Rubio de Francia's extrapolation theory: estimates for the distribution function

TL;DR

This paper extends Rubio de Francia's extrapolation theory, showing that the distribution function of an operator's output can be controlled by the Hardy-Littlewood maximal function, simplifying proofs of known and new extrapolation results.

Contribution

It provides a simple proof technique for extrapolation theorems, including multilinear and two-weight cases, and introduces new applications in rearrangement invariant spaces.

Findings

Distribution function of $Tf$ is majorized by that of $Mf$ plus an integral term.

Simplifies proofs of classical and multilinear extrapolation results.

Establishes new applications in two-weight problems and rearrangement invariant spaces.

Abstract

Let $T$ be an arbitrary operator bounded from $L^{p_0}(w)$ into $L^{p_0, \infty}(w)$ for every weight $w$ in the Muckenhoupt class $A_{p_0}$. It is proved in this article that the distribution function of $Tf$ with respect to any weight $u$ can be essentially majorized by the distribution function of $Mf$ with respect to $u$ (plus an integral term easy to control). As a consequence, well-known extrapolation results, including results in a multilinear setting, can be obtained with very simple proofs. New applications in extrapolation for two-weight problems and estimates on rearrangement invariant spaces are established too.