  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 $T$.
