Gradient regularity for nonlinear sub-elliptic systems with the drift term: sub-quadratic growth case

1.   Introduction

In this paper, we consider the following nonlinear sub-elliptic systems with the drift term $Tu$ under sub-quadratic natural growth conditions in the Heisenberg group $\mathbb{H}^{n}$

where $\Omega$ is a bounded domain, and the horizontal gradient $X = \{X_1, \cdots, X_{2n} \}$ with the horizontal vector fields $X_i\: (i = 1, \cdots, 2n)$ and the vertical vector fields $T$ is defined (2.1) in the next section, $u = (u^{1}, \cdots, u^{N}):\Omega\rightarrow \mathbb{R}^{N}, \; A_{i}^{\alpha}(\xi, u, Xu):\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{2n\times N}\rightarrow \mathbb{R}^{2n\times N}$, and $B^{\alpha}(\xi, u, Xu):\Omega\times\mathbb{R}^{N}\times\mathbb{R}^{2n\times N}\rightarrow \mathbb{R}^{N}$.

As is well known, operators with drift terms possess significant importance for research and application. For instance, the Kolmogorov–Fokker–Planck operator frequently arises in transport diffusion equations within physical science, natural science, and statistical models. Following the publication of Lanconelli and Polidoro's fundamental work ^[25], this type of operator has garnered increasing attention. For a comprehensive understanding of the Kolmogorov–Fokker–Planck operator, the readers may refer to ^[26], which reviews the class of Kolmogorov operators with constant coefficients. Within the family of Kolmogorov operators, homogeneous ones occupy a central role. Indeed, any Kolmogorov operator can be approximated, in an appropriate sense, by a homogeneous operator. For more regularity results concerning operators with drift terms, the readers may refer to previous studies ^{[5,14,20,21,24]} and the references therein. In particular, Austin and Tyson ^[1] achieved $C^{\infty}$-smoothness by the geometric analysis method for the following operator

in the Heisenberg groups. Recently, Zhang and Niu ^[35] treated a quasi-linear sub-elliptic equation with drift in the Heisenberg group. For nonlinear discontinuous sub-elliptic systems with drift, Zhang and Wang ^[36,37] proved the partial $C^{0, \gamma}(0 < \gamma < 1)$ Hölder regularity of weak solutions.

The findings in the study of weak solutions for sub-elliptic equations and systems without the drift term include several notable regularity results. These results are significant because they provide insights into the behavior of solutions under various conditions. For a comprehensive understanding, the readers are encouraged to consult the works of Domokos ^[17]; Capogna ^[6,7]; Manfredi and Mingione ^[27]; Mingione, Zatorska-Goldstein, and Zhong^[29]; Mukherjee and Zhong ^[28]; and Citti and Mukherjee ^[9] for sub-elliptic equations, as well as the studies ^{[8,19,30,33,34]} for sub-elliptic systems. Among these contributions, a particularly noteworthy development is the extension of the $\mathcal{A}$-harmonic approximation technique to noncommutative nil-potent Lie groups. This technique involves constructing approximate solutions that satisfy certain harmonic-like properties, which can then be used to deduce the regularity properties of the original solutions. By applying this method in the context of noncommutative nil-potent Lie groups, researchers have been able to establish optimal partial regularity for nonlinear sub-elliptic systems, involving different growth rates and variant structure coefficients. It is worth pointing out that the $\mathcal{A}$-harmonic approximation method was introduced by Simon ^[31], and developed by Duzaar and Steffen ^[18] in the Euclidean space, and we refer the readers to ^{[3,4,11,15,16]} and the references therein for more results concerning nonlinear elliptic and parabolic systems.

Therefore, we examine the technique of $\mathcal{A}$-harmonic approximation to achieve $C^1$ regularity for nonlinear sub-elliptic systems with the drift term $Tu$ in the Heisenberg group. The primary novel aspect of this paper is our capacity to tackle the systems (1.1) that incorporate the drift term $Tu$, featuring a sub-quadratic growth rate, while relaxing the assumption on the principal coefficients to Dini continuity. We note that the first new challenge emerges due to the presence of the drift term $Tu$ without any assumption of integrability. Then, we adopt a clever approach to avoid the requirement of integrability. In fact, we subtly employ the relationship

