Heisenberg-Modulation Spaces at the Crossroads of Coorbit Theory and Decomposition Space Theory

Véronique Fischer, David Rottensteiner, Michael Ruzhansky

Research output: Contribution to journalArticlepeer-review


We show that generalised time-frequency shifts on the Heisenberg group Hn – R2n`1 give rise to a novel type of function spaces on R2n`1. Similarly to classical modulation spaces and Besov spaces on R2n`1, these spaces can be characterised in terms of specific frequency partitions of the Fourier domain Rp2n`1 as well as decay of the matrix coefficients of specific Lie group representations. The representations in question are the generic unitary irreducible representations of the 3-step nilpotent Dynin-Folland group, also known as the Heisenberg group of the Heisenberg group or the meta-Heisenberg group. By realising these representations as nonstandard time-frequency shifts on the phase space R4n`2 – Hn x R2n`1, we obtain a Fourier analytic characterisation, which from a geometric point of view locates the spaces somewhere between modulation spaces and Besov spaces. A conclusive comparison with the latter and some embeddings are given by using novel methods from decomposition space theory.

Original languageEnglish
Pages (from-to)51-92
Number of pages42
JournalJournal of Lie Theory
Issue number1
Publication statusPublished - 1 Apr 2024


  • Besov space
  • coorbit theory
  • decomposition space
  • Dynin-Folland group
  • flat orbit condition
  • Heisenberg group
  • Kirillov theory
  • meta-Heisenberg group
  • modulation space
  • Nilpotent Lie group
  • square-integrable representation

ASJC Scopus subject areas

  • Algebra and Number Theory


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