9
$\begingroup$

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.

$\endgroup$
5
$\begingroup$

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).

$\endgroup$
2
  • $\begingroup$ Thank you Giorgio, this is very helpful. Any clue regarding L1 bounds? $\endgroup$ – Amir Sagiv Dec 6 '20 at 13:39
  • 1
    $\begingroup$ No, I am sorry. Apart from the estimate from below which follows by duality. $\endgroup$ – Giorgio Metafune Dec 6 '20 at 16:46

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.