TITLE: Spherical harmonics – pointwise and L1 bounds
QUESTION [9 upvotes]: Let $\{ \phi _{d,m}\}_{m\geq 1}$ be multi-dimensional spherical harmonics, i.e., solutions of $\Delta \phi = E\phi$ on the sphere $S^d$ for $d>1$, arranged in an increasing order $E_1 \leq E_2 \leq \cdots \leq E_m \leq \cdots $, and normalized in $L^2(S^d)$, i.e., $\int_{S^d} |\phi_{d,m}(x)|^2 \, dx = 1 $.
Question: What are the best known upper and lower bounds on the $L^1$ and $L^{\infty}$ norms of $\phi_{d,m}$? What are good references for these bounds?
I am specifically interested in the dependence on $d$, so if the results are only known for the first few $m$ values, it would be of interest.

REPLY [5 votes]: 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 or in Stein-Weiss: Introduction to Fourier Analysison Euclidean spaces, Chapter IV, Corollary 2(b).