Inversion formulas for complex Radon transform on projective varieties and boundary value problems for systems of linear PDEs

Let $G\subset\mathbb C\mathrm P^n$ be a linearly convex compact set with smooth boundary, $D=\mathbb C\mathrm P^n\setminus G$, and let $D^*\subset(\mathbb C\mathrm P^n)^*$ be the dual domain. Then for an algebraic, not necessarily reduced, complete intersection subvariety $V$ of dimension $d$ we construct an explicit inversion formula for the complex Radon transform $R_V\colon H^{d,d-1}(V\cap D)\to H^{1,0}(D^*)$ and explicit formulas for solutions of an appropriate boundary value problem for the corresponding system of differential equations with constant coefficients on $D^*$.