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

Abstract

We develop a potential theory approach for some degenerate parabolic operators in non-divergence form and with non-smooth coefficients, which are modeled on smooth Hörmander vector fields. We prove necessary and sufficient Wiener-type tests for the regularity of boundary points. As a consequence we obtain, in particular, a conetype criterion. We also investigate the related boundary value problem and the Hölder regularity at the boundary.

A Wiener test à la Landis for evolutive Hörmander operators

“…We now turn back to the sub-Riemannian setting in order to put our result in perspective with respect to the state of the art already mentioned. In [17,26] we found necessary and sufficient conditions (different from each other) which are uniform in the class of operators (1.2). Such conditions were expressed in terms of a series of capacities of compact sets involving only the underlying metric, whereas in the true characterization (1.3) of the present paper we express the condition with balayages of super-level sets of the fundamental solution Γ(•, •) of each operator H in the class.…”

“…In all these papers, the precise knowledge of the fundamental solution plays a crucial role. A different approach has been carried out in [17,26] for Hörmander operators, but the necessary and the sufficient condition for the regularity are different.…”

On the regularity of vector fields underlying a degenerate-elliptic PDE

“…[6,22,38]) that the existence of vector fields with suitable properties allowing us to write L as a sum of squares (possibly up to first order terms) can be crucial for studying qualitative and quantitative properties for the solutions or subsolutions to Lu = 0: the properties of the metric space related to such vector fields have been widely investigated and successfully exploited. Moreover, motivated by the studies on certain nonlinear degenerate-elliptic equations of sub-Riemannian type and on linear subelliptic equations with nonsmooth coefficients (see [1,11,12,14,42,45] and the monographs [7,8,43], with the references therein), there have been recent investigations concerning the minimal regularity assumptions for having vector fields with some Hörmander-type properties [9,10,24,[31][32][33][34][35][36]40]. For these reasons we think it is worthwhile to focus on conditions under which we can guarantee the existence in Ω of vector fields with the desired regularity just looking at the quadratic form A(x).…”