Inversion of the Penrose transform and the Cauchy-Fantappie formula

We discuss the construction of an explicit inversion of the Penrose transform with the focus on connections with the Radon transform, multi-dimensional residues and the Cauchy-Fantappie integral formula following to results [1,2]. The focus is on the new representation (M) of the inverse Penrose transform as a residue. The proof of this formula can be extracted from [1]. This proof includes an explicit computation of this residue (D). In this formula not the exact values of all coefficients but the existence of a differential operator, inverting the Penrose transform (we call this Leibnitz-Newton's phenomenon) is important. It is similar to local inversion formulas in integral geometry. (C) 2014 Elsevier By. All rights reserved.