The inhomogeneous fractional Dirichlet problem

We study boundary regularity for the inhomogeneous Dirichlet problem for $2s$-stable operators in generalized H\"older spaces. Moreover, we provide explicit counterexamples that showcase the sharpness of our results. Our approach directly addresses the inhomogeneous Dirichlet problem, rather than subtracting an appropriate extension of the exterior data. Even for the fractional Laplacian, our result is new.