Harnack’s inequality for a class of non-divergent equations in the Heisenberg group

Abstract

Abstract. We prove an invariant Harnack's inequality for operators in nondivergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach from [DGL08] to obtain Harnack's inequality.

Harnack inequality for a class of Kolmogorov–Fokker–Planck equations in non-divergence form

“…In this work, we prove Harnack's inequality for non-negative solutions to Ku = 0 under either a Cordes-Landis condition or a continuity assumption on the coefficient matrix A (see subsection 1.1, hypotheses H1 and H2). Similar results have been obtained for other Hörmander type operators, namely for non-divergence form operators structured on Heisenberg vector fields [1,13,32]. The techniques we employ in the present work are inspired by the insightful contributions of Landis from the '60s [24], where he obtained what is nowadays referred to as the growth lemma for nonnegative subsolutions of uniformly elliptic equations, assuming that the eigenvalue ratio is close to 1.…”

“…The proof of Lemma 4.2 can then be carried out in exactly the same way. This is in contrast with operators in groups of Heisenberg type considered in [1] and [32], where it is not clear how to establish the analogue of Lemma 4.2 under the more general condition (4.13) without making additional structural assumptions on the coefficient matrix. A similar obstruction arises when attempting to prove the analogue of Lemma 4.4 (see [1,Section 3]).…”

“…Let δ > 0 be the number from Lemma 6.2 corresponding to the choice of M = 2 1+(n+ 1 2 )(Q+2) , and define the structural constant…”

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).…”

“…We would like at first to give the details of the statement we already mentioned by Bony in [6, p. 279], since it was the initial motivation for the present study. Suppose we have a second order operator L as in (1), where the nonnegative matrix A(x) and the first order terms (b h (x)) N h=1 are C ∞ -smooth functions. Then our vector fields X 1 , .…”