Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T12:57:28.112Z Has data issue: false hasContentIssue false
  • English
  • Français

Real roots of polynomials and right inverses for partial differential operators in the space of tempered distributions

Published online by Cambridge University Press:  14 November 2011

Michael Langenbruch
Michael Langenbruch
Affiliation:
Mathematisches Institut der Universität, Einsteinstrasse 62, D-4400 Munster, Germany
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let P(D) be a partial differential operator with constant coefficients. If P(D) has a continuous linear right inverse in the space of tempered distributions, then P is the product of a polynomial without real roots and a real polynomial admitting a right inverse. If the polynomial P is real and irreducible, then P(D) admits a right inverse in the tempered distributions if and only if P(×) has the property of zeros of R. Thorn.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 3-4 , 1990 , pp. 169 - 179
Copyright
Copyright © Royal Society of Edinburgh 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aytuna, A., Krone, J. and Terzioǧlu, T.. Complemented infinite type power series subspaces of nuclear Fréchet spaces (preprint, 1987).Google Scholar
2Bierstone, E. and Schwarz, G.. Continuous linear division and extension of C∞-functions. Duke Math. J. 50 (1983), 233–271.CrossRefGoogle Scholar
3Bochak, J.. Sur le theoreme des zeroes de Hilbert “differentiable”. Topology 12 (1973), 417–424.CrossRefGoogle Scholar
4Bochnak, J., Coste, M. and Roy, M.-F.. Géométrie algibriquereelle (Berlin: Springer, 1987).Google Scholar
5Hörmander, L.. On the division of distributions by polynomials. Ark. Mat. 3 (1958), 555–568.CrossRefGoogle Scholar
6Hörmander, L.. The analysis of linear partial differential operators I + II (Berlin: Springer, 1983).Google Scholar
7Köthe, G.. Topologische lineare Räume I (Berlin: Springer, 1966).CrossRefGoogle Scholar
8Langenbruchu, M.. Partial differential operators without solution operatorsin weighted spaces of (generalized) functions. Manuscripta Math. 56 (1986), 353–374.CrossRefGoogle Scholar
9Langenbruch, M.. Complemented kernels of partial differential operators inweighted spaces of (generalized) functions. Studio Math. 89 (1988), 37–63.CrossRefGoogle Scholar
10Langenbruch, M.. Solution operators for partial differential operators inweighted Gevrey spaces. Michigan Math. J. 37 (1990).CrossRefGoogle Scholar
11Langenbruch, M.. Tame right inverses for partial differential equations. In Advances in the theory of Fréchet spaces, ed. Terzioglu, T., Nato ASI-Series (C) 287 (1989), 79–114.CrossRefGoogle Scholar
12Malgrange, B.. Ideals of differentiable functions (London: Oxford University Press, 1966).Google Scholar
13Meise, R., Taylor, B. A. and Vogt, D.. Partial differential operators with continuous linear right inverse. In Advances in the theory of Fréchet spaces, ed. Terzioǧlu, T., Nato ASI-Series (C) 287 (1989), 47–62.CrossRefGoogle Scholar
14Shiota, M.. Equivalence of differentiable mappings and analytic mappings. Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 237–322.CrossRefGoogle Scholar
15Vogt, D.. Subspaces and quotients of (s). In Functional analysis: Surveys and recent results, eds Bierstedt, K. D. and Fuchssteiner, B.. North-Holland Math. Stud. 27 (1977), 167–187.Google Scholar