  In this paper we study the dynamics of the stellar interior of the early
red-giant star KIC 4448777 by asteroseismic inversion of 14 splittings of the
dipole mixed modes obtained from {\it Kepler} observations. In order to
overcome the complexity of the oscillation pattern typical of red-giant stars,
we present a procedure which involves a combination of different methods to
extract the rotational splittings from the power spectrum. We find not only
that the core rotates faster than the surface, confirming previous inversion
results generated for other red giants (Deheuvels et al. 2012,2014), but we
also estimate the variation of the angular velocity within the helium core with
a spatial resolution of $\Delta r=0.001R$ and verify the hypothesis of a sharp
discontinuity in the inner stellar rotation (Deheuvels et al. 2014). The
results show that the entire core rotates rigidly with an angular velocity of
about $\langle\Omega_c/2\pi\rangle=748\pm18$~nHz and provide evidence for an
angular velocity decrease through a region between the helium core and part of
the hydrogen burning shell; however we do not succeed to characterize the
rotational slope, due to the intrinsic limits of the applied techniques. The
angular velocity, from the edge of the core and through the hydrogen burning
shell, appears to decrease with increasing distance from the center, reaching
an average value in the convective envelope of
$\langle\Omega_s/2\pi\rangle=68\pm22$~nHz. Hence, the core in KIC~4448777 is
rotating from a minimum of 8 to a maximum of 17 times faster than the envelope.
We conclude that a set of data which includes only dipolar modes is sufficient
to infer quite accurately the rotation of a red giant not only in the dense
core but also, with a lower level of confidence, in part of the radiative
region and in the convective envelope.
