Surjectivity of Euler operators on temperate distributions

Euler operators are partial differential operators of the form P(θ) where P is a polynomial and θj=xj∂/∂xj. We show that every non-trivial Euler operator is surjective on the space of temperate distributions on Rd. This is in sharp contrast to the behaviour of such operators when acting on spaces of...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2018-10, Vol.466 (2), p.1393-1399
Main Author: Vogt, Dietmar
Format: Article
Language:English
Subjects:
[formula omitted]-functions of exponential decay
Euler differential operators
Global solvability
Temperate distributions
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Summary:Euler operators are partial differential operators of the form P(θ) where P is a polynomial and θj=xj∂/∂xj. We show that every non-trivial Euler operator is surjective on the space of temperate distributions on Rd. This is in sharp contrast to the behaviour of such operators when acting on spaces of differentiable or analytic functions.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2018.06.063