Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains

We prove Besov regularity estimates for the solution of the Dirichlet problem involving the integral fractional Laplacian of order $s$ in bounded Lipschitz domains $\mathrm{\Omega }$: $$\begin{array}{rlr}{\displaystyle \mathrm{\parallel }u{\mathrm{\parallel }}_{{\dot{B}}_{2,\mathrm{\infty }}^{s+r}(\mathrm{\Omega })}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })},}& {\displaystyle r=\min \{s,1\mathrm{/}2\},}& {\displaystyle ifs\mathrm{\ne }1\mathrm{/}2,}\\ {\displaystyle \mathrm{\parallel }u{\mathrm{\parallel }}_{{\dot{B}}_{2,\mathrm{\infty }}^{1-ϵ}(\mathrm{\Omega })}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })},}& {\displaystyle ϵ\in (0,1),}& {\displaystyle ifs=1\mathrm{/}2,}\end{array}$$ with explicit dependence of $C$ on $s$ and $ϵ$. 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.