Modulation spaces on the Heisenberg group

TL;DR

The paper develops modulation spaces on the Heisenberg group by leveraging twisted representations derived from the twisted Heisenberg group $\mathbb{H}^n_\lambda(\mathbb{C})$ and the group $G_n$. It constructs twisted modulation spaces $M_\lambda^{p,q}(\mathbb{R}^{2n})$ via matrix coefficients of $\Pi_\lambda$, and then builds global spaces $M^{p,q}(\mathbb{H}^n)$ by integrating over $\lambda$, establishing invariance under Heisenberg translations and modulations. The work analyzes matrix coefficients, Bargmann-type transforms, and proves completeness and duality results, situating these spaces within a coorbit-theoretic framework for nilpotent groups. Overall, it provides a natural noncommutative generalization of classical modulation spaces, enabling time-frequency analysis on the Heisenberg group and its twisted variants. The framework connects twisted Fock spaces, Bargmann transforms, and group representations to give robust Banach spaces with practical invariance and duality properties.

Abstract

In this article we show how certain irreducible unitary representation $ Π_λ$ of the twisted Heisenberg group $ \He_λ^n(\C)$ leads to the twisted modulation spaces $ M_λ^{p,q}(\R^{2n}).$ These $ Π_λ$ 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 $ Π_λ$ we obtain another unitary representation $ Π$ 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 $ Π.$ 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_λ^{p,q}(\R^{2n})$ and $ M^{p,q}(\He^n) $ such as completeness and invariance under suitable Fourier transforms.

Theorems & Definitions (43)

• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• Lemma 3.4
• proof
• Corollary 3.5
• proof
• Remark 3.6