A proof of Łojasiewicz’s theorem. (English) Zbl 1474.46080

Kim, Namhoon

Abstr. Appl. Anal. 2014, Article ID 695617, 6 p. (2014).

Summary: We give a necessary and sufficient condition for a primitive of a distribution to have the value at a point in the sense of Łojasiewicz. A formula defining the indefinite integral of a distribution with a basepoint is introduced, and further structural results are discussed.

MSC:

46F25	Distributions on infinite-dimensional spaces
46F10	Operations with distributions and generalized functions

References:

[1]	Łojasiewicz, S., Sur la valeur et la limite d’une distribution en un point, Studia Mathematica, 16, 1-36 (1957) · Zbl 0086.09405
[2]	Estrada, R.; Vindas, J., A general integral, Dissertationes Mathematicae, 483, 1-49 (2012) · Zbl 1245.26006
[3]	Vindas, J.; Pilipović, S., Structural theorems for quasiasymptotics of distributions at the origin, Mathematische Nachrichten, 282, 11, 1584-1599 (2009) · Zbl 1189.46032
[4]	Gel’fand, I. M.; Shilov, G. E., Generalized Functions. Vol. 1: Properties and Operations (1964), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0115.33101
[5]	Antosik, P.; Mikusiński, J.; Sikorski, R., Theory of Distributions. The Sequential Approach (1973), Amsterdam, The Netherlands: Elsevier Scientific, Amsterdam, The Netherlands · Zbl 0267.46028
[6]	Łojasiewicz, S., Sur la fixation des variables dans une distribution, Studia Mathematica, 17, 1-64 (1958) · Zbl 0086.09501
[7]	Shiraishi, R.; Itano, M., On the multiplicative products of distributions, Journal of Science of the Hiroshima University A-I: Mathematics, 28, 223-235 (1964) · Zbl 0141.31901
[8]	Itano, M., On the multiplicative products of distributions, Journal of Science of the Hiroshima University A-I: Mathematics, 29, 51-74 (1965) · Zbl 0141.31902
[9]	Shiraishi, R., On the value of distributions at a point and the multiplicative products, Journal of Science of the Hiroshima University A-I: Mathematics, 31, 89-104 (1967) · Zbl 0153.16404
[10]	Cohen, M. D., On the value of a distribution at a point, Mathematische Zeitschrift, 122, 101-103 (1971) · Zbl 0213.40003
[11]	Mikusiński, J., On the value of a distribution at a point, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 8, 681-683 (1960) · Zbl 0198.46402
[12]	Vernaeve, H.; Vindas, J., Characterization of distributions having a value at a point in the sense of Robinson, Journal of Mathematical Analysis and Applications, 396, 1, 371-374 (2012) · Zbl 1276.46032
[13]	Trèves, F., Topological Vector Spaces, Distributions and Kernels (1967), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0171.10402