Let \(\mathbb{S}^d_{R}\) be the \(d\)-sphere with radius \(R>0\), and let \(\nabla\) denote the Levi-Civita connection on \(\mathbb{S}^{d}_R\). Let \(\nabla^ku\) denote the \(k\)-th total covariant derivative tensor of \(u\), and let \(\langle \nabla^ku, \nabla^ku\rangle_g\) be the smooth fiber metric induced by \(g\).
Suppose \(u\in 𝟙_{(-\infty ,E]}(-\Delta_{\mathbb{S}^d_R})\), for \(E>0\). Then
\begin{equation}\label{eq:bernstein} \lVert \nabla^k u\rVert_{L^2(\mathbb{S}^d_{R})}^2 \le \bigl(E+\tfrac{d}{R^2} k^2\bigr)^{k} \lVert u\rVert_{L^2(\mathbb{S}^d_{R})}^2 \end{equation}for every \(k\in \mathbb{N}\).
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