Partial differential operators with continuous linear right inverse. (English) Zbl 0709.46018

Meise, R.; Taylor, B. A.; Vogt, D.

Advances in the theory of Fréchet spaces, Proc. NATO Adv. Res. Workshop, Istanbul/Turkey 1988, NATO ASI Ser., Ser. C 287, 47-62 (1989).

[For the entire collection see Zbl 0699.00024.]
Let \(\Omega\) be an open convex set in \({\mathbb{R}}^ n\), \({\mathcal E}(\Omega)\), \({\mathcal D}'(\Omega)\) the usual Schwartz’ spaces of functions and distributions and P a polynomial on \({\mathbb{C}}^ n\); it is proved that the operator P(D) has a continuous right inverse in \({\mathcal E}(\Omega)\) iff it has a continuous right inverse in \({\mathcal D}'(\Omega)\); moreover the existence of such a right inverse in both of these spaces is equivalent to the validity of a certain “analytic Phragmen-Lindelöf principle” for the variety \(V=\{P(-z)=0\}\). It is also shown that in the case \(\Omega ={\mathbb{R}}^ n\) this principle is very efficient to handle examples. Finally an extensive discussion on Phragmen-Lindelöf conditions on algebraic varieties is presented.

Reviewer: I.Ciorănescu

MSC:

46F10	Operations with distributions and generalized functions
47A05	General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
35A05	General existence and uniqueness theorems (PDE) (MSC2000)

Keywords:

Schwartz’ spaces of functions and distributions; continuous right inverse; analytic Phragmen-Lindelöf principle; algebraic varieties