Hello, I am looking for boundary regularity of solutions of $(-\Delta)^s u= f(x)$ in $\Omega$ with Dirichlet boundary conditions and where $f $ is nice enough say $f\in C^{1,\alpha}(\overline\Omega)$. I have seen a result by Cabre for $s=1/2$ and $f=0$ on the boundary. He showed $u\in C^{2,\alpha}(\overline\Omega)$. So, my guess is in the fractional case for $f=0$ on the boundary we should be able to get $u\in C^{1,\alpha+2s}(\overline\Omega)$ for $s<1/2$ and $u\in C^{1+2s,\alpha}(\overline\Omega)$ for $s>=1/2$. But, I cannot prove it or find it in literature. Is anyone aware of that?

Moreover, I have seen similar type results on the whole space. But, I am not sure how I can use them to prove boundary regularity for bounded domains.

Thanks.