Boundary regularity type results of fractional laplacian Let $u$ be a solution of $-\Delta u= f \text{ in } \Omega$ with $u=0 \text{ on } \partial \Omega.$ If $f\in L^2(\Omega)$, then $u\in H^2(\Omega)\cap H^{1}_{0}(\Omega)$provided $\Omega$ is a smooth bounded domain. 
Does this type of theorem holds true for the equation $(-\Delta)^s u= f \text{ in } \Omega$ with $u=0 \text{ in } \mathbb R^N -\Omega$ with $0<s<1.$ 
 A: In your case, $u$ is locally in the fractional Sobolev space $H^{2s}$ (inside $\Omega$), and globally in $H^s$. The latter is either an assumption (in the weak formulation of the problem) or a proposition (when a different notion of a solution is used). The former follows already from the result given in Stein's book:

E. M. Stein, Singular Integrals and Differentiability Properties Of Functions, Princeton University Press, Princeton, 1970.

Similar questions were studied in much more detail by Xavier Ros-Oton and Joaquim Serra (with smoother $f$, though) in their paper:

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl. 101 (2014): 275–302.

You may also have a look at the survey paper:

X. Ros-Oton, Nonlocal equations in bounded domains: a survey, Publ. Mat. 60(1) (2016): 3–26.

The first paper which deals with boundary regularity of $u$ is likely due to Krzysztof Bogdan:

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Stud. Math. 123(1) (1997): 43–80.

You can also find these references in my survey:

M. Kwaśnicki, Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019.

