Extrapolation for multilinear Muckenhoupt classes and applications to the bilinear Hilbert transform

TL;DR

This paper advances the theory of weighted inequalities for multilinear operators, solving a longstanding extrapolation problem and applying it to improve bounds for the bilinear Hilbert transform and related operators.

Contribution

It solves the multivariable Rubio de Francia extrapolation problem for multilinear Muckenhoupt classes and extends results to broader classes and applications.

Findings

Extended weighted inequalities for bilinear Hilbert transform.

Reproved and extended Marcinkiewicz-Zygmund estimates.

Derived new bounds for commutators with BMO functions.

Abstract

In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes $A_{\vec{p}}$, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calder\'on-Zygmund operators. Furthermore, we go beyond the classes $A_{\vec{p}}$ and extrapolate within the classes $A_{\vec{p},\vec{r}}$ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several applications which can be easily obtained using extrapolation. First, for the bilinear Hilbert transform one can extrapolate from the recent result of Culiuc et al. who considered the Banach range and extend the estimates to the quasi-Banach range. As a direct consequence, we obtain weighted vector-valued inequalities reproving some of the results by Benea and Muscalu. We also extend recent results of Carando et al. on Marcinkiewicz-Zygmund estimates for multilinear Calder\'on-Zygmund operators. Finally, our last application gives new weighted estimates for the commutators of multilinear Calder\'on-Zygmund operators and for the bilinear Hilbert transform with BMO functions using ideas from B\'enyi et al.