  We calculate the superfluid fraction of an interacting Fermi gas, in the
presence of a one-dimensional periodic potential of strength $V_0$ and
wave-vector $q$. Special focus is given to the unitary Fermi gas, characterized
by the divergent behavior of the s-wave scattering length. Comparison with the
Leggett's upper bound $(\langle n_{1D}\rangle <1/n_{1D}>)^{-1}$, with $n_{1D}$
the 1D column density, explicitly shows that, differently from the case of a
dilute interacting Bose gas, the bound significantly overestimates the value of
the superfluid fraction, except in the phonon regime of small $q$. Sum rule
arguments show that the combined knowledge of the Leggett bound and of the
actual value of the superfluid fraction allows for the determination of
curvature effects providing the deviation of the dispersion of the
Anderson-Bogoliubov mode from the linear phonon dependence. The comparison with
the predictions of the weakly interacting BCS Fermi gas points out the crucial
role of two-body interactions. The implications of our predictions on the
anisotropic behavior of the sound velocity are also discussed.
