$C^{1,\alpha}$-Regularity of Quasilinear equations on the Heisenberg Group

In this article, we reproduce results of classical regularity theory of quasilinear elliptic equations in the divergence form, in the setting of Heisenberg Group. The conditions encompass a very wide class of equations with isotropic growth conditions, which are a generalization of the p-Laplace type equations in this respect; these also include all equations with polynomial or exponential type growth. In addition, some even more general conditions have also been explored.


Introduction
Regularity theory for weak solutions of second order quasilinear elliptic equations in the Eucledean spaces, has been well-developed over a long period of time since the pioneering work of De Giorgi [9] and has involved significant contributions of many authors. For more details on this topic, we refer to [38,10,39,20,18,42,15,27], etc. and references therein. A comprehensive study of the subject can be found in the nowadays classical books by Gilbarg-Trudinger [22], Ladyzhenskaya-Ural'tseva [26] and Morrey [33].
The goal of this paper is to obtain regularity results in the setting of Heisenberg Group H n , that are previously known in the Eucledean setting. We consider the equation (1.1) Qu = div H A(x, u, Xu) + B(x, u, Xu) = 0 in a domain Ω ⊂ H n for any n ≥ 1, where Xu = (X 1 u, . . . , X 2n u) is the horizontal gradient of a function u : Ω → R and div H is the horizontal divergence of a vector field (see Section 2 for details). Here A : Ω × R × R 2n → R 2n and B : Ω × R × R 2n → R are given locally integrable functions. We also assume that A is differentiable and the (2n × 2n) matrix D p A(x, z, p) = (∂A i (x, z, p)/∂p j ) ij is symmetric for every x ∈ Ω, z ∈ R and p = (p 1 , . . . , p 2n ) ∈ R 2n . Thus, the results of this setting can also be applied to minimizers of a variational integral for a smooth scalar function f : Ω×R×R 2n → R; the Euler-Lagrange equation corresponding to the functional I, would be an equation of the form (1.1). The equations in settings similar to ours, are often referred as sub-elliptic equations.
The function g shall be used in the hypothesis of growth and ellipticity conditions satisfied by A and B, as given below. The condition (1.2) appears in the work of Lieberman [29], in the Eucledean setting. In the case of Heisenberg Groups, a special class of quasilinear equations with growth conditions involving (1.2), has been recently studied in [35]. We remark that the special case g(t) = t p−1 for 1 < p < ∞, would correspond to equations with p-laplacian type growth. For a more detailed discussion on the relevance of the condition (1.2) and more examples of such function g, we refer to [29,31,1,35] etc. The study of regularity theory for sub-elliptic equations goes back to the fundamental work of Hörmander [24]. We refer to [5,6,8,16,13,14,32,30,12] and references therein, for earlier results on regularity of weak solutions of quasilinear equations.
The structure conditions for the equation (1.1) used in this paper, have been introduced in [29], which are generalizations of the so called natural conditions for elliptic equations in divergence form; these have been extensively studied by Ladyzhenskaya-Ural'tseva in [26] for equations in the Eucledean setting. The first structure condition is as follows.
For weak solutions of equation (1.1) with the above structure conditions, the appropriate domain is the Horizontal Orlicz-Sobolev space HW 1,G (Ω) (see Section 2 for the definition), where G(t) = t 0 g(s)ds. The following is the first result of this paper. Theorem 1.1. Let u ∈ HW 1,G (Ω) ∩ L ∞ (Ω) be a weak solution of the equation (1.1), with G(t) = t 0 g(s)ds and |u| ≤ M in Ω. Suppose the structure condition (1.3) holds for some χ ≥ 0, 0 < R ≤ R 0 and a function g satisfying (1.2) with δ > 0, then there exists c > 0 and α ∈ (0, 1) dependent on n, δ, g 0 , a 1 , a 2 , a 3 , b 0 M, b 1 such that u ∈ C 0,α loc (Ω) and (1.4) osc Br u ≤ c r R α osc B R u + χR , whenver B R 0 ⊂⊂ Ω and B r , B R are concentric to B R 0 with 0 < r < R ≤ R 0 .
The above theorem follows as a consequence of Harnack inequalities, Theorem 3.4 and Theorem 3.5 in Section 3. Similar Harnack inequalities in the sub-elliptic setting, has also been shown in [6] for the special case of polynomial type growth. The proof of these are standard imitations of the corresponding classical results due to Serrin [37], see also [40,29]. Theorem 1.1 is necessary for our second result, the C 1,α -regularity of weak solutions. This is new and relies on some recent development in [35], which in turn is based on the work of Zhong [43]. The structure conditions considered for this, are as follows.
Pertaining to the growth conditions involving (1.2), local Lipshcitz continuity for the class of equations of the form div H A(Xu) = 0, has been shown in [35]. As a follow up, here we show the C 1,α -regularity for this case as well, with a robust gradient estimate unlike (1.6). Theorem 1.3. Let u ∈ HW 1,G (Ω) be a weak solution of the equation div H A(Xu) = 0, where A : R 2n → R 2n , the matrix DA is symmetric and the following structure condition holds, |p| |ξ| 2 ≤ DA(p) ξ, ξ ≤ L g(|p|) |p| |ξ| 2 ; |A(p)| ≤ L g(|p|).
for every p, ξ ∈ R 2n , L ≥ 1 is a given constant and g satisfies (1.2) with δ > 0. Then Xu is locally Hölder continuous and there exists σ = σ(n, g 0 , L) ∈ (0, 1) and c = c(n, δ, g 0 , L) > 0 such that for any B r 0 ⊂ Ω and 0 < r < r 0 /2, we have The proof of the above theorem, follows similarly along the line of that in [34]. It involves Caccioppoli type estimates of the horizontal and vertical vector fields along with the use of an integrability estimate of [43] and a double truncation of [39] and [28].
We remark that the spaces C 0,α and C 1,α considered in this paper, are in the sense of Folland-Stein [17]. In other words, the spaces are defined with respect to the homogeneous metric of the Heisenberg Group, see Section 2 for details. No assertions are made concerning the regularity of the vertical derivative.
This paper is organised as follows. In Section 2, we provide a brief review on Heisenberg Group and Orlicz spaces. Then in Section 3, first we prove a global maximum principle exploring some generalised growth conditions along the lines of [29]; then we prove the Harnack inequalities, thereby leading to the proof of Theorem 1.1. The whole of Section 4 is devoted to the proof of Theorem 1.3. Finally in Section 5, the proof of Theorem 1.2 is provided and some possible extensions of the structure conditions are discussed.

