  We report on observations of redshifted CO(1-0) line emission and
observed-frame $\rm\sim$ 30GHz radio continuum emission from five
ultra-luminous, mid-IR selected hot, Dust-Obscured Galaxies (Hot DOGs) at
$z\rm\gtrsim$ 3 using the Karl G. Jansky Very Large Array. We detect CO(1-0)
line emission in all five Hot DOGs, with one of them at high signal to noise.
We analyse FIR-radio spectral energy distributions, including dust, free-free
and synchrotron emission for the galaxies. We find that most of the 115 GHz
rest-frame continuum is mostly due to synchrotron or free-free emission, with
only a potentially small contribution from thermal emission. We see a deficit
in the rest-frame 115 GHz continuum emission compared to dusty star-forming
galaxies (DSFGs) and sub-millimetre galaxies (SMGs) at high redshift,
suggesting that Hot DOGs do not have similar cold gas reserves compared with
star-forming galaxies. One target, W2305-0039, is detected in the FIRST 1.4 GHz
survey, and is likely to possess compact radio jets. We compare to the
FIR-radio correlation, and find that at least half of the Hot DOGs in our
sample are radio-quiet with respect to normal galaxies. These findings suggest
that Hot DOGs have comparably less cold molecular gas than star-forming
galaxies at lower, $z\rm\sim$ 2 redshifts, and are dominated by powerful, yet
radio-quiet AGN.
