Smooth bumps, a Borel theorem and partitions of unity on p.c.f. fractals.∗

Recent years have seen considerable developments in the theory of analysis on certain fractal sets from both probabilistic and analytic viewpoints [1, 10, 19]. In this theory, either a Dirichlet energy form or a diffusion on the fractal is used to construct a weak Laplacian with respect to an appropriate measure, and thereby to define smooth functions. As a result the Laplacian eigenfunctions are well understood, but we have little knowledge of other basic smooth functions except in the case where the fractal is the Sierpinski Gasket [15, 5, 16]. At the same time the existence of a rich collection of smooth functions is crucial to several aspects of classical analysis, where tools like smooth partitions of unity, test functions and mollifications are frequently used. In this work we give two proofs of the existence of smooth bump functions on fractals, one taking the probabilistic and the other the analytic approach. The probabilistic result (Theorem 2.1) is valid provided the fractal supports a heat operator with sub-Gaussian bounds, as is known to be the case for many interesting examples [1, 2, 3] that include non-post-critically finite (non-p.c.f.) fractals such as certain Sierpinski carpets. By contrast the analytic method (Theorem 3.8) is applicable to self-similar p.c.f. fractals with a regular harmonic structure and Dirichlet energy in the sense of Kigami [10]. For p.c.f. fractals we use our result on the existence of bump functions to prove a Borel-type theorem, showing that there are compactly supported smooth functions with prescribed jet at a junction point (Theorem 4.3). This gives a very general answer to a question raised in [15, 5], and previously solved only for the Sierpinski Gasket [16]. We remark, however, that even in this special case the results of [16] neither contain nor are contained in the theorem proven here, as the functions in [16] do not have compact support, while those here do not deal with the tangential derivatives at a junction point. Finally we apply our Borel theorem to the problem of partitioning smooth functions. Multiplication does not generally preserve smoothness in the fractal setting [4],