Boundedness for Weyl-Pedersen calculus on flat coadjoint orbits

TL;DR

This paper investigates the boundedness and compactness of operators from the Weyl-Pedersen calculus on flat coadjoint orbits of nilpotent Lie groups, extending classical pseudo-differential operator results.

Contribution

It provides new criteria for boundedness, compactness, and Schatten class membership of Weyl-Pedersen operators using invariant differential operators on coadjoint orbits.

Findings

Characterization of bounded and compact operators in the Weyl-Pedersen calculus.

Conditions for operators to belong to Schatten ideals.

Recovery of classical properties like Calderón-Vaillancourt theorem for the Heisenberg group.

Abstract

We describe boundedness and compactness properties for the operators obtained by the Weyl-Pedersen calculus in the case of the irreducible unitary representations of nilpotent Lie groups that are associated with flat coadjoint orbits. We use spaces of smooth symbols satisfying appropriate growth conditions expressed in terms of invariant differential operators on the coadjoint orbit under consideration. Our method also provides conditions for these operators to belong to one of the Schatten ideals of compact operators. In the special case of the Schr\"odinger representation of the Heisenberg group we recover some classical properties of the pseudo-differential Weyl calculus, as the Calder\'on-Vaillancourt theorem, and the Beals characterization in terms of commutators.