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We study the zero exterior problem for the elliptic equation $$\Delta^{\alpha/2}u-\lambda u=f, \quad x\in D\,; \quad u|_{D^c}=0 $$ as well asfor the parabolic equation $$ u_t=\Delta^{\alpha/2}u+f, \quad t>0,\, x\in D \,;\quad u(0,\cdot)|_D=u_0, \,u|_{[0,T]\times D^c}=0. $$ Here, $\alpha\in (0,2)$,$\lambda \geq 0$ and $D$ is a $C^{1,1}$ open set. We prove uniqueness andexistence of solutions in weighted Sobolev spaces, and obtain global Sobolevand H\"older estimates of solutions and their arbitrary order derivatives. Wemeasure the Sobolev and H\"older regularities of solutions and their arbitraryderivatives using a system of weights consisting of appropriate powers of thedistance to the boundary. The range of admissible powers of the distance to theboundary is sharp.

Sobolev regularity theory for the non-local elliptic and parabolic equations on $C^{1,1}$ open sets