Should Type Ia Supernova Distances be Corrected for their Local
  Environments?
  Recent analyses suggest that distance residuals measured from Type Ia
supernovae (SNe Ia) are correlated with local host galaxy properties within a
few kpc of the SN explosion. However, the well-established correlation with
global host galaxy properties is nearly as significant, with a shift of 0.06
mag across a low to high mass boundary (the mass step). Here, with 273 SNe Ia
at $z<0.1$, we investigate whether stellar masses and rest-frame $u-g$ colors
of regions within 1.5 kpc of the SN Ia explosion site are significantly better
correlated with SN distance measurements than global properties or properties
measured at random locations in SN hosts. At $\lesssim2\sigma$ significance,
local properties tend to correlate with distance residuals better than
properties at random locations, though despite using the largest low-$z$ sample
to date we cannot definitively prove that a local correlation is more
significant than a random correlation. Our data hint that SNe observed by
surveys that do not target a pre-selected set of galaxies may have a larger
local mass step than SNe from surveys that do, an increase of $0.071\pm0.036$
mag (2.0$\sigma$). We find a $3\sigma$ local mass step after global mass
correction, evidence that SNe Ia should be corrected for their local mass, but
we note that this effect is insignificant in the targeted low-$z$ sample. Only
the local mass step remains significant at $>2\sigma$ after global mass
correction, and we conservatively estimate a systematic shift in H$_0$
measurements of -0.14 $\textrm{km}\,\textrm{s}^{-1}\textrm{Mpc}^{-1}$ with an
additional uncertainty of 0.14 $\textrm{km}\,\textrm{s}^{-1}\textrm{Mpc}^{-1}$,
$\sim$10\% of the present uncertainty.
