Vanishing Theorems in Colombeau Algebras of Generalized Functions

Valmorin, V.

Canadian mathematical bulletin, Tome 51 (2008) no. 4, pp. 618-626

Résumé

Using a canonical linear embedding of the algebra ${{G}^{\infty }}\left( \Omega\right)$ of Colombeau generalized functions in the space of $\overline{\mathbb{C}}$ -valued $\mathbb{C}$ -linear maps on the space $D\left( \Omega\right)$ of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class ${{G}^{\infty }}$ . These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

Détail

DOI : 10.4153/CMB-2008-061-3

Mots-clés : 32A60, 45P05, 46F30, Colombeau generalized functions, linear integral operators, generalized holomorphic functions

Bibliographie

[1] [1] Aragona, J., On existence theorems for the operator on generalized differential forms. Proc. London Math. Soc. 53(1986), no. 3, 474–488. Google Scholar

[2] [2] Bernard, S., Colombeau, J-F., and Delcroix, A., Generalized integral operators and applications. Math. Proc. Cambridge Philos. Soc. 141(2006), no. 3, 521–546. Soc. Google Scholar

[3] [3] Colombeau, J-F., New Generalized Functions and Multiplication of Distributions. North-Holland Math. Stud. 84, North-Holland, Amsterdam, 1984. Google Scholar

[4] [4] Colombeau, J-F., Elementary Introduction to New Generalized Functions. North-Holland Math. Stud. 113, North-Holland, Amsterdam, 1985. Google Scholar

[5] [5] Colombeau, J-F. and Galé, J.E., Holomorphic generalized functions. J. Math. Anal. Appl. 103(1984), no. 1, 117–133. Google Scholar

[6] [6] Colombeau, J-F. and Galé, J.E., Analytic continuation of generalized functions. Acta Math. Hung. 52(1988), no. 1-2, 57–60. Google Scholar

[7] [7] Garetto, C., Topological structures in Colombeau algebras: investigation of the duals of . Monatshf. Math. 146(2005), no. 3, 203–226. Google Scholar

[8] [8] Grosser, M., Kunzinger, M., Oberguggenberger, M., and Steinbauer, R., Geometric Theory of Generalized Functions with Applications to General relativity. Mathematics and Its Applications 537, Kluwer Academic Publishers, Dordrecht, 2001. Google Scholar

[9] [9] Khelif, A. and Scarpalezos, D., Zeros of generalized holomorphic functions. Monatsh. Math 149(2006), no. 4, 323–335. Google Scholar

[10] [10] Oberguggenberger, M., Multiplication of distributions and application to partial differential equations. Pitman Res. Notes Math. Ser. 259, Longman, Harlow, New York, 1992. Google Scholar

[11] [11] Oberguggenberger, M. and Kunzinger, M., Characterization of Colombeau generalized functions by their pointvalues. Math. Nachr. 203(1999), 147–157. Google Scholar

[12] [12] Oberguggenberger, M., Pilipovic, S., and Valmorin, V., Global representatives of Colombeau holomorphic generalized functions. Monatsh. Math. 151(2007), no. 1, 67–74. Google Scholar

[13] [13] Pilipović, S., Scarpalezos, D., and Valmorin, V., Equalities in algebras of generalized functions. Forum Math. 18(2006), no. 5, 789–801. Google Scholar