  We present SMA and NOEMA observations of the host galaxy of FRB 121102 in the
CO 3-2 and 1-0 transitions, respectively. We do not detect emission from either
transition. We set $3\sigma$ upper limits to the CO luminosity $L_{CO} < 2.5
\times 10^7\,{\rm K\,km\,s}^{-1} {\, \rm pc^{-2}}$ for CO 3-2 and $L_{CO} < 2.3
\times 10^9\, {\rm K\,km\,s}^{-1} {\, \rm pc^{-2}}$ for CO 1-0. For
Milky-Way-like star formation properties, we set a $3\sigma$ upper limit on the
$H_2$ mass of $2.5 \times 10^8 \rm\ M_{\odot}$, slightly less than the
predictions for the $H_2$ mass based on the star formation rate. The true
constraint on the $H_2$ mass may be significantly higher, however, because of
the reduction in CO luminosity that is common forlow-metallicity dwarf galaxies
like the FRB host galaxy. These results demonstrate the challenge of
identifying the nature of FRB progenitors through study of the host galaxy
molecular gas. We also place a limit of 42 $\mu$Jy ($3\sigma$) on the continuum
flux density of the persistent radio source at 97 GHz, consistent with a
power-law extrapolation of the low frequency spectrum, which may arise from an
AGN or other nonthermal source.

High Energy Astrophysical Phenomena;Astrophysics of Galaxies