Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds (Q2851612)

Let \(M\) be a non-compact \(n\)-dimensional Riemannian manifold of finite volume.NEWLINENEWLINE The authors are concerned with estimates for Lebesgue norms of eigenfunctions of the Laplacian and exhibit minimal assumptions on \(M\) to ensure \(L^q\) bounds. The eigenfunction estimates are new even for the Neumann Laplacian on open subsets of finite volume.NEWLINENEWLINE Section 2 outlines the main results of the paper. In Section 2.1, the authors provide eigenfunction estimates via the isocapacity function of \(M\) and give \(L^q\) bounds and a sharpness condition established. An \(L^\infty\) bound is also given as is a sharpness condition. In Section 2.2, eigenfunction estimates are given via isoperimetric function. \(L^q\) bounds for eigenfunctions and a sharpness condition are given. Section 3 contains background and preliminaries. Section 4 treats manifolds of revolution. Section 5 deals with \(L^q\) bounds for eigenfunctions and Section 6 deals with boundedness of eigenfunctions. Applications are treated in Section 7. Section 7.1 deals with a family of manifolds of revolution with borderline decay and Section 7.2 presents a family of manifolds with clustering submanifolds.