TOPOLOGY OF THE SPACE OF CONORMAL DISTRIBUTIONS
Résumé
Given a closed manifold M and a closed regular submanifold L, consider the corresponding locally convex space I = I(M, L) of conormal distributions, with its natural topology, and the strong dual I = I (M, L) = I(M, L; Ω) of the space of conormal densities. It is shown that I is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and I is a complete Montel space, which is a projective limit of bornological barreled spaces. In the case of codimension one, similar properties and additional descriptions are proved for the subspace K ⊂ I of conormal distributions supported in L and for its strong dual K. We construct a locally convex Hausdoff space J and a continuous linear map I → J such that the sequence 0 → K → I → J → 0 as well as the transpose sequence 0 → J → I → K → 0 are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that I ∩ I = C ∞ (M) in the space of distributions. These results will be applied to prove a trace formula for foliated flows, involving the reduced cohomologies of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by the flow.
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