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The aim of this paper is to establish a pseudo-differential Weyl calculus on graded nilpotent Lie groups $G$ which extends the celebrated Weyl calculus on $\mathbb{R}^n$. To reach this goal, we develop a symbolic calculus for a very general class of quantization schemes, following [Doc. Math., 22, 1539--1592, 2017], using the H\"{o}rmander symbol classes $S^m_{\rho, \delta}(G)$ introduced in [Progress in Mathematics, 314. Birkh\"{a}user/Springer, 2016]. We particularly focus on the so-called symmetric calculi, for which quantizing and taking the adjoint commute, among them the Euclidean Weyl calculus, but we also recover the (non-symmetric) Kohn-Nirenberg calculus, on $\mathbb{R}^n$ and on general graded groups [Progress in Mathematics, 314. Birkh\"{a}user/Springer, 2016]. Several interesting applications follow directly from our calculus: expected mapping properties on Sobolev spaces, the existence of one-sided parametrices and the G\r{a}rding inequality for elliptic operators, and a generalization of the Poisson bracket for symmetric quantizations on stratified groups. In the particular case of the Heisenberg group $\mathbb{H}_n$, we are able to answer the fundamental questions of this paper: which, among all the admissible quantizations, is the natural Weyl quantization on $\mathbb{H}_n$? And which are the criteria that determine it uniquely? The surprisingly simple but compelling answers raise the question whether what is true for $\mathbb{R}^n$ and $\mathbb{H}_n$ also extends to general graded groups, which we answer in the affirmative in this paper. Among other things, we discuss and investigate an analogue of the symplectic invariance property of the Weyl quantization in the setting of graded groups, as well as the notion of the Poisson bracket for symbols in the setting of stratified groups, linking it to the symbolic properties of the commutators., Comment: 70 pages, no figures

This note gives a wide-ranging update on the multiplier theorems by Akylzhanov and the second author [J. Funct. Anal., 278 (2020), 108324]. The proofs of the latter crucially rely on $L^p$-$L^q$ norm estimates for spectral projectors of left-invariant weighted subcoercive operators on unimodular Lie groups, such as Laplacians, sub-Laplacians and Rockland operators. By relating spectral projectors to heat kernels, explicit estimates of the $L^p$-$L^q$ norms can be immediately exploited for a much wider range of (connected unimodular) Lie groups and operators than previously known. The comparison with previously established bounds by authors show that the heat kernel estimates are sharp. As an application, it is shown that several consequences of the multiplier theorems, such as time asymptotics for the $L^p$-$L^q$ norms of the heat kernels and Sobolev-type embeddings, are then automatic for the considered operators., Comment: 10 pages

We prove strict necessary density conditions for coherent frames and Riesz sequences on homogeneous groups. Let $N$ be a connected, simply connected nilpotent Lie group with a dilation structure (a homogeneous group) and let $(\pi, \mathcal{H}_{\pi})$ be an irreducible, square-integrable representation modulo the center $Z(N)$ of $N$ on a Hilbert space $\mathcal{H}_{\pi}$ of formal dimension $d_\pi $. If $g \in \mathcal{H}_{\pi}$ is an integrable vector and the set $\{ \pi (\lambda )g : \lambda \in \Lambda \}$ for a discrete subset $\Lambda \subseteq N / Z(N)$ forms a frame for $\mathcal{H}_{\pi}$, then its density satisfies the strict inequality $D^-(\Lambda )> d_\pi $, where $D^-(\Lambda )$ is the lower Beurling density. An analogous density condition $D^+(\Lambda) < d_{\pi}$ holds for a Riesz sequence in $\mathcal{H}_{\pi}$ contained in the orbit of $(\pi, \mathcal{H}_{\pi})$. The proof is based on a deformation theorem for coherent systems, a universality result for $p$-frames and $p$-Riesz sequences, some results from Banach space theory, and tools from the analysis on homogeneous groups.

