# Journal of Operator Theory

Volume 90, Issue 2, Autumn 2023 pp. 605-623.

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

**Authors**:
Somnath Ghosh (1), R.K. Srivastava (2)

**Author institution:** (1) Department of Mathematics, Indian Institute of Technology Guwahati, 781039, India

(2) Department of Mathematics, Indian Institute of Technology Guwahati, 781039, India

**Summary: ** In this article, we define the Weyl transform on a 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 $L^p(G\times \widehat{G})$ with $1\leqslant p\leqslant 2.$ Further, for the Euclidean motion group and
Heisenberg motion group, we prove that the Weyl transform cannot be extended as a bounded
operator for the symbol belongs to $L^p(G\times \widehat{G})$ with $p$ between $2$ and $\infty$. To carry out this,
we construct positive, square integrable and compactly supported function, on the respective groups,
such that the $L^{p'}$ norm of its Fourier transform is infinite, where $p'$ is the conjugate index of $p.$

**DOI: **http://dx.doi.org/10.7900/jot.2022jan21.2393

**Keywords: **Euclidean motion group, Fourier transform, Heisenberg motion group, Weyl transform

Contents
Full-Text PDF