Gradient estimates for heat kernels and harmonic functions

Riesz transform under perturbations via heat kernel regularity

“…by [6,18], one further sees that the gradient estimates for heat kernels and harmonic functions are also stable under such metric perturbations. Coulhon-Dungey [16] has addressed the stability issue of Riesz transform under perturbations.…”

“…Zhang [38] had derived a sufficient condition on the perturbation of a manifold with nonnegative Ricci curvature for the stability of Yau's estimate (equivalent to (RH p ) with p = ∞, cf. [15,18,37]), which implies the boundedness of the Riesz transform for all p ∈ (1, ∞) by [18, Theorem 1.9]. We did not prove the stability of Yau's estimate, but the advantage of our result is that our condition (GD) is much more explicit and, it is convenient for applications.…”

“…The case p > 2 is more involved. It was shown in [18] that on a Riemannian manifold (M, g), under (D) together with a scale invariant L 2 -Poincaré inequality, ∇L −1/2 is bounded on L p (M, µ), p ∈ (2, ∞), if and only if, it holds for every ball B(x, r) and every solution to Lu = 0 on B(x, r) that…”

“…The estimates for the heat kernel and its gradient (cf. [6,18]), together with (GD) are essential in our proofs.…”

Heat kernel analysis on diamond fractals

“…[10,49] and references therein. One of the main reasons to investigate this type of questions in the particular setting of generalized diamond fractals is that these spaces, which arise as inverse limits of metric measure graphs, see Figure 1, lack regularity properties such as volume doubling or uniformly bounded degree, that are often assumed in the literature [13,14,23,26]. In this regard, the investigations carried out here provide the starting point of a larger research program, where diamond fractals may be considered as model spaces towards a classification of inverse limit spaces by means of the heat semigroup.…”