Does there exist a theory of fractional laplacian on exterior domains such as $$ \ \ \left\{\begin{aligned} (-\Delta)^{s} u&= 0 &&\text{in } \mathbb R^N\setminus \mathbb B \\ u & =g &&\text{ in } \mathbb B \end{aligned} \right. $$ where $\mathbb B$ is a unit ball, $N\geq 1$ and $s\in (0, 1).$ Does there exist a Poisson kernel for this type of problem. Is the solution for the problem be expressible as the combination of the Poisson kernel and $g.$


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