The sup-norm of a (normalized) spherical harmonic of degree $k$ is bounded by $C \sqrt d_k$, where $d_k$ is the dimension of all spherical harmonics of order $k$ and $C$ is also given. The estimate is precise. Everything is explicit and can be found, for example, in the notes by P. Garrett: Harmonic analysis on spheres.
Giorgio Metafune
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