There is a finite cover \((K_i)_{i=1,\ldots ,N}\) of \(\mathbb{S}^d_R\) consisting of spherical caps with radius \(r\), and the covering multiplicity is bounded by some constant \(C_\kappa> 0\) depending on \(d\).

Let \(\nabla\) denote the levi-civita connection on \(\mathbb{S}^d_R\) and let \(g\) denote the round metric on \(\mathbb{S}^d_R\).

We call a spherical cap \(K_i\) bad if there exists a number \(k>1\) such that

\begin{equation*} \lVert \nabla^k u\rVert_{L^2(K_i)}^2 \le C^k\bigl(E+\tfrac{k^2}{R^2}\bigr)^{k} \lVert u\rVert_{L^2(K_i)}^2, \end{equation*}

with

\[ C=d(2C_\kappa + 1). \]

Note, that \(C\) depends only on \(d\), since \(C_\kappa\) also does, and that every \(K_i\) is either good or bad.

Then, by the Bernstein inequality on the sphere ,

\begin{align*} \int_{\bigcup_{K_i \text{ is bad }}} u^2\, dV_g &\le \sum_{K_i \text{ is bad}} \int_{K_i} u^2 \,dV_g \\ &\le \sum_{K_i \text{ is bad}} \int_{K_i} \sum_{k=1}^{\infty} \frac{1}{C^k\bigl(E+\tfrac{k^2}{R^2}\bigr)^{k}} \langle \nabla^k u, \nabla^k u\rangle_g \,dV_g \\ &= \sum_{k=1}^{\infty} \frac{1}{C^k\bigl(E+\tfrac{k^2}{R^2}\bigr)^{k}} \sum_{K_i \text{ is bad}} \int_{K_i} \langle \nabla^k u, \nabla^k u\rangle_g \,dV_g \\ &\le \sum_{k=1}^{\infty} \frac{C_\kappa d^k}{C^k\bigl(E+\tfrac{k^2}{R^2}\bigr)^{k}} \int_{\mathbb{S}^d_R} \langle \nabla^k u, \nabla^k u\rangle_g \,dV_g \\ &\le C_\kappa \sum_{k=1}^{\infty} \frac{d^k}{C^{k}} \int_{\mathbb{S}^d_R} u^2 \,dV_g\\ &= \frac{C_\kappa}{C/d-1} \int_{\mathbb{S}^d_R} u^2\,dV_g \\ &= \frac{1}{2} \int_{\mathbb{S}^d_R} u^2 \,dV_g. \end{align*}

Since every \(K_i\) is either good or bad, we deduce

\[ \int_{\mathbb{S}^d_R} u^2 \,dV_g \le \int_{\bigcup_{K_i \text{ is bad }}} u^2\, dV_g + \int_{\bigcup_{K_i \text{ is bad }}} u^2\, dV_g \le \frac{1}{2} \int_{\mathbb{S}^d_R} u^2 \,dV_g + \int_{\cup_{K_i \text{ is good}} } u^2 \,dV_g, \]

which implies \eqref{eq:good_cap_ineq}.