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 $0<s<1.$ 



Boundary regularity type results of fractional laplacian
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.
> 
> 
> 


