Optimal Hölder Regularity for Discontinuous Sub-Elliptic Systems Structured on Hörmander’s Vector Fields

Abstract

This paper studies discontinuous quasilinear sub-elliptic systems associated with Hörmander’s vector fields under controllable and natural growth conditions. By a new $\mathcal{A}$-harmonic approximation reformulation for bilinear forms $\mathcal{A}\in \mathrm{Bil}({\mathbb{R}}^{kN},{\mathbb{R}}^{kN})$, we obtain optimal partial Hölder continuity with exact exponents for weak solutions with vanishing mean oscillation coefficients.

1. Introduction and Main Results

Assume that $X_1,\cdots ,X_k$ are a family of smooth vector fields defined in a bounded domain $\mathsf{\Omega }\subset {\mathbb{R}}^n$ ($n\ge k$) with the form

$$X_i=\sum _{j=1}^n{b}_{ij}\left(\xi \right){\displaystyle \frac{\partial }{\partial {\xi }_j}},b_{ij}\left(\xi \right)\in C^{\infty }\left(\mathsf{\Omega }\right),i=1,\cdots ,k.$$

In his seminal work [1], Hörmander showed that the sum of squares of such vector fields is hypo-elliptic under the finite rank condition. Trivial cases include the Euclidean setting ($X_i={\displaystyle \frac{\partial }{\partial x_i}},i=1,\cdots ,n$), while non-Abelian examples include Heisenberg and Carnot groups.

Partial regularity for elliptic systems has been widely studied since the work of De Giorgi [2]. Various methods have been developed, including the direct and indirect approaches ([3,4]). A significant development in this area is the introduction of the $\mathcal{A}$-harmonic approximation techniques, associated with bilinear forms $\mathcal{A}$ on ${\mathbb{R}}^{nN}$ by Duzaar and Steffen [5] and refined in [6], later extended to p-Laplacian systems in [7]. Weak solutions under general assumptions on coefficients have been extensively studied in both standard growth and non-standard growth: see [8,9,10,11,12,13] for standard growth and [14,15,16,17] for non-standard cases.

The $\mathcal{A}$-harmonic approximation method is further adapted to Heisenberg and Carnot groups, leading to optimal partial regularity to sub-elliptic systems under various structural conditions, such as [18,19,20,21,22,23]. For the case of sub-elliptic equations, we also refer to [24,25,26] and the references therein.

In general, sub-elliptic equations and systems formed by Hörmander’s vector fields pose greater challenges due to non-commutativity and the lack of homogeneity. After Hörmander’s pioneering work [1], these systems have attracted broad interest, as seen in [22,27,28,29,30] and the references therein. In particular, using the classical direct method, Gao, Niu, and Wang [30] treated the following quasilinear systems in u with vanishing mean oscillation (VMO) coefficients

$$\sum _{i,j=1}^k\sum _{\beta =1}^N{X}_i^{*}\left(A_{ij}^{\alpha \beta }(\xi ,u)X_j{u}^{\beta }\right)=B^{\alpha }(\xi ,u,Xu),\xi \in \mathsf{\Omega },\alpha =1,2,\cdots ,N,$$

(1)

where $X_i^{*}$ denotes the formal adjoint of $X_i$. It should be pointed out that their proof methodology, relying on reverse Hölder inequalities and perturbation arguments, tends to be very technical.

As is known, the idea of $\mathcal{A}$-harmonic approximation serves as an effective approach for examining the regularity of weak solutions of systems; see [8,9,10,12,16,21,23,31]. This work addresses a gap in the literature by providing a novel application of the $\mathcal{A}$-harmonic approximation to system (1). The main contribution is an affirmative answer to the question of whether a unified and simplified proof for optimal regularity is possible under both controlled and natural growth conditions, which we achieve by introducing a modified $\mathcal{A}$-harmonic approximation method. We consider weak solutions of (1) in the following sense:

$${\int }_{\mathsf{\Omega }}A(\xi ,u)XuX\phi d\xi ={\int }_{\mathsf{\Omega }}B(\xi ,u,Xu)\phi d\xi ,\forall \phi \in C_0^{\infty }(\mathsf{\Omega },{\mathbb{R}}^N),$$

(2)

where we denote $A:=\left(A_{ij}^{\alpha \beta }\right):\mathsf{\Omega }\times {\mathbb{R}}^N\to {\mathbb{R}}^{k^2{N}^2}$, and $B:=\left(B^{\alpha }\right):\mathsf{\Omega }\times {\mathbb{R}}^N\times {\mathbb{R}}^{kN}\to {\mathbb{R}}^N$. Unlike previous studies relying on reverse Hölder inequalities, our approach simplifies the analysis by reformulating the $\mathcal{A}$-harmonic approximation. In fact, our tool of choice is to establish and apply the modification of the $\mathcal{A}$-harmonic approximation argument: see Lemma 7 and the details of its proofs below. The proposed approach offers several key advantages. First, our proof circumvents the need for both reverse Hölder inequalities and $L^q$-$L^p$ estimates [30]. Second, it streamlines the proof of partial regularity, as noted in the work of Duzaar and Grotowski [6]. Most notably, our results—which establish an exact Hölder exponent—generalize the theory from coefficients that are continuous in $\xi$ and u to those that are merely of vanishing mean oscillation in the $\xi$-variable.

The strategy of this paper is as follows. We begin by establishing the modification of the $\mathcal{A}$-harmonic approximation, which is the most fundamental tool for proving the optimal partial regularity in this paper. Second, we derive Caccioppoli-type inequalities for weak solutions of the system (1), which serves as a crucial tool in our analysis. Next, using the refined $\mathcal{A}$-harmonic approximation argument, we obtain an energy type estimate for a functional that quantifies the local oscillation of the weak solution, up to a small excess. Finally, via an iteration lemma, we prove the boundedness of this functional, which implies the desired Hölder regularity.

We shall now state the following precise hypotheses (H1, H2, HC, and HN) governing the coefficients $A_{ij}^{\alpha \beta }$ and $B^{\alpha }$, which are assumed throughout this work. First, we proceed with the definition of the vanishing mean oscillation function space.

Definition 1

(BMO and VMO space). A function $u\in L_{\mathrm{loc}}^1\left(\mathsf{\Omega }\right)$ is said to be of Bounded Mean Oscillation (BMO) in Ω—denoted by $u\in BMO\left(\mathsf{\Omega }\right)$—if for any $0<\rho <\infty$, the following condition holds

$$M_{\rho }(u,\mathsf{\Omega })={\parallel u\parallel }_{\mathrm{BMO}\left(\mathsf{\Omega }\right)}:=\underset{\xi \in \mathsf{\Omega }}{sup}\underset{0<r<\rho }{sup}{⨏}_{\mathsf{\Omega }(\xi ,r)}|u\left(\eta \right)-u_{\xi ,r}|d\eta <+\infty ,$$

where $\mathsf{\Omega }(\xi ,r)=\mathsf{\Omega }\cap B(\xi ,r)$, and $u_{\xi ,r}={⨏}_{\mathsf{\Omega }(\xi ,r)}u\left(\eta \right)d\eta ={|\mathsf{\Omega }(\xi ,r)|}^{-1}{\int }_{\mathsf{\Omega }(\xi ,r)}u\left(\eta \right)d\eta$. The space $\mathrm{BMO}\left(\mathsf{\Omega }\right)$ is a Banach space modulo constant equipped with the seminorm ${\parallel ·\parallel }_{\mathrm{BMO}\left(\mathsf{\Omega }\right)}$. Moreover, $u\in \mathit{BMO}\left(\mathsf{\Omega }\right)$ is said to belong to VMO$\left(\mathsf{\Omega }\right)$ if and only if

$$M_0\left(u\right)=\underset{\rho \to 0}{lim}{M}_{\rho }(u,\mathsf{\Omega })=0.$$

The space $\mathrm{VMO}\left(\mathsf{\Omega }\right)$ is a closed subspace of $\mathrm{BMO}\left(\mathsf{\Omega }\right)$.

H1 

(Uniform ellipticity). There exist two constants $0<\lambda \le \mathsf{\Lambda }$ such that

$${\lambda |\eta |}^2\le A_{ij}^{\alpha \beta }(\xi ,u){\eta }_i^{\alpha }{\eta }_j^{\beta }\le \mathsf{\Lambda }{|\eta |}^2,\forall \xi \in \mathsf{\Omega },u\in {\mathbb{R}}^N,\eta \in {\mathbb{R}}^{kN}.$$

(3)

H2 

(Minimal regularity on $A_{ij}^{\alpha \beta }$). The leading coefficient $A_{ij}^{\alpha \beta }(\xi ,u)$ is VMO in ξ with uniform respect to $u\in {\mathbb{R}}^N$ and is continuous in u with uniform respect to $\xi \in \mathsf{\Omega }$. That is, $\underset{\rho \to 0}{lim}{M}_{\rho }\left(A_{ij}^{\alpha \beta }(\xi ,u)\right)=0$, and there exist constants $L\in [1,\infty )$ and a bounded, concave, and non-decreasing modulus of continuity $\omega :[0,\infty )\to [0,1]$ with $\underset{s\to 0}{lim}\omega \left(s\right)=0=\omega \left(0\right)$ such that

$$\left|A_{ij}^{\alpha \beta }(\xi ,u)-A_{ij}^{\alpha \beta }(\xi ,u_0)\right|\le L\omega \left(|u-u_0{|}^2\right),\forall u,u_0\in {\mathbb{R}}^N,\xi \in \mathsf{\Omega }.$$

(4)

HC 

(Controllable growth). For some $C>0$, the inhomogeneity $B^{\alpha }(\xi ,u,Xu)$ satisfies

$$|B^{\alpha }(\xi ,u,Xu)|\le C({|Xu|}^{2(1-{\displaystyle \frac{1}{\gamma }})}+{|u|}^{\gamma -1}+f^{\alpha }),$$

(5)

where $f^{\alpha }\in L^q\left(\mathsf{\Omega }\right)$ for $q>{\displaystyle \frac{2Q}{Q+2}}$, and $\gamma ={\displaystyle \frac{2Q}{Q-2}}$ if $2<Q$, with the number Q being a locally homogeneous dimension related to Hörmander’s vector fields.

HN 

(Natural growth). For $|u|\le M=\mathit{ess}{\mathit{sup}}_{\xi \in \mathsf{\Omega }}|u\left(\xi \right)|$, the inhomogeneity $B^{\alpha }$ satisfies

$$|B^{\alpha }{(\xi ,u,Xu)|\le \mu \left(M\right)(|Xu|}^2+f^{\alpha }),$$

(6)

where $\mu \left(M\right)$ is a constant possibly depending on M, and $f^{\alpha }\in L^q\left(\mathsf{\Omega }\right),q>max\{{\displaystyle \frac{Q}{2}},2\}.$

This work is devoted to establishing optimal partial Hölder regularity with exact exponents by a new $\mathcal{A}$-harmonic approximation reformulation. Our main theorems are stated below.

Theorem 1.

