A note on the weak regularity theory for degenerate Kolmogorov equations

On the fundamental solution for degenerate Kolmogorov equations with rough coefficients

“…Now, given the notation of ( 27), we introduce the upper and lower cylinders and we introduce some preliminary results. Quite recently, there have been various developments in the study of weak solutions to Lu = 0 , and in particular the following Harnack inequality proved in [4,Theorem 1.3], the first one in the weak framework considered in this work, holds true in our setting.…”

“…. In a standard manner, see for instance [4,5,28], we let W(Ω) denote the closure of C ∞ c (Ω) in the norm where the previous norm can explicitly be computed as follows:…”

“…where x = (x (1) , … , x ( ) ) and x = (x (0) , x) ∈ ℝ N . In particular, W(Ω) is a Banach space, and it was firstly introduced in [4] as an extension of the natural functional setting that arises in the study of the weak regularity theory for the kinetic Kolmogorov-Fokker-Planck equation [5,[17][18][19]. For further properties of the space W , we refer the reader to [28], where the authors provide a characterization of this space in the kinetic Kolmogorov-Fokker-Planck setting, i.e.…”

“…The authors of [25] already suggested this type of result could be extended to the general non-homogeneous Kolmogorov operator of step in (1), once a suitable Harnack inequality is established. The present work is a first step in this direction as it handles the homogeneous case, the only one for which a Harnack inequality is available, see [4,Theorem 1.3].…”

“…The argument relies on the combination of a Caccioppoli type inequality and a Sobolev type inequality. In order to handle the more general case, we need to replace the Caccioppoli and the Sobolev inequality contained in [24] (Theorem 2.3 and 2.5, respectively) with the ones given in [4]. More precisely, we consider the Caccioppoli inequality [4, Theorem 3.4] and we focus on the new term involving the coefficient a ∈ L q loc (Ω) m 0 , which is handled as follows where = q q−1 , and q is the integrability exponent introduced in (H3B).…”