Covering Lemmas and an Application to Nodal Geometry on Riemannian Manifolds

The main part of this note is to show a general covering lemma in Rn, n 2 2 , with the aim to obtain the estimate for BMO norm and the volume of a nodal set of eigenfunctions on Riemannian manifolds. This article is a continuation of our previous work [L]. In [L] we proved a covering lemma in R2 and applied it to the BMO norm estimates for eigenfunctions on Riemannian surfaces. The principal part of this article is to prove a general covering lemma in Rn for n 2 2 . As applications, we can obtain the BMO estimate for eigenfunctions and the volume estimate for the nodal set. Let M n be a smooth, compact, and connected Riemannian manifold with no boundary. Let A denote the Laplacian on M n . Let -Au = ilu, u an eigenfunction with eigenvalue il , il > 1 . Our main results can be stated as follows Theorem A (BMO estimate for log lul) . For u , il as above and n 2 3 , where C is independent of il and u and is only dependent on n and M n Theorem B (geometry of nodal domains). Let n 2 3 and u , il as above. Let B c M n be any ball, and let R c B be any of the connected components of {x E B :u(x) # 0) . If R intersects the middle halfof B , then where C is independent of il and u Donnelly and Fefferman [DFl, DF2] and Chanillo and Muckenhoupt [CM] proved Theorem A with (10gil)~ replaced by iln(n+2)/4 and iln log2 , respectively, and Theorem B with A-2n2-n14 replaced by il-(n+n2(n+2))/2 and il-2n2-n/2 (log il)-2n , respectively. Received by the editors June 28, 1991. 1991 Mathematics Subject Class$cation. Primary 35B05, 58G03. @ 1993 American Mathematical Society 0002-9939193 $1.00 + $.25 per page