Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals

TL;DR

This paper introduces two methods for constructing smooth bump functions on p.c.f. fractals, enabling smooth decompositions and a Borel theorem, thus extending classical analysis tools to fractal spaces.

Contribution

It develops probabilistic and analytic techniques for smooth function manipulation on p.c.f. fractals, including a Borel theorem and partitions of unity.

Findings

Probabilistic method applies to a broad class of fractals.

Established a Borel theorem for p.c.f. fractals.

Smooth functions can be partitioned without loss of smoothness.

Abstract

We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat operator, and the other using the analytic theory for p.c.f. fractals and a fixed point argument. The heat semigroup (probabilistic) method is applicable to a more general class of metric measure spaces with Laplacian, including certain infinitely ramified fractals, however the cut off technique involves some loss in smoothness. From the analytic approach we establish a Borel theorem for p.c.f. fractals, showing that to any prescribed jet at a junction point there is a smooth function with that jet. As a consequence we prove that on p.c.f. fractals smooth functions may be cut off with no loss of smoothness, and thus can be smoothly decomposed subordinate to an open cover. The latter result provides a replacement for classical partition of unity arguments in the p.c.f. fractal setting.