and introduce a horizontal affine function $l$ defined in (2.6) to derive a suitable estimate for the drift term $Tu$. In contrast to the sub-quadratic sub-elliptic systems without any drift term, as examined in ^[33] for the range $1 < m < 2$, our scenario requires more stringent constraints $\frac{2Q}{Q+2} < m < 2$. This is primarily attributed to the fact that the estimates stemming from the drift term $Tu$ cannot be incorporated into the existing estimates during the formulation of Caccioppoli-type inequalities; see Lemma 3.2. The second challenge arises from the sub-quadratic growth rate, which prevents us from utilizing $L^2$-theory for functions in the horizontal Sobolev space $HW^{1, m}$ with $\frac{2Q}{Q+2} < m < 2$. For this reason, we choose the following excess functional

with $\rlap{-} \int_{B_\rho(\xi_0)}u(\xi)d\xi = \left|B_\rho(\xi_0)\right|_{\mathbb{H}^n}^{-1}\int_{B_\rho(\xi_0)}u(\xi)d\xi$ and $V(A) = (1+|A|^{2})^{\frac{4}{m-2}}$, and establish decay estimates for $\Phi$ by a generalization of the $\mathcal{A}$-harmonic approximation (Lemma 3.1) with the auxiliary function $V$ in the Heisenberg groups.

Now we are in the position to introduce the following structural assumptions for the coefficients $A_{i}^{\alpha}$ and $B^{\alpha}$ that are essential for our analysis throughout the paper.

(H1) The leading coefficient $A_i^\alpha(\xi, u, p)$ is differentiable in $p$, and there exists a constant $C$ such that

where $A_{i, p_\beta^j}^{\alpha}(\xi, u, p) = \frac{\partial{A}_{i}^{\alpha} (\xi, u, p)}{\partial{p}_\beta^j}$.

(H2) The term $A_i^\alpha(\xi, u, p)$ satisfies the following ellipticity condition

where $\lambda$ is a positive constant.

(H3) There exists a modulus of continuity $\mu:(0, \infty)\rightarrow [0, \infty)$ such that

where $K(\cdot):[0, \infty)\rightarrow [0, \infty)$ is monotonously nondecreasing. Without loss of generality, it is convenient to take $K(\cdot)\ge1$.

(HN) (Natural growth condition) For $|u|\leq{M} = \sup_{\Omega}|u|$, the nonhomogeneous term $B^{\alpha}(\xi, u, p)$ satisfies the following growth condition

where the positive constants $a = a(M)$ and $b = b(M)$ possibly depend on $M > 0$.

Without loss of generality, we can assume that

($\mu$1) $\mu$ is nondecreasing with $\mu(0+) = 0, \; \mu(1) = 1$;

($\mu$2) $\mu$ is concave, and $r\longrightarrow r^{-\gamma}\mu(r)$ is nonincreasing for some exponent $\gamma\in(0, 1)$;

($\mu$3) Dini's condition $M(r) = \int_{0}^{r}\frac{\mu(\rho)}{\rho}d\rho < \infty$ holds for some $r > 0$.

Furthermore, (H1) implies that

In addition, there exists a continuously non-negative and bounded function $\omega(s, t):[0, \infty)\times[0, \infty)\rightarrow [0, \infty)$, satisfying $\omega(s, 0) = 0$ for all $s$. Furthermore $\omega(s, t)$ is monotonously nondecreasing in $s$ for a fixed $t$ and monotonously nondecreasing in $t$ for a fixed $s$ such that

By the method of $\mathcal{A}$-harmonic approximation to establish $C^1$ regularity, the key point is to establish a certain excess decay estimate for the excess functional $\Phi$. In the case where $m\geq2$, this functional is given by

However, in the case of the sub-quadratic $\frac{2Q}{Q+2} < m < 2$, one should establish the excess decay estimate for the following functional:

It is shown that if $\Phi(\xi_0, \rho, Xl)$ is small enough on a ball $B_{\rho}(\xi_0)$, then for some fixed $\theta\in(0, 1)$, one has the excess improvement

where $\sigma = \min\{(2-m)(m-1)/m, (m-1)/2\}$ and $K^{\ast}(s, t) = C_{7}H^{2/(m-1)^{2}(2-m)}(s, M+t)$ with positive constants $C_7$. Iteration of this result leads to the excess decay estimate, which implies the regularity result.

