$L^p$-Poincar\'e inequalities on nested fractals

TL;DR

This paper establishes scale-invariant $L^p$-Poincaré inequalities on nested fractals for $1 \\le p \\le 2$, using heat kernel methods to develop local $L^p$-theory of Sobolev spaces on fractals.

Contribution

It introduces a new approach to $L^p$-Poincaré inequalities on fractals via heat kernel techniques and extends the theory to Sobolev spaces and maximal functions.

Findings

Proves $L^p$-Poincaré inequalities on nested fractals for $1 \\le p \\le 2$.

Develops local $L^p$-theory of Korevaar-Schoen-Sobolev spaces on fractals.

Illustrates results with the Vicsek set.

Abstract

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.