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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Elements of the spectral subspace of the Laplace-Beltrami operator \(-\Delta_{\mathbb{S}^d_R}\) are spherical polynomials .
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Employing the estimates (0x68dcfef9) and (0x68bbccca) , yields that there is constant depending on \(d\), such that
\begin{equation*} \lVert \nabla^k u\rVert_{L^2(\mathbb{S}^d_{R})}^2 \le C^k (E^k + R^{-2k} (k!)^2)). \end{equation*}This estimate is applicable in Kovrijkine’s Lemma if the radius in the expansion is small enough.
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In the proof we heavily use the fact that the curvature tensor of \(\mathbb{S}^d_R\) is covariantly constant. By the Cartan-Ambrose-Hicks theorem , the curvature tensor of every locally symmetric Riemannian manifold is covariantly constant. Therefore, it looks feasible to generalize the Bernstein inequality on those spaces. If we can control the boundary term in the divergence theorem, we can also drop the compactness assumption. In fact, the curvature tensor of the hyperbolic space is the same as the one of the sphere, except for a sign.
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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.
For \(k=0\), the statement holds trivially. Assume relation \eqref{eq:bernstein} holds for \(k\ge 0\).
We now prove the claim for \(k+1\). Using the divergence theorem , we deduce
\begin{align*} \langle \nabla^{k+1} u, \nabla^{k+1} u\rangle_{L^2(\mathbb{S}^d_R)}^2 &= \int_{\mathbb{S}^d_R} \langle \nabla^{k+1} u, \nabla^{k+1} u\rangle_g \,dV_g \\ &=-\langle \div (\nabla^{k+1} u), \nabla^k u\rangle_{L^2(\mathbb{S}^d_R)}. \end{align*}We derive a local estimate of \(\langle \div (\nabla^{k+1} u), \nabla^{k} u\rangle_g\). Let \(u_{i_1\cdots i_{k+2}}\) denote the local representation of \(\nabla^{k+2} u\). Then,
\[ u_{i_1\cdots i_{k}p}{}^p\quad \text{and}\quad u_{i_1\cdots i_kp}{}^p u^{i_1\cdots i_k} \]are local representations of \(\div (\nabla^{k+1} u)\) and \(\langle \div (\nabla^k u), \nabla^k u\rangle_g\), respectively. The summation convention implies summation over corresponding indices, and indices are raised via musical isomorphisms .
By the Ricci identity and the fact, that the curvature tensor of \(\mathbb{S}^d_R\) is covariantly constant,
\begin{equation}\label{eq:div} \begin{aligned} u_{i_1 \cdots i_k p}{}^{p} &= (\Delta u)_{i_1 \cdots i_k} + \sum_{s=1}^{k} \Bigl( R_{i_s p}{}^{pm} \, u_{i_1 \cdots i_{s-1} m\, i_{s+1} \cdots i_k} + \\ &\quad +2 \sum_{r < s} R_{i_s}{}^p{}_{i_r}{}^m \, u_{i_1 \cdots i_{r-1} m\, i_{r+1} \cdots i_{s-1} p\, i_{s+1} \cdots i_k}\Bigr), \end{aligned} \end{equation}where \(R_{ijkl}\) and \((\Delta u)_{i_1\cdots i_k}\) denote the Riemann curvature endomorphism and \(\nabla^k (\Delta u)\) in local coordinates, respectively (see (0x68dce40d) ).
Since \(R_{i_sp}{}^{pm}=R_{i_s}{}^m\), where \(R_{ij}\) is the Ricci curvature in local coordinates, and
\[ R_{ij} = \frac{d-1}{R^2} g_{ij} \]on the sphere \(\mathbb{S}^d_R\), where \(g_{ij}\) denotes the metric tensor, it follows that
\begin{equation}\label{eq:ricci} R_{i_s p}{}^{pm} \, u_{i_1 \cdots i_{s-1} m\, i_{s+1} \cdots i_k} u^{i_1\cdots i_k} = \frac{d-1}{R^2} u_{i_1\cdots i_k}u^{i_1\cdots i_k}. \end{equation}Since
\[ R_{i j k l} = \frac{1}{R^2} \big( g_{i l} g_{j k} - g_{i k} g_{j l} \big) \]on the sphere \(\mathbb{S}^d_R\), we deduce
\begin{align*} R_{i_s}{}^p{}_{i_r}{}^m \, u_{i_1 \cdots i_{l-1} m\, i_{l+1} \cdots i_{s-1} p\, i_{s+1} \cdots i_k} \, u^{i_1\cdots i_k} = \frac{1}{R^2} ( u_{i_1 \cdots i_{r-1}\, i_s\, i_{r+1} \cdots i_{s-1}\,i_r\, i_{s+1} \cdots i_k} \, u^{i_1\cdots i_k} + \\ + u_{i_1 \cdots i_{r-1} \,p\, i_{r+1} \cdots i_{s-1}}{}^p{}_{ i_{s+1} \cdots i_k} \, u^{i_1 \cdots i_{r-1} \,m\, i_{r+1} \cdots i_{s-1}}{}_m{}^{ i_{s+1} \cdots i_k} ). \end{align*}The right-hand side is the local representation of
\[ \frac{1}{R^2} \bigl( \langle S_{rs}(\nabla^k u), \nabla^k u\rangle_g + \langle T_{rs} (\nabla^k u), T_{rs}(\nabla^k u)\rangle_g \bigr), \]where \(S_{rs}\) and \(T_{rs}\) denote the switch operator and the trace operator with respect to the \(r\)-th and \(s\)-th index.
Application of the Cauchy-Schwartz inequality and basic properties of \(S_{rs}\) and \(T_{rs}\) yields
\begin{equation}\label{eq:curv} \biggl\lvert \frac{1}{R^2} \bigl( \langle S_{rs}(\nabla^k u), \nabla^k u\rangle_g + \langle T_{rs} (\nabla^k u), T_{rs}(\nabla^k u)\rangle_g \bigr)\biggr\rvert \le \frac{2}{R^2} \langle \nabla^k u, \nabla^k u\rangle \end{equation}(see (0x68dcfaac) ).
Combining \eqref{eq:div}, \eqref{eq:ricci} and \eqref{eq:curv} yields
\[ \bigl\lvert \langle -\div (\nabla^k u), \nabla^k u\rangle_g\bigr\rvert \le \bigl\lvert\langle \nabla^k (- \Delta u), \nabla^k u\rangle_g\bigr\rvert +\Bigl(k \tfrac{d-1}{R^2} + \tfrac{k (k-1)}{R^2} \Bigr) \langle \nabla^k u, \nabla^k u\rangle_g. \]Since
\[ \bigl\lvert\langle \nabla^k (- \Delta u), \nabla^k u\rangle_g\bigr\rvert \le E \langle \nabla^k u, \nabla^k u\rangle_g \](see (0x68dcfc53) ), we obtain
\begin{equation}\label{eq:local_bernstein} \langle \nabla^{k+1} u , \nabla^{k+1} u \rangle_g \leq \Bigl( E + \tfrac{d-1}{R^2} (k+1)^2 \Bigr) \langle \nabla^k u , \nabla^k u \rangle_g. \end{equation}By \eqref{eq:local_bernstein} and the induction hypothesis, we finally deduce
\[ \lVert \nabla^{k+1} u\rVert^2_{L^2(\mathbb{S}^d_R)} \leq \Bigl( E + \tfrac{d}{R^2} (k+1)^2 \Bigr) \lVert \nabla^k u\rVert^2_{L^2(\mathbb{S}^d_R)} \le \bigl(E+\tfrac{d}{R^2} (k+1)^2\bigr)^{k+1} \lVert u\rVert_{L^2(\mathbb{S}^d_{R})}^2, \]what was to be shown.