Sobolev spaces and Poincar\'e inequalities on the Vicsek fractal

TL;DR

This paper demonstrates that various natural definitions of Sobolev spaces on the Vicsek fractal are equivalent and establishes Poincaré inequalities, advancing the understanding of analysis on fractal structures.

Contribution

It proves the equivalence of different Sobolev space constructions on the Vicsek fractal and shows they form a real interpolation scale, including Poincaré inequalities.

Findings

Different Sobolev space approaches coincide on the Vicsek fractal.

Sobolev spaces form a real interpolation scale.

Established $L^p$-Poincaré inequalities for all $p \,\ge 1$.

Abstract

In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of discrete $p$-energies and the approach by limit of Sobolev spaces on cable systems all yield the same functional space with equivalent norms for $p>1$. As a consequence we prove that the Sobolev spaces form a real interpolation scale. We also obtain $L^p$-Poincar\'e inequalities for all values of $p \ge 1$.