  We exploit the DustPedia sample of galaxies within approximately 40 Mpc,
selecting 388 sources, to investigate the correlations between IR luminosity
(L$_{\rm IR}$), the star formation rate (SFR), and the CO(1-0) luminosity
(L$_{\rm CO}$) down to much lower luminosities than reached by previous
analyses. We find a sub-linear dependence of the SFR on L$_{\rm IR}$. Below
$\log(\hbox{L}_{\rm IR}/\hbox{L}_\odot)\simeq 10$ or $\hbox{SFR}\simeq
1\,\hbox{M}_\odot\,\hbox{yr}^{-1}$, the SFR/L$_{\rm IR}$ ratio substantially
exceeds the standard ratio for dust-enshrouded star formation, and the
difference increases with decreasing L$_{\rm IR}$ values. This implies that the
effect of unobscured star formation overcomes that of dust heating by old
stars, at variance with results based on the $\textit{Planck}$ ERCSC galaxy
sample. We also find that the relations between the L$_{\rm CO}$ and L$_{\rm
IR}$ or the SFR are consistent with those obtained at much higher luminosities.
