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

Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains