Let \(n\ge 2\), \(N\ge 0\), \(S\in \Pi_N\) be spherical polynomial of order at most \(N\) . Then for \(\gamma>0\) and \(p \in [1,\infty]\), we have

\begin{equation*} \lVert S\rVert_{H^{\gamma, p}(\mathbb{S}^n)} \le C_{n,\gamma} N^{\gamma} \lVert S\rVert_{L^p(\mathbb{S}^n)}. \end{equation*}

[1, Theorem 4.10]

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  1. H. Mhaskar, F. Narcowich, J. Prestin, and J. Ward, $L^p$ Bernstein estimates and approximation by spherical basis functions, Mathematics of Computation, vol. 79, no. 271, pp. 1647–1679, 2009. doi:10.1090/S0025-5718-09-02322-9