Let \((M,g)\) be a compact smooth Riemannian \(d\)-manifold and \(\nabla\) the corresponding Levi-Civita connection . Furthermore, let \(u_1,\ldots ,u_n\in C^\infty(M)\) be eigenfunctions of the Laplacian with the respective eigenvalues \(\lambda_1,\ldots ,\lambda_n\). We denote the maximal eigenvalue with \(E\). Let \(u=\sum_{i=1}^{n} a_i u_i\) with \(a_i\in \mathbb{R}\). The for every good ball \(B\) with radius \(r>0\) there is a point \(p\in B\) and a constant \(C>0\) depending only on \(M\) such that
\begin{equation*} \langle \nabla^ku , \nabla^ku\rangle_{g_p} \le \frac{2}{\lvert B\rvert} (CE)^k \lVert u\rVert_{L^2(B)}^2 \end{equation*}for every \(k\ge 0\).
We call such a point good.
Since \(B\) is good we have
\begin{equation*} \lVert \nabla^k u\rVert_{L^2(B)}^2\le (C'E)^k \lVert u\rVert_{L^2(B)}^2, \end{equation*}for some numerical constant \(C>0\) and all \(k\in \mathbb{N}\). If no good point exists, then for every \(x\in B\) there exists a \(k\in \mathbb{N}\) such that
\begin{equation*} \langle \nabla^ku , \nabla^ku\rangle_{g_p} \ge \frac{2}{\lvert B\rvert} (C'E)^k \lVert u\rVert_{L^2(B)}^2. \end{equation*}Let \(C'=\alpha C\) for \(\alpha>1\). Then
\begin{equation*} \frac{2}{\lvert B\rvert} \int_{B} u^2 \le \sum_{k=0}^{n} \frac{1}{(\alpha CE)^k} \langle \nabla^ku , \nabla^ku\rangle_{g_p}. \end{equation*}Integrating both sides over \(B\) leads to
\begin{align} 2 \int_{B} u^2 &\le \sum_{k=0}^{n} \frac{1}{(\alpha CE)^k} \int_{B} \langle \nabla^ku , \nabla^ku\rangle \\ &\le \sum_{k=0}^{n} \frac{1}{\alpha^k} \int_{B} u^2 \\ &\le \frac{\alpha}{\alpha -1} \int_{B} u^2, \end{align}where we used the good ball property and \(\alpha>1\). Set \(\alpha=3\) and we get
\begin{equation*} 2 \int_{B} u^2 \le \frac{3}{2} \int_{B} u^2. \end{equation*}This contradiction proves our claim.
Questions Link to heading
- Is \(\lvert B\rvert\) constant? How it depends on \(a\)?