I would like to ask if there is a holder esitimate for eigenfunctions of Laplace operator on sphere? I mean the esimate for \begin{equation} \|\partial^{\alpha}h_n\|_{L^{\infty}(S^{d-1})} \end{equation} in term of $n$, where $h_n$ is the $n^{th}$ eigenfunction (in the orthonormal basic of $L^2(S^{d-1})$ composed of eigenfunctions).\ If $\alpha=0$, I've found the following result \begin{equation} \|h_n\|_{L^{\infty}(S^{d-1})}\le C_1\lambda_n^{\frac{d-1}{2}}\le C_2 n^{\frac{d-1}{2d}}. \end{equation} Thanks in advance for your help!
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