Suppose \((M,g)\) is a compact Riemannian \(d\)-manifold, and \(u\) is a solution of
\[ -\Delta_g u = \lambda u. \]For a measurable \(E\subseteq B\) with non-zero measure, the following inequality hold
\[ \max_B \lvert u\rvert \le \Bigl(\frac{C \Vol B}{\Vol E}\Bigr)^{C\lambda^K} \max_E \lvert u\rvert, \]where
\[ K=\frac{d(d+1)}{4} \][1].
Links Link to heading
- eigenfunctions of the Laplace-Beltrami operator on compact manifolds
- BMO estimate for analytic functions
Questions Link to heading
- Why it is called logarithmic BMO estimate?
- Is it possible to use the BMO estimate for analytic functions to find a similar result?
Related Notes Link to heading
References Link to heading
- H. Donnelly and C. Fefferman,
Growth and geometry of eigenfunctions of the Laplacian,
in Analysis and partial differential equations, Dekker, New York, 1990, vol. 122, p. 635–655.