Integral representation of solutions to higher-order fractional Dirichlet problems on balls

We provide closed formulas for (unique) solutions of nonhomogeneous Dirichlet problems on balls involving any positive power s > 0 of the Laplacian. We are able to prescribe values outside the domain and boundary data of different orders using explicit Poisson-type kernels and a new notion of higher-order boundary operator, which recovers normal derivatives if s is an element of N. Our results unify and generalize previous approaches in the study of polyharmonic operators and fractional Laplacians. As applications, we show a novel characterization of s-harmonic functions in terms of Martin kernels, a higher-order fractional Hopf Lemma, and examples of positive and sign-changing Green functions.