# Are spherical harmonics uniformly bounded?

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 $$\left\Vert Y^m_l \right\Vert\leq C^m_l <\infty.$$

There is the general fact that when a compact topological group acts (continuously) transitively on a topological space with an invariant probability measure, a stable finite-dimensional vector space of functions has a sharp estimate for the ratio of sup-norm to $L^2$ norm: square root of the dimension of the space of functions.