Integral geometry and \(D\)-modules
Let \({\mathcal M}\) be a system of linear partial differential equations on one function on a smooth manifold \(X\), and denote by \(\text{Sol} ({\mathcal M}, C^\infty (X))\) the space of smooth solutions to \({\mathcal M}\). The author addresses the following two questions: 1) What is a natural realization for the continuous functionals on the space \(\text{Sol} ({\mathcal M}, C^\infty (X)) \)? -- 2) What is the natural realization for the continuous linear maps from \(\text{Sol} ({\mathcal M}, C^\infty (X))\) to \(\text{Sol} ({\mathcal N}, C^\infty (Y))\)? -- The interpretation of the problem is made in terms of \(D\)-modules, allowing to relate it to some examples appearing in integral geometry, particularly to those exposed in [\textit{I. M. Gelfand}, \textit{S. G. Gindikin} and \textit{M. I. Graev}, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 16, 53-226 (1980; Zbl 0465.52005)] and [\textit{I. M. Gelfand}, \textit{M. I. Graev} and \textit{Z. Ya. Shapiro}, Funct. Anal. Appl. 1, No. 1, 14-27 (1967); translation from Funkts. Anal. Prilozh. 1, No. 1, 15-31 (1967; Zbl 0164.23103)]. The construction of a class of natural linear maps between solution spaces is followed by a detailed exposition showing how it works for the particular case of spheres in Euclidean space. Finally, a bicategory of \(D\)-modules is introduced, which is an algebraic model of integral geometry. This is used to discuss some further examples.
scientific article; zbMATH DE number 817766