Boundedness for Weyl–Pedersen Calculus on Flat Coadjoint Orbits

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 Schrödinger representation of the Heisenberg group, we recover some classical properties of the pseudo-differential Weyl calculus, as the Calderón–Vaillancourt theorem, and the Beals characterization in terms of commutators.