Integral transforms for divisors in \(\mathbf P_n(\mathbb C)\) and solutions of systems of PDE’s. (English) Zbl 1079.53116

The paper is devoted to the study of two integral transforms in complex analysis, namely the analytic Radon transform and the Andreotti-Norguet transform. There is already a whole series of papers investigating these transforms, but most of them deal only with linear cycles. In this situation the description of both the transforms (kernel, image) is relatively simple. The aim of the authors is more difficult. They characterize the image of these transforms essentially in the case od compact hypersurfaces (not necessarily smooth) of arbitrary fixed degree which are contained in a linearly concave open subset of \(\mathbf P_n(\mathbb C)\). They show that the image is the set of all solutions of a certain system of partial differential equations. In fact, they generalize in two directions: instead of hyperplanes they consider hypersurfaces, and they do not assume that the hypersurfaces are smooth. In the end they even extend their results from the analytic category to the \(C^{\infty }\)-category.


53C65 Integral geometry
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