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

TL;DR

This paper develops explicit inversion formulas for the complex Radon transform on projective varieties and applies them to boundary value problems for systems of linear PDEs, advancing integral geometry and PDE theory.

Contribution

It introduces new explicit inversion formulas for the complex Radon transform on algebraic subvarieties and solves boundary value problems for related linear PDE systems.

Findings

Constructed explicit inversion formulas for the complex Radon transform.

Derived formulas for solutions to boundary value problems for PDE systems.

Extended the applicability of Radon transform techniques to projective varieties.

Abstract

Let $G\subset \C P^n$ be a linearly convex compact with smooth boundary, $D={\C}P^n\setminus G$, and let $D^* \subset (\C P^n)^*$ 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 $R_V:\ H^{d,d-1}(V\cap D)\to H^{1,0}(D^*)$, and explicit formulas for solutions of an appropriate boundary value problem for the corresponding system of differential equations with constant coefficients on $D^*$.