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

We show that generalised time-frequency shifts on the Heisenberg group H_n – R²ⁿ`1 give rise to a novel type of function spaces on R<sup>2n</sup>`¹. Similarly to classical modulation spaces and Besov spaces on R²ⁿ`<sup>1</sup>, these spaces can be characterised in terms of specific frequency partitions of the Fourier domain R<sup>p2n</sup>`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`<sup>2</sup> – H<sub>n</sub> x R<sup>2n</sup>`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.