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.