Potential analysis for a class of diffusion equations: A Gaussian bounds approach

Abstract

MSC: 35H10 35K65 31E05 35H20 35A08 Keywords: Gaussian bounds Potential analysis Boundary behavior of PW solutions Non-divergence Hörmander operators Harnack inequalityWe axiomatically develop a potential analysis for a general class of hypoelliptic diffusion equations under the following basic assumptions: doubling condition and segment property for an underlying distance and Gaussian bounds of the fundamental solution. Our analysis is principally aimed to obtain regularity criteria and uniform boundary estimates for the Perron-Wiener solution to the Dirichlet problem. As an example of application, we also derive an exterior cone criterion of boundary regularity and scale-invariant Harnack inequality and Hölder estimate for an important class of operators in non-divergence form with Hölder continuous coefficients, modeled on Hörmander vector fields.

A Wiener test à la Landis for evolutive Hörmander operators

“…In the following lemma we finally determine the required bound for the ratio Γ(z,ζ) Γ(z0,ζ) for z ∈ F i h and ζ ∈ F i k . We do this by exploiting the Hölder continuity of the solutions to Hu = 0 proved in [18]. It is not surprising to infer estimates for the fundamental solution or for the relevant Green kernel by using Hölder-type estimates (see the related results in [18,Proposition 7.4] and [17,Lemma 3.3], see also [27,25]).…”

“…We do this by exploiting the Hölder continuity of the solutions to Hu = 0 proved in [18]. It is not surprising to infer estimates for the fundamental solution or for the relevant Green kernel by using Hölder-type estimates (see the related results in [18,Proposition 7.4] and [17,Lemma 3.3], see also [27,25]). The novelty in the present situation is due to the special regions F i k , and it is strictly related with the careful choices for q and p in (3.5) and (3.8).…”

“…Here and in what follows we agree to let lim inf ζ→z w(ζ) = sup V ∈Uz (inf V w) being U z a basis of neighborhoods of z. We know from [18,Proposition 8.3] that…”

“…It turns out that in our approach we can use essentially only two-sided Gaussian estimates for Γ with respect to a well-behaved distance. For this reason we decided to present the results for a more general class of diffusion operators by using an axiomatic approach in the spirit of [18,17]. In the following subsection, we proceed by fixing precisely the class of operators under consideration.…”

Wiener criterion for X-elliptic operators

“…A nonhomogeneous Harnack inequality is also proved in [41]. We also refer to [5,6,11,24,32,43] for other related papers.…”