Regularity for sub-elliptic systems with VMO-coe cients in the Heisenberg group : the sub-quadratic structure case

Abstract: We consider nonlinear sub-elliptic systems with VMO-coe cients for the case 1 < p < 2 under controllable growth conditions, as well as natural growth conditions, respectively, in the Heisenberg group. On the basis of a generalization of the technique ofA-harmonic approximation introduced by DuzaarGrotowski-Kronz, and an appropriate Sobolev-Poincaré type inequality established in the Heisenberg group, we prove partial Hölder continuity results for vector-valued solutions of discontinuous sub-elliptic problems. The primarymodel covered by our analysis is the non-degenerate sub-elliptic p-Laplacian systemwith VMOcoe cients, involving sub-quadratic growth terms.


Introduction and statements of main results
In this paper, we consider discontinuous sub-elliptic systems with sub-quadratic growth coe cients that belong to the space of functions with vanishing mean oscillation (VMO, for short) in the Heisenberg group H n . We establish optimal partial Hölder continuity for vector-valued weak solutions in the sense that the solution is Hölder continuous on an open subset of its domain with full measure. More precisely, let Ω be a bounded domain, and horizontal gradient X = {X , · · · X n } with the horizontal vector elds X i (i = , · · · , n) in H n , we consider sub-elliptic systems of the type − n i= X i A α i (ξ , u, Xu) = B α (ξ , u, Xu), in Ω, α = , , · · · , N, (1.1) where the primary coe cient A α i ∈ VMO and satis es some standard ellipticity and growth conditions with polynomial growth rate p ∈ ( , ), and the inhomogeneous term B α conforms to either controllable growth conditions, or natural growth conditions under an additional smallness assumption on the weak solutions. For the precise statement of the assumptions, and more details about the Heisenberg group, we refer to (H1)-(H4)-(HC) and (HN) below, and Section , respectively.
The main new aspect of this paper is the fact that we are able to deal with the inhomogeneity B α : R n+ × R N × R n×N → R N that satis es the sub-quadratic controllable growth conditions, as well as sub-quadratic natural growth conditions, respectively, and the primary coe cient A α i : R n+ × R N × R n×N → R n×N that satis es only a VMO-condition in ξ and is continuous in u. More precisely, we assume that the partial mapping ξ → A α i (ξ , u, P)/( + |P|) p− is VMO uniformly in (u, P), in the sense of (1.5) below and, moreover, u → A α i (ξ , u, P)/( + |P|) p− is continuous in the sense of (1.3) below. Our tool of choice is the use of an appropriate Sobolev-Poincaré inequality, and the harmonic approximation lemma; see Lemma 3.1, Lemma 3.3 below, respectively. The method of proof employed here will avoid the use of L q − L p -estimates for the horizontal gradient and reverse Hölder inequalities. Our results essentially extend those results that the coe cients are continuous with respect to variables ξ and u to the case of the coe cients being VMO in the rst variable ξ . We point out that partial Hölder continuity is the best one can expect under such weak assumptions concerning regularity of the structural functions A α i and B α in the (ξ , u)-variables. We now impose the precise structure assumptions for coe cients A α i and B α we are dealing with. (H1). The primary coe cient A α i satis es following ellipticity and growth conditions for a growth exponent < p < : With respect to the dependence on the rst variable ξ , we do not impose a continuity condition, but we merely assume the following VMO-condition. (H4). The mapping ξ → A α i (ξ , u, P)/( + |P|) p− satis es the following VMO-condition uniformly in u and P: A α i (ξ , u, P) − A α i (·, u, P) ξ ,r ≤ v ξ (ξ , r)( + |P|) p− , for all ξ ∈ Br(ξ ), < p < . (1.5) where Here we have used the short-hand notation where C is a positive constant. We note that Q ≥ is the homogeneous dimension in non-Abelian Heisenberg groups (see (2.1) below), and the exponent p ∈ ( , ). So those infer that p < Q, and then, p * = pQ Q−p in our setting.
