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.
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