Gelfand Triples for the Kohn–Nirenberg Quantization on Homogeneous Lie Groups

Abstract

We study the action of the group Fourier transform and of the Kohn–Nirenberg quantization (Fischer and Ruzhansky, Quantization on Nilpotent Lie Groups. Birkäuser, Boston, 2016) on certain Gelfand triples for homogeneous Lie groups G. Even for the Heisenberg group $G=\mathbb H$ there seems to be no simple intrinsic characterization for the Fourier image of the Schwartz space of rapidly decreasing smooth functions $\mathcal S(G)$, see (Geller, J Funct Anal 36(2), 205–254, 1980; Astengo et al., Stud Math 214, 201–222, 2013). But we may derive a simple characterization of the Fourier image for a certain subspace $\mathcal S_*(G)$ of $\mathcal S(G)$. We restrict our considerations to the case, where G admits irreducible unitary representations, that are square integrable modulo the center Z(G) of G, and where $\dim Z(G)=1$. This enables us to use an especially applicable characterization of these irreducible unitary representations that are square integrable modulo Z(G) (Moore and Wolf, Trans Am Math Soc 185, 445–445, 1973; Măntoiu and Ruzhansky, J Geom Anal 29(2), 2823–2861, 2018; Gröchenig and Rottensteiner, J Funct Anal 275(12), 3338–3379, 2018). Also, Pedersen’s machinery (Pedersen, Invent. Math. 118, 1–36, 1994) combines very well with this setting (Măntoiu and Ruzhansky, J Geom Anal 29(2), 2823–2861, 2018). Starting with $\mathcal S_*(G)$, we are able to construct distributions and Gelfand triples around L ²(G, μ) and its Fourier image $L^2(\widehat G,\widehat \mu )$, such that the Fourier transform becomes a Gelfand triple isomorphism. In this context we show for the Fourier side, that multiplication of distributions with a large class of vector valued smooth functions is possible and well behaved. Furthermore, we rewrite the Kohn–Nirenberg quantization as an isomorphism for our new Gelfand triples and prove a formula for the Kohn–Nirenberg symbol, which is known from the compact group case (Ruzhansky and Turunen, Pseudo-Differential Operators and Symmetries. Birkäuser, Boston, 2010).