Does the evolution of the radio luminosity function of star-forming
  galaxies match that of the star-formation rate function?
  The assessment of the relationship between radio continuum luminosity and
star formation rate (SFR) is of crucial importance to make reliable predictions
for the forthcoming ultra-deep radio surveys and to allow a full exploitation
of their results to measure the cosmic star formation history. We have
addressed this issue by matching recent accurate determinations of the SFR
function up to high redshifts with literature estimates of the 1.4 GHz
luminosity functions of star forming galaxies (SFGs). This was done considering
two options, proposed in the literature, for the relationship between the
synchrotron emission ($L_{\rm synch}$), that dominates at 1.4 GHz, and the SFR:
a linear relation with a decline of the $L_{\rm synch}$/SFR ratio at low
luminosities or a mildly non-linear relation at all luminosities. In both cases
we get good agreement with the observed radio luminosity functions but, in the
non-linear case, the deviation from linearity must be small. The luminosity
function data are consistent with a moderate increase of the $L_{\rm
synch}$/SFR ratio with increasing redshift, indicated by other data sets,
although a constant ratio cannot be ruled out. A stronger indication of such
increase is provided by recent deep 1.4 GHz counts, down to $\mu$Jy levels.
This is in contradiction with models predicting a decrease of that ratio due to
inverse Compton cooling of relativistic electrons at high redshifts.
Synchrotron losses appear to dominate up to $z\simeq 5$. We have also updated
the Massardi et al. (2010) evolutionary model for radio loud AGNs.