Assume H1–H2 and HC hold for coefficients $A_{ij}^{\alpha \beta }$ and $B^{\alpha }$. Then for any weak solution $u\in H{W}_{\mathrm{loc}}^{1,2}(\mathsf{\Omega },{\mathbb{R}}^N)$ of system (1), there exists an open ${\mathsf{\Omega }}_0\subset \mathsf{\Omega }$ with ${\mathcal{H}}^{Q-2}(\mathsf{\Omega }\setminus {\mathsf{\Omega }}_0)=0$ such that

• $u\in {\mathsf{\Gamma }}_{\mathrm{loc}}^{0,2-{\displaystyle \frac{Q}{q}}}({\mathsf{\Omega }}_0,{\mathbb{R}}^N)$, ${\displaystyle \frac{2Q}{Q+2}}<q<Q$.
• $u\in {\mathsf{\Gamma }}_{\mathrm{loc}}^{0,\alpha }({\mathsf{\Omega }}_0,{\mathbb{R}}^N)$ for all $\alpha \in (0,1)$, $q\ge Q$.

where the regular set is characterized by ${\mathsf{\Omega }}_0=\left\{{\xi }_0\in \mathsf{\Omega }:\underset{\rho \to 0}{lim}sup\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi \right)=0\right\}.$ Particularly, when $X_i={\displaystyle \frac{\partial }{\partial x_i}}$, these results reduce to the classical Euclidean case for quasilinear elliptic systems.

Remark 1.

Theorem 1 implies that the weak solution u possesses Hölder regularity with the exact exponent $2-{\displaystyle \frac{Q}{q}}$ in the sub-elliptic setting. This indicates that the Hölder exponent increases as the integrability parameter q (from the condition $f\in L_{\mathrm{loc}}^q\left(\mathsf{\Omega }\right)$) increases, which in turn implies the higher regularity of the solution.

Theorem 2.

Assume H1–H2 and HN hold for coefficients $A_{ij}^{\alpha \beta }$ and $B^{\alpha }$. Then for any weak solution $u\in H{W}_{\mathrm{loc}}^{1,2}(\mathsf{\Omega },{\mathbb{R}}^N)\cap L^{\infty }(\mathsf{\Omega },{\mathbb{R}}^N)$ of system (1), there exists an open ${\mathsf{\Omega }}_0\subset \mathsf{\Omega }$ with ${\mathcal{H}}^{Q-2}(\mathsf{\Omega }\setminus {\mathsf{\Omega }}_0)=0$ such that

• $u\in {\mathsf{\Gamma }}_{loc}^{0,1-{\displaystyle \frac{Q}{2q}}}({\mathsf{\Omega }}_0,{\mathbb{R}}^N)$, $q>max\{{\displaystyle \frac{Q}{2}},2\}$.
• $Xu\in L^{2,\lambda }({\mathsf{\Omega }}_0,{\mathbb{R}}^{kN})$ with $\lambda =Q-{\displaystyle \frac{Q}{q}}$.

In particular, when $X_i={\displaystyle \frac{\partial }{\partial x_i}}$, these results reduce to the classical Euclidean case for quasilinear elliptic systems.

Remark 2.

It is noted that under natural growth conditions, the Campanato estimate gives the exponent $\lambda =Q+2-{\displaystyle \frac{Q}{q}}$, and Campanato’s characterization leads to Hölder regularity with exponent $1-{\displaystyle \frac{Q}{2q}}$. Under controlled growth conditions, the Morrey estimate for the gradient of the weak solution yields the Morrey exponent $\lambda =Q+2-{\displaystyle \frac{2Q}{q}}$. Applying Morrey’s lemma then implies Hölder continuity with exponent $2-{\displaystyle \frac{Q}{q}}=2(1-{\displaystyle \frac{Q}{2q}})$. This clearly shows that the regularity result in the controlled growth case is stronger than that in the natural growth case.

Theorem 3.

Under assumptions H1–H2 and assuming that the inhomogeneity $|B^{\alpha }{(\xi ,u,Xu)|\le \mu \left(M\right)(|Xu|}^{2-ϵ}+f^{\alpha })$ for $ϵ>0$ is sufficiently small, the regularity conclusions of Theorem 2 hold without requiring the condition $2\mu \left(M\right)M<\lambda$.

Remark 3.

Generally, the condition $2\mu \left(M\right)M<\lambda$ in Theorem 2 is essential. However, Theorem 3 shows that it can be removed by slightly strengthening the constraint on $B^{\alpha }$ while preserving the same regularity result.

This paper is structured as follows. Section 2 provides a brief introduction to Hörmander’s vector fields, along with a summary of known results related to these vector fields. In Section 3, we present a reformulation of the $\mathcal{A}$-harmonic approximation technique, building upon the work of Wang and Liao [32] in the context of vector fields. Section 4 is devoted to the proof of the partial regularity result stated in Theorem 1 under the controllable structure assumptions H1–H2 and HC. This is achieved in several steps: we first give a priori estimates for weak solutions to homogeneous systems with constant coefficients; then establish a Caccioppoli-type inequality, which serves as a fundamental tool in partial regularity theory; and finally prove the main result using the adapted $\mathcal{A}$-harmonic approximation argument. The final section presents the proofs of Theorems 2 and 3 under the natural growth conditions H1–H2 and HN.

2. Preliminaries

Let $\mathsf{\Omega }\subset {\mathbb{R}}^n$ be a bounded, open, and path-connected domain, and let ${\left\{X_i\right\}}_{i=1}^k$ ( $k\le n$) be a family of $C^{\infty }$ real-valued vector fields defined in a neighborhood of the closure $\overline{\mathsf{\Omega }}$ of $\mathsf{\Omega }$. For a multi-index $\alpha =(i_1,i_2,\cdots ,i_k)$ where each $i_j\in \{1,\cdots ,k\}$, we denote by $X_{\alpha }$ the commutator

$$[X_{i_1},[X_{i_2},\cdots ,[X_{i_{k-1}},X_{i_k}]]]$$

of length $l=|\alpha |:={\displaystyle \sum _{j=1}^k}{i}_j=i_1+i_2+\cdots +i_k$. The vector fields are said to satisfy Hörmander’s condition if there exists a positive integer s such that the set ${\left\{X_{\alpha }\right\}}_{|\alpha |\le s}$ spans the tangent space of ${\mathbb{R}}^n$ at every point of $\mathsf{\Omega }$; that is, rank Lie $[X_1,\cdots ,X_s]\equiv n$.

Now, we introduce a metric as follows. An admissible curve $\gamma$ is a Lipschitz continuous path $\gamma :[0,b]\to \mathsf{\Omega },$ if there exist measurable functions $c_i\left(t\right)$, $i=1,\cdots ,k$, defined on $[0,b]$ such that

$$\sum _{i=1}^k{c}_i{\left(t\right)}^2\le 1\mathrm{and}{\gamma }^{\prime }\left(t\right)=\sum _{i=1}^k{c}_i\left(t\right)X_i\left(\gamma \left(t\right)\right)\mathrm{for}\mathrm{almost}\mathrm{all}t\in [0,b].$$

The Carnot–Carathéodory (C–C) metric on $\mathsf{\Omega }$ associated with the vector fields ${\left\{X_i\right\}}_{i=1}^k$ is then defined by

$${\varrho }_{cc}(\xi ,\eta )=inf\left\{b\ge 0:\exists \mathrm{an}\mathrm{admissible}\mathrm{curve}\gamma :[0,b]\to \mathsf{\Omega }\mathrm{such}\mathrm{that}\gamma \left(0\right)=\xi ,\gamma \left(b\right)=\eta \right\}.$$

(7)

From definition (7), the C–C ball of radius r centered at $\xi$ is defined as $B_r\left(\xi \right)=\left\{\eta \in \mathsf{\Omega }:{\varrho }_{cc}(\xi ,\eta )<r\right\}.$ A fundamental doubling property of the Lebesgue measure with respect to these metric balls was established by Nagel, Stein, and Wainger in [33]. Specifically, for any bounded set $\mathsf{\Omega }\subset {\mathbb{R}}^n$, there exist positive constants $R_0$ and $C_0$ such that

$$\left|B_{2R}\left(\xi \right)\right|\le C_0\left|B_R\left(\xi \right)\right|,\mathrm{for}\mathrm{all}\xi \in {\mathsf{\Omega }}^{\prime }\mathrm{and}0<R<R_0.$$

(8)

If $Q={log}_2{C}_0$, then the number Q is referred to as the local homogeneous dimension relative to $\mathsf{\Omega }$ and the system ${\left\{X_i\right\}}_{i=1}^k$.

In the special case when $X={\left\{{\displaystyle \frac{\partial }{\partial x_i}}\right\}}_{i=1}^n$ is a family of the standard basis of the Euclidean space ${\mathbb{R}}^n$, then $C_0=2^n$, and $Q=n$.

We now present the definitions of several relevant function spaces.

Definition 2

(Sobolev space). Let $1\le p<+\infty$. We denote by

$$H{W}^{1,p}\left(\mathsf{\Omega }\right)=\left\{u\in L^p\left(\mathsf{\Omega }\right)|X_{i}u\in L^p\left(\mathsf{\Omega }\right),i=1,\cdots ,k\right\}$$

(9)

the Sobolev space. Then $H{W}^{1,p}\left(\mathsf{\Omega }\right)$ is a Banach space under the norm

$${∥u∥}_{H{W}^{1,p}\left(\mathsf{\Omega }\right)}={∥u∥}_{L^p\left(\mathsf{\Omega }\right)}+\sum _{i=1}^k{∥X_{i}u∥}_{L^p\left(\mathsf{\Omega }\right)}.$$

The local Sobolev space $H{W}_{\mathrm{loc}}^{1,p}\left(\mathsf{\Omega }\right)$ is defined as the set of all functions $u:\mathsf{\Omega }\to \mathbb{R}$ such that for every open set U with compact closure $\overline{U}\subset \mathsf{\Omega }$, we have $u\in H{W}^{1,p}\left(U\right)$.

