  We study the superfluid behavior of a population imbalanced ultracold atomic
Fermi gases with a short range attractive interaction in a one-dimensional (1D)
optical lattice, using a pairing fluctuation theory. We show that, besides
widespread pseudogap phenomena and intermediate temperature superfluidity, the
superfluid phase is readily destroyed except in a limited region of the
parameter space. We find a new mechanism for pair hopping, assisted by the
excessive majority fermions, in the presence of continuum-lattice mixing, which
leads to an unusual constant BEC asymptote for $T_c$ that is independent of
pairing strength. In result, on the BEC side of unitarity, superfluidity, when
it exists, may be strongly enhanced by population imbalance.
