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 $B \subset \Omega$ centered at $x$ 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 $B$.*

*Edit:* The original question asked about the case where the balls $B$ are not necessarily centered at $x$. This has been solved in the comments - the "if" direction holds, but "only if" does not.

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