Applying the divergence theorem several times, yields
\[ \langle \nabla^k u, \nabla^k u\rangle_{L^2(\mathbb{S}^d_R)} = \langle P_k(-\Delta) u, u\rangle_{\mathbb{S}^d_R}, \]where \(P_k\) is a polynomial. We can express \(P_k\) by the algebraic description in this note , which can be derived using the Ricci identity . We claim
\begin{equation*} \lvert P_k(t)\rvert \le (t+\frac{C}{R^2}k)^k. \label{eq:1} \end{equation*}for all \(t\in [0,E]\), which implies
\[ \lVert \nabla^k u\rVert_{L^2(\mathbb{S}^d_{R})} \le \bigl(E+\tfrac{C}{R^2} k\bigr)^{\tfrac{k}{2}} \lVert u\rVert_{L^2(\mathbb{S}^d_{R})}. \]Examples of \(P_k\) up to \(k=11\) support this claim.