Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T15:49:07.268Z Has data issue: false hasContentIssue false
  • English
  • Français

Kernel theorems in spaces of tempered generalized functions

Published online by Cambridge University Press:  01 May 2007

ANTOINE DELCROIX
ANTOINE DELCROIX*
Affiliation:
Laboratoire AOC, Faculté des Sciences, Université des Antilles et de la Guyane, Campus de Fouillole, 97159 Pointe-à-Pitre, Guadeloupe (France). e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In analogy to the classical isomorphism between and , we show that a large class of moderate linear mappings acting between the space of Colombeau rapidly decreasing generalized functions and the space of temperate ones admits generalized integral representations, with kernels belonging to . Furthermore, this result contains the classical one.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 3 , May 2007 , pp. 557 - 572
Copyright
Copyright © Cambridge Philosophical Society 2007

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

REFERENCES

[1]Antonevich, A. B. and Radyno, Ya. V.. On a general method for constructing algebras of generalized functions. Soviet Math. Dokl. 43 (3) (1992), 680–684.Google Scholar
[2]Bernard, S., Colombeau, J.-F. and Delcroix, A.. Generalized integral operators and applications. Math. Proc. Camb. Phil. Soc. 141/3, (2006), 521–546.CrossRefGoogle Scholar
[3]Colombeau, J.-F.. New Generalized Functions and Multiplication of Distributions (North-Holland, 1984).Google Scholar
[4]Colombeau, J.-F.. Elementary Introduction to New Generalized Functions (North-Holland, 1985).Google Scholar
[5]Delcroix, A.. Generalized integral operators and Schwartz kernel theorem. J. Math. Anal. Appl. 306 (2) (2005), 481–501.CrossRefGoogle Scholar
[6]Delcroix, A.. Regular nonlinear generalized functions and applications. Bull. Cl. Sci. Math. Nat. Sci. Math 311 (2006), 163–174.Google Scholar
[7]Delcroix, A.. Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity. J. Math. Anal. Appl 327 (2007), 564–584.CrossRefGoogle Scholar
[8]Delcroix, A., Hasler, M., Pilipović, S. and Valmorin, V.. Generalized function algebras as sequence space algebras. Proc. Amer. Math. Soc. 132 (2004), 2031–2038.CrossRefGoogle Scholar
[9]Garetto, C.. Pseudo-differential Operators in Algebras of Generalized Functions and Global Hypoellipticity. Acta Appl. Math. 80 (2) (2004), 123–174.CrossRefGoogle Scholar
[10]Garetto, C., Gramchev, T. and Oberguggenberger, M.. Pseudo-Differential operators and regularity theory. Electron J. Differential Equations 116 (2005), 1–43.Google Scholar
[11]Grosser, M., Kunzinger, M., Oberguggenberger, M. and Steinbauer, R.. Geometric Theory of Generalized Functions with Applications to General Relativity (Kluwer Academic Press, 2001).CrossRefGoogle Scholar
[12]Hörmander, L.. The Analysis of Linear Partial Differential Operators I, Distribution Theory and Fourier Analysis. Grundlehren der mathematischen Wissenchaften 256. (Springer Verlag, 2nd edition, 1990).Google Scholar
[13]Marti, J.-A.. Non linear Algebraic analysis of delta shock wave to Burgers' equation. Pacific J. Math. 210 (1), (2003) 165–187.CrossRefGoogle Scholar
[14]Nedeljkov, M., Pilipović, S. and Scarpalézos, D.. The linear theory of Colombeau generalized functions. Pitman Research Notes in Mathematics Series. 385 (Longman, 1998).Google Scholar
[15]Oberguggenberger, M.. Multiplication of Distributions and Applications to Partial Differential Equations (Longman Scientific and Technical, 1992).Google Scholar
[16]Pilipović, S. and Scarpalézos, D.. Colombeau generalized ultradistributions. Math. Proc. Camb. Phil. Soc. 130 (2001), 541–553.CrossRefGoogle Scholar
[17]Radyno, Ya.-V., Tkhan, Ngo Fu and Ramadan, S.. The Fourier transform in an algebra of new generalized functions. Russian acad. Sci. Dokl. Math. 46 (3) (1992), 414–417.Google Scholar
[18]Scarpalézos, D.. Colombeau's generalized functions: Topological structures; Microlocal properties. A simplified point of view. Prépublication Mathématiques de Paris 7/CNRS, URA212 (1993).Google Scholar
[19]Scarpalézos, D.. Colombeau's generalized functions: Topological structures; Microlocal properties. A simplified point of view. Part I. Bull. Cl. Sci. Math. Nat. Sci. Math. 25 (2000), 89–114.Google Scholar
[20]Scarpalézos, D.. Colombeau's generalized functions: topological structures; microlocal properties. A simplified point of view. Part II. Publ. Inst. Math. (Beograd) (N.S.) 76 (90) (2004), 111–125.CrossRefGoogle Scholar
[21]Schwartz, L.. Théorie des Distributions (Hermann, 3rd print, 1965).Google Scholar
[22]Treves, F.. Topological Vector Spaces, Distributions and Kernels (Academic Press, 1967).Google Scholar
[23]Valmorin, V.. Schwartz kernel theorem in Colombeau algebras. Preprint AOC (2005).Google Scholar