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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.

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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).

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  • $\begingroup$ Thank you Giorgio, this is very helpful. Any clue regarding L1 bounds? $\endgroup$
    – Amir Sagiv
    Commented Dec 6, 2020 at 13:39
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    $\begingroup$ No, I am sorry. Apart from the estimate from below which follows by duality. $\endgroup$
    – Giorgio Metafune
    Commented Dec 6, 2020 at 16:46
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The L1 bound follows immediately from Cauchy-Schwarz inequality applied to the functions $|Y^m_l|$ ($m$ is a multi-index)and (the constant function) $1$, and gives $\int_{S^d}|Y^m_l|\cdot 1 d\Omega\leq \sqrt{Vol(S^d)}$

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  • $\begingroup$ but is this sharp? $\endgroup$
    – Amir Sagiv
    Commented Oct 9, 2023 at 17:49

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