Several regularity results were focused on sub-elliptic systems in Heisenberg groups, or Hörmander vector elds; see Bramanti [2]. Xu and Zuily [34], Capogna and Garofalo [5], and Shores [25] showed partial regularity for quasi-linear sub-elliptic systems with quadratic growth p = . Their methods depend on generalization of classical freezing coe cient method. Then, by the generalization of the method of A-harmonic approximation, Föglein [16] treated homogeneous nonlinear sub-elliptic systems with Hölder continuous coe cients, under super-quadratic growth conditions p ≥ in the Heisenberg group, and established partial Hölder continuity for the horizontal gradient Xu. Later Wang and Liao [30] considered the case of < p < for inhomogeneous systems in Carnot groups. Furthermore, Wang, Liao and Gao [31] weakened assumptions on coe cients A α i with Hölder continuity in the variables (ξ , u) to the assumptions of Dini continuity, and proved partial regularity result with optimal estimates for the modulus of continuity for the horizontal derivative Xu.
Regularity results for discontinuous sub-elliptic systems with VMO coe cients instead of continuous coe cients have been established in the work [7] by Di Fazio and Fanciullo, and [19] by Gao, Niu and Wang for the case of quadratic growth; [32] by Wang and Manfredi, [14] by Dong and Niu, [36] by Zheng and Feng,and [33] by Wang, Zhang and Yang for non-quadratic growth conditions. We note that the regularity results in [14] and [36] have a limitation of p near , and the result in [19] holds only under a strong smallness condition for the dimension. In contrast, our partial Hölder continuity result stated below, is valid for the full range < p < in any dimension. The typical strategy in partial regularity depends on decay estimates for certain excess functionals, which measure the oscillations of the solution or its gradient in a suitable sense. In this paper, we are working with a combination of a zero-order excess functional Cγ and a rst-order excess functional Ψ. For the case p ≥ , the functional Ψ is de ned by with the horizontal a ne functions l : R n → R N de ned in the subsection 2.2 below. It is straightforward to adapt the standard A-harmonic approximation lemma by utilizing L -theory combined with the standard Sobolev inequality; see Wang and Manfredi [32] for the super-quadratic natural growth case. However, in the present situation, we treat the case of sub-quadratic controllable growth, and sub-quadratic natural growth, respectively. So one should establish the decay estimate for the following excess functional On the other hand, we de ne the Campanato type excess functional Cγ by which provides a measure of the oscillations in the weak solutions u itself. It is remarkable that the excess functionals de ned above involve only u, which simpli es the proofs of our partial regularity results. It is shown that if Ψ is small enough on a ball B ξ (ρ) ⊂⊂ U, then, for some xed θ ∈ ( , ), one obtain an excess improvement Ψ(ξ , θr, l ξ ,θr ) ≤ C θ Ψ * (ξ , r, l ξ ,r ) under smallness condition assumptions; see for example, Lemma 4.3. At this point, one has to assume smallness on the Ψ * -excess. Also we note that such an excess improvement estimate has two di erent quantities Ψ and Ψ * on the left, and the right hand side, respectively. Therefore, in contrast to the standard proof of partial regularity, the excess improvement cannot be iterated directly to yield an excess-decay estimate for