$L^p$-Poincaré inequalities on nested fractals

Abstract

We prove on some nested fractals scale invariant $L^p$-Poincaré 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 Hajłasz-Sobolev spaces on fractals are given. Results are illustrated and further developed in the case of the Vicsek set.

Paper is outdated in terms of scope and methods. Also requires several revisions. See in particular 2207.02949

References in corpus (1)

• Reflected Brownian motion on simple nested fractals