$p$-Regularity and Weights for Operators Between $L^p$-Spaces

Abstract

We explore the connection between p -regular operators on Banach function spaces and weighted p -estimates. In particular, our results focus on the following problem. Given finite measure spaces \mu and \nu , let T be an operator defined from a Banach function space X(\nu) and taking values on L^p (v d \mu) for v in certain family of weights V\subset L^1(\mu)_+ we analyze the existence of a bounded family of weights W\subset L^1(\nu)_+ such that for every v\in V there is w \in W in such a way that T:L^p(w d \nu) \to L^p(v d \mu) is continuous uniformly on V . A condition for the existence of such a family is given in terms of p -regularity of the integration map associated to a certain vector measure induced by the operator T .