Preliminaries
In this section, we fix the notations used and provide a brief introduction of the Heisenberg Group H n . Also, we provide some essential facts on Orlicz spaces and the Horizontal Sobolev spaces, which are required for the purpose of this setting.

Heisenberg Group.
Here we provide the definition and properties of Heisenberg group that would be useful in this paper. For more details, we refer the reader to [2], [7], etc.
Thus, H n with the group operation (2.1) forms a non-Abelian Lie group, whose left invariant vector fields corresponding to the canonical basis of the Lie algebra, are for every 1 ≤ i ≤ n and the only non zero commutator T = ∂ t . We have We call X 1 , . . . , X 2n as horizontal vector fields and T as the vertical vector field. For a scalar function f : H n → R, we denote Xf = (X 1 f, . . . , X 2n f ) and XXf = (X i (X j f )) i,j as the Horizontal gradient and Horizontal Hessian, respectively. From (2.2), we have the following trivial but nevertheless, an important inequality |T f | ≤ 2|XXf |. For a vector valued function The Euclidean gradient of a function g : R k → R, shall be denoted by ∇g = (D 1 g, . . . , D k g) and the Hessian matrix by D 2 g.
The Carnot-Carathèodory metric (CC-metric) is defined as the length of the shortest horizontal curves, connecting two points. This is equivalent to the Korànyi metric, denoted as d H n (x, y) = y −1 · x H n , where the Korànyi norm for x = (x 1 , . . . , x 2n , t) ∈ H n is Throughout this article we use CC-metric balls denoted by B r (x) = {y ∈ H n : d(x, y) < r} for r > 0 and x ∈ H n . However, by virtue of the equivalence of the metrics, all assertions for CC-balls can be restated to Korànyi balls. The Haar measure of H n is just the Lebesgue measure of R 2n+1 . For a measurable set E ⊂ H n , we denote the Lebesgue measure as |E|. For an integrable function f , we denote The Hausdorff dimension with respect to the metric d is also the homogeneous dimension of the group H n , which shall be denoted as Q = 2n + 2, throughout this paper. Thus, for any CC-metric ball B r , we have that |B r | = c(n)r Q . For 1 ≤ p < ∞, the Horizontal Sobolev space HW 1,p (Ω) consists of functions u ∈ L p (Ω) such that the distributional horizontal gradient Xu is in L p (Ω , R 2n ). HW 1,p (Ω) is a Banach space with respect to the norm We define HW 1,p loc (Ω) as its local variant and HW 1,p 0 (Ω) as the closure of C ∞ 0 (Ω) in HW 1,p (Ω) with respect to the norm in (2.4). The Sobolev Embedding theorem has the following version in the setting of Heisenberg group (see [6], [7]). Theorem 2.2 (Sobolev Embedding). Let B r ⊂ H n and 1 < q < Q. For all u ∈ HW 1,q 0 (B r ), there exists constant c = c(n, q) > 0 such that Hölder spaces with respect to homogeneous metrics have appeared in Folland-Stein [17] and therefore, are sometimes called are known as Folland-Stein classes and denoted by Γ α or Γ 0,α in some literature. However, here we maintain the classical notation and define for 0 < α ≤ 1, which are Banach spaces with the norm These have standard extensions to classes C k,α (Ω) for k ∈ N, which consists of functions having horizontal derivatives up to order k in C 0,α (Ω). The local counterparts are denoted as C k,α loc (Ω). Now, the definition of Morrey and Campanato spaces in sub-elliptic setting differs in different texts. Here, we adopt the definition similar to the classical one.
For any domain Ω ⊂ H n and λ > 0, we define the Morrey space as Br |u| dx < c r λ ∀ B r ⊂ Ω, r > 0 and the Campanato space as where in both definitions B r represents balls with metric d. These spaces are Banach spaces and have properties similar to the classical spaces in the Eucledean setting. We shall use the fact that for every 0 < α < 1 and Q = 2n + 2, we have where the inclusion is to be understood as taking continuous representatives. For details on classical Morrey and Campanato spaces, we refer to [25] and for the sub-elliptic setting we refer to [7].

