  We study the heat transport along an edge state of a two-dimensional electron
gas in the quantum Hall regime, in contact to two reservoirs at different
temperatures. We consider two exactly solvable models for the edge state
coupled to the reservoirs. The first one corresponds to filling $\nu=1$ and
tunneling coupling to the reservoirs. The second one corresponds to integer or
fractional filling of the sequence $\nu=1/m$ (with $m$ odd), and capacitive
coupling to the reservoirs. In both cases we solve the problem by means of
non-equilibrium Green function formalism. We show that heat propagates chirally
along the edge in the two setups. We identify two temperature regimes, defined
by $\Delta$, the mean level spacing of the edge. At low temperatures, $T<
\Delta$, finite size effects play an important role in heat transport, for both
types of contacts. The nature of the contacts manifest themselves in different
power laws for the thermal conductance as a function of the temperature. For
capacitive couplings a highly non-universal behavior takes place, through a
prefactor that depends on the length of the edge as well as on the coupling
strengths and the filling fraction. For larger temperatures, $T>\Delta$,
finite-size effects become irrelevant, but the heat transport strongly depends
on the strength of the edge-reservoir interactions, in both cases. The thermal
conductance for tunneling coupling grows linearly with $T$, whereas for the
capacitive case it saturates to a value that depends on the coupling strengths
and the filling factors of the edge and the contacts.
