The boundary Harnack principle and the 3G principle in fractal‐type spaces

Anthony Graves‐McCleary and Laurent Saloff‐Coste

Mathematische Nachrichten, 2025, vol. 298, issue 11, 3554-3575

Abstract: We prove a generalized version of the 3G$3G$ 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 Rn$\mathbf {R}^n$, n≥3$n\ge 3$, 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$3G$ principle for these spaces. We also discuss applications to Schrödinger operators.

Date: 2025
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