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

Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic function taking boundary value $F(x,\cdot)$. Poisson's kernel is defined as $P(x,y)=-\frac{\partial}{\partial n(y)}G(x,y)$ with $x\in E$ and $y\in \partial E$ and $n(y)$ is the outer normal at $y$.

Question: is $\int_E P(x,y)\,dx$ always finite for all $y\in \partial E$?