Consider a function $f:\mathrm{SU}(N)\to \mathbb{C}$, where $\mathrm{SU}(N)$ is equipped with the Haar measure $\mu$. Assume that $f$ is Lipshitz continuous with Lipshitz constant $K$. Here continuity is with respect to the Hilbert-Schmidt norm.
Is it possible to bound the maximal value $\sup_{g\in G} |f(g)|$ by the $L^2$ norm and the Lipshitz constant?
My intuition is that the Lipshitz continuity rules out "too peaked" functions $f$.