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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

David Rottensteiner
Department of Mathematics, Analysis, Logic and Discrete Mathematics, Ghent University, Belgium

Michael Ruzhansky
(1) Dept. of Mathematics, Ghent University, Belgium, Ghent University, Belgium
(2) School of Mathematical Sciences, Queen Mary University, London, United Kingdom


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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