  We study the relationship between the gas column density (derived from GMRT
21 cm data) and the star formation rate surface density (derived from publicly
available GALEX data) for a sample of 23 extremely faint dwarf irregular
galaxies drawn from the Faint Irregular Galaxy GMRT Survey (FIGGS). Our sample
galaxies have a median HI mass of 2.8e07 solar masses and a median blue
magnitude -13.2. We find that gas column density averaged over the star forming
region of the disk lies below most estimates of the "threshold density" for
star formation, and that the average star formation rate surface density for
most of the galaxies is also lower than would be expected from the
"Kennicutt-Schmidt" law (Kennnicutt 1998}. We also use our data to look for
small scale (400 pc and 200 pc) correlations. At 400 pc linear resolution, for
18 of our 23 galaxies, we find that star formation rate surface density can be
parametrized as having a power law dependence on gas column density, which
varies accross the sample and is in general steeper than "Kennicutt-Schmidt"
law. The power law relation holds until one reaches the sensitivity limit of
the GALEX data, i.e. we find no evidence for a "threshold density" below which
star formation is completely cut off. For the 5 galaxies for which a power law
does not provide a good parametrization, there are substantial offsets between
the UV bright regions and the HI high column density maps. At 200 pc
resolution, the offsets between the peaks in the HI and UV images are more
pronounced, and a power law parametrization is possible for only 5 of 10
galaxies.
