Regularity properties of distributions and ultradistributions

Pilipović, S.; Scarpalezos, D.

doi:10.1090/S0002-9939-01-06013-0

Regularity properties of distributions and ultradistributions
HTML articles powered by AMS MathViewer

by S. Pilipović and D. Scarpalezos PDF

Proc. Amer. Math. Soc. 129 (2001), 3531-3537 Request permission

Abstract:

We give necessary and sufficient conditions for a regularized net of a distribution in an open set $\Omega$ which imply that it is a smooth function or $C^k$ function in $\Omega$. We also give necessary and sufficient conditions for an ultradistribution to be an ultradifferentiable function of corresponding class.

References

Hebe A. Biagioni, A nonlinear theory of generalized functions, 2nd ed., Lecture Notes in Mathematics, vol. 1421, Springer-Verlag, Berlin, 1990. MR 1049623, DOI 10.1007/BFb0089552
Jean-François Colombeau, New generalized functions and multiplication of distributions, North-Holland Mathematics Studies, vol. 84, North-Holland Publishing Co., Amsterdam, 1984. Notas de Matemática [Mathematical Notes], 90. MR 738781
Yu. V. Egorov, On the theory of generalized functions, Uspekhi Mat. Nauk 45 (1990), no. 5(275), 3–40, 222 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 5, 1–49. MR 1084986, DOI 10.1070/RM1990v045n05ABEH002683
Ricardo Estrada and Ram P. Kanwal, Asymptotic analysis, Birkhäuser Boston, Inc., Boston, MA, 1994. A distributional approach. MR 1254657, DOI 10.1007/978-1-4684-0029-8
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
Hikosaburo Komatsu, Ultradistributions. III. Vector-valued ultradistributions and the theory of kernels, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 3, 653–717. MR 687595
H. Komatsu, Microlocal Analysis in Gevrey Classes and in Convex Domains, Springer, Lec. Not. Math. 1726(1989), 426 - 493.
M. Oberguggenberger, Multiplication of distributions and applications to partial differential equations, Pitman Research Notes in Mathematics Series, vol. 259, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1187755
Hans-Joachim Petzsche, Generalized functions and the boundary values of holomorphic functions, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 2, 391–431. MR 763428
Stevan Pilipović, Microlocal analysis of ultradistributions, Proc. Amer. Math. Soc. 126 (1998), no. 1, 105–113. MR 1451826, DOI 10.1090/S0002-9939-98-04357-3
S. Pilipović, D. Scarpalezos, Colombeau Generalized Ultradistributions, Math. Proc. Camb. Phil. Soc., 130(2001), 541-553.