Orlicz-Sobolev Spaces.
In this subsection, we recall some basic facts on Orlicz-Sobolev functions, which shall be necessary later. Further details can be found in textbooks e.g. [25], [36]. There are several different definitions available in various references. However, within a slightly restricted range of functions (as in our case), all of them are equivalent. We refer to the book of Rao-Ren [36], for a more general discussion. Definition 2.4 (Conjugate). The generalised inverse of a montone function ψ is defined as ψ −1 (t) := inf{s ≥ 0 | ψ(s) > t}. Given any Young function Ψ(t) = t 0 ψ(s)ds, its conjugate A Young function Ψ is convex, increasing, left continuous and satisfies Ψ(0) = 0 and lim t→∞ Ψ(t) = ∞. The generalised inverse of Ψ is right continuous, increasing and coincides with the usual inverse when Ψ is continuous and strictly increasing. In general, the inequality is satisfied for all t ≥ 0 and equality holds when Ψ(t) and Ψ −1 (t) ∈ (0, ∞). It is also evident that that the conjugate function Ψ * is also a Young function, Ψ * * = Ψ and for any constant c > 0, we have (c Ψ) * (t) = c Ψ * (t/c).
Here are two standard examples of complementary pair of Young functions.
A Young function Ψ is called doubling if there exists a constant C 2 > 0 such that for all t ≥ 0, we have Ψ(2t) ≤ C 2 Ψ(t). By virtue of (1.2), the structure function g is doubling with the doubling constant C 2 = 2 g 0 and hence, we restrict to Orlicz spaces of doubling functions. Definition 2.6. Let Ω ⊂ R m be Borel and ν be a σ-finite measure on Ω. For a doubling Young function Ψ, the Orlicz space L Ψ (Ω, ν) is defined as the vector space generated by the set {u : Ω → R | u measurable, Ω Ψ(|u|) dν < ∞}. The space is equipped with the following Luxemburg norm If ν is the Lebesgue measure, the space is denoted by L Ψ (Ω) and any u ∈ L Ψ (Ω) is called a Ψ-integrable function.
The function u → u L Ψ (Ω,ν) is lower semi continuous and L Ψ (Ω, ν) is a Banach space with the norm in (2.15). The following theorem is a generalised version of Hölder's inequality, which follows easily from the Young's inequality (2.14), see [36] or [41].
The Orlicz-Sobolev space W 1,Ψ (Ω) can be defined similarly by L Ψ norms of the function and its gradient, see [36], that resembles W 1,p (Ω). But here for Ω ⊂ H n , we require the notion of Horizontal Orlicz-Sobolev spaces, analoguous to the horizontal Sobolev spaces defined in the previous subsection. The spaces HW 1,Ψ loc (Ω), HW 1,Ψ 0 (Ω) are defined, similarly as earlier. We remark that, all these notions can be defined for a general metric space, equipped with a doubling measure. We refer to [41] for the details.
The following theorem, so called (Ψ, Ψ)-Poincaré inequality, has been proved (see Proposition 6.23 in [41]) in the setting of a general metric space with a doubling measure and metric upper gradient. We provide the statement in the setting of Heisenberg Group. Theorem 2.10. Given any doubling N-function Ψ with doubling constant c 2 > 0, every u ∈ HW 1,Ψ (Ω) satisfies the following inequality for every B r ⊂ Ω and some c = c(n, c 2 ) > 0, In case of Ψ(t) = t p , the inequality is referred as (p, p)-Poincaré inequality. The following corrollary follows easily from (2.17) and the (1, 1)-Poincaré inequality on H n . Corollary 2.11. Given a convex doubling N-function Ψ with doubling constant c 2 > 0, there exists c = c(n, c 2 ) such that for every B r ⊂ Ω and u ∈ HW 1,Ψ (Ω) ∩ HW 1,1 0 (Ω), we have Given a domain Ω ⊂ H n , using (2.18) and arguments with chaining method (see [23]), it is also possible to show that for u, Ψ and c = c(n, c 2 ) > 0 as in Corrollary 2.11, we have Now we enlist some important properties of the function g that satisfies (1.2).
Since G is convex, an easy application of Jensen's inequality yields All the above properties hold even if δ = 0 in (1.2) and they are purposefully kept that way. However, the properties corresponding to δ > 0, shall be required in some situations. For this case, (2.21) and (2.22) becomes (2.28) and hence t → g(t)/t g 0 is decreasing and t → g(t)/t δ is increasing.

Hölder continuity of weak solutions
In this section, we show that weak solutions of quasilinear equations in the Heisenberg Group satisfy the Harnack inequalities, which leads to the Hölder continuity, thereby proving Theorem 1.1. The techniques are standard, based on appropriate modifications of similar results in the Eucledean setting, by Trudinger [40] and Lieberman [29].
On a domain Ω ⊂ H n , we consider the prototype quasilinear operator in divergence form throughout this paper, where A : Ω × R × R 2n → R 2n and B : Ω × R × R 2n → R are given functions. Appropriate additional hypothesis on structure conditions satisfied by A and B, shall be assumed in the following subsections, accordingly as required.
Here onwards, throughout this paper, we fix the notations We remark that the conditions chosen for A, always ensure some sort of ellipticity for the operator (3.1) and the existence of weak solutions u ∈ HW 1,G (Ω) for Qu = 0 is always assured. Any pathological situation, where this does not hold, is avoided.

