
Journal of Lie Theory 34 (2024), No. 1, 051092 Copyright Heldermann Verlag 2024 HeisenbergModulation 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 [Abstractpdf] We show that generalised timefrequency 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 DyninFolland group, also known as the Heisenberg group of the Heisenberg group or the metaHeisenberg group. By realising these representations as nonstandard timefrequency 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, metaHeisenberg group, DyninFolland group, squareintegrable representation, Kirillov theory, flat orbit condition, modulation space, Besov space, coorbit theory, decomposition space. MSC: 42B35, 22E25, 22E27. [ Fulltextpdf (397 KB)] for subscribers only. 