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

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 $p.$