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

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