Ψ-excess. In the present situation, however, iteration of the excess improvement yields that the Ψ-excess in (1.9) and also the Cγ-excess remain bounded. Finally, the boundedness of the Cγ-excess on any scale leads immediately to desired Hölder continuity of weak solutions u via the integral characterization of continuity by Campanato. We point out that the idea of such a combination of two excess functionals has its origin by Foss and Mingione [17] for continuous vector elds and integrands, and then, adapted to discontinuous problems with VMO coe cients for p ≥ by Bögelein-Duzaar-Habermann-Scheven [1]. It is worth mentioning that we obviously do not have access to use L -theory for functions in the horizontal Sobolev space HW ,p with < p < . Therefore, we have to establish the following Sobolev-Poincaré inequality with the function V (see Lemma 3.1 below), with the constant C P dependence only on N, p, Q. This inequality is an essential tool in order to get the regularity result. It is also one technique point where our case di ers from the case p ≥ in [32]. Under the previous assumptions (H1)-(H4) and (HC), and (H1)-(H4) and (HN), respectively, we establish the following two partial Hölder continuity results. Then, there exists a relatively closed singular set Ω ⊂ Ω such that u ∈ C ,γ loc (Ω\Ω , R N ) for every γ ∈ ( , ). Moreover, for any λ ∈ ( , Q) we have Xu ∈ L p,λ loc (Ω\Ω , R n×N ) with the Morrey parameter λ = Q − p( − γ). Finally, we have that the singular set satis es Ω ⊂ Σ ∪ Σ , where Σ = ξ ∈ Ω : lim r→ sup (Xu) ξ ,r = ∞ , with the functional V de ned in (2.3), and the singular set has ( n + )-Lebesgue measure zero |Ω | = and its complement Ω \ Ω is a set of full measure in Ω.  Xu) just as a special case of (1.1), where A α i (ξ ) ∈ VMO, and < p < . So, combining the result for ≤ p < ∞ established by Wang and Manfredi in [32], our partial Hölder continuity results covers the model case of subelliptic p-Laplacian system with < p < ∞. It is remarkable that Zheng and Feng [36] showed everywhere regularity for weak solutions of sub-elliptic p-harmonic systems while p is very closed to .
The organization of this paper is as follows. In Section 2, we collect some basic notions and facts associated to Heisenberg groups, involving quasi-distance, horizontal Sobolev spaces, and horizontal a ne function and some estimates. In Section 3, rstly an appropriate Sobolev-poincaré inequality which plays an important part on proving Hölder regularity is established. Then, an A-harmonic approximation lemma, and a prior estimate for weak solution h ∈ HW , to the constant coe cient homogeneous sub-elliptic systems are given. In Section 4, we prove partial regularity results of Theorem 1.1 under sub-quadratic controllable structure assumptions (H1)-(H4) and (HC) by several steps.
Step is to gain a suitable Caccioppoli-type inequality which is an essential tool to get partial regularity. An appropriate linearization strategy is given in the second step. Then, one can achieve that solutions are approximately A-harmonic by the linearization procedure, and an excess improvement estimate for the functional Ψ is obtained under two smallness condition assumptions, by combining with A-harmonic approximation lemma in the third steps. Once the excess improvement is established, the iteration for the Ψ-excess and the Cγ-excess can be acquired in Step . Finally, we show boundedness of the Campanato-type excess which leads immediately to desired Hölder continuity and Morrey regularity of Theorem 1.1. The last section shows the results of Theorem 1.2 under sub-quadratic natural structure assumptions (H1)-(H4) and (HN). In such a case, we establish appropriate estimates just for the natural growth term, and the rest procedure is similar to the proof of Theorem 1.1.

