Quantitative estimates and extrapolation for multilinear weight classes

TL;DR

This paper develops a quantitative multilinear extrapolation theorem that extends previous results to include cases with infinite exponents, enabling new vector-valued and endpoint estimates in harmonic analysis.

Contribution

It introduces a novel multilinear extrapolation framework that handles infinite exponents and extends sparse domination estimates to quasi-Banach spaces.

Findings

Extended multilinear extrapolation to include infinite exponents

Reproved vector-valued bounds for the bilinear Hilbert transform

Extended quantitative estimates to quasi-Banach spaces

Abstract

In this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite. This extends the recent extrapolation result of Li, Martell, and Ombrosi. We also obtain vector-valued estimates including $\ell^\infty$ spaces and, in particular, we are able to reprove all the vector-valued bounds for the bilinear Hilbert transform obtained through the helicoidal method of Benea and Muscalu. Moreover, our result is quantitative and, in particular, allows us to extend quantitative estimates obtained from sparse domination in the Banach space setting to the quasi-Banach space setting. Our proof does not rely on any off-diagonal extrapolation results and we develop a multilinear version of the Rubio de Francia algorithm adapted to the multisublinear Hardy-Littlewood maximal operator. As a corollary, we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak-type and BMO spaces.