  We extend and survey results in the theory of analysis on fractal sets from
the standard Laplacian on the Sierpi\'nski gasket to the energy Laplacian,
which is defined weakly by using the Kusuoka energy measure. We also extend
results from the Sierpi\'nski gasket to level-$k$ Sierpi\'nski gaskets, for all
$k\geq 2$. We observe that the pointwise formula for the energy Laplacian is
valid for all level-$k$ Sierpi\'nski gaskets, $SG_k$, and we provide a proof of
a known formula for the renormalization constants of the Dirichlet form for
post-critically finite self-similar sets along with a probabilistic
interpretation of the Laplacian pointwise formula. We also provide a vector
self-similar formula and a variable weight self-similar formula for the Kusuoka
measure on $SG_k$, as well as a formula for the scaling of the energy
Laplacian.