Global Maximum principle.
Given weak solution u ∈ HW 1,G (Ω) for Qu = 0, here we show global L ∞ estimates of u under appropriate boundary conditions. The method and techniques are adaptations of similar classical results in [29] for quasilinear equations in the Eucledean setting.
Additional conditions on f 1 and f 2 , yields apriori integral estimates as in the following lemma. Similar results in Eucledean setting, can be found in [22] and [26].
for some a 1 ≥ 1, R > 0 and every t > M/R, then there exists c(n) > 0 such that for Q = 2n + 2 and c = c(n)[(1 + a 1 )(1 + 2b 0 )] Q , we have sup Proof. The proof is similar to that of Lemma 2.1 in [29] (see also Lemma 10.8 in [22]) and follows from standard Moser's iteration. We provide a brief outline. Note that, we can assume |u| ≥ M without loss of generality, as otherwise we are done; we provide the proof for u ≥ M , the proof for u ≤ −M is similar. The test function ϕ = h(u) is used for the equation Qu = 0, where letting G = G(|u|/R) and τ = G(M/R), we choose for β ≥ 2b 0 and Q = 2n + 2. Thus ϕ/u ≥ 0 and ϕ = 0 on ∂Ω, since M ≥ u 0 L ∞ . Hence, applying ϕ as a test function and using (3.4), we get Note that Xϕ = h (u)Xu and we have For every β ≥ 2b 0 , we obtain that (3.9) where we have used h (u) ≥ 2b 0 ϕ/u and (3.3) for the first inequality and (3.8) for the second inequality of the above. From (3.9) and (3.6), we obtain (3.10) 1 2 Ω h (u)g(|Xu|)|Xu| dx ≤ a 1 (β + 1)(2n + 4) Then, we use (2.24) of Lemma 2.12 with t = |Xu| and s = |u|/R, to obtain for some c(n) > 0, where for the last inequality of the above, we have used (3.10) and (3.5).
Remark 3.3. With minor modifications of the above arguments, the global bound can also be shown corresponding to u + for weak supersolutions u i.e. for Qu ≥ 0.

Harnack Inequality.
Here we show that weak solutions of Qu = 0, satisfy Harnack inequality. The proofs are standard modifications of those in [40] and [29] for the Eucledean setting. We also refer to [6] for the Harnack inequalities on special cases, in the sub-elliptic setting.
In this subsection, we consider for given non-negative constants a 1 , a 2 , a 3 , and χ, R > 0.
Hence, we useū = u + + χR for the proof. Given any σ ∈ (0, 1), we choose a standard cutoff function η ∈ C ∞ 0 (B R ) such that 0 ≤ η ≤ 1, η = 1 in B σR and |Xη| ≤ 2/(1 − σ)R. Then, for some γ ∈ R and β ≥ 1 + |γ| which are chosen later, we use as a test function for Qu ≥ 0, to get (3.30) Now we use the structure condition (3.27) for the left hand side and (3.28),(3.29) for the right hand side of the above inequality. Then, we use (2.21) and (2.22) of Lemma 2.12 and also the fact that Here onwards, we use c = c(n, g 0 , a 1 , a 2 , a 3 , b 0 M, b 1 ) > 0 as a large enough constant, throughout the rest of the proof. Now we estimate both I 1 and I 2 as follows.
From Theorem 3.4 and Theorem 3.5, the following corrollary is immediate.
and with the structure conditions (3.23), (3.24) and for given non-negative constants a 1 , a 2 , Thus, bounded weak solutions satisfy the Harnack inequality (3.42), which implies the Hölder continuity of weak solutions. By standard arguments, it is possible to show that there for every 0 < r < R and B R ⊂ Ω. This is enough to prove Theorem 1.1. ) holds if f 1 (|z|), f 2 (|z|) ∼ g(|z|/R)|z|/R + g(χ)χ. Therefore, it is not restrictive to assume |u| ≤ M since we have Theorem 3.2 for the above cases. Furthermore, (3.41) can be relaxed to so that, in this case (3.29) can be obtained immediately.

Hölder continuity of Horizontal gradient
In this section, we consider a homogenous quasilinear equation where the operator does not depend on x and u. Estimates for this equation shall be necessary in Section 5. However, all results in this section are obtained independently, without any reference to the rest of this paper, apart from the usage of the structure function g in (1.2).
We warn the reader that in this section z is used as a variable in R 2n , unlike the other sections. This is done to maintain continuity with [35].
In a domain Ω ⊂ H n , we consider for all z ∈ R 2n and DA(z) as the 2n × 2n Jacobian matrix (∂A i (z)/∂z j ) ij . We assume that DA(z) is symmetric and satisfies for every z, ξ ∈ R 2n and L ≥ 1, where we denote F(t) = g(t)/t maintaining the notation (3.2). Here g : [0, ∞) → [0, ∞) is a given C 1 function satisfying (1.2) and g(0) = 0. The above equation has been considered previously in [35] where local boundedness of Xu for a weak solution u of (4.1), has been established. The goal of this section is to prove the local Hölder continuity of Xu. We restate Theorem 1.3 here, which is the main result of this section.

