  We prove on some nested fractals scale invariant $L^p$-Poincar\'e
inequalities on metric balls in the range $1 \le p \le 2$. Our proof is based
on the development of the local $L^p$-theory of Korevaar-Schoen-Sobolev spaces
on fractals using heat kernel methods. Applications to scale invariant Sobolev
inequalities and to the study of maximal functions and Haj\l{}asz-Sobolev
spaces on fractals are given. Results are illustrated and further developed in
the case of the Vicsek set.
