Theorem 13.1.1 in Jost's Partial Differential Equations asserts that if $f \in L^\infty(\Omega)$, with $\Omega$ a bounded open set in $\mathbb{R}^2$, then $$ u(x) = \int_\Omega \log |x-y| f(y)\ dy $$ is in $C^{1,\alpha}(\Omega)$.

But I think there is an error in the proof. Equation 13.1.7 says that for fixed $x_1$ and $x_2$ there exist a constant $c_3$ and a point $x_3$ on the line connecting $x_1$ and $x_2$ such that

\begin{equation*} \left| \frac{x^i_1 - y^i}{|x_1-y|^2} - \frac{x^i_2 - y^i}{|x_2-y|^2} \right| \leq c_3 \frac{|x_1-x_2|}{|x_3-y|^2} \end{equation*}

Here, $x_1^i$ refers to the $i^{th}$ component of $x_1 \in \mathbb{R}^2$. However, either the constant or $x_3$ must depend on $y$. Otherwise, we could send $y$ to $x_1$ and get the left-hand side of the inequality to blow up, while keeping the right-hand side finite.

But in the next step, Jost splits an integral in $y$ over a domain $D$ into an integral over $D \setminus B_\delta(x_3)$ and $B_\delta(x_3)$ and treats $c_3$ as a constant independent of $y$. I don't see how this works if one of $c_3$ or $x_3$ depends on $y$.

My questions are:

(1) Is this an error in Jost or am I missing something? Perhaps there is an obvious way to fix $x_3$ that I don't see.

(2) If it is an error in Jost, is the result still true? I don't see it in, for instance, Gilbarg and Trudinger. They have a similar estimate but require Holder continuity for $f$.