The Boundary Harnack Principle and the 3G Principle in Fractal-Type Spaces

TL;DR

This paper extends the 3G Principle for Green's functions to fractal and fractal-like spaces, including higher-dimensional Sierpinski carpets, with implications for Schrödinger operators.

Contribution

It generalizes the 3G Principle to a broad class of fractal-type spaces lacking traditional dimensional measures.

Findings

Validates the 3G Principle in higher-dimensional fractals

Applies results to Schrödinger operators in fractal spaces

Extends Green's function estimates to non-standard fractal geometries

Abstract

We prove a generalized version of the $3G$ Principle for Green's functions on bounded inner uniform domains in a wide class of Dirichlet spaces. In particular, our results apply to higher-dimensional fractals such as Sierpinski carpets in $\mathbf{R}^n$, $n\geq 3$, as well as generalized fractal-type spaces that do not have a well-defined Hausdorff dimension or walk dimension. This yields new instances of the $3G$ Principle for these spaces. We also discuss applications to Schr\"odinger operators.