Pseudo-differential Operators on Homogeneous Spaces

Abstract

In this chapter we discuss pseudo-differential operators on homogeneous spaces. The main question addressed here is how operators on such a space are related to pseudo-differential operators on the group that acts on the space. Once such a correspondence is established, one can use it to map the whole construction developed earlier from the group to the homogeneous space. We also note that among other things, this chapter provides an application to the characterisation of pseudo-differential operators in terms of Σ ^m -classes in Theorem 10.9.6. An important class of examples to keep in mind here are the spheres \( \mathbb{S}^n \cong SO(n)\backslash SO(n + 1) \cong SO(n + 1)/SO(n) \).