  Phonon modes and their dispersion relations in ultrathin homogenous
dielectric membranes are calculated using elasticity theory. The approach
differs from the previous ones by a rigorous account of the effect of the film
surfaces on the modes with different polarizations. We compute the heat
capacity of membranes and the heat conductivity of narrow bridges cut out of
such membranes, in a temperature range where the dimensions have a strong
influence on the results. In the high temperature regime we recover the
three-dimensional bulk results. However, in the low temperature limit the heat
capacity, $C_V$, is proportional with $T$ (temperature), while the heat
conductivity, $\kappa$, of narrow bridges is proportional to $T^{3/2}$, leading
to a thermal cut-off frequency $f_c=\kappa/C_V\propto T^{1/2}$.
