# Eigenfunctions of the fractional Laplacian are smooth?

Let $$\Omega\subset\mathbb{R}^n$$ open, bounded with smooth boundary, let $$s\in(0,1)$$. I know that the fractional Laplacian has a sequence of eigenfunctions $$\{e_k\}_{k\in\mathbb{N}}\subset H^s(\mathbb{R}^N)$$, $$e_k=0$$ a.e. on $$\mathbb{R}^n\setminus\Omega$$, $$\forall k\in\mathbb{N}$$. Moreover I know that these eigenfunctions are continuous on the whole of $$\mathbb{R}^n$$. My question is: are these eigenfunctions smooth in $$\Omega$$? Can you give me some reference on this result (if they exist)?

This is consequence of the theory of Pseudo-differential operators. Your fractional Laplacian $$P_s$$ is a PDO of order $$2s$$. Above all, it is elliptic. This is a classical result (certainly in Hörmander) that if $$P_su\in H^r_{\rm loc}$$, then $$u\in H^{r+2s}_{\rm loc}$$, where $$H^{\cdots}$$ is the class of Sobolev spaces.
Apply this to an eigenfunction $$e_k$$, which you know is continuous, hence is $$H^0_{\rm loc}=L^2_{\rm loc}$$. Bootstrapping, you find that $$e_k$$belongs to $$\bigcap_{\ell\in\mathbb N}H^{2\ell s}_{\rm loc}\subset C^\infty.$$