Let $M = (M, g)$ be a Riemannian manifold, and for $f \in L_{loc}^1(M)$ (with respect to the volume measure $V$), write $$A_{x, r} f : = V(B(x, r))^{-1} \int_{B(x, r)} f \; \mathrm{d} V .$$ My question: Under what conditions on $f$ is the function $X \times \mathbb{R}^+ \to \mathbb{C}$ given by $(x, r) \mapsto A_{x, r} f$ a $C^1$ function? I've tried a few "toy examples" like when $M = \mathbb{R}, \mathbb{R}^2$ where this average admits an explicit formula, but I haven't been able to extrapolate to any greater generality. I admittedly don't know a lot about this area, so any relevant references would be greatly appreciated.
EDIT: $\mathbb{R}^+ : = (0, \infty)$
EDIT 2: Thorgott alerted me tothis post which pointed out that this function needn't be well-defined for large $r$, since even balls of finite radius can have infinite volume in the right metric. I'm okay limiting my attention to the case where $M$ is compact, which if I'm not mistaken should alleviate this issue.