  We have exploited LOFAR deep observations of the Lockman Hole field at 150
MHz to investigate the relation between the radio luminosity of star-forming
galaxies (SFGs) and their star formation rates (SFRs), as well as its
dependence on stellar mass and redshift. The adopted source classification,
SFRs and stellar masses are consensus estimates based on a combination of four
different SED fitting methods. We note a flattening of radio spectra of a
substantial minority of sources below $\sim 1.4 $ GHz. Such sources have thus a
lower "radio-loudness" level at 150 MHz than expected from extrapolations from
1.4 GHz using the average spectral index. We found a weak trend towards a lower
$\hbox{SFR}/L_{150 \rm MHz}$ ratio for higher stellar mass, $M_\star$. We argue
that such a trend may account for most of the apparent redshift evolution of
the $L_{150 \rm MHz}/\hbox{SFR}$ ratio, in line with previous work. Our data
indicate a weaker evolution than found by some previous analyses. We also find
a weaker evolution with redshift of the specific star formation rate than found
by several (but not all) previous studies. Our radio selection provides a view
of the distribution of galaxies in the $\hbox{SFR}$-$M_\star$ plane
complementary to that of optical/near-IR selection. It suggests a higher
uniformity of the star formation history of galaxies than implied by some
analyses of optical/near-IR data. We have derived luminosity functions at 150
MHz of both SFGs and radio-quiet (RQ) AGN at various redshifts. Our results are
in very good agreement with the T-RECS simulations and with literature
estimates. We also present explicit estimates of SFR functions of SFGs and RQ
AGN at several redshifts derived from our radio survey data.
