Given a function $f \in L^1(\mathbb R)$, suppose that
$$ \Big|\frac{1}{\text{vol}(B(x,r))}\int_{B(x,r)} f(z)dz -f(x) \Big| \le C r^\alpha $$ uniformly over $x \in K$ some bounded domain of $\mathbb R^d$ for some $C>0,\alpha \in (0,1) $. Do we have that $f \in C^\alpha(K)$? The converse is simple to get, but I could not prove whether this is true.