A direct proof of the cutoff Sobolev inequality on the Sierpiński gasket

TL;DR

The paper addresses the problem of providing a direct proof of the cutoff Sobolev inequality on the Sierpiński gasket, a key ingredient for sub-Gaussian heat-kernel estimates on fractals. It adopts a constructive, cell-based approach exploiting the gasket's strongly recurrent structure, energy measures, and a Morrey-Sobolev framework to derive explicit cutoff functions with energy control, then uses localization and the Markov property to establish CSS($d_w$) with $d_w=\\frac{\\log 5}{\\log 2}$ and $\\Psi(r)=r^{d_w}$. The main contribution is the first direct verification of CSS on a fractal, providing explicit energy bounds and a transparent methodology. This result strengthens the link between geometric/analytic conditions (VD, PI) and function-space inequalities on fractals and offers a blueprint for extending to other fractals and to $p$-energies under compatible conditions.

Abstract

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

Figures (2)

• Figure 1: The Sierpiński gasket
• Figure 2: $\mathcal{N}(K)$

Theorems & Definitions (7)

• Lemma 1.1: Mur24a
• Theorem 1.2
• Lemma 2.1: Morrey-Sobolev inequality, FH99
• Lemma 2.2
• proof
• Proposition 2.3
• proof