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On boundary Hilbert differential complexes. (English) Zbl 0606.58046

For a compact convex set K in \({\mathbb{R}}^ n\), we denote by \({\mathcal D}_ K\) the space of smooth functions with support in K, with the Schwartz topology. If A(D) is a \(p\times q\) matrix of l.p.d.o.’s with constant coefficients, then A(D), as a linear map from \({\mathcal D}^ q_ K\) to \({\mathcal D}^ p_ K\), has a closed image and therefore is a topological homomorphism. In particular, under the action \(z_ j\cdot u=\frac{1}{i}\partial u/\partial x_ j\), \({\mathcal D}_ K\) is a flat \({\mathcal P}={\mathbb{C}}[z_ 1,...,z_ n]\)-module. This theorem solves a question posed by V. Palamodov [Linear differential operators with constant coefficients (1967; Zbl 0191.434)]. By the same action, the space \(W_ K\) of Whitney functions on K is an injective \({\mathcal P}\)- module. By duality it follows that \({\mathcal E}'_ K\) (distributions with support in K) and \(\check {\mathcal D}'_ K\) (extendible distributions on the interior of K) are respectively a flat and an injective \({\mathcal P}\)- module. Using these results, we study the cohomology groups of the complexes of l.p.d.o.’s induced on the boundary of a convex set by complexes of l.p.d.o.’s with constant coefficients in \({\mathbb{R}}^ n\). If M is a finitely generated \({\mathcal P}\)-module, we set \(E^ j(M)=Ext^ j_{{\mathcal P}}(M,{\mathcal P})\). If M is associated to the complex above, local and global cohomology groups of the tangential complex vanish when \(E^{j+1}_{(M)}=0\) and \(j\neq 0\), while the existence of nonvanishing local cohomology classes in dimension j is related to the nonhyperbolicity, in the direction normal to the boundary, of the characteristic variety of the module \(E^{j+1}(M)\). These results are also extended to the case of unbounded convex and concave subsets of \({\mathbb{R}}^ n\) and of their boundaries.

MSC:

58J10 Differential complexes
32V40 Real submanifolds in complex manifolds

Citations:

Zbl 0191.434
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