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

Abstract

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 cutoff 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.

Boundary Harnack inequality for Markov processes with jumps

“…Assumption D and scaling property (e) follow from the transition density estimate (5.1) by Proposition 5.3; see also [27,Lemmas 5.3 and 5.6]. Finally, Assumption B with scaling property (f) follows by the construction of [71,Section 2]. Roughly speaking, the method of [71] yields smooth bump functions in the domain of the generator of the diffusion Z t with appropriate scaling.…”

“…In this part we repeat the construction of smooth bump functions of [71]. We adopt the setting of Example 5.7: Z t is a diffusion process on an Ahlfors-regular n-space X, the transition semigroup T Z t of Z t satisfies sub-Gaussian bounds (5.5), and X t is defined to be the process Z t subordinated by an independent α/d w -stable subordinator, α ∈ (0, d w ).…”

“…Let h = T Z t g for some t > 0 and g ∈ L 2 (X). One of the main results of [71], Theorem 2.2, states that given any compact K and ε > 0, there is a function f such that f ∈ D(∆ l ) for all l > 0, f (x) = h(x) on K and f (x) = 0 when dist(x, K) ≥ ε. There are at least three issues when one tries to apply this result in our setting.…”

“…Second, to get Assumption B, we need to apply the above theorem with h(x) = 1 for x ∈ K, where h = T Z t g. This condition is satisfied when g(x) = 1 for all x ∈ X. However, such a function g is in L 2 (X) only when m is a finite measure, and the general case is not covered by [71]. For that reason, we choose to repeat the construction of [71] in L ∞ (X) (instead of L 2 (X)) setting.…”

Pseudodifferential Operators Associated with a Semigroup of Operators

“…Then we may localize, so if x, y are in a ball B of radius r, then we choose a bump function φ (which equals 1 on B and 0 in (2B) c ). We refer the reader to [32] for an abstract construction and existence of such functions. Then we apply (2.1) to f = f φ and we can prove local estimates with a an extra term in d(x, y) 1 2 .…”