$p$-regularity and weights for operators between $L^p$-spaces

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 $T$.