The main result in this paper is as follows:

Theorem 1.1. Assume that the coefficients $A_{i}^{\alpha}$ and $B^{\alpha}$ satisfy (H1)–(H3) and (HN) with $(\mu1)-(\mu3)$. Let $u\in{HW}^{1, m}\left(\Omega, \mathbb{R}^N\right)\cap{L^{\infty}}\left(\Omega, \mathbb{R}^N\right)$ be a weak solution to the system (1.1) with $2a(1+3M)/[3^{m-2}{C(M, m, n)}] < \lambda$ and the constant $C(M, m, n)$ in line with Lemma $2.2,$ i.e., for $\forall\varphi\in C_0^\infty\left(\Omega, \mathbb{R}^N\right)$,

Then, there exists an open subset $\Omega_0\subset\Omega$, such that $u\in C^{1}\left(\Omega_0, \mathbb{R}^N\right)$. Moreover, $\Omega\backslash\Omega_0 = \Sigma_1\cup\; \Sigma_2$ and the Haar measure $(\Omega\backslash\Omega_0) = 0$, where

In addition, for $\tau\in[\gamma, 1)$ and $\xi_{0}\in\Omega_0$, the derivative $Xu$ has the modulus of continuity $r\longrightarrow r^{\tau}+M(r)$ in the neighborhood of $\xi_{0}$.

It is worth pointing out that the Haar measure in the Heisenberg groups with the underlying manifold $\mathbb{R}^{2n+1}$ is just the Lebesgue measure in $\mathbb{R}^{2n+1}$. Our result is optimal in the sense that when $\mu (\rho) = \rho^\gamma, 0 < \gamma < 1$, we have $M(r) = \int_0^r {\frac{{\mu (\rho)}}{\rho }} d\rho = \gamma ^{ - 1}r^\gamma$, and we obtain $C^{1, \gamma }$ optimal Hölder regularity of weak solutions in that case; see ^[32] by Wang, Liao, and Yu.

Remark 1.1. When we replace the condition in (1.6) with

with $\varepsilon > 0$ and omit the requirement $2a(1+3M)/[3^{m-2}{C(M, m, n)}] < \lambda$ in Theorem 1.1, the conclusion of Theorem 1.1 remains valid.

2.   Preliminaries

In this section, we introduce the Heisenberg group $\mathbb{H}^n$, some definitions of function spaces, and several elementary estimates which will be used later.

The Heisenberg group $\mathbb{H}^n$ is defined as $\mathbb{R}^{2n+1}$ with the following group multiplication:

for all $\xi = (\xi^1, t) = (x_1, x_2, \cdots, {x_n}, y_1, y_2, \cdots, y_n, t)$, $\tilde{\xi} = (\tilde{\xi}^1, \tilde t) = (\tilde x_1, \tilde x_2, \cdots, \tilde x_n, \tilde y_1, \tilde y_2, \cdots, \tilde y_n, \tilde t)$. Its neutral element is ${0}$, and its inverse of $(\tilde{\xi}^1, t)$ is given by $(-\tilde{\xi}^1, -t)$. In particular, the mapping $(\xi, \tilde{\xi})\mapsto\xi\cdot\tilde{\xi}^{-1}$ is smooth; therefore $(\mathbb{H}^n, \cdot)$ is a Lie group.

The basic vector fields corresponding to its Lie algebra can be explicitly calculated, given by

The special structure of the commutators is

that is, $\left({\mathbb{H}^n, \; \cdot } \right)$ is a nil-potent Lie group of step $2$. $X = (X_1, \cdots, X_{2n})$ is called the horizontal gradient, and $T$ is the vertical derivative.

The homogeneous norm is defined by $\left|\left|\left(\xi^1, t\right)\right|\right|_{\mathbb{H}^{n}} = \left(\left|\xi^1\right|^4+t^2\right)^{1/4},$ and the metric induced by this homogeneous norm is given by

The measure used on $\mathbb{H}^n$ is the Haar measure (the Lebesgue measure in ${\mathbb{R}}^{2n+1}$), and the volume of the homogeneous ball $B_R (\xi _0) = \left\{ {\xi \in \mathbb{H}^n:d(\xi _0, \xi) < R}\right\}$ is given by

where the number

is called the homogeneous dimension of $\mathbb{H}^n$, and the quantity $\omega _n$ is the volume of the homogeneous ball of radius 1.

Definition 2.1. Let $\Omega\subset{\mathbb{H}^n}$ be an open set, where the horizontal Sobolev space $HW^{1, m}(\Omega)\; (1\le\; m < \infty)$ is defined as:

which is a Banach space under the norm

and the spaces $HW_0^{1, m}(\Omega)$ is the completion of $C_0^{\infty}(\Omega)$ under the norm (2.5).

