  In this paper we present the results announced in the recent work by the
first, second, and fourth authors of the current paper concerning Rubio de
Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here
we consider the situations where some of the exponents of the Lebesgue spaces
appearing in the hypotheses and/or in the conclusion can be possibly infinity.
The scheme we follow is similar, but, in doing so, we need to develop a
one-variable end-point off-diagonal extrapolation result. This complements the
corresponding ``finite'' case obtained by Duoandikoetxea, which was one of the
main tools in the aforementioned paper. The second goal of this paper is to
present some applications. For example, we obtain the full range of mixed-norm
estimates for tensor products of bilinear Calder\'on-Zygmund operators with a
proof based on extrapolation and on some estimates with weights in some
mixed-norm classes. The same occurs with the multilinear Calder\'on-Zygmund
operators, the bilinear Hilbert transform, and the corresponding commutators
with BMO functions. Extrapolation along with the already established weighted
norm inequalities easily give scalar and vector-valued inequalities with
multilinear weights and these include the end-point cases.
