Wiener-type tests from a two-sided Gaussian bound

Abstract

In this paper, we are concerned with hypoelliptic diffusion operators H. Our main aim is to show, with an axiomatic approach, that a Wiener-type test of H-regularity of boundary points can be derived starting from the following basic assumptions: Gaussian bounds of the fundamental solution of H with respect to a distance satisfying doubling condition and segment property. As a main step toward this result, we establish some estimates at the boundary of the continuity modulus for the generalized Perron–Wiener solution to the relevant Dirichlet problem. The estimates involve Wiener-type series, with the capacities modeled on the Gaussian bounds. We finally prove boundary Hölder estimates of the solution under a suitable exterior cone condition

A Wiener test à la Landis for evolutive Hörmander operators

“…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).…”

“…Let us mention that other definitions of capacities related to H are possible and they are discussed, e.g., in [17,Section 2]. For example, one can deal with capacities with respect to Gaussian kernels G a (·, ·) which, in turn, can be estimated in terms of the Lebesgue measure (see [17,Proposition 2.5]).…”

“…Let us mention that other definitions of capacities related to H are possible and they are discussed, e.g., in [17,Section 2]. For example, one can deal with capacities with respect to Gaussian kernels G a (·, ·) which, in turn, can be estimated in terms of the Lebesgue measure (see [17,Proposition 2.5]). We can then obtain the following sufficient condition for the regularity which is more geometric and easier to be tested with respect to condition (i) in Corollary 4.1 (see also [13,Corollary 1.3]).…”

“…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.…”

On the Dirichlet problem in cylindrical domains for evolution Oleı̌nik–Radkevič PDE's: A Tikhonov-type theorem

“…We note that condition (5.9) holds if Ω satisfies the exterior cone-type condition introduced in [Kog19]. Geometric boundary regularity criteria for wide classes of hypoelliptic evolution operators are also established in [Man97], [LU10], [LTU17] and [Kog17].…”