Previous Results.
Here we provide some results that are known and previously obtained, which would be essential for our purpose. For more details, we refer to [35] and references therein.
The following monotonicity and ellipticity inequalities follow easily from (4.2).
for all z, w ∈ R 2n and some constant c(g 0 ) > 0. These are essential to show the existence of a weak solution u ∈ HW 1,G (Ω) of the equation (4.1). We refer to [35] for a brief discussion on existence and uniqueness for (4.1). The following theorem is Theorem 1.1 of [35], which shows the local Lipschitz continuity of the weak solutions.
Theorem 4.2. Let u ∈ HW 1,G (Ω) be a weak solution of equation (4.1) satisfying structure condition (4.2) and g satisfies (1.2) with δ > 0. Then Xu ∈ L ∞ loc (Ω, R 2n ); moreover for any B r ⊂ Ω, we have Now, we also require the following apriori assumption as considered in [35], in order to temporarily remove possible singularities of the function F. Here onwards, this shall be assumed until the end of this section. This combined with the local boundedness of Xu from Theorem 4.2, makes the equation (4.1) to be uniformly elliptic and enables us to conlcude (4.8) Xu ∈ HW 1,2 loc (Ω, R 2n ) ∩ C 0,α loc (Ω, R 2n ), T u ∈ HW 1,2 loc (Ω) ∩ C 0,α loc (Ω) from Theorem 1.1 and Theorem 3.1 of Capogna [5]. However, every estimates in this section, are independent of the constants m 1 and m 2 and (4.7) shall be ultimately removed.
The regularity (4.8) is necessary to differentiate the equation (4.1) and obtain the equations satisfied by X l u and T u, as shown in the following two lemmas. The proofs are simple and omitted here, we refer to [35] and [43] for details.
Lemma 4.4. If u ∈ HW 1,G (Ω) is a weak solution of (4.1), then for any l ∈ {1, . . . , n}, we have that X l u is weak solution of (4.10) and similarly, X n+l u is weak solution of We enlist some Caccioppoli type inequalitites, that are very similar to those in [43] and [34]. They will be essential for the estimates in the next subsection.
The following lemma is similar to Lemma 3.3 in [43], the proof is trivial and omitted here.
Lemma 4.5. For any β ≥ 0 and all η ∈ C ∞ 0 (Ω), we have, for some c = c(n, g 0 , L) > 0, that The following lemma is similar to Corollary 3.2 of [43] and Lemma 2.5 of [34]. This is crucial for the proof of the Hölder continuity of the horizontal gradient. The proof of the lemma is similar to that in [43] and involves few other Caccioppoli type estimates. An outline is provided in Appendix II, for the reader's convenience.

