  A strong effort has been devoted to understand the physical origin of radio
emission from low-luminosity AGN (LLAGN), but a comprehensive picture is still
missing. We used high-resolution ($\le$1 arcsec), multi-frequency (1.5, 5.5, 9
and 14 GHz) NSF's Karl G. Jansky Very Large Array (VLA) observations to
characterise the state of the nuclear region of ten Seyfert nuclei, which are
the faintest members of a complete, distance-limited sample of 28 sources. With
the sensitivity and resolution guaranteed by the VLA-A configuration, we
measured radio emission for six sources (NGC3185, NGC3941, NGC4477, NGC4639,
NGC4698 and NGC4725), while for the remaining four (NGC0676, NGC1058, NGC2685
and NGC3486) we put upper limits at tens uJy/beam level, below the previous
0.12 mJy/beam level of Ho&Ulvestad (2001), corresponding to luminosities down
to L$\le10^{19}$ W/Hz at 1.5 GHz for the highest RMS observation. Two sources,
NGC4639 and NGC4698, exhibit spectral slopes compatible with inverted spectra
($\alpha\le$0, $S_{\nu}\,\propto\,{\nu}^{-\alpha}$), hint for radio emission
from an optically-thick core, while NGC4477 exhibits a steep (+0.52$\pm$0.09)
slope. The detected sources are mainly compact on scales $\le$ arcseconds,
predominantly unresolved, except NGC3185 and NGC3941, in which the resolved
radio emission could be associated to star-formation processes. A significant
X-ray - radio luminosities correlation is extended down to very low
luminosities, with slope consistent with inefficient accretion, expected at
such low Eddington ratios. Such sources will be one of the dominant Square
Kilometre Array (SKA) population, allowing a deeper understanding of the
physics underlying such faint AGN.
