Poincar\'e Inequality and Hajlasz-Sobolev spaces on nested fractals

TL;DR

This paper establishes a Poincaré inequality for functions on nested fractals and explores Hajlasz-Sobolev spaces, linking weak gradients to upper gradients in metric space analysis.

Contribution

It introduces a Poincaré inequality for functions on nested fractals and clarifies the relationship between weak and upper gradients in Hajlasz-Sobolev spaces.

Findings

Proved a Poincaré-type inequality on nested fractals.

Described the relation between weak gradients and upper gradients.

Extended the analysis of Sobolev spaces to fractal structures.

Abstract

Given a nondegenerate harmonic structure, we prove a Poincar\'e-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Hajlasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.