H-type group, a-Weyl transform and pseudo-differential operators

Abstract

This paper is about the pseudo-differential operator on the H-type group, denoted by H. We present the trace formula for the a-Weyl transform. Also, we give the characterization of Hilbert-Schmidt class and trace class pseudo-differential operators on H.

Pseudo-Differential Operators, Wigner Transform, and Weyl Transform on the Affine Poincaré Group

“…In particular, the resulting operator T σ is in Hilbert-Schmidt class as explained in [29]. Similar kinds of results were proved in [10] for pseudo-differential operators on the Heisenberg group, and for the H-type group in [30]. Recently, a necessary and sufficient condition on the symbols such that the corresponding pseudo-differential operators on the affine group and similitude group (polar affine group) are in Hilbert-Schmidt class has been obtained in [6,7].…”

Trace class and Hilbert-Schmidt pseudo differential operators on step two nilpotent Lie groups

“…In this paper, we will only consider G to be a step two nilpotent Lie group without MW-condition. However, for MW-condition, the calculation will be similar and one can look at [31].…”

“…Dasgupta and Wong in [3,5] provided necessary and sufficient conditions on the symbols such that the corresponding pseudo-differential operators on the Heisenberg group are in Hilbert-Schmidt class. Mingkai and Jianxun [31] studied the properties of pseudo-differential on H-type group. Recently, a similar result was established by Dasgupta and Kumar for pseudo-differential operators on the abstract Heisenberg group [4].…”

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

“…Dasgupta and Wong [7] obtained necessary and sufficient conditions on symbols such that the corresponding pseudo-differential operators on the Heisenberg group belong to the Hilbert-Schmidt class, which was further extended by Dasgupta and the first author for abstract Heisenberg groups [8]. Such type of properties for pseudo-differential on H-type groups and on the Affine groups are given in [35] and [10], respectively. Recently, trace class and Hilbert-Schmidt pseudo differential operators on step two nilpotent Lie groups were discussed by the authors in [22].…”