Harmonic analysis on the rank-$2$ value group of a two-dimensional local field

TL;DR

This paper develops harmonic analysis on rank-2 free Abelian groups and connects it to harmonic analysis on two-dimensional local fields, including Fourier transforms and group actions.

Contribution

It constructs harmonic analysis tools for rank-2 groups and links them to two-dimensional local fields, extending previous work.

Findings

Constructed spaces of functions and distributions on rank-2 groups

Developed Fourier transform theory for these groups

Connected harmonic analysis on rank-2 groups with two-dimensional local fields

Abstract

In this work we construct harmonic analysis on free Abelian groups of rank $2$, namely: we construct and investigate spaces of functions and distributions, Fourier transforms, actions of discrete and extended discrete Heisenberg groups. In case of the rank-$2$ value group of a two-dimensional local field with finite last residue field we connect this harmonic analysis with harmonic analysis on the two-dimensional local field, where the latter harmonic analysis was constructed in earlier works by the authors.