L²-L^p estimates and Hilbert–Schmidt pseudo differential operators on the Heisenberg motion group

Abstract

In this paper, we study some operator theoretical properties of pseudo-differential operators with operator-valued symbols on the Heisenberg motion group. Specifically, we investigate L 2 -L p boundedness of pseudo-differential operators on the Heisenberg motion group for the range 2 ≤ p ≤ ∞. We also provide a necessary and sufficient condition on the operatorvalued symbols in terms of λ-Weyl transforms such that the corresponding pseudo-differential operators on the Heisenberg motion group are in the class of Hilbert-Schmidt operators. As a consequence, we obtain a characterization of the trace class pseudo-differential operators on the Heisenberg motion group and provide a trace formula for these trace class operators.