# Rescaled extrapolation for vector-valued functions

## TL;DR

This paper extends Rubio de Francia's extrapolation theorem to vector-valued functions in UMD Banach spaces, providing new proofs and results for Littlewood-Paley estimates and variational Carleson operators.

## Contribution

It introduces an extension of extrapolation theorems to vector-valued functions in UMD spaces, enabling new and simplified proofs of key harmonic analysis results.

## Key findings

- Proved Littlewood-Paley-Rubio de Francia-type estimates.
- Established boundedness of variational Carleson operators in UMD Banach spaces.
- Provided simplified proofs for known results in harmonic analysis.

## Abstract

We extend Rubio de Francia's extrapolation theorem for functions valued in UMD Banach function spaces, leading to short proofs of some new and known results. In particular we prove Littlewood-Paley-Rubio de Francia-type estimates and boundedness of variational Carleson operators for Banach function spaces with UMD concavifications.