Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^\infty$ boundary, $n\ge 3$. Define $$V(z)=\int_\Omega \frac{1}{|z-y|^{n-2}}dy$$ Is it true that $V(z) \in C^{\infty}(\partial \Omega)$?

Motivation: Classical Holder estimate says that if $f \in C_c^\alpha(\mathbb{R}^n)$, then the Newtonian potential given by $$V(z)=\int_{\mathbb{R}^n} \frac{f(y)}{|z-y|^{n-2}}dy$$ is $C^2$ with respect to $z$ variable. If $f$ is merely bounded, then $V$ is $C^1$ and we cannot expect better regulartiy. This can be seen from the classical example that $f=\chi_{B_1}$, where $\chi$ is the characteristic function and $B_1$ is the unit ball.

However, for this example, one can see that $V\in C^{\infty}(\partial B_1)$. From this example, it is natural to study the smoothness of $V$ along smooth boundaries. This motivates the original question. Of course, one can consider similar problems for more general Riesz potentials.

**Further Update**:

The question actually was motivated from a talk yesterday given by Jian Lu from South China Normal University. The talk is on chord log Minkowski problem. The speaker actually have already proved the smoothness of $V(z)$ along boundaries of any smooth convex bodies, and the details can be seen in http://arxiv.org/abs/2304.14220 (2nd version). Hence I was led to ask about the regularity on generic smooth sets.