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Partial differential equations on nilpotent groups. (English) Zbl 0561.35015
Lie group representations III, Proc. Spec. Year, College Park/Md. 1982-83, Lect. Notes Math. 1077, 210-253 (1984).
[For the entire collection see Zbl 0535.00011.]
This is a survey of recent results on hypoellipticity, analyticity and local solvability for left invariant differential operators on graded nilpotent groups, fields to which the author has made significant contributions. The exposition is very clear and proofs of the basic theorems are sketched. Particular emphasis is given to a group theoretical criterion on Helffer and Nourrigat for the hypoellipticity of homogeneous operators on homogeneous spaces and to results of P. Levy- Bruhl on local solvability. In order to provide nontrivial examples of the results, the author gives rather complete details for the simply- connected, connected nilpotent Lie group of dimension 4 and rank of nilpotency 3.
Reviewer: W.Miller

35H10 Hypoelliptic equations
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
22E35 Analysis on \(p\)-adic Lie groups
43A80 Analysis on other specific Lie groups
47F05 General theory of partial differential operators
35A20 Analyticity in context of PDEs