  We prove a limited range, off-diagonal extrapolation theorem that generalizes
a number of results in the theory of Rubio de Francia extrapolation, and use
this to prove a limited range, multilinear extrapolation theorem. We give two
applications of this result to the bilinear Hilbert transform. First, we give
sufficient conditions on a pair of weights $w_1,\,w_2$ for the bilinear Hilbert
transform to satisfy weighted norm inequalities of the form \[ BH :
L^{p_1}(w_1^{p_1}) \times L^{p_2}(w_2^{p_2}) \longrightarrow L^p(w^p), \] where
$w=w_1w_2$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}<\frac{3}{2}$. This
improves the recent results of Culiuc et al. by increasing the families of
weights for which this inequality holds and by pushing the lower bound on $p$
from $1$ down to $\frac{2}{3}$, the critical index from the unweighted theory
of the bilinear Hilbert transform. Second, as an easy consequence of our method
we obtain that the bilinear Hilbert transform satisfies some vector-valued
inequalities with Muckenhoupt weights. This reproves and generalizes some of
the vector-valued estimates obtained by Benea and Muscalu in the unweighted
case. We also generalize recent results of Carando, et al. on
Marcinkiewicz-Zygmund estimates for multilinear Calder\'on-Zygmund operators.
