On regularization of vector distributions on manifolds (Q339419)

Let \(E\to M\) and \(F\to N\) be vector bundles, denote by \({\mathcal D}'(M,E)\) the space of \(E\)-valued distributions, and by \(\Gamma(M,E)\) the space of smooth sections of \(E\). The main result is a proof of the isomorphismsNEWLINENEWLINENEWLINE\[NEWLINE {\mathcal D}'(M,E) \cong \Gamma(M,E)\otimes_{C^\infty(M)} {\mathcal D}'(M) \cong {\mathcal L}_{C^\infty(M)}(\Gamma(M,E^*),{\mathcal D}'(M)) NEWLINE\]NEWLINE and NEWLINE\[NEWLINE {\mathcal L}({\mathcal D}'(M,E),\Gamma(N,F)) \cong \Gamma(M\times N,E^*\boxtimes F)\otimes_{C^\infty(M\times N)} {\mathcal L}({\mathcal D}'(M),C^\infty(N)) NEWLINE\]NEWLINE NEWLINE\[NEWLINE \cong {\mathcal L}_{C^\infty(M\times N)}(\Gamma(M\times N,E\boxtimes F^*),{\mathcal L}({\mathcal D}'(M),C^\infty(N))). NEWLINE\]NEWLINENEWLINENEWLINEMore precisely, the isomorphisms are shown to hold in the category of locally convex modules.NEWLINENEWLINESince, for example, the first isomorphism corresponds to the intuitive picture of identifying vector valued distributions with sections of the bundle with distributional coefficients, these results have important consequences for a general theory of regularizing vector-valued distributions in a geometrical setting.NEWLINENEWLINEThe proofs are based, on the one hand, on the notion of additive category, and, on the analytical side, on L.\ Schwartz' theory of \(\varepsilon\)-tensor products.