Geodesic
Multiplicative representations of Bruhat--Schwartz distributions on the additive group of $p$-adic numbers
Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 12-21.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study various decompositions of Bruhat–Schwartz distributions on the additive group of $p$-adic numbers related to the group action of the multiplicative group of $p$-adic numbers. For regular distributions, we establish an identity which defines an equivalent distribution on the multiplicative $p$-adic group. We then establish some relations to rewrite or decompose distributions using the Mellin transform. The main result of our paper is a decomposition of Bruhat–Schwartz functions into finite sums of radial functions with quasi-character coefficients. This decomposition allows us to expand distributions into discrete series of ray-wise projections. The group action of the multiplicative $p$-adic integer group on the set of distributions corresponds to element-wise coefficient multiplication in the aforementioned series expansion. 
@article{TIMB_2022_30_1_a1,
     author = {N. V. Guletskii and Ya. M. Radyna},
     title = {Multiplicative representations of {Bruhat--Schwartz} distributions on the additive group of $p$-adic numbers},
     journal = {Trudy Instituta matematiki},
     pages = {12--21},
     publisher = {mathdoc},
     volume = {30},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMB_2022_30_1_a1/}
}
TY  - JOUR
AU  - N. V. Guletskii
AU  - Ya. M. Radyna
TI  - Multiplicative representations of Bruhat--Schwartz distributions on the additive group of $p$-adic numbers
JO  - Trudy Instituta matematiki
PY  - 2022
SP  - 12
EP  - 21
VL  - 30
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TIMB_2022_30_1_a1/
LA  - ru
ID  - TIMB_2022_30_1_a1
ER  - 
%0 Journal Article
%A N. V. Guletskii
%A Ya. M. Radyna
%T Multiplicative representations of Bruhat--Schwartz distributions on the additive group of $p$-adic numbers
%J Trudy Instituta matematiki
%D 2022
%P 12-21
%V 30
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TIMB_2022_30_1_a1/
%G ru
%F TIMB_2022_30_1_a1
N. V. Guletskii; Ya. M. Radyna. Multiplicative representations of Bruhat--Schwartz distributions on the additive group of $p$-adic numbers. Trudy Instituta matematiki, Tome 30 (2022) no. 1, pp. 12-21. http://geodesic.mathdoc.fr/item/TIMB_2022_30_1_a1/

[1] Riemann B., “Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse”, Ges. Math. Werke und Wissenschaftlicher Nachlaß, 2 (1859), 145–155

[2] Iwasawa K., “Letter to J. Dieudonné”, Zeta functions in geometry, Kinokuniya Company Ltd., Tokyo, 1992, 445–450 | DOI | MR | Zbl

[3] Haran S., “Analytic potential theory over the p-adics”, Annales de l'Institut Fourier, 43, no. 4, Institut Fourier, Grenoble, 1993, 905–944 | DOI | MR | Zbl

[4] Burnol J.-F., “On Fourier and Zeta(s)”, Forum Mathematicum De Gruyter, 16:6 (2004), 789–840 | DOI | MR | Zbl

[5] Burnol J.-F., The explicit formula and the conductor operator, 1999, arXiv: math/9902080

[6] Davenport H., Multiplicative number theory, Graduate Texts in Mathematics, 74, Springer Science Business Media, 2013 | MR

[7] Gelfand I., Graev M., Pyatetskii-Shapiro I., Teoriya predstavlenii i avtomorfnye funktsii, Nauka, M., 1966 | MR

[8] Vladimirov V., Volovich I., Zelenov E., p-adicheskii analiz i matematicheskaya fizika, Nauka, M., 1994 | MR

[9] Bruhat F., “Distributions sur un groupe localement compact et applications à l‘’étude des représentations des groupes p-adiques”, Bulletin de la Société Mathématique de France, 89 (1961), 43–75 | DOI | MR | Zbl

[10] Eiter T., Kyed M., “Time-Periodic Linearized Navier-Stokes Equations: An Approach Based on Fourier Multipliers”, Particles in Flows, eds. T. Bodnár, G. P. Galdi, Š. Nečasová, Springer International Publishing, Cham, 2017, 77–137 | DOI | MR | Zbl

[11] Schikhof W., Ultrametric Calculus: An Introduction to P-adic Analysis, Cambridge University Press, 1984 | MR | Zbl

[12] Halmos P., Measure Theory, Springer, 1950 | DOI | MR