Some Properties of the Derivatives on Sierpinski Gasket Type Fractals

TL;DR

This paper investigates derivatives on p.c.f. fractals, establishing properties like weak continuity, decay rates, and boundedness, and provides counterexamples to existing conjectures, advancing understanding of analysis on fractals.

Contribution

It introduces new results on the behavior of derivatives on p.c.f. fractals, including decay rates, boundedness, and counterexamples to conjectures, enriching fractal analysis theory.

Findings

Derivatives exhibit weak continuity on p.c.f. fractals.

Normal derivatives at vertices decay to zero under certain conditions.

Counterexample disproves a conjecture on higher order weak tangents.

Abstract

In this paper, we focus on Strichartz's derivatives, a family of derivatives including the normal derivative, on p.c.f. (post critically finite) fractals, which are defined at vertex points in the graphs that approximate the fractal. We obtain a weak continuity property of the derivatives for functions in the domain of the Laplacian. For a function with zero normal derivative at any fixed vertex, the derivatives, including the normal derivatives of the neighboring vertices will decay to zero. The optimal rates of approximations are described and several non-trivial examples are provided to illustrate that our estimates are sharp. We also study the boundness property of derivatives for functions in the domain of the Laplacian. A necessary condition for a function having a weak tangent of order one at a vertex point is provided. Furthermore, we give a counter-example of a conjecture of Strichartz on the existence of higher order weak tangents.