  We revisit the study of $\omega$-hypercontractions corresponding to a single
weight sequence $\omega=\{\omega_k\}_{k\geq0}$ introduced by Olofsson in
\cite{O} and find an analogue of Nagy-Foias characteristic function in this
setting. Explicit construction of characteristic functions is obtained and it
is shown to be a complete unitary invariant. By considering a multi-weight
sequence $\clw$ and $\clw$-hypercontractions we extend Olofsson's work \cite{O}
in the multi-variable setting. Model for $\clw$-hypercontractions is obtained
by finding their dilations on certain weighted Bergman spaces over the polydisc
corresponding to the multi-weight sequence $\clw$. This recovers and provides a
different proof of the earlier work of Curto and Vasilescu \cite{CVPoly, CV}
for $\gamma$-contractive multi-operators through a particular choice of
multi-weight sequence.
