Uncertainty Principle Estimates for Vector Fields

We derive weighted norm estimates for integral operators of potential type and for their related maximal operators. These operators are generalizations of the classical fractional integrals and fractional maximal functions. The norm estimates are derived in the context of a space of homogeneous type. The conditions required of the weight functions involve generalizations of the Fefferman–Phong “r-bump” condition. The results improve some earlier ones of the same kind, and they also extend to homogeneous spaces some estimates that were previously known to hold only in the classical Euclidean setting.