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On second order hypoelliptic differential operators and the \(\overline\partial\)-Neumann problem. (English) Zbl 0747.58045

Complex analysis, Proc. Int. Workshop Ded. H. Grauert, Aspects Math. E 17, 134-142 (1991).
[For the entire collection see Zbl 0722.00013.]
Let \({\mathbf X}=\{X_ 1,\dots,X_ m\}\) be linearly independent vector fields in \(U\in\mathbb{R}^ n\), \(m\leq n\), and assume that finite many of \[ \{X_ i,[X_ j,X_ k],[X_ l,[X_ p,X_ q]],\dots\} \] generates \({\mathbf T}_ PU\) for all \(P\in U\). By Hörmander’s theorem the \({\mathbf X}\)- Laplacian \[ \Delta_{\mathbf X}=X^ 2_ 1+\cdots+X^ 2_ m \] is hypoelliptic. In the paper under review, the author computes the fundamental solutions for some examples and study the geometric meaning of the distance function yielded by the fundamental solutions.

MSC:

58J10 Differential complexes
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs

Citations:

Zbl 0722.00013
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