Limited range extrapolation with quantitative bounds and applications

TL;DR

This paper develops a quantitative multilinear limited range extrapolation method to analyze weighted inequalities for operators beyond classical Calderón-Zygmund theory, with applications to various multilinear operators and inequalities.

Contribution

It introduces a new quantitative multilinear limited range extrapolation framework that extends previous results and applies to a broader class of operators and spaces.

Findings

Established a multilinear limited range extrapolation with quantitative bounds.

Extended quantitative estimates from Banach to quasi-Banach spaces.

Applied results to bilinear Bochner-Riesz means, rough singular integrals, and Fourier multipliers.

Abstract

In recent years, sharp or quantitative weighted inequalities have attracted considerable attention on account of $A_2$ conjecture solved by Hyt\"{o}nen. Advances have greatly improved conceptual understanding of classical objects such as Calder\'{o}n-Zygmund operators. However, plenty of operators do not fit into the class of Calder\'{o}n-Zygmund operators and fail to be bounded on all $L^p(w)$ spaces for $p \in (1, \infty)$ and $w \in A_p$. In this paper we develop Rubio de Francia extrapolation with quantitative bounds to investigate quantitative weighted inequalities for operators beyond the (multilinear) Calder\'{o}n-Zygmund theory. We mainly establish a quantitative multilinear limited range extrapolation in terms of exponents $p_i \in (\mathfrak{p}_i^-, \mathfrak{p}_i^+)$ and weights $w_i^{p_i} \in A_{p_i/\mathfrak{p}_i^-} \cap RH_{(\mathfrak{p}_i^+/p_i)'}$, $i=1, \ldots, m$, which refines a result of Cruz-Uribe and Martell. We also present an extrapolation from multilinear operators to the corresponding commutators. Additionally, our result is quantitative and allows us to extend special quantitative estimates in the Banach space setting to the quasi-Banach space setting. Our proof is based on an off-diagonal extrapolation result with quantitative bounds. Finally, we present various applications to illustrate the utility of extrapolation by concentrating on quantitative weighted estimates for some typical multilinear operators such as bilinear Bochner-Riesz means, bilinear rough singular integrals, and multilinear Fourier multipliers. In the linear case, based on the Littlewood-Paley theory, we include weighted jump and variational inequalities for rough singular integrals.