Correlations between the radio continuum, infrared and CO emission are known to exist for several types of galaxies and across several orders of magnitude. However, the low-mass, low-luminosity and low-metallicity regime of these correlations is not well known. A sample of metal-rich and metal-poor dwarf galaxies from the literature has been assembled to explore this extreme regime. The results demonstrate that the properties of dwarf galaxies are not simple extensions of those of more massive galaxies; the different correlations reflect different star-forming conditions and different coupling between the star formation and the various quantities. It is found that dwarfs show increasingly weaker CO and infrared emission for their luminosity, as expected for galaxies with a low dust content, slower reaction rates, and a hard ionizing radiation field. In the higher-luminosity dwarf regime (L_1.4GHz > 10^27 W, where L_1.4GHz ~ 10^29 W for a Milky Way star formation rate of ~1 M_sun yr^-1), the total and non-thermal radio continuum emission appear to adequately trace the star formation rate. A breakdown of the dependence of the (Halpha-based) thermal, non-thermal, and, hence, total radio continuum emission on star formation rate occurs below L_1.4GHz ~ 10^27 W, resulting in a steepening or downturn of the relations at extreme low luminosity. Below L_FIR ~ 10^36 W ~ 3 x 10^9 L_sun, the infrared emission ceases to adequately trace the star formation rate. A lack of a correlation between the magnetic field strength and the star formation rate in low star formation rate dwarfs suggests a breakdown of the equipartition assumption. As extremely metal-poor dwarfs mostly populate the low star formation rate and low luminosity regime, they stand out in their infrared, radio continuum and CO properties.
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