The first eigenfunction of fractional laplacian Let $\Omega$ be bounded and smooth domain in $\mathbb{R}^n$, $s\in(0,1)$, $e_1\in \mathbb{H}^s(\Omega)$ the first eigenfunction of fractional laplacian $(-\Delta)^s$ with eigenvalue $\lambda_1>0$, in weak formulation, that is:
$$ 
\frac{C(n,s)}{2}\int_{\mathbb{R}^n\times\mathbb{R}^n}\frac{(e_1(x)-e_1(y))(\phi(x)-\phi(y))}{|x-y|^{n+2s}}\,dx\,dy=\lambda_1\int_\Omega e_1(x)\phi(x)\,dx,
\quad\forall\phi\in \mathbb{H}^s(\Omega).
 $$
I know that $e_1$ is continuous on whole $\mathbb{R}^n$. I want to prove that:
$$ (-\Delta)^se_1(x)=\lambda_1e_1(x), \quad\forall x\in\Omega,$$
but i have no idea on how to proceed. Any help would be appreciated.

Here:
$$ \mathbb{H}^s(\Omega)=\{u\in H^s(\mathbb{R}^n): u=0\,\, \text{ q.o. }\in \mathbb{R}^n\setminus\Omega\},$$
and:
$$ 
(-\Delta)^su(x):=\frac{C(n,s)}{2}\int_{\mathbb{R}^n}\frac{2u(x)-u(x+y)-u(x-y)}{|y|^{n+2s}}\,dy,\quad\forall x\in\mathbb{R}^n, \forall u\in\mathcal{S}(\mathbb{R}^n).
$$
Moreover, how i can define $(-\Delta)^s$ for less regular function?
 A: Just an extended comment.

*

*Theorem 4.1 in The Cauchy process and the Steklov problem by Rodrigo Bañuelos and Tadeusz Kulczycki (JFA 2004, DOI:10.1016/j.jfa.2004.02.005) shows that the eigenfunctions $e_n$ are even real-analytic for $s = \tfrac12$, and the authors write that the proof carries over to general $s$ at the price of additional technical difficulties.


*The eigenfunctions are $C^\infty$, as can be easily proved directly using potential-theoretic methods: we have
$$ \lambda_n e_n(x) = \int_B G_B(x, y) e_n(y) dy = I_{2s} e_n(x) - \int_{B^c} I_{2s} e_n(z) P_B(x, z) dz , $$
where $B$ is a ball contained in $\Omega$, $G_B(x,y)$ is the Green function, $P_B(x,z)$ is the harmonic measure (a.k.a. the Poisson kernel), and $I_{2s}$ is the Riesz potential operator. Now it is well-known that if $f$ is of class $C^\beta$ near a point $x$, then $I_{2s} f$ is of class $C^{\beta + 2s}$ near $x$ (see, for example, Stein's book). Furthermore $P_\Omega(\cdot, z)$ is known explicitly and it is smooth (even real-analytic). Thus the above display is self-improving, and shows that if $e_n$ is merely bounded in $B$, then it is automatically $C^\infty$ in $B$. A similar argument is given in my survey Fractional Laplace Operator and its Properties, DOI:10.1515/9783110571622-007.


*Alternatively, one can use the PDE-flavoured regularity theory, developed in the last decade by Caffarelli, Silvestre, Serra, Ros-Oton and others. In any case, however, this is not a trivial


*Once we know that $e_n$ is smooth enough, all that remains is to use Fubini's theorem to rearrange the integrals, and use a density argument.
