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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 39, Issue 1, 2020, pp. 41–65**

**DOI: 10.4171/ZAA/1650**

Published online: 2020-01-24

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

Enrique A. Sánchez Pérez^{[1]}and Pedro Tradacete

^{[2]}(1) Universitat Politècnica de València, Spain

(2) Instituto de Ciencias Matemáticas (ICMAT), Madrid, Spain

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 $T$.

*Keywords: *Banach function space, $p$-regular operator, weighted $p$-estimate

Sánchez Pérez Enrique, Tradacete Pedro: $p$-Regularity and Weights for Operators Between $L^p$-Spaces. *Z. Anal. Anwend.* 39 (2020), 41-65. doi: 10.4171/ZAA/1650