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 generalization s of the classical fractional integrals and fractional maximal functions. T he norm estimates are derived in the context of a space of homogeneous type . The conditions required of the weight functions involve generalizations o f the Fefferman-Phong "r-bump" condition. The results improve some earlier ones of the same kind. nad they also extend to homogeneous spaces some esti mates that were previously. known to hold only in the classical Euclidean s etting. (C). 2001 Academic Press.