The truncation argument.
In this subsection, we follow the technique of [34] and prove Caccioppoli type inequalities invovling a double truncation of horizontal derivatives. In the setting of Euclidean spaces, similar ideas have been implemented previously by Tolksdorff [39] and Lieberman [28].
In the setting of equations with p-laplace type growth, the above lemma has been shown previously in [34](see Lemma 1.1). The proof is going to be similar. Hence, we would require two auxillary lemmas, similarly as in [34].
We also remark that the inequality (4.20) also holds corresponding to the truncation v = min µ(r)/8, max(µ(r)/4 + X l u, 0) , and the proof can be carried out in the same way as that of Lemma 4.8.
The following lemma is the analogue of Lemma 3.1 of [34]. The proof is similar and lengthy, which we provide in the Appendix I. Lemma 4.9. For any β ≥ 0 and all non-negative η ∈ C ∞ 0 (Ω), we have that where v is as in (4.14) and c = c(n, g 0 , L) > 0.
The following technical lemma, that is required for the proof of Lemma 4.8, is a weighted Caccioppoli inequality for T u involving v similar to that in Lemma 3.2 of [34]. We provide the proof here for sake of completeness. Lemma 4.10. Let B r ⊂ Ω be a ball and η ∈ C ∞ 0 (B r ) be a cut-off function satisfying (4.22) and (4.23). Let τ ∈ (1/2, 1) and γ ∈ (1, 2) be two fixed numbers. Then, for any β ≥ 0, we have the following estimate, where c = c(n, g 0 , L, τ, γ) > 0 and Proof. We denote the left hand side of (4.24) by M , where 1/2 < τ < 1. Now we use ϕ = η τ (β+2)+4 v τ (β+4) |Xu| 4 T u as a test function for the equation (4.9). We obtain that where the integrals in the right hand side of (4.27) are denoted by K 1 , K 2 , K 3 in order. To prove the lemma, we estimate both sides of (4.27) as follows.
For the left hand side, we have by the structure condition (4.2) that and for the right hand side of (4.27), we estimate each item K i , i = 1, 2, 3, one by one.
To this end, we denote First, we estimate K 1 by the structure condition (4.2) and Hölder's inequality, to get (4.30) where c = c(n, g 0 , L, τ ) > 0. Second, we estimate K 2 also by the structure condition (4.2) and Hölder's inequality, (4.31) Finally, we estimate K 3 . In the following, the first inequality follows from the structure condition (4.2), the second from Hölder's inequality and the third from Lemma 4.9. We have (4.32) where I is the right hand side of (4.21) in Lemma 4.9 (4.33) where c = c(n, g 0 , L) > 0. Notice that the integrals on the right hand side of (4.30) and (4.31) are both controlled from above by I. Hence, we can combine (4.30), (4.31) and (4.32) to obtain |K 1 | + |K 2 | + |K 3 | ≤ cK , where c = c(n, g 0 , L, τ ) > 0. Now, we estimateK by Hölder's inequality as follows.
where M is as in (4.26) and we denote by H the second integral on the right hand side of (4.35) Combining (4.35) and (4.34), we get for some c = c(n, g 0 , L, τ ) > 0. To estimate M , we estimate H and I from above. We estimate H by Corollary 4.7 with q = 2/(1 − τ ) and monotonicity of g, to obtain (4.38) where c = c(n, g 0 , L, τ ) > 0. Now, we fix 1 < γ < 2 and estimate each term of I in (4.33) as follows. For the first term of I, we have by Hölder's inequality and monotonicity of g that For the second term of I, we similarly have For the third term of I, we have that where c = c(n, g 0 , L, γ) > 0. Here in the above inequalities, the first one follows from Hölder's inequality and the second from Lemma 4.6 and monotonicity of g. Combining the estimates for three items of I above (4.39), (4.40) and (4.41), we get the following estimate for I, Proof of Lemma 4.8. First, notice that we may assume γ < 3/2, since otherwise we can apply Hölder's inequality to the integral in the right hand side of the claimed inequality (4.20). Also, we recall from (4.14), that for some l ∈ {1, . . . , 2n}, v = min µ(r)/8 , max (µ(r)/4 − X l u, 0) .
For the right hand side of (4.45), we claim that each item I 1 , I 2 , I 3 satisfies (4.47) where m = 1, 2, 3, 1 < γ < 3/2 and c is a constant depending only on n, g 0 , L and γ. Then the lemma follows from the estimate (4.46) for the left hand side of (4.45) and the above claim (4.47) for each item in the right. Thus, we are only left with proving the claim (4.47).
In the rest of the proof, we estimate I 1 , I 2 , I 3 one by one. First for I 1 , using integration by parts, we have that where c = c(n, g 0 , L) > 0. For the latter inequality of (4.48), we have used the fact that g(t) = tF(t) is monotonically increasing. Now we apply Young's inequality to the last term of (4.48) to end up with (4.49) where c = c(n, g 0 , L) > 0 and c 0 is the same constant as in (4.46). The claimed estimate (4.47) for I 1 , follows from the above estimate (4.49) and Hölder's inequality.
To estimate I 2 , we have by the structure condition (4.2) that from which it follows by Hölder's inequality that (4.50) where q = 2γ/(γ − 1). The fact that the integrals are on the set E, is crucial since we can use (4.19) and the following estimates can not be carried out unless the function F is increasing. We have the following estimates for the first two integrals of the above, using (4.19).
where c = c(n, g 0 , L) > 0. We estimate the last integral in the right hand side of (4.50) by (4.12) of Lemma 4.6 and monotonicity of g, to obtain (4.53) where c = c(n, g 0 , L, γ) > 0. Now combining the above three estimates (4.51), (4.52) and (4.53) for the three integrals in (4.50) respectively, we end up with the following estimate for I 2 from which, together with Young's inequality, the claim (4.47) for I 2 follows. Finally, we prove the claim (4.47) for I 3 . Recall that By virtue of the regularity (4.15) for v, integration by parts yields (4.54) where we denote the last two integrals in the above equality by I 1 3 and I 2 3 , respectively. The estimate for I 1 3 easily follows from the structure condition (4.2) and monotonicity of g, as (4.55) Thus by Hölder's inequality, I 1 3 satisfies estimate (4.47). To estimate I 2 3 , note that by (4.17) and the structure condition (4.2) we have where the set E is as in (4.16). For 1 < γ < 3/2, we continue to estimate I 2 3 by Hölder's inequality as follows, Since, we have (4.19) on the set E, hence Now we can apply Lemma 4.10 to estimate M from above. Note that Lemma 4.10 with τ = 2 − γ, gives us that (4.59) M ≤ c(β + 2) 2(2−γ) |B r | γ−1 r 2γ F(µ(r))µ(r) 6 J 2−γ where c = c(n, g 0 , L, γ) > 0 and J is defined as in (4.25) Now, it follows from (4.59) and (4.57) that By Young's inequality, we end up with where c 0 > 0 is the same constant as in (4.47). Note that, with J as in (4.60), I 2 3 satisfies an estimate similar to (4.47). Now the desired claim (4.47) for I 3 follows, since both I 1 3 and I 2 3 satisfy similar estimates. This concludes the proof of the claim (4.47), and hence the proof of the lemma.
The following corollary follows from Lemma 4.8 by Moser's iteration. We refer to [34] for the proof.   We clearly have ω(r) ≤ 2µ(r). For any function w, we define is similarly defined. The following lemma is similar to Lemma 4.1 of [34] and Lemma 4.3 of [43]. For sake of completeness, we provide a proof in Appendix I. Lemma 4.12. Let B r 0 ⊂ Ω be a ball and 0 < r < r 0 /2. Suppose that there is τ > 0 such that |Xu| ≥ τ µ(r) in A + k,r (X l u) for an index l ∈ {1, 2, ..., 2n} and for a constant k ∈ R. Then for any q ≥ 4 and any 0 < r < r ≤ r, we have r 0 )) and c = c(n, p, L, q, τ ) > 0. Remark 4.13. Similarly, we can obtain an inequality, corresponding to (4.64), with (X l u−k) + replaced by (X l u − k) − and A + k,r (X l u) replaced by A − k,r (X l u). Lemma 4.14. There exists a constant s = s(n, g 0 , L) ≥ 0 such that for every 0 < r ≤ r 0 /16, we have the following, where α = 1/2 when 0 < g 0 < 1 and α = 1/(1 + g 0 ) when g 0 ≥ 1.
Proof of Theorem 4.1.