Definition 3

(Morrey space). Let $1⩽p<\infty$, $\lambda >0$ and write

$$L^{p,\lambda }\left(\mathsf{\Omega }\right)=\left\{u\in L^p\left(\mathsf{\Omega }\right)|\underset{\xi \in \mathsf{\Omega },\rho <diam\mathsf{\Omega }}{sup}{\rho }^{-\lambda }{\int }_{B_{\rho }\left(\xi \right)\cap \mathsf{\Omega }}{\left|u\left(\zeta \right)\right|}^{p}d\zeta <\infty \right\}.$$

We say that $L^{p,\lambda }(\mathsf{\Omega },{\mathbb{R}}^N)$ is a Morrey space with the norm

$${∥u∥}_{L^{p,\lambda }\left(\mathsf{\Omega }\right)}={∥u∥}_{L^p\left(\mathsf{\Omega }\right)}+{\left(\underset{\xi \in \mathsf{\Omega },\rho <diam\mathsf{\Omega }}{sup}{\rho }^{-\lambda }{\int }_{B_{\rho }\left(\xi \right)\cap \mathsf{\Omega }}{\left|u\left(\zeta \right)\right|}^{p}d\zeta \right)}^{{\displaystyle \frac{1}{p}}}.$$

Definition 4

(Campanato space). Let $1⩽p<\infty$, $\lambda >0$ and write

$${\mathcal{L}}^{p,\lambda }\left(\mathsf{\Omega }\right)=\left\{u\in L^p\left(\mathsf{\Omega }\right)|\underset{\xi \in \mathsf{\Omega },\rho <diam\mathsf{\Omega }}{sup}{\rho }^{-\lambda }{\int }_{B_{\rho }\left(\xi \right)\cap \mathsf{\Omega }}{\left|u\left(\zeta \right)-u_{\xi ,\rho }\right|}^{p}d\zeta <\infty \right\},$$

where $u_{\xi ,\rho }$ denotes the average of u over the ball $B_{\rho }\left(\xi \right)$ intersected with Ω. The Campanato space ${\mathcal{L}}^{p,\lambda }\left(\mathsf{\Omega }\right)$ is equipped with the norm

$${∥u∥}_{{\mathcal{L}}^{p,\lambda }\left(\mathsf{\Omega }\right)}={∥u∥}_{L^p\left(\mathsf{\Omega }\right)}+{\left(\underset{\xi \in \mathsf{\Omega },\rho <diam\mathsf{\Omega }}{sup}{\rho }^{-\lambda }{\int }_{B_{\rho }\left(\xi \right)\cap \mathsf{\Omega }}{\left|u\left(\zeta \right)-u_{\xi ,\rho }\right|}^{p}d\zeta \right)}^{{\displaystyle \frac{1}{p}}}.$$

Definition 5

(Folland–Stein space). Let $\alpha \in (0,1)$. The Folland–Stein space ${\mathsf{\Gamma }}^{0,\alpha }\left(\mathsf{\Omega }\right)$ is defined as

$${\mathsf{\Gamma }}^{0,\alpha }\left(\mathsf{\Omega }\right)=\left\{u\in L^{\infty }\left(\mathsf{\Omega }\right)|\underset{\xi ,\eta \in \mathsf{\Omega },\xi \ne \eta }{sup}{\displaystyle \frac{\left|u\left(\xi \right)-u\left(\eta \right)\right|}{{\varrho }_{cc}^{\alpha }(\xi ,\eta )}}<\infty \right\},$$

where ${\varrho }_{cc}$ denotes the Carnot–Carathéodory metric defined in (7) below. It is equipped with the norm

$$\begin{array}{ccc}\hfill {∥u∥}_{{\mathsf{\Gamma }}^{0,\alpha }\left(\mathsf{\Omega }\right)}& =& {∥u∥}_{L^{\infty }\left(\mathsf{\Omega }\right)}+\underset{\xi ,\eta \in \mathsf{\Omega },\xi \ne \eta }{sup}{\displaystyle \frac{\left|u\left(\xi \right)-u\left(\eta \right)\right|}{{\varrho }_{cc}^{\alpha }(\xi ,\eta )}}.\hfill \end{array}$$

(10)

The following Poincaré-type inequality related to Hörmander’s vector fields was established by Lu in [34].

Lemma 1

(Poincaré-type inequality). There exist constants $C_P>0$ and $R_0>0$ such that for any ball $B_R\left({\xi }_0\right)$ with $0<R<R_0$ and any function $u\in H{W}^{1,q}\left(B_R\left({\xi }_0\right)\right)$ with $1<q<Q$, the following inequality holds for all $1\le p\le {\displaystyle \frac{qQ}{Q-q}}$:

$${\left({⨏}_{B_R\left({\xi }_0\right)}{|u\left(\xi \right)-u_{{\xi }_0,R}|}^{p}d\xi \right)}^{{\displaystyle \frac{1}{p}}}\le C_{P}R{\left({⨏}_{B_R\left({\xi }_0\right)}{|Xu\left(\xi \right)|}^{q}d\xi \right)}^{{\displaystyle \frac{1}{q}}}.$$

(11)

where the constant $C_P$ depends only on q and the homogeneous dimension Q. Without loss of generality, we may assume $C_P>1$.

We now introduce the integral form of Jensen’s inequality for concave functions, and we refer to [3] for details.

Lemma 2

(Jensen’s inequality). Let $\omega :\mathbb{R}\to \mathbb{R}$ be a concave function, and $f\in L^1\left(\mathsf{\Omega }\right)$. Then the following inequality holds:

$${⨏}_{\mathsf{\Omega }}\omega \left(f\right)d\mu \le \omega \left({⨏}_{\mathsf{\Omega }}fd\mu \right).$$

(12)

The following iteration lemma will be used in the proof of the Hölder estimate; see [4].

Lemma 3

(Iteration lemma). Let $\mathsf{\Phi }\left(\rho \right)$ be a non-negative and non-decreasing function defined on the interval $(0, R)$. Suppose there exist non-negative constants a, b, α, and β with $\alpha >\beta$, such that

$$\mathsf{\Phi }\left(\rho \right)\le a\left[{\left({\displaystyle \frac{\rho }{R}}\right)}^{\alpha }+\epsilon \right]\mathsf{\Phi }\left(R\right)+b{R}^{\beta },\forall 0<\rho <R<dist({\xi }_0,\partial \mathsf{\Omega }).$$

Then there exist constants ${\epsilon }_0={\epsilon }_0(a,\alpha ,\beta )>0$ and $c=c(a,\alpha ,\beta )>0$ such that if $0<\epsilon <{\epsilon }_0$, we have

$$\mathsf{\Phi }\left(\rho \right)\le c\left[{\left({\displaystyle \frac{\rho }{R}}\right)}^{\beta }\mathsf{\Phi }\left(R\right)+b{\rho }^{\beta }\right],\forall 0<\rho <R<dist({\xi }_0,\partial \mathsf{\Omega }).$$

The following Morrey’s lemma will be used to establish the Hölder regularity of weak solutions. Readers may refer to the literature [3].

Lemma 4

(Morrey’s lemma). Let $u\in W^{1,2}(B_R\left({\xi }_0\right),{\mathbb{R}}^N)$ and $B_R\left({\xi }_0\right)\subset \mathsf{\Omega }$. If there exist a positive constant C and some $\alpha \in (0, 1)$ such that

$${\int }_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi \le C{\rho }^{Q-2+2\alpha },$$

then $u\in {\mathsf{\Gamma }}^{0,\alpha }(B_R\left({\xi }_0\right),{\mathbb{R}}^N)$.

Finally, we state a lemma concerning the estimate for Hausdorff dimensional measure, which concludes this section; see [35].

Lemma 5

(Hausdorff measure estimate). Let Ω be an open subset of ${\mathbb{R}}^n$ and $u\in L_{loc}^1(\mathsf{\Omega },{\mathbb{R}}^N)$. Then for $0\le s<Q$ and setting

$$E_s=\left\{\xi \in \mathsf{\Omega }:\underset{\rho \to 0}{lim}sup{\rho }^{-s}{\int }_{B_{\rho }\left(\xi \right)}|u(\eta )|d\eta >0\right\},$$

(13)

the s-dimensional Hausdorff measure of $E_s$ satisfies

$${\mathcal{H}}^s\left(E_s\right)=0.$$

3. 𝒜-Harmonic Approximation Reformulation

For a bilinear form $\mathcal{A}$ on ${\mathbb{R}}^{kN}$ (i.e., $\mathcal{A}\in \mathrm{Bil}({\mathbb{R}}^{kN},{\mathbb{R}}^{kN})$, in this section we establish an $\mathcal{A}$-harmonic approximation reformulation. Throughout this discussion, we say that a map $h\in C^{\infty }(B_{\rho }\left({\xi }_0\right),{\mathbb{R}}^N)$ is $\mathcal{A}$-harmonic if and only if

$${\int }_{B_{\rho }\left({\xi }_0\right)}\mathcal{A}(Xh,X\phi )d\xi =0$$

holds for every test function $\phi \in C_0^{\infty }(B_{\rho }\left({\xi }_0\right),{\mathbb{R}}^N)$.

Remark 4.

When $\mathcal{A}(p,p)={|p|}^2$ for $p\in {\mathbb{R}}^{kN}$, an $\mathcal{A}$-harmonic function h corresponds to a classical harmonic function satisfying

$${\Delta }_{H}h=\sum _{i=1}^k{X}_i^{2}h=0,$$

where ${\Delta }_H$ denotes the sub-Laplacian operator.

The following $\mathcal{A}$-harmonic approximation lemma for Hörmander’s vector fields was established by Wang and Liao; see [32]. Lemma 6 implies that if a function g is an approximate solution to a constant-coefficient system with coefficient matrix $\mathcal{A}$, then there exists an exact $\mathcal{A}$-harmonic function h that approximates g closely in the $L^2$ sense.

Lemma 6

($\mathcal{A}$-harmonic approximation). Consider fixed positive λ and L, as well as $k,N\in \mathbb{N}$ with $k\ge 2$. Assume that

$$\mathcal{A}(\nu ,\nu )\ge \lambda {\left|\nu \right|}^2,\mathcal{A}(\mu ,\nu )\le L\left|\mu \right|\left|\nu \right|,\mu ,\nu \in {\mathbb{R}}^{kN},\mathcal{A}\in \mathit{Bil}({\mathbb{R}}^{kN},{\mathbb{R}}^{kN})$$

(14)

and

$${⨏}_{B_{\rho }\left({\xi }_0\right)}{\left|Xg\right|}^{2}d\xi \le 1,g\in H{W}^{1,2}(B_{\rho }\left({\xi }_0\right),{\mathbb{R}}^N).$$

(15)

Then for any given $\epsilon >0$, there exists $\delta =\delta (k,N,\lambda ,\epsilon )\in (0,1]$ such that if

$$|{⨏}_{B_{\rho }\left({\xi }_0\right)}\mathcal{A}(Xg,X\phi )d\xi |\le \delta \underset{B_{\rho }\left({\xi }_0\right)}{sup}\left|X\phi \right|,\forall \phi \in C_0^1(B_{\rho }\left({\xi }_0\right),{\mathbb{R}}^N)$$

(16)

is satisfied, then

$${\rho }^{-2}{⨏}_{B_{\rho }\left({\xi }_0\right)}{\left|h-g\right|}^{2}d\xi \le ϵ$$

(17)

holds, whenever h is an $\mathcal{A}$-harmonic function in $H{W}^{1,2}(B_{\rho }\left({\xi }_0\right),{\mathbb{R}}^N)$ satisfying ${⨏}_{B_{\rho }\left({\xi }_0\right)}{\left|Xh\right|}^{2}d\xi \le 1$.

Based on the aforementioned $\mathcal{A}$-harmonic approximation lemma, we now proceed to establish the following reformulated version.

