Modulation spaces of symbols for representations of nilpotent Lie groups

TL;DR

This paper introduces modulation spaces for irreducible representations of nilpotent Lie groups, analyzing operator continuity and connecting to classical time-frequency analysis in special cases.

Contribution

It develops a new framework of modulation spaces for nilpotent Lie group representations, extending classical time-frequency analysis tools.

Findings

Established basic properties of these modulation spaces.

Analyzed continuity of Weyl correspondence operators.

Connected special cases to classical modulation spaces.

Abstract

We investigate continuity properties of operators obtained as values of the Weyl correspondence constructed by N.V. Pedersen (Invent. Math. 118 (1994), 1--36) for arbitrary irreducible representations of nilpotent Lie groups. To this end we introduce modulation spaces for such representations and establish some of their basic properties. The situation of square integrable representations is particularly important and in the special case of the Schr\"odinger representation of the Heisenberg group we recover the classical modulation spaces used in the time-frequency analysis.