Preliminaries
In this section, we will give introduction of the Heisenberg group H n and de nitions of several function spaces, and some elementary estimates which will be used later.

. Introduction of the Heisenberg group H n
The Heisenberg group H n is de ned as R n+ endowed with the following group multiplication: x , · · · , x n , y , y , · · · , y n , t),ξ = (η,t) = (x ,x , · · · ,x n ,ỹ ,ỹ , · · · ,ỹ n ,t). Its neutral element is 0, and its inverse to (ξ , t) is given by (−ξ , −t). The basic vector elds corresponding to its Lie algebra can be explicitly calculated, and are given by for i = , , · · · , n, and note that the special structure of the commutators: that is, H n , · is a nilpotent Lie group of step . X = (X , · · · , X n ) is said to be the horizontal gradient, and T vertical derivative.
The homogeneous norm is de ned by (ξ , t) H n = ξ + t / , and the metric induced by this homogeneous norm is given by The measure used on H n is the Haar measure (Lebesgue measure in R n+ ), and the volume of the homoge- is called the homogeneous dimension of H n , and the quantity ωn is the volume of the homogeneous ball of radius . Let ≤ p ≤ +∞. We denote by the horizontal Sobolev space. Then, HW ,p (Ω) is a Banach space under the norm For u ∈ HW ,q (B R (ξ )), < q < Q and ≤ p ≤ qQ Q−q , Lu [29] showed the following Poincaré type inequality associated with Hörmander vector elds, which is naturally valid for H n : The inequality (2.2) is valid for p = q (≥ ). Throughout the paper, we shall use the functions V , W : for each ς ∈ R k , k ∈ N and p > . The functions V and W are locally bi-Lipschitz bijection on R k . The following inequality The purpose of introducing W is the fact that in contrast to |V| m , the function |W| m is convex. In fact, rstly by direct computation yields that W p (t) = t p ( + t −p ) − p is a convex and monotone increasing function on [ , ∞) with W p ( ) = ; secondly we have The following lemma includes some useful properties of the function V. The proof can be found in Lemma 2.1 of [4]. For simplicity, here, we replace the coe cient (p− )/ with √ in the left of the rst inequality ( ) below, since the fact that − / < (p− )/ for p > . Lemma 2.1. Let p ∈ ( , ) and V : R k → R k be the functions de ned in (2.3). Then, for any ς , ς ∈ R k and t > , the following inequalities hold:
If the function among horizontal a ne function l : R n → R N , then, we have and Here, we use the notation The proof of the results above can be found in [32] by Wang and Manfredi. On the basis of this formula, elementary calculations yield the following estimates.
We denote by l ξ ,ρ and l ξ ,θρ , the horizontal a ne function de ned as above for the radii ρ and θρ. Then, we have and, more generally, for every horizontal a ne function l : R n → R N .
Proof. By the identity (2.5) and Hölder's inequality, we obtain where we have used the fact that ffl According to the de nition of the function l ξ ,ρ , the following version of the Poincaré inequality (2.2) is true, that is,

Sobolev-Poincaré type inequality and A-harmonic approximation
We know that L -theory cannot be directly used to obtain appropriate estimates for solutions u ∈ HW ,p with < p < , so in this section, we rst establish a suitable version of Sobolev-Poincaré type inequality with functions V. This inequality is an essential tool in proving partial regularity. Then, we give a prior estimate for A-harmonic functions h ∈ HW , , and introduce an A-harmonic approximation lemma which plays an important part in getting excess improvement estimates.
with p * = pQ Q−p the Sobolev critical exponent of p; here the constant C P depends only on Q, N and p. In particular, the inequality also holds if we substitute for p * p .
Proof. We introduce the operator of fractional integration on Ω of order as follows Based on Theorem 2.7 in [3] by Capogna, Danielli and Garofalo, we deduce for < p < +∞ where we denote by p * = pQ Q−p the Sobolev critical exponent of p, and the number Q the homogeneous dimension in H n .
Lu [21] gave a representation formula for a function on graded nilpotent Lie groups for the left invariant vector elds; see Lemma 3.1 there. One form of the representation states that there exist constants c > and C > such that Noting that W /p (t) is monotone increasing and convex, we apply W /p (t) to both sides of the last inequality and have by Jensen's inequality We obtain the assertion of the theorem, rst for W, and then, also for V by (2.4).
Let A ∈ Bil(Ω × R N × R n×N , R n×N ) be a bilinear form with constant tensorial coe cients. We say that a map holds for all testing function φ ∈ C ∞ (Bρ(ξ ), R N ). Shores in [25] showed that weak solutions h ∈ HW , (Ω, R N ) of the constant coe cient homogeneous sub-elliptic systems belongs to C ∞ in the subset Ω ⊂ Ω. Then, the following estimate holds for the solution h ∈ HW , (Ω, R N ), Therefore, we can argue as the proof of Proposition . in [4] to obtain the same estimate for any A-harmonic function h ∈ HW , (Ω, R N ).
Similarly to [10], one can establish the following version of A-harmonic approximation for the case < p < in Heisenberg groups.
Then, there exists an A-harmonic map h ∈ C ∞ (Bρ(ξ ), R N ) which satis es

Partial Hölder continuity for sub-quadratic controllable growth
In this section, we prove the partial regularity result of Theorem 1 under the assumptions of sub-quadratic controllable structure conditions. Now we begin with the following.

. Caccioppoli-type inequality
We know that Caccioppoli-type inequality is a preliminary tool to prove partial regularity for systems. So in this subsection, we shall prove a Caccioppoli-type inequality for weak solutions to the sub-elliptic systems (1.1) with sub-quadratic controllable growth conditions. Then, for any ξ = (x , · · · , x n , y , · · · , y n , t) ∈ Ω with Br(ξ ) ⊂⊂ Ω, and any horizontal a ne functions l : R n → R N with |l(ξ )| + |Xl| ≤ M , we have the estimate Proof. We choose a standard cut-o function ϕ ∈ C ∞ (Br(ξ ), [ , ]) with ϕ ≡ on B r (ξ ) and |Xϕ| ≤ r . Then, φ = ϕ (u − l) can be taken as a testing function for sub-elliptic systems (1.1). Hence, we have where we have used the fact that Xφ = ϕ (Xu − Xl) + ϕ(u − l)Xϕ.
In view of the identities ffl Br(ξ ) A α i (·, l(ξ ), Xl) ξ ,r Xφdξ = , and − Br(ξ ) It follows for weak solutions u of systems (1.1) that with the obvious labelling for I − I . We rst estimate the left-hand side of (4.1). By the rst inequality of (1.2), Young's inequality and de nition of the function V (2.3), we have where we have used the elementary inequality + |a| + |b − a| + |a| + |b| , and < p < . Now, we are going to estimate the terms I − I on the right-hand side of (4.1). For small positive ϵ < appearing in lines, it will be xed later.
Estimate for I . We shall decompose the ball Br(ξ ) into four subsets: and Case 1: Using the second inequality of (1.2), |Xϕ| ≤ r , Young's inequality, and Lemma 2.1, we derive the following bound for I on the subset Ω .
where we have used the inequality ( + |Xl + s(Xu − Xl)|) p− ≤ for < p < . Case 2: Similarly to the case , there is here, we have used the smallness assumption Φ(ξ , r, l) := ffl Br(ξ ) |V(Xu − Xl)| dξ ≤ and ϕ p p− ≤ ϕ . From (4.3), (4.4), (4.5) and (4.6), we have the estimate for the term I as follows where we have used the inequality ε −p ≥ ε − ≥ ε −p for small positive constant ε < . Estimate for I . By the rst inequality of (1.3), we get where we have used in turn Young's inequality, ω ≤ ω, the concavity of ω and Jensen's inequality. Case 2: On the part Ω where |Xu − Xl| > , we nd where we have used the inequality ω p p− ≤ ω. Combining (4.9) with (4.10) leads to where we have use the fact ε −p ≥ ε − for < ε < . The term I can be estimated similarly as I above. Here, we split the ball Br(ξ ) into two subsets Ω := Joining (4.8), (4.11) and (4.14), we obtain We can argue the terms I and I as the same way treating the terms I and I .

Case 1:
On the set Ω where |Xu − Xl| ≤ , we use v ξ ≤ L and (1.6) to infer the following estimate Case 2: On the part Ω where |Xu − Xl| > , we use (1.6) and the fact that v (4.18) Using (4.17) and (4.18), we get Similarly, the term I can be estimated as follows To obtain an appropriate estimate for I , we take the domain Br(ξ ) into four parts as the same way of I .

+ Cp
, we can absorb the rst integral of the right-hand side into the left. Keeping in mind the properties of ϕ, we have thus shown In the sequel, when the choice of ξ or l is clear, we frequently write Φ(r, l) or Φ(r) respectively, as a replacement of Φ(ξ , r, l).

. Approximate A-harmonicity of weak solutions
To apply A-harmonic approximation lemma, we need to establish the following lemma, which provides a linearization strategy for non-linear sub-elliptic systems (1.1).

