$p$-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 $L^p(vd\mu )$ 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 $T:L^p(wd\nu )\to L^p(vd\mu )$ 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 $T$.