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 
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/
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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.

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