A direct proof of the cutoff Sobolev inequality on the Sierpi\'nski gasket

TL;DR

This paper provides a direct proof of the cutoff Sobolev inequality on the Sierpiński gasket, addressing a complex problem crucial for understanding heat kernel estimates on fractal structures.

Contribution

The paper introduces a novel direct proof technique for the cutoff Sobolev inequality on the Sierpiński gasket, simplifying previous approaches and advancing fractal analysis.

Findings

Established the cutoff Sobolev inequality directly on the Sierpiński gasket.

Enhanced understanding of heat kernel estimates on fractals.

Simplified proof method for complex inequalities on fractal spaces.

Abstract

We present a direct proof of the cutoff Sobolev inequality on the Sierpi\'nski gasket, which has long been regarded as highly non-trivial in the context of heat kernel estimates.