  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.