Lemma 7

($\mathcal{A}$-harmonic approximation reformulation). Let $0<\lambda \le \mathsf{\Lambda }<\infty$. Then, for any given $\epsilon >0$, there exist positive constants $\kappa =\kappa (n,N,\lambda ,\mathsf{\Lambda },\epsilon )>0$ with the following property: For any $\mathcal{A}\in Bil\left({\mathbb{R}}^{k^2{N}^2}\right)$ satisfying (14), and any $u\in H{W}^{1,2}(B_R\left({\xi }_0\right),{\mathbb{R}}^N)$, there exists an $\mathcal{A}$-harmonic function $h\in H{W}^{1,2}(B_R\left({\xi }_0\right),{\mathbb{R}}^N)$ such that

$${⨏}_{B_R\left({\xi }_0\right)}{|Xh|}^{2}d\xi \le {⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi ,$$

(18)

and moreover, there exists a test function $\phi \in C_0^{\infty }(B_R\left({\xi }_0\right),{\mathbb{R}}^N)$ satisfying

$${\parallel X\phi \parallel }_{L^{\infty }}\le {\displaystyle \frac{1}{R}},$$

(19)

such that the following estimate holds:

$${⨏}_{B_R\left({\xi }_0\right)}{|u-h|}^{2}d\xi \le \epsilon R^2{⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi +\kappa {\left(R^2{⨏}_{B_R\left({\xi }_0\right)}\mathcal{A}(Xu,X\phi )d\xi \right)}^2.$$

(20)

Proof. 

For any given $\epsilon >0$, we choose $\delta =\delta (n,N,\lambda ,\mathsf{\Lambda },\epsilon )$ as in Lemma 6 above. For $u\in H{W}^{1,2}(B_R\left({\xi }_0\right),{\mathbb{R}}^N)$, we define

$$g=u{\left({⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)}^{-1/2},$$

and then we obtain ${⨏}_{B_R\left({\xi }_0\right)}{|Xg|}^{2}d\xi =1$, which implies that condition (15) is satisfied. We now consider two cases.

Case 1. If the function g satisfies inequality (16), then by Lemma 6, there exists an $\mathcal{A}$-harmonic function $\omega$ such that

$${⨏}_{B_R\left({\xi }_0\right)}{|X\omega |}^{2}d\xi \le 1\mathrm{and}{⨏}_{B_R\left({\xi }_0\right)}{|\omega -g|}^{2}d\xi \le \epsilon R^2.$$

We take $h={\left({⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)}^{1/2}\omega$. Then it is easy to see that h is $\mathcal{A}$-harmonic and

$${⨏}_{B_R\left({\xi }_0\right)}{|Xh|}^{2}d\xi ={⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi {⨏}_{B_R\left({\xi }_0\right)}{|X\omega |}^{2}d\xi \le {⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi .$$

This implies (18).

Noting the fact that $u={\left({⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)}^{1/2}g$, we have

$${|u-h|}^2={|g-\omega |}^2{⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi ,$$

which yields

$${⨏}_{B_R\left({\xi }_0\right)}{|u-h|}^{2}d\xi \le {⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi {⨏}_{B_R\left({\xi }_0\right)}{|g-\omega |}^{2}d\xi \le \epsilon R^2{⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi .$$

So inequality (20) holds.

Case 2. If inequality (16) fails for the function g, then there exists a nonconstant function $\psi \in C_0^{\infty }(B_R\left({\xi }_0\right),{\mathbb{R}}^N)$ such that

$$|{⨏}_{B_R\left({\xi }_0\right)}\mathcal{A}(Xg,X\psi )d\xi |>\delta \underset{B_R\left({\xi }_0\right)}{sup}|X\psi |.$$

By taking ${\displaystyle \phi =\frac{\psi }{R\underset{B_R\left({\xi }_0\right)}{sup}|X\psi |}}$, we get ${\displaystyle {\parallel X\phi \parallel }_{L^{\infty }}=\frac{1}{R}}$, which implies

$${\displaystyle \frac{1}{\delta }}|{⨏}_{B_R\left({\xi }_0\right)}\mathcal{A}(Xg,X\phi )d\xi |>{\displaystyle \frac{1}{R}}.$$

By selecting $h=u_{{\xi }_0,R}$, it follows that by Poincaré-type inequality (11),

$$\begin{array}{cc}\hfill {⨏}_{B_R\left({\xi }_0\right)}{|u-h|}^{2}d\xi =& {⨏}_{B_R\left({\xi }_0\right)}|u-u_{{\xi }_0,R}{|}^{2}d\xi \hfill \\ \hfill \le & C_p^2{R}^2{⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi \hfill \\ \hfill \le & {\displaystyle \frac{C_p^2{R}^4}{{\delta }^2}}{⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi |{⨏}_{B_R\left({\xi }_0\right)}\mathcal{A}(Xg,X\phi )d\xi {|}^2\hfill \\ \hfill \le & {\displaystyle \frac{C_p^2{R}^4}{{\delta }^2}}|{⨏}_{B_R\left({\xi }_0\right)}\mathcal{A}(Xu,X\phi )d\xi {|}^2,\hfill \end{array}$$

where we used the fact that $Xg={\left({⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)}^{-1/2}Xu$ in the last inequality. Taking ${\displaystyle \kappa =\frac{C_p^2}{{\delta }^2}}$, and combining the conclusions from both Case 1 and Case 2, we obtain the desired inequality (20), which completes the proof. □

4. Partial Hölder Regularity for Controllable Growth

In this section, we proceed to the proof of Theorem 1 under the controllable growth conditions H1–H2 and HC. Xu and Zuily in [22] establish a priori estimates for weak solutions to homogeneous systems with constant coefficients of the form

$$\sum _{i,j=1}^k\sum _{\beta =1}^N{A}_{ij}^{\alpha \beta }{X}_i{X}_j{u}^{\beta }=0,\alpha =1,\cdots ,N.$$

(21)

Since $\mathcal{A}\in \mathrm{Bil}({\mathbb{R}}^{kN},{\mathbb{R}}^{kN})$ is a bilinear form with constant tensorial coefficients, the following estimate holds for any $\mathcal{A}$-harmonic function h.

Lemma 8.

Let $h\left(\xi \right)\in H{W}^{1,2}(B_R\left({\xi }_0\right),{\mathbb{R}}^N)$ be weak solutions of the homogeneous system with constant coefficients. Then there exists a constant $C_{\alpha }$ such that, for any $0<\rho <R\le dist({\xi }_0,\partial \mathsf{\Omega })$ with ${\xi }_0\in \mathsf{\Omega }$, we have

$${⨏}_{B_{\rho }\left({\xi }_0\right)}{|Xh|}^{2}d\xi \le C_{\alpha }{⨏}_{B_R\left({\xi }_0\right)}{|Xh|}^{2}d\xi ,$$

(22)

and

$${⨏}_{B_{\rho }\left({\xi }_0\right)}{|Xh-\left(Xh\right)}_{{\xi }_0,\rho }{|}^{2}d\xi \le C_{\alpha }{\left({\displaystyle \frac{\rho }{R}}\right)}^2{⨏}_{B_R\left({\xi }_0\right)}{|Xh-{\left(Xh\right)}_{{\xi }_0,R}|}^{2}d\xi .$$

(23)

4.1. Caccioppoli-Type Inequality for Controllable Growth

The Caccioppoli-type inequality serves as a fundamental and essential tool in the study of regularity for weak solutions. Now let us establish the following Caccioppoli-type inequality for weak solutions of system (1) under assumptions H1–H2 and HC.

Lemma 9

(Caccioppoli-type inequality for controllable growth). Let $u\in H{W}_{\mathrm{loc}}^{1,2}(\mathsf{\Omega },{\mathbb{R}}^N)$ be a weak solution of the quasilinear sub-elliptic system (1) under assumptions H1–H2 and HC. Then for any ${\xi }_0\in \mathsf{\Omega }$ and $0<r\le 1$ such that $B_r\left({\xi }_0\right)⋐\mathsf{\Omega }$, we have the estimation

$${⨏}_{B_{r/2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \le C_c\left[{⨏}_{B_r\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,r}}{r}}{|}^{2}d\xi +r^2{\left({⨏}_{B_r\left({\xi }_0\right)}{(|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}})d\xi \right)}^{2(1-{\displaystyle \frac{1}{\gamma }})}\right].$$

(24)

Proof. 

We test the sub-elliptic system (2) with the test function $\phi ={\varphi }^2(u-u_{{\xi }_0,r}),$ where $\varphi \in C_0^{\infty }\left(B_r\left({\xi }_0\right)\right)$ is a cut-off function satisfying $0\le \varphi \le 1,|X\varphi |\le {\displaystyle \frac{1}{r}},\mathrm{and}\varphi \equiv 1\mathrm{on}{B}_{r/2}\left({\xi }_0\right).$ This leads to the following identity:

$${\int }_{B_r\left({\xi }_0\right)}A(\xi ,u){\varphi }^{2}XuXud\xi =-2{\int }_{B_r\left({\xi }_0\right)}A(\xi ,u)Xu\left(\varphi (u-u_{{\xi }_0,r})X\varphi \right)d\xi +{\int }_{B_r\left({\xi }_0\right)}B(\xi ,u,Xu)\phi d\xi .$$

Applying ellipticity $\left(\mathbf{H}\mathbf{1}\right)$ and controllable growth $\left(\mathbf{HC}\right)$, one gets

$$\begin{array}{cc}\hfill \lambda & {\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi \hfill \\ \hfill \le & 2\mathsf{\Lambda }{\int }_{B_r\left({\xi }_0\right)}|\varphi Xu||(u-u_{{\xi }_0,r})X\varphi |d\xi +C{\int }_{B_r\left({\xi }_0\right)}{(|Xu|}^{2(1-{\displaystyle \frac{1}{\gamma }})}+{|u|}^{\gamma -1}+|f|)|\phi |d\xi \hfill \\ \hfill :=& I+II\hfill \end{array}$$

(25)

with the obvious integral labeling for I and $II$.

By Young’s inequality with $\epsilon >0$, it follows that

$$I\le \epsilon {\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi +{\displaystyle \frac{C\left(\epsilon \right)}{r^2}}{\int }_{B_r\left({\xi }_0\right)}{|u-u_{{\xi }_0,r}|}^{2}d\xi .$$

(26)

By Hölder’s inequality, Sobolev-type inequality, and Young’s inequality with $\epsilon >0$ in turn, the following is yielded:

$$\begin{array}{cc}\hfill II=& C{\int }_{B_r\left({\xi }_0\right)}{(|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}}{)}^{1-{\displaystyle \frac{1}{\gamma }}}|\phi |d\xi \hfill \\ \hfill \le & C{\left({\int }_{B_r\left({\xi }_0\right)}{|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}}d\xi \right)}^{1-{\displaystyle \frac{1}{\gamma }}}{\left({\int }_{B_r\left({\xi }_0\right)}{|\phi |}^{\gamma }d\xi \right)}^{{\displaystyle \frac{1}{\gamma }}}\hfill \\ \hfill \le & C{\left({\int }_{B_r\left({\xi }_0\right)}{|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}}d\xi \right)}^{1-{\displaystyle \frac{1}{\gamma }}}{\left({\int }_{B_r\left({\xi }_0\right)}{|X\phi |}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill \le & \epsilon {\int }_{B_r\left({\xi }_0\right)}{|X\phi |}^{2}d\xi +C\left(\epsilon \right){\left({\int }_{B_r{\xi }_0)}{|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}}d\xi \right)}^{2(1-{\displaystyle \frac{1}{\gamma }})}\hfill \\ \hfill \le & \epsilon {\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi +C\left(\epsilon \right){\int }_{B_r\left({\xi }_0\right)}{|X\varphi |}^2{|u-u_{{\xi }_0,r}|}^{2}d\xi +C\left(\epsilon \right){\left({\int }_{B_r\left({\xi }_0\right)}{|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}}d\xi \right)}^{2(1-{\displaystyle \frac{1}{\gamma }})},\hfill \end{array}$$

(27)

where we used the fact that $X\phi =2\varphi X\varphi (u-u_{{\xi }_0,r})+{\varphi }^{2}Xu$ in the last inequality.

