  We analyze the local elliptic regularity of weak solutions to the Dirichlet
problem associated with the fractional Laplacian $(-\Delta)^s$ on an arbitrary
bounded open set $\Omega\subset\mathbb{R}^N$. For $1<p<2$, we obtain regularity
in the Besov space $B^{2s}_{p,2,\textrm{loc}}(\Omega)$, while for $2\leq
p<\infty$ we show that the solutions belong to
$W^{2s,p}_{\textrm{loc}}(\Omega)$. The key tool consists in analyzing carefully
the elliptic equation satisfied by the solution locally, after cut-off, to
later employ sharp regularity results in the whole space. We do it by two
different methods. First working directly in the variational formulation of the
elliptic problem and then employing the heat kernel representation of
solutions.