C 1,α -regularity of weak solutions
In this section, we prove Theorem 1.2. In a fixed subdomain Ω compactly contained in Ω, we show that the weak solutions are locally C 1,β in Ω . The proof is standard, based on the results of the preceeding section and a Campanato type perturbation technique. Similar arguments in the Eucledean setting, can be found in [10,18,29], etc.

The perturbation argument.
Given Ω ⊂⊂ Ω, we fix x 0 ∈ Ω and a ball B R = B R (x 0 ) ⊂ Ω for R ≤ R 0 = 1 2 dist(Ω , ∂Ω) and consider u ∈ HW 1,G (B R ) ∩ L ∞ (B R ) as weak solution of Qu = 0 in B R , where Q is defined as in (3.1). We recall the structure conditions for Theorem 1.2, as follows; for all (x, z, p) ∈ Ω × R × R 2n and the matrix D p A(x, z, p) is symmetric. In addition, we recall the hypothesis of Theorem 1.2 that, there exists M 0 > 0 such that |u| ≤ M 0 in Ω .
From structure condition (5.1), it is not difficult to check that A(x, z, p) satisfies conditions reminiscent of (3.23) and (3.24); the condition on variable z for (3.23) and (3.24) are absolved in the constants L and L , since the solution u is bounded. However, the condition (5.3) on B is more relaxed than (3.41) and (3.44), which is necessary for C 1,β -regularity.
Thus, this allows us to apply Theorem 1.1 and conclude u is Hölder continuous with for some γ = γ(M 0 , dist(Ω , ∂Ω)) > 0 and τ ∈ (0, 1) can be chosen to be as small as required.
Here onwards, we suppress the dependence of the data n, δ, g 0 , α, L, L , M 0 , dist(Ω , ∂Ω); all positive constants depending on these shall be denoted as c, throughout this subsection, until the end of the proof of theorem 1.2. Let us denote A : R 2n → R 2n as so that from (5.1), A satisfies the structure condition (4.2) and hence also the monotonicity and ellipticity conditions (4.4) and (4.5) (with possible dependence on g 0 and δ). Hence, for the problem (5.6) div H (A(Xũ)) = 0 in B R ; u − u ∈ HW 1,G 0 (B R ). we can use the monotonicity inequalities and uniform estimates from Section 4.
is given, then there exists a unique weak solutioñ u ∈ HW 1,G (B R ) ∩ C(B R ) for the problem (5.6), which satisfies the following: Proof. Existence and uniqueness is standard from monotonicity of A, we refer to [35] for more details. Also, (5.7) follows easily from Comparison principle and the fact that which is easy to show by considering ϕ = (ũ − sup ∂B R u) + (and similarly the other case) as a test function for (5.6), see Lemma 5.1 in [10].
To proceed with the proof of Theorem 1.2, we shall need the following technical lemma which is a variant of a lemma of Campanato [4]. This is elementary but a fundamental lemma. We refer to [21] or [19, Lemma 2.1] for a proof. where A is as in (5.5) andũ ∈ HW 1,G (B R ) ∩ C(B R ) is the weak solution of (5.6). Since u =ũ in ∂B R , the function u −ũ can be used to test the equations satisfied by u andũ, which shall be used to estimate I to obtain both lower and upper bounds. First, using u −ũ as test function for Qu = 0, we obtain (5.11) with θ(R) as in (5.4), where we have used structure condition (5.2) and (5.3) for the first term and (5.7) for the second term of the right hand side of (5.11). Now we use (2.24) of Lemma 2.12 and (5.8) of Lemma 5.1 to estimate the first term of the above and obtain that Secondly, to obtain the upper bound for I, we shall use the monotonicity inequality (4.4). Let us denote S 1 = {x ∈ B R : |Xu − Xũ| ≤ 2|Xu|} and S 2 = {x ∈ B R : |Xu − Xũ| > 2|Xu|}. Taking u −ũ as test function for (5.6) and using (4.4), we obtain (5.13) Recalling G(t) ≤ t 2 F(t) from (2.21), we have from (5.12) and (5.13), that (5.14) Now since |Xu − Xũ| ≤ 2|Xu| in S 1 by definition, we obtain the following from (2.21), monotonicity of g and Hölder's inequality; where the latter inequality of the above follows from (5.12) and (5.13). Now, we add (5.14) and (5.15) to obtain the estimate of the integral over whole of B R , Recalling (4.6) and (5.8), note that for any 0 < r ≤ R/2, we have where Q = 2n + 2. Combining the above with (5.16), we obtain Now, we follow the bootstrap technique of Giaquinta-Giusti [18]. Here onwards the constants dependent on g(1) in addition to the aforementioned data, shall be denoted as C.

