Let \(\mathbb{S}^{d}_R\) be 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 \(\mathbb{S}^d_R\). Let \(\gamma \in (0,1]\), \(r>0\) (small enough) and \(S \subset M\) be a \((\gamma, r)\)-thick set .
Then are two constants \(c_1, c_2>0\) depending on \(d\) such that
\begin{equation*} \lVert u \rVert_{L^2(\mathbb{S}^d_R)} \le \Bigl(\frac{c_1}{\gamma}\Bigr)^{c_2(r\sqrt{E} + 1/2)}\lVert u\rVert_{L^2(S)}. \end{equation*}