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Differential equations and integral geometry. (English) Zbl 0931.53038

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 [I. M. Gelfand, S. G. Gindikin and M. I. Graev, Moscow, Itogi Nauki Tech. 16, 53-226 (1980; Zbl 0465.52005) and I. M. Gelfand, M. I. Graev and 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 A. B. Goncharov, Math. Res. Lett. 2, 415-435 (1995; Zbl 0846.53051)].

MSC:

53C65 Integral geometry
35G05 Linear higher-order PDEs
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References:

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