Definition 2.2. (Horizontal affine function). Let $u\in\; L^{2}\left(B_{\rho}(\xi_0), \; \mathbb{R}^{N}\right)$, $\xi_0\in\; \mathbb{R}^{2n+1}$, and denote the horizontal components as ${\xi^1} = \left(x^{1}, \; \cdots, \; x^{n}, \; y^{1}, \; \cdots, \; y^{n}\right)$ and ${\xi_{0}^1} = \left(x_{0}^{1}, \; \cdots, \; x_{0}^{n}, \; y_{0}^{1}, \; \cdots, \; y_{0}^{n}\right)$. We call

the horizontal affine function.

If the horizontal affine function $l_{\xi_0, \rho}({\xi^1})$ is a minimizer of the functional

we then have

and

Lemma 2.1. (from ^[2]) For every $1 < m < 2$, it holds that

for any $p, \tilde{p}\in R^{2n\times N}$.

From the fact that $\mu$ is nondecreasing, we conclude that $s\mu(t)\leq s\mu(s)$ for all $0\leq t\leq s$. We also note that $s\mu(t)\leq t\mu(s)\leq t$ for $0\leq s\leq t$ and $0\leq s\leq1$ by the nonincreasing property of $r\longmapsto\frac{\mu(r)}{r}$ and $\mu(1)\leq1$. Combining both cases, we get

From $(\mu2)$, we deduce for $\theta\in(0, 1), t > 0$, and $j\in\mathbb{N}$ that

which implies

This shows particularly that $\mu(t)\leq\frac{\gamma^{2}}{4}H^{2}(t)$ for all $t\leq0$, and $t\longmapsto t^{-\gamma}H(t)$ is also nonincreasing.

Throughout the paper, we use the functions $V = V_{m}, W = W_{m}:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$ defined by

for each $\zeta\in\mathbb{R}^{n}$ and $m > 1$.

The purpose of introducing $W$ is the fact that in contrast to $|V|^{\frac{2}{m}}$, the function $|W|^{\frac{2}{m}}$ is convex. In fact, direct computation shows that $W^{\frac{2}{m}}(t) = t^{\frac{2}{m}}(1+t^{2-m})^{-\frac{1}{m}}$ is a convex and monotone increasing function on $[0, \infty)$ with $W^{\frac{2}{m}}(0) = 0$. Moreover, we have

for any $\zeta_{1}, \zeta_{2}\in\mathbb{R}^{n}$.

The following lemma includes some useful properties of the function V, as outlined in Lemma 2.1 from ^[22].

Lemma 2.2. Let $m\in(1, 2)$ and $V, W:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$ be the functions defined above. Then, the following statements hold true for any $\zeta_{1}, \zeta_{2}\in\mathbb{R}^{n}$ and $t > 0:$

$(1)\; \frac{1}{\sqrt{2}}\min\left(|\zeta_{1}|, |\zeta_{1}|^{\frac{m}{2}}\right)\leq|V(\zeta_{1})|\leq\min\left(|\zeta_{1}|, |\zeta_{1}|^{\frac{m}{2}}\right)$;

$(2)\; |V(t\zeta_{1})|\leq\max\left(t, t^{\frac{m}{2}}\right)|V(\zeta_{1})|$;

$(3)\; |V(\zeta_{1}+\zeta_{2})|\leq C(m)(|V(\zeta_{1})|+|V(\zeta_{2})|)$;

$(4)\; \frac{m}{2}|\zeta_{1}-\zeta_{2}|\leq|V(\zeta_{1})-V(\zeta_{2})|/(1+|\zeta_{1}|^{2}+|\zeta_{2}|^{2})^{\frac{m-2}{4}}\leq C(m, n)|\zeta_{1}-\zeta_{2}|$;

$(5)\; |V(\zeta_{1})-V(\zeta_{2})|\leq C(m, n)|V(\zeta_{1}-\zeta_{2})|$;

$(6)\; |V(\zeta_{1}-\zeta_{2})|\leq C(m, M)|V(\zeta_{1})-V(\zeta_{2})|$ for all $\zeta_{1}$ with $|\zeta_{2}|\leq M$.

The inequalities (1)–(3) also hold if $V$ is replaced by $W$.

