Extension and tangential CRF conditions in quaternionic analysis

We prove some extension theorems for quaternionic holomorphic functions in the sense of Fueter. Starting from the existence theorem for the nonhomogeneous Cauchy-Riemann-Fueter Problem, we prove that an $\mathbb{H}$-valued function $f$ on a smooth hypersurface, satisfying suitable tangential conditions, is locally a jump of two $\mathbb{H}$-holomorphic functions. From this, we obtain, in particular, the existence of the solution for the Dirichlet Problem with smooth data. We extend these results to the continous case. In the final part, we discuss the octonian case.