Let $G$ be the Green function of the Fractional Laplacian $(-\Delta)^s$ in a domain $\Omega$ (which is known explicitly for the special case of the ball: link). Let $\bar x \in \Omega$ be fixed. Does $\nabla_x G(\bar x,z)$ satisfy $$(-\Delta)^s\nabla_x G(\bar x,z) = 0 \text{ in } \Omega \ ?$$ In other words I'm asking if $\nabla_x$ and $(-\Delta)^s$ commute on a bounded domain $\Omega$, which was answered in $\mathbb R^n$ in Derivative of fractional Laplacian is the fractional Laplacian of the derivative
