Gelfand triples for the Kohn-Nirenberg quantization on homogeneous Lie groups

TL;DR

This paper develops a framework using Gelfand triples to analyze the Kohn-Nirenberg quantization on certain homogeneous Lie groups, providing new characterizations and explicit formulas within this setting.

Contribution

It introduces a novel Gelfand triple approach for the Kohn-Nirenberg quantization on homogeneous Lie groups with specific representation properties.

Findings

Characterization of the Fourier transform range on a subspace of Schwartz functions.

Construction of Gelfand triples around $L^2$ spaces where Fourier transform is an isomorphism.

Explicit formula for the Kohn-Nirenberg symbol of an operator.

Abstract

In this paper, we study the group Fourier transform and the Kohn-Nirenberg quantization for homogeneous Lie groups as mappings between certain Gelfand triples. For this, we restrict our considerations to the case, where the homogeneous Lie group $G$ admits irreducible unitary representations, that are square integrable modulo the center $Z(G)$ of $G$, and where $\dim Z(G)=1$. Replacing the Schwartz space by a certain subspace $\mathcal S_*(G) \hookrightarrow \mathcal S(G)$, we characterise the range of the group Fourier transform on $\mathcal S_*(G)$ and construct distributions and Gelfand triples around $L^2(G,\mu)$ and its Fourier image $L^2(\hat G,\hat \mu)$, such that the Fourier transform becomes a Gelfand triple isomorphism. We give results on the multiplication of distributions with a large class of vector valued smooth functions and use this to establish the Kohn-Nirenberg quantization as an isomorphism for our Gelfand triples and provide an explicit formula for the Kohn-Nirenberg symbol of an operator.