Limited Range Extrapolation with Quantitative Bounds and Applications

In recent years, sharp or quantitative weighted inequalities have attracted considerable attention on account of the \(A_2\) conjecture solved by Hytönen. Advances have greatly improved conceptual understanding of classical objects such as Calderón–Zygmund operators. However, plenty of operators do not fit into the class of Calderón–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) Calderón–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.