Localization and vector spherical harmonics

This paper establishes the following localization property for vector spherical harmonics: a wide class of non-local, vector-valued operators reduce to local, multiplication-type operations when applied to a vector spherical harmonic. As localization occurs in a very precise, quantifiable and explicitly computable fashion, the localization property provides a set of useful formulae for analyzing vector-valued fractional diffusion and non-local differential equations defined on Sd−1Sd−1. As such analyses require a detailed understanding of operators for which localization occurs, we provide several applications of the result in the context of non-local differential equations.