Let \(\mathbb{S}^{d}_R\) denote the \(d\)-sphere with radius \(R> 0\) and dimension \(d\ge 2\). Suppose \(u\in 𝟙_{(-\infty ,E]}(-\Delta_{\mathbb{S}^d_R})\), for \(E>0\), where \(\Delta_{\mathbb{S}^{d}_R}\) denotes the Laplace-Beltrami operator on \(\mathbb{S}^d_R\).
On every good spherical cap \(K\) there is a point \(x\in K\) such that, for every \(y\in \mathbb{S}^{d}_R\) with \(d_{R}(x,y)\le \rho < \tfrac{R}{C}\),
\[ \lvert u(y)\rvert \le \Bigl[\exp (\rho C \sqrt{E}) + \tfrac{R}{R-\rho C}\Bigr] \biggl(\frac{2}{\lvert K\rvert}\biggr)^{1/2} \lVert u\rVert_{L^2(K)}, \]where \(C>0\) is a universal constant that depends on \(d\).
Let \(K\) be a good spherical cap. Then, according to (0x68e77817) , there exist a constant \(C'>0\) depending on \(d\) and a (good) point \(p\in K\) satisfying
\begin{equation*} \langle \nabla^ku , \nabla^ku\rangle_{g_p} \le \frac{2}{\lvert K\rvert} (C')^k\bigl(E+\tfrac{k^2}{R^2}\bigr)^{k} \lVert u\rVert_{L^2(K)}^2. \end{equation*}Suppose \(\hat{u}\) is the local representation of \(u\) with respect to normal coordinates centered at \(p\).
Let \(\gamma\colon [0,r]\) denote a unit-speed (?) geodesic starting in \(p\). Together with (0x66e293b5) and (0x6731eeba) , we deduce that
\[ \lvert (u\circ \gamma)^{(k)}(0)\rvert^2 = |\nabla^k_{\gamma'(0)} u|^2 \le \langle \nabla^k u, \nabla^k u\rangle_{g_{p}}, \]for \(k\in \mathbb{N}\). Let \(t< \rho\), then
\begin{align*} |(u\circ \gamma)(t)| \le \sum_{k=0}^{\infty} \frac{\lvert (u\circ \gamma)^{(k)}(0)\rvert}{k!}t^k \le \sum_{k=0}^{\infty} \frac{\Bigl( \rho \sqrt{C'}\sqrt{E+\tfrac{k^2}{R^2}}\Bigr)^k}{k!} \biggl(\frac{2}{\lvert K\rvert}\biggr)^{1/2} \lVert u\rVert_{L^2(K)}. \end{align*}Using the inequalities (0x68dcfef9) and (0x68bbccca) , we find another constant \(C>0\), depending on \(d\), such that
\begin{align*} \Bigl( \rho \sqrt{C'}\sqrt{E+\tfrac{k^2}{R^2}}\Bigr)^k &\le (\rho C)^k \Bigl(\sqrt{E}^{\,k} + \tfrac{k!}{R^k}\Bigr). \end{align*}If \(\rho<\frac{R}{C}\), we conclude
\begin{align*} |(u\circ \gamma)(t)| &\le \biggl[\sum_{k=0}^{\infty} \frac{(\rho C \sqrt{E})^k}{k!} + \sum_{k=0}^{\infty }\Bigl(\frac{\rho C}{R}\Bigr)^k\biggr] \biggl(\frac{2}{\lvert K\rvert}\biggr)^{1/2} \lVert u\rVert_{L^2(K)}\\ &= \Bigl[\exp (\rho C \sqrt{E}) + \tfrac{R}{R-\rho C}\Bigr] \biggl(\frac{2}{\lvert K\rvert}\biggr)^{1/2} \lVert u\rVert_{L^2(K)}. \end{align*}This completes the proof.