If $f$ is a complex-valued function on a compact Lie group $G$, we have a decomposition $f = \sum_\mu f_\mu$ corresponding to the Peter-Weyl decomposition $L^2(G) = \oplus_\mu (\dim \mu) V_\mu$.
For $f \in C^\infty(G)$, it can be shown that for any $n > 0$, there are $k, C > 0$ so that: $$\|f_\mu\| \leq C\frac{\|f\|_{C^k}}{\|\mu\|^n}$$ where $\|\mu\|$ is defined to be the infimum of the Lipschitz constants of the orbit maps $g \mapsto \mu(g) \cdot v$, taken over the vectors $v \in V_\mu^1$ of norm $1$. This is proven using the Weyl dimension formula and some representation theory (see lemma 4.4.2.2/3 in Garth Warner's Harmonic Analysis on Semisimple Lie Groups).
My question: is something along these lines true for Hölder functions? A bound involving $\|f\|_{C^\alpha}$ and $\|\mu\|$ as above would be ideal, but I think most things vaguely resembling the inequality above would be helpful for what I'm doing. I suspect something like this is probably well-known, but I haven't had much luck searching for this or asking around.
