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. We selected two samples with 5 GHz VLA radio flux measurements down to 1mJy, reaching levels of radio luminosity as low as 1036
erg/s. In CapettI & Balmaverde (2005A&A...440...73C
) 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, following the Nukers scheme. Here we focus on the 29 core galaxies (hereafter CoreG). We used HST and Chandra data to isolate their optical and X-ray nuclear emission. The CoreG invariably host radio-loud nuclei, with an average radio-loudness parameter of Log R=L5GHz
∼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 (LLRG) toward even lower power, by a factor of ∼1000, covering a combined range of 6 orders of magnitude. 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/LEdd
in both the optical and X-ray band, on any emission from the accretion process. The similarity of CoreG and LLRG when considering the distributions host galaxies luminosities and black hole masses, as well as of the surface brightness profiles, 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. We do not find any relationship between radio-power and black hole mass. A minimum black hole mass of MBH
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. In the unifying model for BL Lacs and radio-galaxies, CoreG likely represent the counterparts of the large population of low luminosity BL Lac now emerging from the surveys at low radio flux limits. This suggests the presence of relativistic jets also in these quasi-quiescent early-type ``core'' galaxies.