## 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

Zbl 0191.434
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