Substituting (26)–(27) into (25), we obtain

$$(\lambda -2\epsilon ){\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi \le {\displaystyle \frac{C\left(\epsilon \right)}{r^2}}{\int }_{B_r\left({\xi }_0\right)}{|u-u_{{\xi }_0,r}|}^{2}d\xi +C\left(\epsilon \right){\left[{\int }_{B_r\left({\xi }_0\right)}{(|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}})d\xi \right]}^{2(1-{\displaystyle \frac{1}{\gamma }})}.$$

(28)

Taking some positive $\epsilon <{\displaystyle \frac{\lambda }{2}}$, we conclude the desired result. □

4.2. Proof of Theorem 1

Based on the aforementioned $\mathcal{A}$-harmonic approximation reformulation (Lemma 7) and the Caccioppoli-type inequality (Lemma 9), we now proceed to the proof of Theorem 1.

Proof. 

By given ${\xi }_0\in \mathsf{\Omega }$, and fixed ${\displaystyle R:0<R<\frac{1}{2}dist({\xi }_0,\partial \mathsf{\Omega })}$, for $0<\rho <R$ we define the following:

$$\mathcal{A}:={\left(A_{ij}^{\alpha \beta }(\xi ,u_{{\xi }_0,\rho })\right)}_{{\xi }_0,{\displaystyle \frac{\rho }{2}}}={⨏}_{B_{\rho /2}\left({\xi }_0\right)}{A}_{ij}^{\alpha \beta }(\xi ,u_{{\xi }_0,\rho })d\xi .$$

Applying the Caccioppoli-type inequality (43), it follows that

$$\begin{array}{cc}\hfill & {⨏}_{B_{{\displaystyle \frac{\rho }{2}}\left({\xi }_0\right)}}{|Xu|}^{2}d\xi \hfill \\ \hfill \le & {\displaystyle \frac{2C_c}{{\rho }^2}}\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|h-h_{{\xi }_0,\rho }{|}^{2}d\xi +{⨏}_{B_{\rho }\left({\xi }_0\right)}{|u-u_{{\xi }_0,\rho }-(h-h_{{\xi }_0,\rho })|}^{2}d\xi \right)\hfill \\ \hfill & +C_c{\rho }^2{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}{(|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}})d\xi \right)}^{2(1-{\displaystyle \frac{1}{\gamma }})}\hfill \\ \hfill :=& III+IV+V\hfill \end{array}$$

(29)

with the obvious integral labeling for $III$, $IV$, and V.

We are now in a position to estimate the terms $III$–V. By the Poincaré-type inequality (11), the a priori estimate (22), and the $\mathcal{A}$-harmonic approximation reformulation (18), the following is yielded:

$$III={\displaystyle \frac{2C_c}{{\rho }^2}}{⨏}_{B_{\rho }\left({\xi }_0\right)}|h-h_{{\xi }_0,\rho }{|}^{2}d\xi \le 2C_c{C}_p{⨏}_{B_{\rho }\left({\xi }_0\right)}{|Xh|}^{2}d\xi \le 2C_c{C}_p{C}_{\alpha }{⨏}_{B_R\left({\xi }_0\right)}{|Xh|}^{2}d\xi \le C{⨏}_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi .$$

(30)

By Poincaré-type inequality (11) again, and Lemma 7 on the $\mathcal{A}$-harmonic approximation modification, the following is implied:

$$\begin{array}{cc}\hfill IV& \le {\displaystyle \frac{2C_c}{{\rho }^2}}{⨏}_{B_{\rho }\left({\xi }_0\right)}{|u-h|}^{2}d\xi \hfill \\ & \le C\epsilon {⨏}_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi +C\kappa {\rho }^2{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}\mathcal{A}XuX\phi d\xi \right)}^2\hfill \\ & \le C\epsilon {\rho }^{-Q}{\int }_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi +C\kappa {\rho }^2{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}\mathcal{A}XuX\phi d\xi \right)}^2.\hfill \end{array}$$

(31)

Now we estimate the term ${⨏}_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}XuX\phi d\xi$. Noting that u represents weak solutions of the sub-elliptic system (1), we have, for any $\phi \in C_0^{\infty }(B_{\rho /2}\left({\xi }_0\right),{\mathbb{R}}^N)$,

$$\begin{array}{cc}\hfill {⨏}_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}XuX\phi d\xi =& {⨏}_{B_{\rho /2}\left({\xi }_0\right)}\left[\mathcal{A}-A(\xi ,u_{{\xi }_0,\rho })\right]XuX\phi d\xi +{⨏}_{B_{\rho /2}\left({\xi }_0\right)}\left[A(\xi ,u_{{\xi }_0,\rho })-A(\xi ,u)\right]XuX\phi d\xi \hfill \\ \hfill & +{⨏}_{B_{\rho /2}\left({\xi }_0\right)}B(\xi ,u,Xu)\phi d\xi .\hfill \end{array}$$

(32)

This implies

$$\begin{array}{cc}\hfill {\left({⨏}_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}XuX\phi d\xi \right)}^2\le & C{\left({⨏}_{B_{\rho /2}\left({\xi }_0\right)}[\mathcal{A}-A(\xi ,u_{{\xi }_0,\rho })]XuX\phi d\xi \right)}^2\hfill \\ \hfill & +C{\left({⨏}_{B_{\rho /2}\left({\xi }_0\right)}[A(\xi ,u_{{\xi }_0,\rho })-A(\xi ,u)]XuX\phi d\xi \right)}^2+C{\left({⨏}_{B_{\rho /2}\left({\xi }_0\right)}B(\xi ,u,Xu)\phi d\xi \right)}^2\hfill \\ \hfill :=& C\left(J_1+J_2+J_3\right).\hfill \end{array}$$

(33)

Since $\underset{B_{\rho }\left({\xi }_0\right)}{sup}|X\phi |\le {\displaystyle \frac{1}{\rho }}$ and $A(\xi ,u)\in VMO\left(\mathsf{\Omega }\right)$ are under the assumptions H1–H2, it follows that by Hölder’s inequality,

$$\begin{array}{cc}\hfill J_1\le & {\displaystyle \frac{1}{{\rho }^2}}{⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi {⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|A(\xi ,u_{{\xi }_0,\rho })-\mathcal{A}|}^{2}d\xi \hfill \\ \hfill \le & {\displaystyle \frac{1}{{\rho }^2}}{2}^{Q+1}\mathsf{\Lambda }{⨏}_{B_{\rho }\left({\xi }_0\right)}\left|A(\xi ,u_{{\xi }_0,\rho })-\mathcal{A}\right|d\xi {⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \hfill \\ \hfill \le & {\rho }^{-Q-2}C(Q,\mathsf{\Lambda })M_{\rho }\left(A(\xi ,u_{{\xi }_0,\rho })\right){\int }_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi .\hfill \end{array}$$

(34)

Similarly, according to the continuous assumptions of the principal coefficients $A(\xi ,u)$ in u uniformly with regard to $\xi \in \mathsf{\Omega }$, we can estimate the second term $J_2$ as follows

$$\begin{array}{cc}\hfill J_2\le & {\displaystyle \frac{1}{{\rho }^2}}{2}^{Q+1}\mathsf{\Lambda }{⨏}_{B_{\rho }\left({\xi }_0\right)}|A(\xi ,u_{{\xi }_0,\rho })-A(\xi ,u)|d\xi {⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \hfill \\ \hfill \le & {\rho }^{-2}C(Q,\mathsf{\Lambda },L)\omega \left({⨏}_{B_{\rho }\left({\xi }_0\right)}{|u-u_{{\xi }_0,\rho }|}^{2}d\xi \right){⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \hfill \\ \hfill \le & {\rho }^{-Q-2}C(Q,\mathsf{\Lambda },L)\omega \left({\rho }^2{⨏}_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right){\int }_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi ,\hfill \end{array}$$

(35)

where we used Jensen’s inequality (12) and the Poincaré-type inequality (11) in the last inequality.

The last term term $J_3$ can be estimated as follows by Hölder’s inequality,

$$J_3\le C{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}{(|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}})d\xi \right)}^{2(1-{\displaystyle \frac{1}{\gamma }})}.$$

(36)

Substituting the estimates (34)–(36) into (33), we get

$$\begin{array}{c}\hfill {\left({⨏}_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}XuX\phi d\xi \right)}^2\le C{\rho }^{-Q-2}\tau \left(\rho \right){\int }_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi +C{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}{(|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}})d\xi \right)}^{2(1-{\displaystyle \frac{1}{\gamma }})},\end{array}$$

(37)

where we denoted $\tau \left(\rho \right)=M_{\rho }\left(A(\xi ,u_{{\xi }_0,\rho })\right)+\omega \left({\rho }^2{⨏}_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right).$ Inserting (37) into the estimate of $IV$ yields

$$\begin{array}{cc}\hfill IV\le & C\left[\epsilon +\tau \left(\rho \right)\right]{\rho }^{-Q}{\int }_{B_R\left({\xi }_0\right)}{|Xu|}^{2}d\xi +C{\rho }^2{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}{(|Xu|}^2+{|u|}^{\gamma }+{|f|}^{{\displaystyle \frac{\gamma }{\gamma -1}}})d\xi \right)}^{2(1-{\displaystyle \frac{1}{\gamma }})}.\hfill \end{array}$$

(38)

The final term V remains to be estimated, which arises from the controllable growth condition.

Taking care to notice that the homogeneous norm $Q\ge 3$ and the assumption $f\in L^q\left(\mathsf{\Omega }\right)$ with $q>{\displaystyle \frac{2Q}{Q+2}}$, by Hölder’s inequality, the following is yielded:

$$\begin{array}{cc}\hfill V\le & C_c\sqrt[Q]{4}{\rho }^2{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}{(|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}})d\xi \right)}^{{\displaystyle \frac{Q+2}{Q}}}+C_c\sqrt[Q]{4}{\rho }^2{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}{|f|}^{{\displaystyle \frac{2Q}{Q+2}}}d\xi \right)}^{{\displaystyle \frac{Q+2}{Q}}}\hfill \\ \hfill \le & C{\rho }^2{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}{(|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}})d\xi \right)}^{1+{\displaystyle \frac{2}{Q}}}+C{\rho }^2{\left({⨏}_{B_{\rho }\left({\xi }_0\right)}{|f|}^{q}d\xi \right)}^{{\displaystyle \frac{2}{q}}},\hfill \end{array}$$

(39)

where we used that $2(1-1/\gamma )=1+2/Q$.