Concluding Remarks.
Here we discuss some possible extensions of the structure conditions that can be included and results similar to the above can be obtained with minor modifications of the arguments.
(1) Any dependence of x in structure conditions for A(x, z, p) and B(x, z, p) has been suppressed so far, for sake of simplicity. However, we remark that for some given non-negative measurable functions a 1 , a 2 , a 4 , a 5 , b 1 , b 2 , the structure condition can also be considered for obtaining the Harnack inequalities. In this case, we would require a 1 , a 2 , a 4 , a 5 , b 1 , b 2 ∈ L q loc (Ω) for some q > Q. Similar arguments can be carried out with a choice of χ > 0, such that a 5 L q (B R ) + b 2 L q (B R ) ≤ g(χ) and a 2 L q (B R ) ≤ g(χ)χ. We refer to [29] and [6] for more details of such cases.
(2) The function g(t)/t in the growth conditions can be replaced by f (t), where f is a continuous doubling positive function on (0, ∞) and t → f (t)t 1−δ is non-decreasing. A C 1 -functiong can be found satisfying (1.2) andg(t) ∼ tf (t)(see [29,Lemma 1.6]), which is sufficient to carry out all of the above arguments.

Appendix I
Proof of Lemma 4.9.
Fix l ∈ {1, 2, ..., n} and β ≥ 0. Let η ∈ C ∞ 0 (Ω) be a non-negative cut-off function. Using as a test-function in equation (4.10), we get Here we denote the integrals in the right hand side of (4.10) by I l 1 , I l 2 , I l 3 and I l 4 in order respectively. Similarly for all l ∈ {n + 1, n + 2, ..., 2n}, from equation (4.11), we have Again we denote the integrals in the right hand side of (5.22) by I l 1 , I l 2 , I l 3 and I l 4 in order respectively. Summing up the above equation (5.21) and (5.22) for all l from 1 to 2n, we end up with (5.23) where all sums for i, j, l are from 1 to 2n.
so that we can write test-function ϕ defined as in (5.20) as ϕ = v β+2 w. Then, for I l 4 in (5.21), we rewrite T = X 1 X n+1 − X n+1 X 1 and use integration by parts to obtain Using Xϕ = (β + 2)v β+1 wXv + v β+2 Xw in (5.30), we get (5.31) Here we denote the first and the second integral in the right hand side of (5.30) by J l and K l , respectively. Now we estimate J l as follows. From structure condition (4.2) and (5.29) from which it follows by Young's inequality, that The above inequality shows that J l satisfies similar estimate as (5.25) for all l = 1, 2, ..., n. Now we estimate K l . Integration by parts again, yields (5.33) For K l 1 , we have by the structure condition (4.2) that from which it follows by Young's inequality that The above inequality shows that K l 1 also satisfies similar estimate as (5.25) for all l = 1, 2, ..., n. We continue to estimate K l 2 in (5.33). Note that T w = (β + 2)η β+1 |Xu| 2 X l uT η + η β+2 |Xu| 2 X l T u + 2n i=1 2η β+2 X l uX i uX i T u.
Therefore we write K l 2 as For the last two integrals in the above equality, we apply integration by parts to get Now we may estimate the integrals in the above equality by the structure condition (4.2), to obtain the following estimate for K l 2 .
By Young's inequality, we end up with the following estimate for K l 2 (5.35) This shows that K l 2 also satisfies similar estimate as (5.25). Now we combine the estimates (5.34) for K l 1 and (5.35) for K l 2 . Recall that K l = K l 1 + K l 2 as denoted in (5.33). We obtain that the following estimate for K l . (5.36) Recall that I l 4 = J l + K l . We combine the estimates (5.32) for J l and (5.36) for K l , and we can see that the claimed estimate (5.25) holds for I l 4 for all l = 1, 2, ..., n. We can prove (5.25) similarly for I l 4 for all l = n + 1, n + 2, ..., 2n. This finishes the proof of the claim (5.25) for I l m for all l = 1, 2, ..., 2n and all m = 1, 2, 3, 4, and hence also the proof of the lemma.
Proof of Lemma 4.12.
In the proof, we only consider l ∈ {1, . . . , n}; the proof is similar for l ∈ {n, . . . , 2n}. In addition, note that we can also assume |k| ≤ µ(r 0 ) without loss of generality, to prove (4.64). This proof is very similar to that of Lemma 4.3 in [43].
The following lemma is similar to Lemma 3.5 of [43].
Proof. Note that have the following identity for any ϕ ∈ C ∞ 0 (Ω), which can be easily obtained using X l ϕ as a test function in equation (4.1) (see the proof of Lemma 3.5 in [35]).
To prove (5.48), we start with I 4 . By structure condition (4.2) and Young's inequality We then apply Lemma 4.5 to estimate the first integral in the right hand side.
The following corollary is easy to prove, by using Hölder's inequality on Lemma 5.4.