For later purposes, we state the following two simple estimates, which can be easily deduced from Lemma 2.2 (1) and (6). For $\zeta_{1}, \zeta_{2}\in\mathbb{R}^{n}$, $|\zeta_{2}|\leq M$, and for $|\zeta_{1}-\zeta_{2}|\leq1$,

When $|\zeta_{1}-\zeta_{2}| > 1$, it yields

We introduce a Sobolev–Poincaré type inequality and a prior estimate specifically for the case of sub-quadratic growth of the Heisenberg groups. Detailed proofs for these assertions can be found in the work by Wang, Liao, and Yu ^[32].

Lemma 2.3. (Sobolev–Poincaré type inequality) Let $m\in(1, 2)$ and $u\in HW^{1, m}(B_{\rho}(\xi_{0}), \mathbb{R}^{N})$ with $B_{\rho}(\xi_{0})\subset\Omega;$ then

where $m^{\ast} = \frac{mQ}{Q-m}$ is the Sobolev critical exponent of $m$. Furthermore, the analogous inequality is valid with $W$ being replaced by $V$, as defined in (2.11). In particular, the inequality also holds if we substitute $2$ for $\frac{2m^{\ast}}{m}$

Lemma 2.4. Let $u\in HW^{1, 1}(\Omega, \mathbb{R}^{N})$ be a weak solution of

for any $\phi\in C_{0}^{1}(\Omega, \mathbb{R}^{N})$, where $A_{i, j}^{\alpha, \beta}$ is a constant matrix satisfying the strong Legendre–Hadamard condition:

Then $u$ is smooth. $C_{0}\geq1$ exists such that for any $B_{\rho}(\xi_{0})\subset\Omega$

We will conclude this section with the following lemma from ^[23], which will be used to establish Caccioppoli-type inequality.

Lemma 2.5. Let $f(t)$ be a non-negative bounded function defined for $0\le T_0\le t\le T_1$. Suppose that for $T_0\le t < s\le T_1$, we have

where $A, \; B, \; \alpha, \; \beta$, and $\theta$ are non-negative constants and $\theta < 1$. Then there exists a constant $\bar{C} = \bar{C}(\theta, \alpha, \beta)$ such that for every $\rho, R: T_0\le \rho < R\le T_1$, we have

3.   Partial $C^1$ regularity

In this section, we mainly prove the Caccioppoli-type inequality for weak solutions of the systems (1.1) with drift. First, we state the result of the $\mathcal{A}$-harmonic approximation lemma, specifically addressing the case of sub-quadratic growth in the Heisenberg group, as exemplified in ^[10] for more general Carnot groups. The proof is similar to that in the Euclidean space ^[13].

3.1.   Caccioppoli-type inequality for sub-elliptic systems

Let $Bil(\mathbb{R}^{2n\times N})$ denote the collection of bilinear forms defined in $\mathbb{R}^{2n\times N}$, and suppose $\mathcal{A}\in Bil(\mathbb{R}^{2n\times N})$. We say that a function $h\in HW^{1, m}(\Omega, \mathbb{R}^{N})$ is $\mathcal{A}$-harmonic if $h$ satisfies:

Lemma 3.1. Let $\lambda$ and $L$ be fixed positive numbers $1 < m < 2$, and $n, N\in\mathbb{N}$ with $n\geq2$. If, for any given $\varepsilon > 0$, there exists $\delta = \delta(n, N, \lambda, L, \varepsilon)\in(0, 1]$ with the following properties:

$(I)$ For any $\mathcal{A}\in Bil(\mathbb{R}^{2n\times N})$ satisfying:

$(II)$ For any $g\in HW^{1, m}(B_{\rho}(\xi_{0}), \mathbb{R}^{N})$ satisfying:

There then exists an $\mathcal{A}$-harmonic function $h$

such that

We point out that Föglein, in ^[19], gave another version of the $\mathcal{A}$-harmonic approximation lemma, which developed the case of quadratic growth in the Euclidean space ^[11] to super-quadratic growth in the Heisenberg group.

In what follows, we assume that $\rho_{1}(s, t) = (1+s+t)^{-1}K(s+t)^{-1}$, and $K_{1}(s, t) = (1+t)^{2m}K^{4}(s+t)$ for $s, t\geq0$, and note that $\rho_{1}\leq1$ and that $s\longrightarrow \rho_{1}(s, t), \; t\longrightarrow\rho_{1}(s, t)$ are nonincreasing functions, where $K(\cdot)$ comes from (H3).

To show Theorem 1.1, our first aim is to establish a suitable Caccioppoli-type inequality.

