It's well known that if $\Omega$ is a bounded set and $f$ is locally Holder continuous on $\Omega$ and bounded, then $u = \int_{\Omega} \Gamma(x-y)f(y) \ dy$ is a classical solution to $\Delta u=f$, where $\Gamma$ here is the fundamental solution of the Laplacian.
I have read the proof of this in Gilbarg and Trudinger. There is a problem in this book to prove that we can replace the assumption $f \in L^\infty(\Omega)$ with the assumption $f \in L^p(\Omega)$ for $p>n/2$. In the proof of the bounded case, the assumption of boundedness only comes in (as far as I can tell) in showing that $$\int_{B_\epsilon(x)} |\nabla \Gamma(x-y)| |f(y)| \to 0$$ as $\epsilon \to 0$. If we assume $f \in L^p(\Omega)$ with $p>n$, this can be shown with Holder's inequality. But just $p>n/2$ is giving me trouble. I'm wondering if this is a mistake, because another book I've glanced through (Analysis, by Lieb and Loss) seems to state that for $p>n$, $u$ is indeed a classical $C^2$ solution, and all they say for $n/2 \le p <n$ is that $u$ is Holder continuous.
So, is the result true? Or is it just a mistake by Gilbarg and Trudinger.