  With a view to prove an Ohsawa-Takegoshi type $L^2$ extension theorem with
$L^2$ estimates given with respect to the log-canonical (lc) measures, a
sequence of measures each supported on lc centres of specific codimension
defined via multiplier ideal sheaves, this article is aiming at providing
evidence and possible means to prove the $L^2$ estimates on compact K\"ahler
manifolds $X$. A holomorphic family of $L^2$ norms on the ambient space $X$ is
introduced which is shown to "deform holomorphically" to an $L^2$ norm with
respect to an lc-measure. Moreover, the latter norm is shown to be invariant
under a certain normalisation which leads to a "non-universal" $L^2$ estimate
on compact $X$. Explicit examples on $\mathbb{P}^3$ with detailed computation
are presented to verify the expected $L^2$ estimates for extensions from lc
centres of various codimensions and to provide hint for the proof of the
estimates in general.
