# Spherical harmonics – pointwise and L1 bounds

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.

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).