Let $E$ be a smooth domain. The Green function $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation and for fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic function in $E$ and takes boundary value $F(x,\cdot)$. Poisson's kernel is defined as $P(x,y)=\frac{\partial}{\partial n(y)}G(x,y)$ for $x\in E$ and $y\in \partial E$. Question: is $\int_E P(x,y)\,dx$ finite for all $y\in \partial E$?
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