Characterizations of Extra-invariant spaces under the left translations on a Lie group

TL;DR

This paper characterizes extra-invariant spaces under left translations on certain Lie groups, including the Heisenberg group, using range functions, extending the understanding of invariant spaces in harmonic analysis.

Contribution

It provides new characterization results for extra-invariant spaces on nilpotent Lie groups, especially the Heisenberg group, under left translations.

Findings

Characterization of extra-invariant spaces via range functions

Extension of theory to the Heisenberg group

Applicable to 2-step nilpotent Lie groups

Abstract

In the context of a connected, simply connected, nilpotent Lie group, whose representations are square-integrable modulo the center, we find characterization results of extra-invariant spaces under the left translations associated with the range functions. Consequently, the theory is valid for the Heisenberg group $\mathbb H^d$, a 2-step nilpotent Lie group.