Substituting the estimates for $III$–V into (29), it implies that

$$\begin{array}{cc}\hfill {\int }_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \le & C\left[{\left({\displaystyle \frac{\rho }{R}}\right)}^Q+\epsilon +\tau \left(\rho \right)+{\left({\int }_{B_{\rho }\left({\xi }_0\right)}\left({|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}\right)d\xi \right)}^{{\displaystyle \frac{2}{Q}}}\right]{\int }_{B_R\left({\xi }_0\right)}\left({|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}\right)d\xi \hfill \\ \hfill & +C{\rho }^{2+Q-{\displaystyle \frac{2Q}{q}}}{\left({\int }_{B_{\rho }\left({\xi }_0\right)}{|f|}^{q}d\xi \right)}^{{\displaystyle \frac{2}{q}}}.\hfill \end{array}$$

(40)

To apply the iteration lemma, we estimate the integral ${\int }_{B_{\rho /2}\left({\xi }_0\right)}{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi$ as follows:

$$\begin{array}{cc}\hfill {\int }_{B_{\rho /2}\left({\xi }_0\right)}{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi \le & 2^{{\displaystyle \frac{Q+2}{Q-2}}}\left({\int }_{B_{\rho }\left({\xi }_0\right)}|u_{{\xi }_0,\rho }{|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi +{\int }_{B_{\rho }\left({\xi }_0\right)}{|u-u_{{\xi }_0,\rho }|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi \right)\hfill \\ \hfill \le & C{\left({\displaystyle \frac{\rho }{R}}\right)}^Q{\int }_{B_R\left({\xi }_0\right)}{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi +C{\left({\int }_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)}^{{\displaystyle \frac{Q}{Q-2}}}\hfill \\ \hfill \le & C\left[{\left({\displaystyle \frac{\rho }{R}}\right)}^Q+{\left({\int }_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)}^{{\displaystyle \frac{2}{Q-2}}}\right]\left({\int }_{B_R\left({\xi }_0\right)}{|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi \right).\hfill \end{array}$$

(41)

Adding the inequality (41) to (40), we get

$$\begin{array}{cc}\hfill {\int }_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi \le & C\left[{\left({\displaystyle \frac{\rho }{R}}\right)}^Q+\epsilon +\tau \left(\rho \right)+\sigma \left(\rho \right)\right]{\int }_{B_R\left({\xi }_0\right)}\left({|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}\right)d\xi \hfill \\ \hfill & +C{\rho }^{2+Q-{\displaystyle \frac{2Q}{q}}}{\left({\int }_{B_{\rho }\left({\xi }_0\right)}{|f|}^{q}d\xi \right)}^{{\displaystyle \frac{2}{q}}},\hfill \end{array}$$

(42)

where the notation $\sigma \left(\rho \right)={\left({\int }_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi \right)}^{{\displaystyle \frac{2}{Q}}}+{\left({\int }_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)}^{{\displaystyle \frac{2}{Q-2}}}$.

Note that by the absolute continuity of the integral, we have that $\sigma \left(\rho \right)\to 0$, as $\rho \to 0$. If we assume that ${\rho }^2{⨏}_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi \to 0$ for $\xi \in {\mathsf{\Omega }}_0\subset \mathsf{\Omega }$, as $\rho \to 0$, then for any $\epsilon >0$, it implies $\tau \left(\rho \right)=M_{\rho }\left(A(\xi ,u_{{\xi }_0,\rho })\right)+\omega \left({\rho }^2{⨏}_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)<\epsilon$, as $\rho \to 0$ due to the VMO property of $A(\xi ,u)$.

Case 1. Notice that $0<Q+2-{\displaystyle \frac{2Q}{q}}<Q$ if ${\displaystyle \frac{2Q}{Q+2}}<q<Q$, and it follows that by iteration Lemma 3,

$$\begin{array}{cc}\hfill {\int }_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi \le & C{\left({\displaystyle \frac{\rho }{R}}\right)}^{Q+2-{\displaystyle \frac{2Q}{q}}}{\int }_{B_R\left({\xi }_0\right)}\left({|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}\right)d\xi +C{\rho }^{Q+2-{\displaystyle \frac{2Q}{q}}}{\left({\int }_{B_{\rho }\left({\xi }_0\right)}{|f|}^{q}d\xi \right)}^{{\displaystyle \frac{2}{q}}},\hfill \end{array}$$

which infers $Xu\in L^{2,\lambda }\left({\mathsf{\Omega }}_0\right)$ with $\lambda =Q+2-{\displaystyle \frac{2Q}{q}}$. By Morrey’s lemma (Lemma 4), it yields $u\in {\mathsf{\Gamma }}_{loc}^{0,\alpha }({\mathsf{\Omega }}_0,{\mathbb{R}}^N)$ with $\alpha =2-{\displaystyle \frac{Q}{q}}$.

Case 2. If $q\ge Q$, also by iteration Lemma 3, it yields that for any $ϵ>0$,

$$\begin{array}{cc}\hfill {\int }_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}d\xi \le & C{\left({\displaystyle \frac{\rho }{R}}\right)}^{Q-ϵ}{\int }_{B_R\left({\xi }_0\right)}\left({|Xu|}^2+{|u|}^{{\displaystyle \frac{2Q}{Q-2}}}\right)d\xi +C{\rho }^{Q-ϵ}{\left({\int }_{B_{\rho }\left({\xi }_0\right)}{|f|}^{q}d\xi \right)}^{{\displaystyle \frac{2}{q}}},\hfill \end{array}$$

which implies $Xu\in L^{2,\lambda }\left({\mathsf{\Omega }}_0\right)$ with $\lambda =Q-ϵ$, and $u\in {\mathsf{\Gamma }}_{loc}^{0,\alpha }({\mathsf{\Omega }}_0,{\mathbb{R}}^N)$ for all $\alpha \in (0,1)$.

Finally, we are in a position to estimate the Hausdorff dimension of the singular set $\mathsf{\Omega }\setminus {\mathsf{\Omega }}_0$. Recall the smallness condition that ${\rho }_{B_{\rho }\left({\xi }_0\right)}^2{|Xu|}^{2}d\xi \to 0$, as $\rho \to 0$ under the definition of ${\mathsf{\Omega }}_0$ in Theorem 1. Applying the Poincaré-type inequality (11), we can express the singular set as

$$\mathsf{\Omega }\setminus {\mathsf{\Omega }}_0=\left\{\xi \in \mathsf{\Omega }:\underset{\rho \to 0}{lim}inf{\rho }^2{⨏}_{B_{\rho }\left({\xi }_0\right)}{|Xu|}^{2}d\xi >0\right\}.$$

According to Lemma 5, it follows that

$${\mathcal{H}}^{Q-2}(\mathsf{\Omega }\setminus {\mathsf{\Omega }}_0)=0.$$

Thus the proof of Theorem 1 is complete. □

5. Partial Hölder Continuity for Natural Growth

In this section, we prove the partial regularity result stated in Theorem 2 and Theorem 3 under the natural structure conditions H1–H2 and HN. In this setting, we restrict our consideration to bounded solutions of (1), where the bound $M={\parallel u\parallel }_{L^{\infty }\left(\mathsf{\Omega }\right)}$ satisfies the smallness condition

$$2\mu \left(M\right)M<\lambda ,$$

with $\mu \left(M\right)$ defined as in (6). A similar smallness assumption is known to be necessary for establishing partial regularity results, even in the classical elliptic case; see [3,4].

5.1. Caccioppolli-Type Inequality for Natural Growth

Similarly to the steps of the proof for controllable growth, we first establish the following Caccioppoli-type inequality under assumptions H1–H2 and HN with $2\mu \left(M\right)M<\lambda$.

Lemma 10

(Caccioppolli-type inequality for natural growth). Let $u\in H{W}_{\mathrm{loc}}^{1,2}(\mathsf{\Omega },{\mathbb{R}}^N)\cap L^{\infty }(\mathsf{\Omega },{\mathbb{R}}^N)$ be a weak solution of the sub-elliptic system (1) under assumptions H1–H2 and HN with $2\mu \left(M\right)M<\lambda$. Then for any ${\xi }_0\in \mathsf{\Omega }\subset {\mathbb{R}}^n$ and any ball $B_r\left({\xi }_0\right)⋐\mathsf{\Omega }$, we have the estimate

$${⨏}_{B_{r/2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \le C_c\left[{⨏}_{B_r\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,r}}{r}}{|}^{2}d\xi +r^{2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right].$$

(43)

Proof. 

We test the sub-elliptic system (2) with the test function $\phi ={\varphi }^2(u-u_{{\xi }_0,r}),$ where $\varphi \in C_0^{\infty }\left(B_r\left({\xi }_0\right)\right)$ is a cut-off function satisfying $0\le \varphi \le 1,|X\varphi |\le {\displaystyle \frac{4}{r}},\mathrm{and}\varphi \equiv 1\mathrm{on}{B}_{r/2}\left({\xi }_0\right).$ Following the same procedure as in the controllable growth case, for weak solutions u of system (1), we have

$$\begin{array}{cc}\hfill \lambda & {\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi \hfill \\ \hfill \le & 2\mathsf{\Lambda }{\int }_{B_r\left({\xi }_0\right)}|\varphi Xu|·|(u-u_{{\xi }_0,r})X\varphi |d\xi +\mu \left(M\right){\int }_{B_r\left({\xi }_0\right)}{\varphi }^2{(|Xu|}^2+|f\left(x\right)|)|u-u_{{\xi }_0,r}|d\xi \hfill \\ \hfill \le & \mu \left(M\right){\int }_{B_r\left({\xi }_0\right)}|u-u_{{\xi }_0,r}{|·|\varphi Xu|}^{2}d\xi +\mu \left(M\right){\int }_{B_r\left({\xi }_0\right)}|u-u_{{\xi }_0,r}|·|f\left(x\right)|d\xi \hfill \\ \hfill & +2\mathsf{\Lambda }{\left({\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{B_r\left({\xi }_0\right)}{|X\varphi |}^2{|u-u_{{\xi }_0,r}|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

(44)

where the last term of the right-hand side is obtained by Hölder’s inequality.

By Hölder’s inequality, Young’s inequality with parameter $\epsilon >0$, and the boundedness of u, it follows that

$$\begin{array}{cc}\hfill \lambda {\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi \le & 2\mu \left(M\right)M{\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi \hfill \\ \hfill & +\mu \left(M\right){\left({\int }_{B_r\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,r}}{r}}{|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{B_r\left({\xi }_0\right)}{|rf\left(x\right)|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +2\mathsf{\Lambda }\epsilon {\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi +C\left(\epsilon \right){\int }_{B_r\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,r}}{r}}{|}^{2}d\xi ,\hfill \end{array}$$

which means

$$\begin{array}{cc}\hfill \lambda {\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi \le & (2\mu \left(M\right)M+2\mathsf{\Lambda }\epsilon ){\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi +C(\mu \left(M\right),\epsilon ){\int }_{B_r\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,r}}{r}}{|}^{2}d\xi \hfill \\ \hfill & +C\left(\epsilon \right)r^2{\int }_{B_r\left({\xi }_0\right)}{|f\left(x\right)|}^{2}d\xi .\hfill \end{array}$$

We take a sufficiently small $\epsilon$ such that $\lambda -2\mu \left(M\right)M+2\mathsf{\Lambda }\epsilon >0$. It follows by Hölder’s inequality that

$$\begin{array}{cc}\hfill {\int }_{B_{r/2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \le & {\displaystyle \frac{C\left(\mu \right(M),\epsilon )}{\lambda -2\mu \left(M\right)M+2\mathsf{\Lambda }\epsilon }}{\int }_{B_r\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,r}}{r}}{|}^{2}d\xi +{\displaystyle \frac{C\left(\epsilon \right)r^2{\left({\omega }_n{r}^Q\right)}^{\frac{q-2}{q}}}{\lambda -2\mu \left(M\right)M+2\mathsf{\Lambda }\epsilon }}{\left({\int }_{B_r\left({\xi }_0\right)}{|f\left(x\right)|}^q\right)}^{{\displaystyle \frac{2}{q}}}\hfill \\ \hfill \le & C_c\left[{\int }_{B_r\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,r}}{r}}{|}^{2}d\xi +r^{Q+2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right],\hfill \end{array}$$

which yields the desired result. □

5.2. Proof of Theorem 2

Based on the fundamental tool of the Caccioppoli-type inequality (Lemma 9) and by employing the $\mathcal{A}$-harmonic approximation reformulation (Lemma 7), we now proceed to the proof of Theorem 2.

