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An extension of the notion of the order of a distribution

  • M. Mišur
  • L. PalleEmail author
Article
  • 22 Downloads

Abstract

In this article we review the notion of the order of a distribution and extend it to the case of positive real numbers. We suggest to use the name Hölder distributions for such distributions. The first part of the paper concerns itself with functional-analytic properties of the Hölder test function spaces and its duals. Of particular interest are the \({C}^{r,\alpha +}_c(\Omega )\) and the \({\mathcal {D}}'_{(r+\alpha )+}(\Omega )\) spaces which have notably better properties such as reflexivity, compared to the classical Hölder spaces. We also give a few examples and some Fourier-analytic properties of distributions of fractional order, and at the end, we note how one can extend classical results where estimates of the order of distributions appear, such as giving a bound on the order of convolution of distributions.

Keywords

Distributions Fractional order Hölder continuous Fourier transform Convolution 

Mathematics Subject Classification

46F05 (primary), 46A04, 46E15 

Notes

Acknowledgements

The idea of extending the notion of the order of distributions from natural numbers to positive reals was suggested to the first author by Luc Tartar during the Generalized Functions 2016 conference in Dubrovnik. The authors are gratefully indebted for his valuable comments and suggestions. The research of the first author was supported in part by the Croatian Science Foundation, Project Number 9780 Weak convergence methods and applications (WeConMApp). A part of the work was performed, while the first author was visiting University Paris-Sud XI under the scholarship of the Government of the French Republic whose support he gratefully acknowledges. A part of the work of the second author was performed while working at the Christian-Albrechts University in Kiel where he was supported by the Deutsche Forschungsgemeinschaft, Project Number 237750060 Fragen der Harmonischen Analysis im Zusammenhang mit Hyperflächen.

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of ScienceUniversity of ZagrebZagrebCroatia
  2. 2.Visage Technologies ABLinköpingSweden
  3. 3.Christian-Albrechts-Universität zu Kiel, Mathematisches SeminarKielGermany

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