Differential equations and integral geometry
Let \(X\) be a smooth manifold of dimension \(n\), and \({\mathcal M}\) a system of linear partial differential equations on \(X\). Denote by \(\text{Sol} ({\mathcal M},C^\infty(X))\) the space of smooth solutions of \({\mathcal M}\). Let \({\mathcal N}\) be a linear system of PDE on a manifold \(Y\), \(K(x,y) dy\) be an \((n,0)\)-form on \(X\times Y\) with compact support along \(X\), and assume it satisfies the system \({\mathcal N}\) along \(Y\). Then, the kernel \(K(x,y) dx\) defines a linear map \(I_K: C^\infty(X)\to \text{Sol} ({\mathcal N},C^\infty(Y))\), \(f(x)\to \int_X f(x) K(x,y) dx\). Its restriction to \(\text{Sol} ({\mathcal M},C^\infty(X))\) gives an operator \(\text{Sol} ({\mathcal M},C^\infty(X))\to \text{Sol} ({\mathcal N},C^\infty(Y))\), and the author considers the problem of finding a natural description for the space of all such linear maps. The discussion is conducted using the language of \({\mathcal D}\)-modules, which occupies Section 3 of the paper, including some general information on the topic and the derived category of these objects, needed for applications to some old problems in integral geometry. Section 4 contains a generalization of the Green class of a \({\mathcal D}\)-module. In Sections 5-6, a construction of operators between solution spaces of linear PDE is presented, while Section 7 develops the relation with integral geometry, particularly with the family of spheres, treated, chiefly in [\textit{I. M. Gelfand, S. G. Gindikin} and \textit{M. I. Graev}, Moscow, Itogi Nauki Tech. 16, 53-226 (1980; Zbl 0465.52005) and \textit{I. M. Gelfand, M. I. Graev} and \textit{Z. J. Shapiro}, Funct. Anal. Appl. 1, 14-27 (1967; Zbl 0164.23103)]. Finally, in Section 8, an algebraic version of the problem is studied. The main results of the present paper were already announced [in \textit{A. B. Goncharov}, Math. Res. Lett. 2, 415-435 (1995; Zbl 0846.53051)].
scientific article; zbMATH DE number 1097660
scientific article