  [Abridged] Physical and evolutionary properties of the sub-mJy radio
population are not entirely known. The radio/optical analysis of the ATESP 5
GHz sample has revealed a significant class of compact flat/inverted
radio-spectrum sources associated to early-type galaxies up to redshift 2. Such
sources are most plausibly triggered by an AGN, but their observational
properties are not entirely consistent with those of standard radio galaxy
populations. In the present work we aim at a better understanding of the radio
spectra of such sources and ultimately of the nature of AGNs at sub-mJy flux
levels. We used the ATCA to get multi-frequency (4.8, 8.6 and 19 GHz)
quasi-simultaneous observations for a representative sub-sample of ATESP radio
sources associated with early-type galaxies (26 objects with S>0.6 mJy). This
can give us insight into the accretion/radiative mechanism that is at work,
since different regimes display different spectral signatures in the radio
domain. From the analysis of the radio spectra, we find that our sources are
most probably jet-dominated systems. ADAF models are ruled out by the high
frequency data, while ADAF+jet scenarios are still consistent with
flat/moderately inverted-spectrum sources, but are not required to explain the
data. We compared our sample with high (>20 GHz) frequency selected surveys,
finding spectral properties very similar to the ones of much brighter (S>500
mJy) radio galaxies extracted from the Massardi et al. (2008) sample. Linear
sizes of ATESP 5 GHz sources associated with early type galaxies are also often
consistent with the ones of brighter B2 and 3C radio galaxies, with possibly a
very compact component that could be associated at least in part to (obscured)
radio-quiet quasar-like objects and/or low power BL Lacs.
