  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.
