A boundary estimate for non-negative solutions to Kolmogorov operators in non-divergence form

A Carleson-type estimate in Lipschitz type domains for non-negative solutions to Kolmogorov operators

“…Our next result improves on the analogous one in [6,Proposition 3.2]. Indeed, as noticed in [6,Remark 4.2], it fails under the weaker Harnack connectivity condition assumed in [6].…”

“…We point out that the strong Harnack connectivity condition is more restrictive than the Harnack connectivity condition used in our previous work [6]. Nevertheless, we are able to prove the validity of this condition near Lip K surfaces.…”

“…In [6,Theorem 1.2] we proved a Carleson type estimate for non-negative solutions u to L u = 0 assuming that u vanishes continuously on some open subset 6 of @ƒ. Furthermore, in [6,Proposition 1.4] we gave an application of [6,Theorem 1.2], assuming that 6 is a N -dimensional C 1 -manifold satisfying either condition (a) or condition (b) in (1.11). The purpose of this paper is to improve on these results by relaxing the regularity assumptions on 6.…”

“…We next follow a classical argument also used in our previous work [6]. We fix a suitably large constant ∏ and we assume, by contradiction, that there exists z 1 ∈ ƒ f,c satisfying u(z 1 ) > ∏.…”

“…In [21] and [32] Frentz, Nyström, Pascucci and Polidoro proved optimal regularity properties of the solution to the obstacle problem, for smooth as well as non-smooth obstacles. Up to date, the only known results about boundary regularity for non-negative solutions to Kolmogorov operators have been proved by the authors in [6], see also [7].…”