Multiplicative representations of Bruhat–Schwartz distributions on the additive group of $p$-adic numbers

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.