Unbounded Weyl transform on the Euclidean motion group and Heisenberg motion group

TL;DR

This paper investigates the properties of the Weyl transform on certain groups, establishing conditions for compactness and unboundedness depending on the symbol's integrability class.

Contribution

It defines the Weyl transform on second countable type I groups and analyzes its boundedness and compactness on Euclidean and Heisenberg motion groups.

Findings

Weyl transform is compact for symbols in L^p with 1 ≤ p ≤ 2.

Weyl transform cannot be extended as a bounded operator for p > 2 on specific groups.

Constructs functions with infinite Fourier transform norms to demonstrate unboundedness.

Abstract

In this article, we define Weyl transform on second countable type - $I$ locally compact group $G,$ and as an operator on $L^2(G),$ we prove that the Weyl transform is compact when the symbol lies in $L^p(G\times \hat{G})$ with $1\leq p\leq 2.$ Further, for the Euclidean motion group and Heisenberg motion group, we prove that the Weyl transform can not be extended as a bounded operator for the symbol belongs to $L^p(G\times \hat{G})$ with $2<p<\infty.$ To carry out this, we construct positive, square integrable and compactly supported function, on the respective groups, such that $L^{p'}$ norm of its Fourier transform is infinite, where $p'$ is the conjugate index of $p.$