  We investigate the upper chromosphere and the transition region of the
sunspot umbra using the radio brightness temperature at 34 GHz (corresponding
to 8.8-mm observations) as observed by the Nobeyama Radioheliograph (NoRH).
Radio free-free emission in the longer millimeter range is generated around the
transition region, and its brightness temperature yields the region's
temperature and density distribution. We use the NoRH data at 34 GHz by
applying the Steer-CLEAN image synthesis. These data and the analysis method
enable us to investigate the chromospheric structures in the longer millimeter
range with high spatial resolution and sufficient visibilities. We also perform
simultaneous observations of one sunspot using the NoRH and the Nobeyama 45-m
telescope operating at 115 GHz. We determine that 115-GHz emission mainly
originates from the lower chromosphere while 34-GHz emission mainly originates
from the upper chromosphere and transition region. These observational results
are consistent with the radio emission characteristics estimated from the
current atmospheric models of the chromosphere. On the other hand, the observed
brightness temperature of the umbral region is almost the same as that of the
quiet region. This result is inconsistent with the current sunspot models,
which predict a considerably higher brightness temperature of the sunspot umbra
at 34 GHz. This inconsistency suggests that the temperature of the region at
which the 34 GHz radio emission becomes optically thick should be lower than
that predicted by the models.
