Generalized Uncertainty Principles on Spaces of Homogeneous Type

We describe a generalized version of Weyl′s principle and of the Heisenberg uncertainty principle under which the geometry and spectral theory of a given operator can be tied together to the geometry of its underlying space. This is achieved by means of multiscale analysis performed on the associated space of homogeneous type. In this paper we consider in detail the case of (−Δ)ϵ/2 on compact manifolds.