  This is the second of a series of three papers exploring the connection
between the multiwavelength properties of AGN in nearby early-type galaxies and
the characteristics of their hosts. In Capetti et al. (2005) we presented a
study of the surface brightness profiles for the 65 objects with available
archival HST images out of the 116 radio-detected galaxies. We classified
early-type galaxies into ``core'' and ``power-law'' galaxies, discriminating on
the basis of the slope of their nuclear brightness profiles. Here we focus on
the 29 core galaxies (hereafter CoreG). We used HST and Chandra data to isolate
their nuclear emission. The CoreG invariably host radio-loud nuclei, with an
average radio-loudness parameter of Log R = 3.6. The optical and X-ray nuclear
luminosities correlate with the radio-core power, smoothly extending the
analogous correlations already found for low luminosity radio-galaxies. This
supports the interpretation of a common non-thermal origin of the nuclear
emission also for CoreG. The luminosities of the nuclear sources, most likely
dominated by jet emission, set firm upper limits, as low as L/L_Edd = 10^{-9}
in both the optical and X-ray band, on any emission from the accretion process.
The similarity of CoreG and LLRG indicates that they are drawn from the same
population of early-type galaxies. LLRG represent only the tip of the iceberg
associated with (relatively) high activity levels, with CoreG forming the bulk
of the population. A minimum black hole mass of M_{BH} = 10^8 M_sun is
apparently associated with the radio-loud nuclei in both CoreG and LLRG, but
this effect must be tested on a sample of less luminous galaxies, likely to
host smaller black holes. ABRIDGED.
