Covering lemmas and BMO estimates for eigenfunctions on Riemannian surfaces (Q1192521)

Let \(M\) be a compact surface. The author gives a better BMO norm estimate of eigenfunctions \(u\) of the equation \(\Delta u+\lambda u=0\), where \(\Delta\) is the Laplacian on \(M\) and \(\lambda>1\): Theorem. For \(\varepsilon>0\), \(\|\log| u|\|_{BMO}\leq c\lambda^{15/8+\varepsilon}\), where \(c\) is independent of \(\lambda\). The proof is based on the covering lemma, which is an improvement of the covering lemma for the two-dimensional case given by \textit{S. Chanillo} and \textit{B. Muckenhoupt} [J. Differ. Geom. 34, No. 1, 85-91 (1991; Zbl 0727.58048)].