Although there is no canonical version of the harmonic oscillator on the Heisenberg group $\mathbf{H}_n$ so far, we make a strong case for a particular choice of operator by using the representation theory of the Dynin-Folland group $\mathbf{H}_{n, 2}$, a $3$-step stratified Lie group, whose generic representations act on $L^2(\mathbf{H}_n)$. Our approach is inspired by the connection between the harmonic oscillator on $\mathbb{R}^n$ and the sum of squares in the first stratum of $\mathbf{H}_n$ in the sense that we define the harmonic oscillator on $\mathbf{H}_n$ as the image of the sub-Laplacian $\mathcal{L}_{\mathbf{H}_{n, 2}}$ under the generic unitary irreducible representation $\pi$ of the Dynin-Folland group which has formal dimension $d_\pi = 1$. This approach, more generally, permits us to define a large class of so-called anharmonic oscillators by employing positive Rockland operators on $\mathbf{H}_{n, 2}$. By using the methods developed in ter Elst and Robinson [tERo], we obtain spectral estimates for the harmonic and anharmonic oscillators on $\mathbf{H}_n$. Moreover, we show that our approach extends to graded $SI/Z$-groups of central dimension $1$, i.e., graded groups which possess unitary irreducible representations which are square-integrable modulo the $1$-dimensional center $Z(G)$. The latter part of the article is concerned with spectral multipliers. By combining ter Elst and Robinson's techniques with recent results in [AkRu18], we obtain useful $L^\mathbf{p}$-$L^\mathbf{q}$-estimates for spectral multipliers of the sub-Laplacian $\mathcal{L}_{\mathbf{H}_{n, 2}}$ and, in fact more generally, of general Rockland operators on general graded groups. As a by-product, we recover the Sobolev embeddings on graded groups established in [FiRu17], and obtain explicit hypoelliptic heat semigroup estimates.

We show that generalised time-frequency shifts on the Heisenberg group $\mathbf{H}_n \cong \mathbb{R}^{2n+1}$, realised as a unitary irreducible representation of a nilpotent Lie group acting on $L^{2}(\mathbf{H}_n)$, give rise to a novel type of function spaces on $\mathbb{R}^{2n+1}$. The representation we employ is the generic unitary irreducible representation of the $3$-step nilpotent Dynin-Folland group. In doing so, we answer the question whether representations of nilpotent Lie groups ever yield coorbit spaces distinct from the classical modulation spaces $M^{\mathbf{p}, \mathbf{q}}_{\mathbf{v}}(\mathbb{R}^{2n+1}), n \in \mathbb{N}$, and also introduce a new member to the zoo of decomposition spaces. As our analysis and proof of novelty make heavy use of coorbit theory and decomposition space theory, in particular Voigtlaender's recent contributions to the latter, we give a complete classification of the unitary irreducible representations of the Dynin-Folland group and also characterise the coorbit spaces related to the non-generic unitary irreducible representations.

Gaussian states are at the heart of quantum mechanics and play an essential role in quantum information processing. In this paper we provide approximation formulas for the expansion of a general Gaussian symbol in terms of elementary Gaussian functions. For this purpose we introduce the notion of a "phase space frame" associated with a Weyl-Heisenberg frame. Our results give explicit formulas for approximating general Gaussian symbols in phase space by phase space shifted standard Gaussians as well as explicit error estimates and the asymptotic behavior of the approximation., Comment: 20 pages, 1 figure, in press, to appear in: Applied and Computational Harmonic Analysis

Let $G$ be a connected, simply connected nilpotent group and $\pi$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2(\mathbf{R}^d)$. We prove that under reasonably weak conditions on $G$ and $\pi$ there exist a discrete subset $\Gamma$ of $G/Z(G)$ and some (relatively) compact set $F \subseteq \mathbf{R}^d$ such that $$\bigl \{ |F|^{-1/2} \hspace{2pt} \pi(\gamma) 1_F \mid \gamma \in \Gamma \bigr\}$$ forms an orthonormal basis of $L^2(\mathbf{R}^d)$. This construction generalizes the well-known example of Gabor orthonormal bases in time-frequency analysis. The main theorem covers graded Lie groups with one-dimensional center. In the presence of a rational structure, the set $\Gamma $ can be chosen to be a uniform subgroup of $G/Z$.