\) for some \(p\). The authors show that \(f\) is bounded and give a uniform estimate \(\|f\|_\infty\leq C(p,\lambda,m)\|f\|_p\). In fact the authors establish this result more generally in the context of Riemannian manifolds which have an isoperimetric function \(H\) satisfying some integrability conditions -- in this setting, \(C=C(p,\lambda,H)\). They exhibit a manifold to which their result applies but for which general inequalities of the form \(\|u\|_p\leq C(p,\tilde p)\|\nabla u\|_{\tilde p}\) do not hold for any \(p\geq1\) and \(\tilde p\geq1\). They impose no constraint on the volume of \((M,g)\) and thus their results generalize previous results of \textit{A. Cianchi} and \textit{V. G. Maz'ya} [Am. J. Math. 135, No. 3, 579--635 (2013; Zbl 1280.58018)] who investigated non-compact manifolds of finite volume.