I am a bit confused with the definition of Green function (see Definition 1.9) here:
https://arxiv.org/pdf/1502.06468.pdf
Forget for a moment that the problem is for the fractional case. Even for the case of the Laplacian, I have never seen Green's formula (for the ball) containing the Poisson kernel. Is instead true also in the local case?
I know (from Evans) that the Green's formula for the ball in the case of the Laplacian is given by $$G(x, y):= \Phi(y-x) -\phi^x(y),$$ with the correction function $\phi^x(y)$ is given by $\Phi(|x|(y-\widetilde{x})$, where $\Phi$ denotes the fundamental solution and $\widetilde{x} = x/|x|^2$. Hence, how to write the Green's formula using the Poisson's kernel?
Let $\Omega \subset \mathbb R^n$ be a bounded domain with smooth boundary with Green’s function and Poisson kernel given by $G$ and $P$ respectively. Then the solution to \begin{align*} \Delta u &=0, \qquad \text{in } \Omega \\ u&=g, \qquad \text{on } \partial \Omega \end{align*} is given by \begin{align} u(x) &= \int_{\partial \Omega} P(x,y)g(y) \, d \mathcal H^{n-1}_y. \tag{$\ast$} \end{align}
The Green’s function is defined as $$ G(x,y) = \Phi(\vert x- y \vert ) -\phi^x(y) $$ with $\phi^x$ the solution to $$ \begin{align*} \Delta_y \phi^x &=0, \qquad \text{in } \Omega \\ \phi^x&=\Phi(\vert x-y\vert ), \qquad \text{on } \partial \Omega . \end{align*} $$ Hence, by $(\ast)$, we have that $$\phi^x(y) = \int_{\partial \Omega} P(y,z)\Phi(\vert x- z \vert )\, d \mathcal H^{n-1}_z .$$ Thus, $$G(x,y) = \Phi(\vert x-y\vert )- \int_{\partial \Omega} P(y,z)\Phi(\vert x- z \vert )\, d \mathcal H^{n-1}_z .$$
The argument for the nonlocal case is completely analogous.