# The Strong Maximum Principle for Schr\"{o}dinger operators on fractals

## TL;DR

This paper establishes a strong maximum principle and a Harnack inequality for Schr"odinger operators on fractals, allowing for weaker regularity conditions and measure-valued potentials, expanding the theoretical understanding of PDEs on fractal spaces.

## Contribution

It introduces a maximum principle for Schr"odinger operators on fractals with measure-valued potentials and weaker regularity assumptions, extending previous results in the field.

## Key findings

- Proved a strong maximum principle for fractal Schr"odinger operators.
- Established a Harnack inequality for solutions on fractals.
- Allowed potentials and Laplacians to be Radon measures, broadening applicability.

## Abstract

We prove a strong maximum principle for Schr\"odinger operators defined on a class of fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.