  We examine the local density of states of an impurity level or a quantum dot
coupled to a fractional quantum Hall edge, or to the end of a single
one-dimensional Luttinger-liquid lead. Effects of an Ohmic dissipative bath are
also taken into account. Using both analytical and numerical techniques we show
that, in general, the density of states exhibits power-law frequency dependence
near the Fermi energy. In a substantial region of the parameter space it simply
reflects the behavior of the tunneling density of states at the end of a
Luttinger-liquid, and is insensitive either to the value of the dot-lead
interaction or to the strength of dissipation; otherwise it depends on these
couplings too. This behavior should be contrasted with the thermodynamic
properties of the level, in particular, its occupancy, which were previously
shown to depend on the various interactions in the system only through the
corresponding Fermi edge singularity exponent, and thus cannot display any
Luttinger-liquid specific power-law. Hence, we can construct different models,
some with and some without interactions in the wire (but with equal Fermi edge
singularity exponents), which would have very different level densities of
states, although they all result in the same level population vs. energy
curves.
