Let $x \in \mathbb{R}^n$ and  $f:\mathbb{R^n} \to \mathbb{R}$ be a non-negative function such that $f(x)=0$. Is it true that (assuming $\alpha,\beta>0$) 
$$\limsup_{r \to 0} r^{-\alpha \beta}\frac{1}{|B_{r}(x)|}\int_{B_{r}(x)} f(y)^\alpha dy < \infty$$
implies $$|f(z)| \le C|z-x|^\beta,$$ for $z$ close enough to $x$ and for some constant $C>0$?