Proof. 

Given ${\xi }_0\in \mathsf{\Omega }$, and fixed $\rho :0<\rho <dist({\xi }_0,\partial \mathsf{\Omega })$, we define

$$\mathcal{A}:={\left(A_{ij}^{\alpha \beta }(\xi ,u_{{\xi }_0,\rho })\right)}_{{\xi }_0,{\displaystyle \frac{\rho }{2}}}={⨏}_{B_{\rho /2}\left({\xi }_0\right)}{A}_{ij}^{\alpha \beta }(\xi ,u_{{\xi }_0,\rho })d\xi .$$

For any $\phi \in C_0^{\infty }(B_{\rho /2}\left({\xi }_0\right),{\mathbb{R}}^N)$, it follows that

$$\begin{array}{cc}\hfill {\int }_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}XuX\phi d\xi =& {\int }_{B_{\rho /2}\left({\xi }_0\right)}(\mathcal{A}-A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho }))XuX\phi d\xi +{\int }_{B_{\rho /2}\left({\xi }_0\right)}(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })-A_{ij}^{\alpha \beta }(\xi ,u))XuX\phi d\xi \hfill \\ \hfill & +{\int }_{B_{\rho /2}\left({\xi }_0\right)}{A}_{ij}^{\alpha \beta }(\xi ,u)XuX\phi d\xi .\hfill \end{array}$$

(45)

Since u is a weak solution of system (1), we obtain

$$\begin{array}{cc}\hfill & \left|{\int }_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}XuX\phi d\xi \right|\hfill \\ \hfill \le & {\int }_{B_{\rho /2}\left({\xi }_0\right)}|\mathcal{A}-A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })||Xu||X\phi |d\xi \hfill \\ \hfill & +{\int }_{B_{\rho /2}\left({\xi }_0\right)}|A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })-A_{ij}^{\alpha \beta }(\xi ,u)||Xu||X\phi |d\xi +{\int }_{B_{\rho /2}\left({\xi }_0\right)}|B(\xi ,u,Xu)||\phi |d\xi .\hfill \end{array}$$

Now we take the test function $\phi$ satisfying ${\displaystyle {\parallel X\phi \parallel }_{L^{\infty }\left(B_{\rho /2}\left({\xi }_0\right)\right)}\le \frac{2}{\rho }.}$ Applying the natural growth condition HN and Hölder’s inequality, we obtain

$$\begin{array}{cc}\hfill & \left|{\int }_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}Xu·X\phi d\xi \right|\hfill \\ \hfill \le & {\displaystyle \frac{2}{\rho }}{\left({\int }_{B_{\rho /2}\left({\xi }_0\right)}|\mathcal{A}-A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho }){|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +{\displaystyle \frac{2}{\rho }}{\left({\int }_{B_{\rho /2}\left({\xi }_0\right)}|A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })-A_{ij}^{\alpha \beta }(\xi ,u){|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +2\mu \left(M\right){\int }_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi +2\mu \left(M\right){\left({\omega }_n{\rho }^Q\right)}^{{\displaystyle \frac{q-1}{q}}}{\parallel f\parallel }_{L^q}\hfill \\ \hfill :=& VI+VII+VIII.\hfill \end{array}$$

(46)

We are in a position to estimate the items $VI$, $VII$, and $VIII$. By using the Caccioppoli-type inequality (43), we find

$$\begin{array}{cc}\hfill VI\le & {\displaystyle \frac{2}{\rho }}{\left({\int }_{B_{\rho }\left({\xi }_0\right)}|\mathcal{A}-A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho }){|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}}{\left[C_c\left({\int }_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{Q+2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill \le & {\displaystyle \frac{2\sqrt{C_c}}{\rho }}{\left({\omega }_n{\rho }^Q{M}_{\rho }\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{Q+2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(47)

Applying assumption H2 and (43), we obtain

$$\begin{array}{cc}\hfill VII\le & {\displaystyle \frac{2L}{\rho }}{\left({\omega }_n{\rho }^Q{⨏}_{B_{\rho }\left({\xi }_0\right)}{\omega }^2(|u-u_{{\xi }_0,\rho }{|}^2)d\xi \right)}^{{\displaystyle \frac{1}{2}}}{\left[C_c\left({\int }_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{Q+2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill \le & {\displaystyle \frac{2L\sqrt{C_c}}{\rho }}{\left({\omega }_n{\rho }^Q\omega \left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi \right)\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{Q+2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)}^{{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

(48)

where we used the concavity of the function $\omega (·)\in [0,1]$ in the last inequality.

For the estimate of term $VIII$, we apply the Caccioppoli-type inequality (43) to deduce

$$\begin{array}{cc}\hfill VIII\le & 2\mu \left(M\right)C_c\left({\int }_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{Q+2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)+2\mu \left(M\right){\left({\omega }_n{\rho }^Q\right)}^{{\displaystyle \frac{q-1}{q}}}{\parallel f\parallel }_{L^q}.\hfill \end{array}$$

(49)

Substituting (47)–(49) into (46), it then follows that

$$\begin{array}{cc}\hfill & \left|{\int }_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}Xu·X\phi d\xi \right|\hfill \\ \hfill \le & 2\sqrt{C_c}{\left({\omega }_n{\rho }^{Q-2}{M}_{\rho }\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)\right)}^{{\displaystyle \frac{1}{2}}}{\left({\omega }_n{\rho }^Q{⨏}_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{Q+2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +2L\sqrt{C_c}{\left({\omega }_n{\rho }^{Q-2}\omega \left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi \right)\right)}^{{\displaystyle \frac{1}{2}}}{\left({\omega }_n{\rho }^Q{⨏}_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{Q+2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +2\mu \left(M\right)C_c\left({\int }_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{Q+2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)+2\mu \left(M\right){\left({\omega }_n{\rho }^Q\right)}^{{\displaystyle \frac{q-1}{q}}}{\parallel f\parallel }_{L^q}\hfill \\ \hfill \le & C(C_c,L,{\omega }_n){\rho }^{Q-2}\left(M_{\rho }^{{\displaystyle \frac{1}{2}}}\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)+{\omega }^{{\displaystyle \frac{1}{2}}}\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi \right)\right)\hfill \\ \hfill & \times {\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +{\rho }^{4-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +C(C_c,\mu \left(M\right),{\omega }_n){\rho }^{Q-2}\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +{\rho }^{4-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2+{\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}\right).\hfill \end{array}$$

(50)

Noting that $2-{\displaystyle \frac{Q}{q}}>0$ due to the assumption $q>{\displaystyle \frac{Q}{2}}$, we choose $\rho >0$ small enough such that ${\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}\le 1$. Then (50) can be rewritten as

$$\begin{array}{cc}\hfill & \left|{⨏}_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}Xu·X\phi d\xi \right|\hfill \\ \hfill \le & C{\rho }^{-2}\left(M_{\rho }^{{\displaystyle \frac{1}{2}}}\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)+{\omega }^{{\displaystyle \frac{1}{2}}}\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi \right)\right)\hfill \\ \hfill & \times {\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +{\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +C{\rho }^{-2}\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +{\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}\right)\hfill \\ \hfill \le & C{\rho }^{-2}\sqrt{J}\left[M_{\rho }^{{\displaystyle \frac{1}{2}}}\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)+{\omega }^{{\displaystyle \frac{1}{2}}}\left(J\right)+\sqrt{J}\right],\hfill \end{array}$$

(51)

where we used the notation $J={⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +{\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}$.

Employing Lemma 7 on the $\mathcal{A}$-harmonic approximation modification, for a given $\epsilon >0$, there exists an $\mathcal{A}$ = ${\left(A_{ij}^{\alpha \beta }(\xi ,u_{{\xi }_0,\rho })\right)}_{{\xi }_0,{\displaystyle \frac{\rho }{2}}}$-harmonic $h\in H{W}^{1,2}(B_{\rho /2}\left({\xi }_0\right),{\mathbb{R}}^N)$ such that

$$\begin{array}{cc}\hfill {⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|u-h|}^{2}d\xi \le & \epsilon {\rho }^2{⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi +K{\rho }^4{\left({⨏}_{B_{\rho /2}\left({\xi }_0\right)}\mathcal{A}Xu·X\phi d\xi \right)}^2\hfill \\ \hfill \le & \epsilon {\rho }^2{C}_c\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)\hfill \\ \hfill & +CK{\rho }^4{\left[{\rho }^{-2}\sqrt{J}\left(M_{\rho }^{{\displaystyle \frac{1}{2}}}\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)+{\omega }^{{\displaystyle \frac{1}{2}}}\left(J\right)+\sqrt{J}\right)\right]}^2\hfill \\ \hfill \le & C_c\epsilon {⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +C_c\epsilon {\rho }^{4-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\hfill \\ \hfill & +4CKJ\left(M_{\rho }\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)+\omega \left(J\right)+J\right)\hfill \\ \hfill \le & C_c\epsilon \left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +{\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}\right)\hfill \\ \hfill & +4CKJ\left(M_{\rho }\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)+\omega \left(J\right)+J\right)\hfill \\ \hfill \le & C\epsilon J+CJ\left(M_{\rho }\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)+\omega \left(J\right)+J\right),\hfill \end{array}$$

(52)

where we used (43), (51), and the fact that ${\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}\le 1$.

