A Wiener test à la Landis for evolutive Hörmander operators

Abstract

In this paper we prove a Wiener-type characterization of boundary regularity, in the spirit of a classical result by Landis, for a class of evolutive Hörmander operators. We actually show the validity of our criterion for a larger class of degenerate-parabolic operators with a fundamental solution satisfying suitable two-sided Gaussian bounds. Our condition is expressed in terms of a series of balayages or, (as it turns out to be) equivalently, Riesz-potentials.bounded and open, the smooth vector fields {X 1 , . . . , X p } satisfy the Hörmander rank condition in a bounded open set D 0 ⊃⊃ D, a i,j , b j are smooth functions in D 0 ×]T 1 , T 2 [, and the matrix (a i,j (·)) i,j is symmetric and uniformly positive definite. Hörmander-type operators arise in many theoretical and applied settings sharing a sub-Riemannian underlying geometry, for instance in mathematical models for finance, control theory, geometric measure theory, pseudohermitian and CR geometry. Relatively to operators in (1.2), our main result (Theorem 1.3 below) reads as follows:2010 Mathematics Subject Classification. 35K65, 35H10, 31C15, 31E05.

Global Gaussian estimates for the heat kernel of homogeneous sums of squares

“…The aim of this section is to show how our global Gaussian estimates for Γ can be used to study the solvability of the H-Dirichlet problem on an arbitrary bounded domain Ω ⊆ R 1+n . All the results we are going to present basically follow by combining the results of the previous sections with the investigations carried out (in an abstract framework) in [20,23,24,35].…”

“…In Section 7 we shall present an application of our global Gaussian estimates to the study of the H-Dirichlet problem. In fact, by crucially exploiting these estimates, we shall show that it is possible to apply to our operators H the axiomatic approach developed in the series of papers [20,23,24,35]; this will lead to some necessary and sufficient conditions for the regularity of boundary points of any bounded open set Ω. Finally, in the last part of the paper we will prove a scale-invariant parabolic Harnack inequality for non-negative solutions of Hu = 0 (see Theorem 8.1 in Section 8).…”

“…Due to the validity of the segment property for d X , the 'good' behavior of Γ in Theorem 3.5, and the validity of global Gaussian estimates for Γ, we are entitled to apply to our context all the abstract results established in [20,23,24,35]. As a consequence, we obtain several necessary/sufficient conditions for a point ω 0 ∈ ∂Ω to be H-regular (in the sense of Definition 7.2).…”

“…[35, Theorem 1.3] Let Ω ⊆ R 1+n be a bounded open set, and let ω 0 = (t 0 , x 0 ) ∈ ∂Ω. For every fixed λ ∈ (0, 1) and every k ∈ N, we consider the setΩ c k (ω 0 ) := ω = (t, x) ∈ R 1+n \ Ω : 1 λ k log(k) ≤ Γ(t 0 , x 0 ; t, x) ≤ 1 λ (k+1) log(k+1) ∪ {(t 0 , x 0 )},where Γ is the global heat kernel of H. Then ω 0 is H-regular if and only if+∞ k=1 V Ω c k (ω0) (ω 0 ) = +∞, where V Ω c k (ω0)is the H-balayage of u 0 ≡ 1 on Ω c k (ω 0 ), see (7.3).…”