Let ${\mathbb H}_l$ be a space of spherical harmonics of order $l$, i.e. $${\mathbb H}_l = \{f: {\mathbb S}^{n-1}\to {\mathbb R} \mid \Delta_{{\mathbb S}^{n-1}}f=-l(l+n-2)f\},$$ where $\Delta_{{\mathbb S}^{n-1}}$ is Laplace-Beltrami operator.
It is known that if $\|f\|_{L_2({\mathbb S}^{n-1})}=1$, then $\|f\|_{L_\infty({\mathbb S}^{n-1})}\leq \sqrt{\frac{N(n,l)}{\omega_{n-1}}}$, where $N(n,k) = {\rm dim}({\mathbb H}_l)$ and $\omega_{n-1}$ is the surface volume of ${\mathbb S}^{n-1}$. Moreover, $$\sup_{f\in {\mathbb H}_l: \|f\|_{L_2({\mathbb S}^{n-1})}=1}\|f\|_{L_\infty({\mathbb S}^{n-1})}=\sqrt{\frac{N(n,l)}{\omega_{n-1}}}.$$ Details can be found at $L^\infty$-bounds on eigenfunctions of Laplace-Beltrami opeator .
Is there any known upper bounds on $$\inf_{f\in {\mathbb H}_l: \|f\|_{L_2({\mathbb S}^{n-1})}=1}\|f\|_{L_\infty({\mathbb S}^{n-1})},$$ i.e. how small supremum norm of harmonics of order $l$ can be?