By the a priori estimate (22), we estimate the integral ${⨏}_{B_{\sigma }\left({\xi }_0\right)}{|u-u_{{\xi }_0,\sigma }|}^{2}d\xi$ for any $0<\sigma <{\displaystyle \frac{\rho }{2}}$ as follows

$$\begin{array}{cc}\hfill {⨏}_{B_{\sigma }\left({\xi }_0\right)}{|u-u_{{\xi }_0,\sigma }|}^{2}d\xi \le & {⨏}_{B_{\sigma }\left({\xi }_0\right)}|u-h_{{\xi }_0,\sigma }{|}^{2}d\xi \le 2{⨏}_{B_{\sigma }\left({\xi }_0\right)}{|u-h|}^{2}d\xi +2{⨏}_{B_{\sigma }\left({\xi }_0\right)}{|h-h_{{\xi }_0,\sigma }|}^{2}d\xi \hfill \\ \hfill \le & 2{⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|u-h|}^{2}d\xi +C{\left({\displaystyle \frac{\sigma }{\rho }}\right)}^2{⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|h-h_{{\xi }_0,\rho /2}|}^{2}d\xi .\hfill \end{array}$$

(53)

To estimate the final term on the right-hand side, we apply the Poincaré-type inequality (11) and the Caccioppoli-type inequality (43), together with (18), to obtain

$$\begin{array}{cc}\hfill {⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|h-h_{{\xi }_0,\rho /2}|}^{2}d\xi \le & C_p^2{\rho }^2{⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|Xh|}^{2}d\xi \hfill \\ \hfill \le & C_p^2{\rho }^2{⨏}_{B_{\rho /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \hfill \\ \hfill \le & C_c{C}_p^2{\rho }^2\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\rho }}{\rho }}{|}^{2}d\xi +{\rho }^{2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)\hfill \\ \hfill \le & C_c{C}_p^2\left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +{\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}\right)=C_c{C}_p^{2}J,\hfill \end{array}$$

(54)

where we use the inequality ${\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}\le 1$ and the definition of $J={⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +{\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}$ in the last step.

Substituting the estimates (52) and (54) into the inequality (53), we derive

$$\begin{array}{cc}\hfill & {⨏}_{B_{\sigma }\left({\xi }_0\right)}{|u-u_{{\xi }_0,\sigma }|}^{2}d\xi \hfill \\ \hfill \le & C\left[{\left({\displaystyle \frac{\sigma }{\rho }}\right)}^2+\epsilon +M_{\rho }\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)+\omega \left(J\right)+J\right]\hfill \\ \hfill & \times \left({⨏}_{B_{\rho }\left({\xi }_0\right)}|u-u_{{\xi }_0,\rho }{|}^{2}d\xi +{\rho }^{2-{\displaystyle \frac{Q}{q}}}{\parallel f\parallel }_{L^q}\right).\hfill \end{array}$$

(55)

Noting the VMO property of $A_{ij}^{\alpha \beta }(\xi ,u)$: $\underset{\rho \to 0}{lim}{M}_{\rho }\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)=0$, and the continuous modulus of $\omega \left(J\right)$, we infer that $\overline{\epsilon }=\epsilon +M_{\rho }\left(A_{ij}^{\alpha \beta }(·,u_{{\xi }_0,\rho })\right)+\omega \left(J\right)+J<{\epsilon }_0,$ where ${\epsilon }_0$ is the constant from Lemma 3 for some sufficiently small $\rho >0$. If we assume $\underset{\rho \to 0}{lim}{B}_{\rho }\left({\xi }_0\right){|u-u_{{\xi }_0,\rho }|}^{2}d\xi =0$ on any $\xi \in {\mathsf{\Omega }}_0$ with some ${\mathsf{\Omega }}_0\subset \mathsf{\Omega }$, then by the iteration of Lemma 3, we have

$${⨏}_{B_{\sigma }\left({\xi }_0\right)}{|u-u_{{\xi }_0,\sigma }|}^{2}d\xi \le C{\sigma }^{2-{\displaystyle \frac{Q}{q}}},$$

(56)

which implies

$$u\in {\mathcal{L}}^{2,Q+2-{\displaystyle \frac{Q}{q}}}({\mathsf{\Omega }}_0,{\mathbb{R}}^N).$$

By Campanato’s characterization of Hölder continuous functions, it yields $u\in {\mathsf{\Gamma }}_{loc}^{0,\alpha }({\mathsf{\Omega }}_0,{\mathbb{R}}^N)$ with $\alpha =1-{\displaystyle \frac{Q}{2q}}$.

Furthermore, by the Caccippoli-type inequality (43), and the estimation (56), we deduce that

$$\begin{array}{cc}\hfill & \underset{\xi \in {\mathsf{\Omega }}_0,\sigma \in (0,\rho /2)}{sup}{\sigma }^{{\displaystyle \frac{Q}{q}}}{⨏}_{B_{\sigma /2}\left({\xi }_0\right)}{|Xu|}^{2}d\xi \hfill \\ \hfill \le & \underset{\xi \in {\mathsf{\Omega }}_0,\sigma \in (0,\rho /2)}{sup}{C}_c{\sigma }^{{\displaystyle \frac{Q}{q}}}\left({⨏}_{B_{\sigma }\left({\xi }_0\right)}|{\displaystyle \frac{u-u_{{\xi }_0,\sigma }}{\sigma }}{|}^{2}d\xi +{\sigma }^{2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)\hfill \\ \hfill \le & \underset{\xi \in {\mathsf{\Omega }}_0,\sigma \in (0,\rho /2)}{sup}{C}_c{\sigma }^{{\displaystyle \frac{Q}{q}}}\left({\displaystyle \frac{C}{{\sigma }^2}}·{\sigma }^{2-{\displaystyle \frac{Q}{q}}}+{\sigma }^{2-{\displaystyle \frac{2Q}{q}}}{\parallel f\parallel }_{L^q}^2\right)\hfill \\ \hfill \le & C_c(1+{\rho }^{2-{\displaystyle \frac{Q}{q}}})<\infty ,\hfill \end{array}$$

which implies that $Xu\in L^{2,\lambda }$ with $\lambda =Q-{\displaystyle \frac{Q}{q}}$.

The Hausdorff dimension estimation for the remaining part proceeds similarly to that in Theorem 1 and is therefore omitted for brevity. This completes the proof of Theorem 2. □

5.3. Proof of Theorem 3

Finally, we present the proof of Theorem 3. We note that the hypothesis of Theorem 3 is a minor modification of that of Theorem 2, i.e., the condition $|B^{\alpha }{(\xi ,u,Xu)|\le \mu \left(M\right)(|Xu|}^2+f^{\alpha })$ is replaced by $|B^{\alpha }{(\xi ,u,Xu)|\le \mu \left(M\right)(|Xu|}^{2-ϵ}+f^{\alpha })$ with $ϵ>0$ sufficiently small, while all other assumptions remain unchanged. Thus, for brevity, we only estimate the distinctly different terms and omit the repetitive parts.

Proof. 

In fact, we need only reevaluate the term $\mu \left(M\right){\int }_{B_r\left({\xi }_0\right)}|u-u_{{\xi }_0,r}{|·|\varphi Xu|}^{2-ϵ}d\xi$ in (44). By Hölder’s inequality and Young’s inequality, it follows that

$$\begin{array}{cc}\hfill & \mu \left(M\right){\int }_{B_r\left({\xi }_0\right)}|u-u_{{\xi }_0,r}{|·|\varphi Xu|}^{2-ϵ}d\xi \hfill \\ \hfill \le & \mu \left(M\right){\left[{\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi \right]}^{{\displaystyle \frac{2-ϵ}{2}}}{\left[{\int }_{B_r\left({\xi }_0\right)}{|u-u_{{\xi }_0,r}|}^{{\displaystyle \frac{2}{ϵ}}}d\xi \right]}^{{\displaystyle \frac{ϵ}{2}}}\hfill \\ \hfill \le & \epsilon \mu \left(M\right)r^{ϵ}{\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi +C\left(\epsilon \right)\mu \left(M\right){\int }_{B_r\left({\xi }_0\right)}{r}^{-2}{|u-u_{{\xi }_0,r}|}^{{\displaystyle \frac{2}{ϵ}}}d\xi \hfill \\ \hfill \le & \epsilon \mu \left(M\right)r^{ϵ}{\int }_{B_r\left({\xi }_0\right)}{|\varphi Xu|}^{2}d\xi +C\left(\epsilon \right)\mu \left(M\right){\left(2M\right)}^{{\displaystyle \frac{2-2ϵ}{2}}}{\int }_{B_r\left({\xi }_0\right)}{\left|{\displaystyle \frac{u-u_{{\xi }_0,r}}{r}}\right|}^{2}d\xi ,\hfill \end{array}$$

(57)

where we used the fact that $ϵ$ is some small positive constant.

Hence, the Caccioppoli-type inequality in Lemma 10 follows by a standard argument. The proof of Theorem 3 is then completed by following the same procedure as for Theorem 2. □

6. Conclusions

This paper establishes optimal partial Hölder regularity for quasilinear sub-elliptic systems under Hörmander’s vector fields via a novel $\mathcal{A}$-harmonic approximation method. To present the main results of this paper more clearly, we provide the following

Table 1. Comparison of regularity results under controlled growth vs. natural growth conditions.

Aspect	Controlled Growth (Theorem 1)	Natural Growth (Theorems 2 and 3)
Hypothesis on $A_{ij}^{\alpha \beta }$ and $B^{\alpha }$	H1–H2 and HC	H1–H2 and HN
Assumptions for u	$u\in H{W}_{\mathrm{loc}}^{1,2}\left(\mathsf{\Omega }\right)$	$u\in H{W}_{\mathrm{loc}}^{1,2}\left(\mathsf{\Omega }\right)\cap L^{\infty }\left(\mathsf{\Omega }\right)$
Regularity results	•$u\in {\mathsf{\Gamma }}_{\mathrm{loc}}^{0,2-{\displaystyle \frac{Q}{q}}}\left({\mathsf{\Omega }}_0\right)$, ${\displaystyle \frac{2Q}{Q+2}}<q<Q$	•$u\in {\mathsf{\Gamma }}_{\mathrm{loc}}^{0,1-{\displaystyle \frac{Q}{2q}}}\left({\mathsf{\Omega }}_0\right)$, $q>max\{{\displaystyle \frac{Q}{2}},2\}$
	•$u\in {\mathsf{\Gamma }}_{\mathrm{loc}}^{0,\alpha }\left({\mathsf{\Omega }}_0\right)$, $\alpha \in (0,1)$, $q\ge Q$	•$Xu\in L^{2,\lambda }\left({\mathsf{\Omega }}_0\right)$, $\lambda =Q-{\displaystyle \frac{Q}{q}}$
Exact Hölder exponent	$2-{\displaystyle \frac{Q}{q}}$	$1-{\displaystyle \frac{Q}{2q}}$
Mathematical insight	This comparison highlights a fundamental trade-off in the theory: the weaker structural hypotheses of HN necessitate stronger a priori assumptions (boundedness of u) and, crucially, lead to weaker regularity outcomes (lower Hölder exponents) compared to HC. This directly illustrates how the regularity of solutions is intrinsically governed by the strength of the underlying structural conditions.

summarizing the conditions, assumptions, and conclusions for both the controlled growth and natural growth cases.

Unlike classical direct approaches relying on reverse Hölder inequalities, our technique simplifies the analysis and unifies the treatment of both controlled and natural growth cases. Key improvements over the existing literature include the following:

• Extending regularity theory from continuous to VMO coefficients in the $\xi$-variable;
• Obtaining exact Hölder exponents: $2-Q/q$ (controlled growth) versus $1-Q/\left(2q\right)$ (natural growth);
• Relaxing the smallness condition $2\mu \left(M\right)M<\lambda$ under slightly stronger inhomogeneity constraints.

The methodology developed in this work not only simplifies existing arguments but also provides a unified framework for studying regularity in sub-elliptic systems. Future research directions include extending this approach to exploring applications to nonlinear sub-elliptic systems with non-standard growth conditions.