An Extrapolation Theorem in Non-Euclidean Geometries and its Application to Partial Differential Equations

We prove a generalization of an extrapolation theorem in the fashion of García-Cuerva and Rubio de Francia towards R -boundedness on weighted Lebesgue spaces over locally compact abelian groups. This result can be applied to show maximal L p regularity for differential operators that correspond to parabolic evolution equations subject to more general spatial geometries, for example the partially periodic Stokes operator. As a main tool, we generalize the classical Muckenhoupt theorem on maximal operators to locally compact abelian groups.