On Radon-Penrose transformation and k-Cauchy-Fueter operator

Abstract

It is known that there is a 1-1 correspondence between the first cohomology of the sheaf \(\mathcal{O}\)(−k−2) over the projective space and the solutions to the k-Cauchy-Fueter equations on the quaternionic space ℍn. We find an explicit Radon-Penrose type integral formula to realize this correspondence: given a \(\bar \partial\)-closed (0, 1)-form f with coefficients in the (−k−2)th power of the hyperplane section bundle H−k−2, there is an integral representation Pf such that ι*(Pf) is a solution to the k-Cauchy-Fueter equations, where ι is an embedding of the quaternionic space ℍn into ℂ4n.