We investigate the relation between gas and star formation in sub-galactic regions, ~360 pc to ~1.5 kpc in size, within the nearby starburst dwarf NGC4449, in order to separate the underlying relation from the effects of sampling at varying spatial scales. Dust and gas mass surface densities are derived by combining new observations at 1.1 mm, obtained with the AzTEC instrument on the Large Millimeter Telescope, with archival infrared images in the range 8-500 micron from the Spitzer Space Telescope and the Herschel Space Observatory. We extend the dynamic range of our mm (and dust) maps at the faint end, using a correlation between the far-infrared/millimeter colors F(70)/F(1100) [and F(160)/F(1100)] and the mid-infrared color F(8)/F(24) that we establish for the first time for this and other galaxies. Supplementing our data with maps of the extinction-corrected star formation rate (SFR) surface density, we measure both the SFR-molecular gas and the SFR-total gas relations in NGC4449. We find that the SFR-molecular gas relation is described by a power law with exponent that decreases from ~1.5 to ~1.2 for increasing region size, while the exponent of the SFR-total gas relation remains constant with value ~1.5 independent of region size. We attribute the molecular law behavior to the increasingly better sampling of the molecular cloud mass function at larger region sizes; conversely, the total gas law behavior likely results from the balance between the atomic and molecular gas phases achieved in regions of active star formation. Our results indicate a non-linear relation between SFR and gas surface density in NGC4449, similar to what is observed for galaxy samples.
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