Covering Lemmas and HMO Estimates for Eigenfunctions on Riemannian Surfaces

Abstract

The principal aim of this note is to prove a covering Lemma in ${\mathbb{R}}^2$. As an application of this covering lemma, we can prove the BMO estimates for eigenfunctions on two-dimensional Riemannian manifolds $(M^2,g)$. We will get the upper bound estimate for $\Vert \log |u\Vert {|}_{BMO}$, where $u$ is the solution to $\Delta u+\lambda u=0$, for $\lambda >1$ and $\Delta$ is the Laplacian on $(M^2,g)$. A covering lemma in homogeneous spaces is also obtained in this note.