Aims. It is now established that the faint radio population is a mixture of star-forming galaxies and faint active galactic nuclei (AGNs), with the former dominating below S1.4 GHz ∼ 100μJy and the latter at larger flux densities. The faint radio AGN component can itself be separated into two main classes, mainly based on the host-galaxy properties: sources associated with red/early-type galaxies (like radio galaxies) are the dominant class down to ∼100 μJy; quasar/Seyfert–like sources contribute an additional 10–20%. One of the major open questions regarding faint radio AGNs is the physical process responsible for their radio emission. This work aims at investigating this issue, with particular respect to the AGN component associated with red/early-type galaxies. Such AGNs show, on average, flatter radio spectra than radio galaxies and are mostly compact (≤30 kpc in size). Various scenarios have been proposed to explain their radio emission. For instance they could be core/core-jet dominated radio galaxies, low-power BL Lacertae, or advection-dominated accretion flow (ADAF) systems. Methods. We used the Australia Telescope Compact Array (ATCA) to extend a previous follow-up multi-frequency campaign to 38 and 94 GHz. This campaign focuses on a sample of 28 faint radio sources associated with early-type galaxies extracted from the ATESP 5 GHz survey. Such data, together with those already at hand, are used to perform radio spectral and variability analyses. Both analyses can help us to disentangle between core- and jet-dominated sources, as well as to verify the presence of ADAF/ADAF+jet systems. Additional high-resolution observations at 38 GHz were carried out to characterise the radio morphology of these sources on kiloparsec scales. Results. Most of the sources (25/28) were detected at 38 GHz, while only one (ATESP5J224547−400324) of the twelve sources observed at 94 GHz was detected. From the analysis of the radio spectra we confirmed our previous findings that pure ADAF models can be ruled out. Only eight out of the 28 sources were detected in the 38-GHz high-resolution (0.6 arcsec) radio images and of those eight only one showed a tentative core-jet structure. Putting together spectral, variability, luminosity, and linear size information we conclude that different kinds of sources compose our AGN sample: (a) luminous and large (≥100 kpc) classical radio galaxies (∼18% of the sample); (b) compact (confined within their host galaxies), low-luminosity, power-law (jet-dominated) sources (∼46% of the sample); and (c) compact, flat (or peaked) spectrum, presumably core-dominated, radio sources (∼36% of the sample). Variability is indeed preferentially associated with the latter.
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