  We consider a model of a two-dimensional interface of the SOS type, with
finite-range, even, strictly convex, twice continuously differentiable
interactions. We prove that, under an arbitrarily weak potential favouring
zero-height, the surface has finite mean square heights. We consider the cases
of both square well and $\delta$ potentials. These results extend previous
results for the case of nearest-neighbours Gaussian interactions in \cite{DMRR}
and \cite{BB}. We also obtain estimates on the tail of the height distribution
implying, for example, existence of exponential moments. In the case of the
$\delta$ potential, we prove a spectral gap estimate for linear functionals. We
finally prove exponential decay of the two-point function (1) for strong
$\delta$-pinning and the above interactions, and (2) for arbitrarily weak
$\delta$-pinning, but with finite-range Gaussian interactions.
