Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$ with smooth boundary. For any $f \in L^1 (\Omega)$, is it true that $f \in L^\infty (\Omega)$ if and only if the following condition holds?

For every $\delta > 0$, there exists some $C > 0$ such that for almost every $x \in \Omega$, and every open ball $B \subset \Omega$ containing $x$ with radius at least $\delta$ we have

$$|f(x)| \leq C \def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_B |f|.$$

*Note: Here $\def \avint{\mathop{\,\rlap{-}\!\int}\nolimits} \avint_B |f|$ denotes the average integral of $|f|$ over $B$.*

*Another note:* The balls $B$ are not necessarily centered at $x$.