Abstract
Sheaf morphisms are considered in sheaves of generalized functions. It is proved that for (ultra)distributions they must be continuous outside discrete points. Contrary to Peetre’s original theorem, which applies to sheaves of test functions, an example makes clear that these points can really be points of discontinuity. Finally, it is shown that in the sheaf of hyper-functions there are more general discontinuous sheaf morphisms.
Peetre’s theorem says that any sheaf morphism in the sheaf of C∞-functions is a differential operator. We shall investigate sheaf morphisms in sheaves of generalized functions, in particular distributions, ultradistributions of the Beurling and of the Roumieu type, and hyperfunctions. All these sheaves are soft so that their sections with a compact support form flabby cosheaves which are the duals, with respect to a certain topology, of the sheaves of their associated test functions. The main point is to investigate the continuity of a cosheaf morphism P (= local operator) in one of these cosheaves. At places where P is continuous its transposed tP is a continuous sheaf morphism in the sheaf of test functions and it follows that tP, and hence P itself, are appropriate differential operators there. In this paper we shall only briefly mention these results, as well as the generalization of Peetre’s theorem to the soft sheaves of test functions. Our main attention will be on the continuity of a local operator in a space of generalized functions and we shall indicate what possibilities there are for a discontinuous sheaf morphism.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. E. Bredon, “Sheaf theory”, McGraw-Hill, New York, (1967).
H. Komatsu, Ultradistributions I, Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo, Sec. IA 20, 25–105 (1973).
J. Peetre, Rectification á l’article “Une caractérisation abstraite des opérateurs différentiels”, Math. Scand. 8, 116–120 (1960).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Plenum Press, New York
About this chapter
Cite this chapter
de Roever, J.W. (1988). Peetre’s Theorem and Generalized Functions. In: Stanković, B., Pap, E., Pilipović, S., Vladimirov, V.S. (eds) Generalized Functions, Convergence Structures, and Their Applications. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1055-6_30
Download citation
DOI: https://doi.org/10.1007/978-1-4613-1055-6_30
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4612-8312-6
Online ISBN: 978-1-4613-1055-6
eBook Packages: Springer Book Archive