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 × Ĝ) with 1 ≤ p ≤ 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 × Ĝ) with 2 < p < ∞. 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.

L2-Lp estimates and Hilbert–Schmidt pseudo differential operators on the Heisenberg motion group

“…Recently, trace class and Hilbert-Schmidt pseudo differential operators on step two nilpotent Lie groups were discussed by the authors in [22]. Motivated by these previous studies as well as the recent developments on λ-Weyl transform on the Heisenberg motion group [18,17], in this paper, we study and extend some of the aforementioned results to the setting of the Heisenberg motion group. The λ-Weyl transform plays an important role in the proof of our results.…”

“…In this subsection, we recall some basics of harmonic analysis on the Heisenberg motion group to make the paper selfcontained. A complete account of representation theory of the Heisenberg motion group can be found in [31,28,2,18,27]. However, we mainly adopt the notation and terminology given in [18].…”

“…A complete account of representation theory of the Heisenberg motion group can be found in [31,28,2,18,27]. However, we mainly adopt the notation and terminology given in [18].…”

“…λ-Weyl transform. In this subsection, we recall some basic definitions and important properties of λ-Weyl transform associated to a symbol on L 2 (G × ), where G × = C n × K. The authors in [18] studied unboundedness properties of λ-Weyl transforms on G. For a detailed study on λ-Weyl transforms on the Heisenberg motion group, we refer to [2,17,28]. We also refer to the book of Wong [33] for Weyl transform on R n .…”