  Let $L=-\operatorname{div}_x(A(x)\nabla_x)$ be a uniformly elliptic operator
in divergence form in a bounded domain $\Omega$. We consider the fractional
nonlocal equations $$\begin{cases} L^su=f,&\hbox{in}~\Omega,\\
u=0,&\hbox{on}~\partial\Omega, \end{cases}\quad \hbox{and}\quad \begin{cases}
L^su=f,&\hbox{in}~\Omega,\\ \partial_Au=0,&\hbox{on}~\partial\Omega.
\end{cases}$$ Here $L^s$, $0<s<1$, is the fractional power of $L$ and
$\partial_Au$ is the conormal derivative of $u$ with respect to the
coefficients $A(x)$. We reproduce Caccioppoli type estimates that allow us to
develop the regularity theory. Indeed, we prove interior and boundary Schauder
regularity estimates depending on the smoothness of the coefficients $A(x)$,
the right hand side $f$ and the boundary of the domain. Moreover, we establish
estimates for fundamental solutions in the spirit of the classical result by
Littman--Stampacchia--Weinberger and we obtain nonlocal integro-differential
formulas for $L^su(x)$. Essential tools in the analysis are the semigroup
language approach and the extension problem.
