The spherical harmonics are given by 
$$Y^m\_l(\phi,\theta):=N^m\_l e^{im\theta}P^m\_l(\cos \phi) $$
where $P^m\_l$ are the associated Legendre Polynomials and $N^m\_l$ is the normalisation.
From inspecting the first few it seems 
$$\left\Vert Y^m\_l \right\Vert \leq C <\infty.$$
**Is this true?** If not is there an easy upper bounded of the form 
$$\left\Vert Y^m\_l \right\Vert\leq C^m\_l <\infty.$$



Are spherical harmonics uniformly bounded?
As mentioned above, your question cannot be answered without knowledge of which norming sequence you are using. However, there are uniform bounds for the derivatives of the Legendre polynomials in the literature (see, e.g., pp. 523, 524 of the article reviewed in MR 0306897 by Guillemont-Tessier, which is available on line) and these immediately give estimates for the functions you are interested in.


