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We have exploited LOFAR deep observations of the Lockman Hole field at 150MHz to investigate the relation between the radio luminosity of star-forminggalaxies (SFGs) and their star formation rates (SFRs), as well as itsdependence on stellar mass and redshift. The adopted source classification,SFRs and stellar masses are consensus estimates based on a combination of fourdifferent SED fitting methods. We note a flattening of radio spectra of asubstantial minority of sources below $\sim 1.4 $ GHz. Such sources have thus alower "radio-loudness" level at 150 MHz than expected from extrapolations from1.4 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 arguethat such a trend may account for most of the apparent redshift evolution ofthe $L_{150 \rm MHz}/\hbox{SFR}$ ratio, in line with previous work. Our dataindicate a weaker evolution than found by some previous analyses. We also finda weaker evolution with redshift of the specific star formation rate than foundby several (but not all) previous studies. Our radio selection provides a viewof the distribution of galaxies in the $\hbox{SFR}$-$M_\star$ planecomplementary to that of optical/near-IR selection. It suggests a higheruniformity of the star formation history of galaxies than implied by someanalyses of optical/near-IR data. We have derived luminosity functions at 150MHz of both SFGs and radio-quiet (RQ) AGN at various redshifts. Our results arein very good agreement with the T-RECS simulations and with literatureestimates. We also present explicit estimates of SFR functions of SFGs and RQAGN at several redshifts derived from our radio survey data.

The LOFAR Two-metre Sky Survey Deep fields: A new analysis of low-frequency radio luminosity as a star-formation tracer in the Lockman Hole region