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

Véronique Fischer¹ ; David Rottensteiner² ; Michael Ruzhansky^3, 4

¹Department of Mathematical Sciences, University of Bath, United Kingdom
²Department of Mathematics, Analysis, Logic and Discrete Mathematics, Ghent University, Belgium
³(1) Dept. of Mathematics, Ghent University, Belgium, Ghent University, Belgium
⁴(2) School of Mathematical Sciences, Queen Mary University, London, United Kingdom

Journal of Lie Theory, Tome 34 (2024) no. 1, pp. 51-92

Résumé

We show that generalised time-frequency shifts on the Heisenberg group $\mathbf{H}_n \cong \mathbb{R}^{2n+1}$ give rise to a novel type of function spaces on $\mathbb{R}^{2n+1}$. Similarly to classical modulation spaces and Besov spaces on $\mathbb{R}^{2n+1}$, these spaces can be characterised in terms of specific frequency partitions of the Fourier domain $\widehat{\mathbb{R}}^{2n+1}$ 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 non-standard time-frequency shifts on the phase space $\mathbb{R}^{4n+2} \cong \H \times \mathbb{R}^{2n+1}$, 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.

Détail

Classification : 42B35, 22E25, 22E27
Mots-clés : Nilpotent Lie group, Heisenberg group, meta-Heisenberg group, Dynin-Folland group, square-integrable representation, Kirillov theory, flat orbit condition, modulation space, Besov space, coorbit theory, decomposition space