  We present Very Large Array observations of the 33 GHz radio continuum
emission from 22 local ultraluminous and luminous infrared (IR) galaxies
(U/LIRGs). These observations have spatial (angular) resolutions of 30--720 pc
(0.07"-0.67") in a part of the spectrum that is likely to be optically thin.
This allows us to estimate the size of the energetically dominant regions. We
find half-light radii from 30 pc to 1.7 kpc. The 33 GHz flux density correlates
well with the IR emission, and we take these sizes as indicative of the size of
the region that produces most of the energy. Combining our 33 GHz sizes with
unresolved measurements, we estimate the IR luminosity and star formation rate
per area, and the molecular gas surface and volume densities. These quantities
span a wide range (4 dex) and include some of the highest values measured for
any galaxy (e.g., $\mathrm{\Sigma_{SFR}^{33GHz} \leq 10^{4.1} M_{\odot} yr^{-1}
kpc^{-2}}$). At least $13$ sources appear Compton thick ($\mathrm{N_{H}^{33GHz}
\geq 10^{24} cm^{-2}}$). Consistent with previous work, contrasting these data
with observations of normal disk galaxies suggests a nonlinear and likely
multi-valued relation between SFR and molecular gas surface density, though
this result depends on the adopted CO-to-H$_{2}$ conversion factor and the
assumption that our 33 GHz sizes apply to the gas. 11 sources appear to exceed
the luminosity surface density predicted for starbursts supported by radiation
pressure and supernovae feedback, however we note the need for more detailed
observations of the inner disk structure. U/LIRGs with higher surface
brightness exhibit stronger [{\sc Cii}] 158$\mu$m deficits, consistent with the
suggestion that high energy densities drive this phenomenon.
