by Luke G. Rogers, Robert S. Strichartz and Alexander Teplyaev
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.