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**Jose27** Let $\displaystyle f: \Omega \subset \mathbb{R} ^n \longrightarrow \mathbb{R}$ such that $\displaystyle f \in L^1 ( \Omega )$ and for all $\displaystyle x \in \Omega$ and $\displaystyle R_x$ such that $\displaystyle \overline{B_{R_x} (x)} \subset \Omega$ we have $\displaystyle f(x)=\frac{1}{ \vert B_{R_x}(x) \vert } \int_{B_{R_x}(x)} \ f$ . Is $\displaystyle f$ harmonic in $\displaystyle \Omega$ ?

I know the result is true if $\displaystyle f \in C^2( \Omega )$, but I don't know how to argue that if $\displaystyle f \in L^1 ( \Omega )$ then it's twice differentiable.