Wiener criterion for X-elliptic operators

Abstract

In this note we prove a Wiener criterion of regularity of boundary points for the Dirichlet problem related to X-elliptic operators in divergence form enjoying the doubling condition and the Poincaré inequality. As a step towards this result, we exhibit some other characterizations of regularity in terms of the capacitary potentials. Finally, we also show that a cone-type criterion holds true in our setting.

The Role of Fundamental Solution in Potential and Regularity Theory for Subelliptic PDE

“…Related results on the regularity of boundary points for the Dirichlet problem are also proved in [76], [105], [106], [104].…”

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).…”

On a non-smooth potential analysis for Hörmander-type operators

“…The last proposition, together with (4.9), says in particular that the d-cone condition ensures the H-regularity of z 0 (see also the recent results in [21,22] for cone-type criteria for some classes of hypoelliptic operators). We recall that, if Ω is a cylinder of the type A×]t 1 , t 2 [, the d-cone condition is implied by the density condition (2.6) for the set A ⊂ R N (see the results in [43,41] for the boundary regularity of some stationary operators under condition (2.6)). To complete the proof of Corollary 1.2, we want also to deduce the hölder regularity of H Ω ϕ at z 0 by assuming more control on the oscillation ω ϕ (z 0 , r).…”