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We introduce a weighted Sobolev space theory for the non-local ellipticequation $$ \Delta^{\alpha/2}u=f, \quad x\in D\,; \quad u|_{\overline{D}^c}=g$$ as well as for the non-local parabolic equation $$ u_t=\Delta^{\alpha/2}u+f,\quad t>0,\, x\in D \,; \quad u(0,\cdot)|_D=u_0, \,u|_{(0,T)\times\overline{D}^c}=g. $$ Here, $\alpha\in (0,2)$ and $D$ is a $C^{1,1}$ open set.We prove uniqueness and existence results in weighted Sobolev spaces. Wemeasure the Sobolev and H\"older regularities of arbitrary order derivatives ofsolutions using a system of weights consisting of appropriate powers of thedistance to the boundary.  One of the most interesting features of our results is that, unlike theclassical result in Sobolev spaces without weights, the weighted regularitiesof solutions in $D$ are barely affected by those of $g$ in $\overline{D}^c$.For instance, even if $g=\delta_{x_0}$, the dirac delta distributionconcentrated at $x_0\in \overline{D}^c $, the solution to the elliptic equationgiven with $f=0$ is infinitely differentiable in $D$, and for any $k=0,1,2,3,\cdots$, $\varepsilon>0$, and $\delta\in (0,1)$, it holds that $$|d_x^{-\frac{\alpha}{2}+\varepsilon+k}D^k_xu|_{C(D)}+|d_x^{-\frac{\alpha}{2}+\varepsilon+k+\delta} D^k_xu|_{C^{\delta}(D)}<\infty,$$ where $d_x=dist(x, \partial D)$.

Weighted Sobolev space theory for non-local elliptic and parabolic equations with non-zero exterior condition on $C^{1,1}$ open sets