  The dominant form of oxygen in cold molecular clouds is gas-phase carbon
monoxide (CO) and ice-phase water (H$_2$O). Yet, in planet-forming disks around
young stars, gas-phase CO and H$_2$O are less abundant relative to their ISM
values, and no other major oxygen-carrying molecules have been detected. Some
astrochemical models predict that gas-phase molecular oxygen (O$_2$) should be
a major carrier of volatile oxygen in disks. We report a deep search for
emission from the isotopologue $^{16}$O$^{18}$O ($N_J=2_1-0_1$ line at 233.946
GHz) in the nearby protoplanetary disk around TW Hya. We used imaging
techniques and matched filtering to search for weak emission but do not detect
$^{16}$O$^{18}$O. Based on our results, we calculate upper limits on the
gas-phase O$_2$ abundance in TW Hya of $(6.4-70)\times10^{-7}$ relative to H,
which is $2-3$ orders of magnitude below solar oxygen abundance. We conclude
that gas-phase O$_2$ is not a major oxygen-carrier in TW Hya. Two other
potential oxygen-carrying molecules, SO and SO$_2$, were covered in our
observations, which we also do not detect. Additionally, we report a
serendipitous detection of the C$^{15}$N $N_J = 2_{5/2}-1_{3/2}$ hyperfine
transitions, $F = 3 - 2$ and $F = 2 - 1$, at 219.9 GHz, which we found via
matched filtering and confirm through imaging.
