Journal of Lie Theory 34 (2024), No. 1, 051--092
Copyright Heldermann Verlag 2024
Heisenberg-Modulation Spaces at the Crossroads of Coorbit Theory and Decomposition Space Theory
Véronique Fischer
Department of Mathematical Sciences, University of Bath, United Kingdom
v.c.m.fischer@bath.ac.uk
David Rottensteiner
Department of Mathematics, Analysis, Logic and Discrete Mathematics, Ghent University, Belgium
david.rottensteiner@ugent.be
Michael Ruzhansky
(1) Dept. of Mathematics, Ghent University, Belgium, Ghent University, Belgium
(2) School of Mathematical Sciences, Queen Mary University, London, United Kingdom
michael.ruzhansky@ugent.be
[Abstract-pdf]
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.
Keywords: 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.
MSC: 42B35, 22E25, 22E27.
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