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 \(\mathbb{S}^d_R\).
Suppose \(K\) is good spherical cap with radius \(r>0\). Let \(\lvert K\rvert\) denote the spherical volume of \(K\).
There exists a point \(p\in K\) and a constant \(C>0\) depending only on \(d\) such that
\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*}for every \(k\ge 0\).
We call such a point good.
- \(r\) is not needed.
Since \(K\) is good, there is a constant \(C'>0\) depending on \(d\), such that
\begin{equation}\label{eq:good_ball_prop} \lVert \nabla^k u\rVert_{L^2(K)}^2 \le (C')^k\bigl(E+\tfrac{k^2}{R^2}\bigr)^{k} \lVert u\rVert_{L^2(K)}^2, \end{equation}for all \(k\in \mathbb{N}\). We set \(C=\alpha C'\) with \(\alpha>1\).
If no good point exists, then for every \(q\in K\) there exists a \(k\in \mathbb{N}\) such that
\begin{equation*} \langle \nabla^ku , \nabla^ku\rangle_{g_q} \ge \frac{2}{\lvert K\rvert} (\alpha C')^k\bigl(E+\tfrac{k^2}{R^2}\bigr)^{k} \lVert u\rVert_{L^2(K)}^2, \end{equation*}Then
\begin{equation*} \frac{2}{\lvert K\rvert} \int_{K} u^2 \,dV_g \le \sum_{k=0}^{\infty} \frac{1}{(\alpha C')^k\bigl(E+\tfrac{k^2}{R^2}\bigr)^{k}} \langle \nabla^ku , \nabla^ku\rangle_{g_q}. \end{equation*}Integrating both sides over \(K\), and employing \eqref{eq:good_ball_prop} yields
\begin{align*} 2 \int_{K} u^2 \,dV_g &\le \sum_{k=0}^{n} \frac{1}{(\alpha C')^k\bigl(E+\tfrac{k^2}{R^2}\bigr)^{k}} \int_{K} \langle \nabla^ku , \nabla^ku\rangle_g \, dV_g \\ &\le \sum_{k=0}^{n} \frac{1}{\alpha^k} \int_{K} u^2 \,dV_g \\ &\le \frac{\alpha}{\alpha -1} \int_{B} u^2, \end{align*}If we set \(\alpha=3\), it follows that
\begin{equation*} 2 \int_{K} u^2 \,dV_g\le \frac{3}{2} \int_{K} u^2\,dV_g. \end{equation*}This contradiction proves our claim.