Modulation spaces on the Heisenberg group

Abstract

In this article we show how certain irreducible unitary representation ${\Pi }_{\lambda }$ of the twisted Heisenberg group $ \He_λ^n(\C)$ leads to the twisted modulation spaces $M_{\lambda }^{p,q}({\mathbb{R}}^{2n}).$ These ${\Pi }_{\lambda }$ also turn out to be irreducible unitary representations of another nilpotent Lie group $G_n$ which contains two copies of the Heisenberg group $ \He^n.$ By lifting ${\Pi }_{\lambda }$ we obtain another unitary representation $\Pi$ of $G_n$ acting on $ L^2(\He^n).$ We define our modulation spaces $ M^{p,q}(\He^n) $ in terms of the matrix coefficients associated to $\Pi .$ These spaces are shown to be invariant under Heisenberg translations and Heisenberg modulations which are different from euclidean modulations. We also establish some of the basic properties of $M_{\lambda }^{p,q}({\mathbb{R}}^{2n})$ and $ M^{p,q}(\He^n) $ such as completeness and invariance under suitable Fourier transforms.

28 pages