We define and study an algebra ψ1,0,ν ∞(M0) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields ν on a compactification M of M0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to ψ1,0,ν ∞(M0). We also consider the algebra Diff ν*(M0) of differential operators on M0 generated by ν and C∞ (M), and show that ψ1,0,ν∞(M0) is a microlocalization of Diffν*(M0). Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and "suspended" versions of the algebra ψ1,0,ν ∞(M0).
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty