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We exploit the DustPedia sample of galaxies within approximately 40 Mpc, selecting 388 sources, to investigate the correlations between IR luminosity (L_IR), the star formation rate (SFR), and the CO(1-0) luminosity (L_CO) down to much lower luminosities than reached by previous analyses. We find a sub-linear dependence of the SFR on L_IR. Below <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo form="prefix">log</mo><mo>(</mo><msub><mi mathvariant="normal">L</mi><mi>IR</mi></msub><mo>/</mo><msub><mi mathvariant="normal">L</mi><mo>⊙</mo></msub><mo>)</mo><mo>≃</mo><mn>10</mn></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>SFR</mi><mo>≃</mo><mn>1</mn><msub><mi mathvariant="normal">M</mi><mo>⊙</mo></msub><msup><mi>yr</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula>, the SFR/L_IR ratio substantially exceeds the standard ratio for dust-enshrouded star formation, and the difference increases with decreasing L_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 <i>Planck</i> ERCSC galaxy sample. We also find that the relations between the L_CO and L_IR or the SFR are consistent with those obtained at much higher luminosities.