On regular harmonics of one quaternionic variable

We prove some results about the Fueter-regular homogeneous polynomials, which appear as components in the power series of any quaternionic regular function. We obtain a differential condition that characterizes the homogeneous polynomials whose trace on the unit sphere extends as a regular polynomial. We apply this result to define an injective linear operator from the space of complex spherical harmonics to the module of regular homogeneous polynomials of a fixed degree $k$.