Besov-Lipschitz norm and $p$-energy measure on scale-irregular Vicsek sets

TL;DR

This paper develops $p$-energy norms and measures for scale-irregular Vicsek sets, linking them to Besov-Lipschitz norms and exploring their properties and convergence behaviors.

Contribution

It introduces the existence and characterization of $p$-energy norms on scale-irregular Vicsek sets, extending analysis beyond self-similar fractals.

Findings

Established existence of $p$-energy norms and measures.

Characterized $p$-energy norms via Besov-Lipschitz norms.

Analyzed weak monotonicity and convergence properties.

Abstract

In this paper, we establish the existence of $p$-energy norms and the corresponding $p$-energy measures for scale-irregular Vicsek sets, which may lack self-similarity. We also investigate the characterizations of $p$-energy norms in terms of Besov-Lipschitz norms, with their weak monotonicity and the corresponding Bourgain-Brezis-Mironescu convergence.