Compactness methods for $Γ^{1,α}$ boundary Schauder estimates in Carnot groups

Abstract

Introduction 1 2. Preliminaries 3 2.1. Horizontal Laplaceans 5 2.2. Intrinsic distance and gauge pseudo-distance 6 2.3. Homogeneous polynomials 7 2.4. The Folland-Stein Hölder classes 8 2.5. The characteristic set 9 3. Some basic regularity estimates 9 4. Proof of Theorem 1.1 14 References 25

Borderline gradient estimates at the boundary in Carnot groups

“…In the last part of this section, some known regularity results also has been presented which will be needed in the proof of Theorem 1.3. Most of the definitions related to the Carnot group have been taken from [4]. Therefore, quite often we will be referring to [4] for the details.…”

“…Very recently in [3], by suitably adapting the Levi's method of parametrix, Baldi, G. Citti and G. Cupini established Γ 2,α type Schauder estimate for non-divergence form operators upto the non-characteristic portion of a C ∞ boundary in more general Carnot groups, see Theorem 1.1 in [3]. Subsequently in [4], by employing an alternate approach based on geometric compactness arguments, the authors showed the validity of Γ 1,α boundary Schauder estimate for divergence form operators as in (1.1) above when boundary is C 1,α regular and when a ij , f i ∈ Γ 0,α , h ∈ Γ 1,α , g ∈ L ∞ see Theorem 1.1 in [4]. We note that such compactness arguments has its roots in the fundamental works of Caffarelli as in [9] and is independent of the method of parametrix.…”

“…We note that such compactness arguments has its roots in the fundamental works of Caffarelli as in [9] and is independent of the method of parametrix. In this article, we consider a similar framework as in [4] and prove the horizontal continuity of the gradient under weaker assumptions on the coefficients and the domain and when the scalar term g belongs to the scaling critical Lorentz space L(Q, 1) with Q being the homogeneous dimension of the Carnot group G. For the precise notion of the function space L(Q, 1), we refer the reader to Definition 4.1.…”

Borderline regularity for fully nonlinear equations in Dini domains

“…We now use an idea similar to that in the proof of Lemma 4.1 in [1]. After flattening the boundary as in the proof of Lemma 2.10, we extend u k to B 1 using (2.2) with k 0 = 1 and we still denote the extended function by u k .…”

“…We also note that unlike what is conventionally done in the divergence form theory, the boundary cannot be flattened in our situation to begin with because of the lower regularity assumption on Ω and the fact that our equation has non-divergence structure. In that sense, our techniques are also partially inspired by that in the recent paper [1], which is on boundary Schauder estimates on Carnot groups where the boundary cannot be flattened either.…”