Endpoint resolvent estimates for compact Riemannian manifolds

Abstract

We prove $L^p\to L^{p^{\prime }}$ bounds for the resolvent of the Laplace-Beltrami operator on a compact Riemannian manifold of dimension $n$ in the endpoint case $p=2(n+1)/(n+3)$. It has the same behavior with respect to the spectral parameter $z$ as its Euclidean analogue, due to Kenig-Ruiz-Sogge, provided a parabolic neighborhood of the positive half-line is removed. This is region is optimal, for instance, in the case of a sphere.

14 pages

References in corpus (1)

• Localized $L^p$-estimates for eigenfunctions: II

Cited by in corpus (1)

• $(L^r,L^s)$ Resolvent Estimate for the Sphere off the Line $\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$