Resumen de Covering lemmas and BMO estimates for eigenfunctions on Riemannian surfaces.

Guozhen Lu

• The principal aim of this note is to prove a covering Lemma in R2. As an application of this covering lemma, we can prove the BMO estimates for eigenfunctions on two-dimensional Riemannian manifolds (M2, g). We will get the upper bound estimate for || log |u| ||BMO, where u is the solution to ?u + ?u = 0, for ? > 1 and ? is the Laplacian on (M2, g). A covering lemma on homogeneous spaces is also obtained in this note.