Lemma 4.2. Under the assumptions of Theorem 1.1 are satis ed, B ρ (ξ )
⊆ Ω with ρ ≤ ρ and an arbitrary horizontal function l : R n → R N , we de ne Proof. Without loss of generality, we assume that sup

then, w is approximately A−harmonic in the sense that
with obvious labelling of J and J . In order to get the bound for the rst term J , we rst use the inequality (1.4) to obtain By the monotonicity of µ and the inequality above, it yields Here, we decompose the ball Bρ(ξ ) into two parts Ω and Ω . Based on the following facts the integral J can be rewritten as with the obvious meaning of J + J + J . Using the assumption of |l(ξ )| + |Xl| ≤ M and VMO-condition (1.5), We nd that where we have used the inequality < p − < in the second line. Now, we discuss it on the domain Ω and Ω , respectively. Case 1: On the set Ω where |Xu − Xl| ≤ , the following estimate holds where we have used the assumption v ξ ≤ L and Lemma 2.1. Then, we get the following estimate for J By rst inequality of (1.3), the term J can be estimated as follows Similarly, for the case of |Xu − Xl| ≤ on Ω , applying Young's inequality, Jensen's inequality and Lemma 2.1, we deduce that Finally, we handle the term J by the same as the way for I to obtain Joining the estimates (4.31)-(4.33) with (4.30), we have |u − l(ξ )| p dξ + V(ρ) Plugging (4.29) and (4.34) into (4.28), we nally arrive at where we have employed the Caccioppoli-type inequality from Lemma 4.1, Ψ * (ρ) ≤ C(n, p)Ψ * ( ρ) in the last step. This yields the claim.

. Excess improvement
The strategy of our proof is to approximate the given solution in the sense of L by A-harmonic functions. Now we are in the position to establish the excess improvement.
First note that, for our choice of the bilinear form Next by Lemma 4.2 with ρ = r and l = l ξ ,r , and the assumptions (i) and (ii), we nd the mapw is approximately A-harmonic in the sense that for all φ ∈ C ∞ (B r (ξ ), R N ), and The estimates (4.35) and (5.2) tell us that the conditions of Lemma 3.3 are satis ed. So, there exists an A- In order to estimate excess functional we now have to handle the integral ffl B θr (ξ ) V(X h(ξ )) dξ . Since the function h(ξ ) is A-harmonic, we know that h(ξ ) ∈ C ∞ (Ω) by Lemma 3.2. Noting that the boundedness |Xh(ξ )| ≤ M in the ball B r (ξ ) ⊂⊂ Ω, and using Hölder's inequality, we have the estimate for θ ∈ ( , ) where we have taken ε = θ Q+ . Scaling back to u, we infer In view of the de ning property of l ξ ,θr , we arrive at here, we have denoted C = C C(C P , C )( + δ − ). Then, it implies excess improvement estimate Ψ(ξ , θr, l ξ ,θr ) ≤ C θ Ψ * (ξ , r, l ξ ,r ).
Proof. We begin by choosing the constants. First, we let where c is de ned in (2.6), and C is determined in Lemma 4.3, respectively. We note that the choice of θ xes the constant δ = δ(Q, N, p, ν, L, θ Q+ ) from Lemma 3.3. Next, we x an ε * small su ciently to ensure (4.39) Then, we choose κ * > so small that ω(κ * ) ≤ ε * .
In view of the well known equivalence of Campanato and Morrey spaces for parameters λ ∈ ( , Q), it yields Xu ∈ L p,λ (U, R n×N ) with λ = Q − p( − γ). In particular, the parameter λ can be chosen arbitrary chose to Q. This concludes the proof of Theorem 1.1.

Partial Hölder continuity for sub-quadratic Natural growth
In this section, we prove the partial regularity result of Theorem 1.2 under the assumptions of sub-quadratic natural structure conditions (H1)-(H4) and (HN). In this case, we will need to restrict ourselves to bounded solution of (1.  (1.8). Such a similar smallness condition is necessary for a partial regularity result even in the elliptic case with quadratic growth (p = ); for example, see [18].
Proof of Theorem1.2. It is enough to use Lemma 5.5, and repeat the procedure for the proof of Theorem 1.1 in the previous Subsection 4.5.