  We prove Besov regularity estimates for the solution of the Dirichlet problem
involving the integral fractional Laplacian of order $s$ in bounded Lipschitz
domains $\Omega$: \[ \begin{aligned} \|u\|_{\dot{B}^{s+r}_{2,\infty}(\Omega)}
\le C \|f\|_{L^2(\Omega)}, &
  \quad r = \min\{s,1/2\}, & \quad \mbox{if } s \neq 1/2, \\
\|u\|_{\dot{B}^{1-\epsilon}_{2,\infty}(\Omega)} \le C \|f\|_{L^2(\Omega)}, &
  \quad \epsilon \in (0,1), & \quad \mbox{if } s = 1/2, \end{aligned} \] with
explicit dependence of $C$ on $s$ and $\epsilon$. These estimates are
consistent with the regularity on smooth domains and show that there is no loss
of regularity due to Lipschitz boundaries. The proof uses elementary
ingredients, such as the variational structure of the problem and the
difference quotient technique.
