Let \((M,g)\) be a compact analytic Riemannian 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 \(E\).
On every good geodesic ball \(B\) there is a point \(x\in B\) such that for every \(y\in B_R(x)\) with \(R>0\) the following inequality holds
\[ \lvert u(y)\rvert\le \biggl(\frac{2}{\Vol(B)}\biggr)^{1/2} \exp(CR\sqrt{E}) \lVert u\rVert_{L^2(B)}, \]where \(C>0\) is a universal constant depending on \(d\)(?).
Let \(B\) be a good ball. Then there is a good point \(p\in B\) such that
\[\langle \nabla^ku , \nabla^ku\rangle_{g_p} \le \frac{2}{\Vol(B)} (CE)^k \lVert u\rVert_{L^2(B)}^2 \]{#eq:eq1} for every geodesic \(\gamma\) starting in \(p\).
Using (0x66e293b5) and (0x6731eeba) we obtain
\[ \lvert (u\circ \gamma)^{(k)}(0)\rvert^2 = |\nabla^n_{\gamma'(0)} u|^2 \le \langle \nabla^k u, \nabla^k u\rangle_{g_{\gamma(0)}}. \]{#eq:eq2}
Let \(t
where we used [@eq:eq2] and [@eq:eq1].
- Due to (0x67504d60) \(u\) is analytic.