Lemma 3.2. (Caccioppoli-type inequality) Let $u\in HW^{1, m}(\Omega, \mathbb{R}^{N})\cap L^{\infty}(\Omega, \mathbb{R}^{N})$ be a weak solution to the system in (1.1) under the conditions (H1)–(H3) and (HN) and $(\mu1)-(\mu3)$ with $2a(1+3M)/[3^{m-2}{C(M, m, n)}] < \lambda$. Then, for every $\xi_{0}\in\Omega$, and $0 < \rho\leq\rho_{1}^{\frac{m}{(2-m)(m-1)}}(|u_{0}|, |Xl|)$, it holds that

with

where we define $K(\cdot) = K(|u_{0}|+|Xl|)$, $\xi^{1} = (x_{1}, x_{2}, \cdots, \; x_{n}, y_{1}, y_{2}, \cdots, y_{n})$ is the horizontal component of $\xi = (x_{1}, x_{2}, \cdots, \; x_{n}, y_{1}, y_{2}, \cdots, y_{n}, t)\in \mathbb{H}^{n}$, and the constant $C_{c} = C_{c}(n, N, m, \lambda, M)$.

Proof. Let $\eta\in C_{0}^{\infty}(B_{\rho}(\xi_{0}))$ be a standard cut-off function satisfying $0\leq\eta\leq1, |X\eta| < \frac{c}{\rho}$, and $\eta\equiv1$ on $B_{\rho/2}(\xi_{0})$. We take $v = u(\xi)-u_{0}-Xl(\xi^{1}-\xi_{0}^{1})$ and $l = u_{0}+Xl(\xi^{1}-\xi_{0}^{1})$, and define the two functions

Then, one has

and

Using (H2), Lemma 2.1, and the elementary inequality

we have

From (3.10), it follows that

Noting that $A_{i}^{\alpha}(\xi_{0}, u_{0}, Xl)$ is a constant, we have

By the condition $\varphi = v$ on $B_{\rho/2}(\xi_{0})$, the elementary inequality $1+|a|^{2}+|b-a|^{2}\leq3(1+|a|^{2}+|b|^{2})$, and the fact that $\frac{m-2}{2} < 0$ holds true for $\frac{2Q}{Q+2} < m < 2$, the left-hand side of (3.11) can be estimated by

where we have applied the fact that $Xv = Xu-Xl$, Lemma 2.2 (4) in the third inequality from the end, Lemma 2.2 (6) in the penultimate inequality.

We are in the position to estimate the term $I$. By the fact that $Tl = X_{i}X_{n+i}l-X_{n+i}X_{i}l = 0$ and the condition $|T\eta|\le \frac{C}{\rho^2}$, it leads to

Applying Hölder's inequality, we infer that

To obtain a suitable estimate for $II$, we need to split the domain $B_{\rho}(\xi_{0})$ into four parts: $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|Xu-Xl|\leq1\}$, $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|Xu-Xl| > 1\}$, $B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|Xu-Xl > 1\}$, and $B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|Xu-Xl|\leq1\}$. We then use Young's inequality, and note that $|v| = |u-u_{0}-Xl(\xi^{1}-\xi_{0}^{1})|\leq2M+|Xl|\rho$ on $B_{\rho}(\xi_{0})$ to have the following estimates.

Case 1: For $B_{\rho}(\xi_{0})\cap\{|Xu-Xl| > 1\}\cap\{|v/\rho|\leq1\}$, it follows that

Case 2: For the set $B_{\rho}(\xi_{0})\cap\{|Xu-Xl| > 1\}\cap\{|v/\rho| > 1\}$, we have

Case 3: For $B_{\rho}(\xi_{0})\cap\{|Xu-Xl|\leq1\}\cap\{|v/\rho|\leq1\}$, we get

Case 4: For the case where $B_{\rho}(\xi_{0})\cap\{|Xu-Xl|\leq1\}\cap\{|v/\rho| > 1\}$, one obtains

Combining these estimates in $II''$, we have

where we have used $|Xl|\leq M+1$.

The condition (H3) yields the following (note that $m-1 < m/2$):

where we have used the inequality $s\mu(t)\leq s\mu(s)+t$ for $s\in[0, 1]$ and $t > 0$.

To obtain a suitable estimate for $III$, we need to split the domain $B_{\rho}(\xi_{0})$ into four parts: $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|X\varphi|\leq1\}, \; B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|X\varphi| > 1\}, \; B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|X\varphi > 1\}$, and $B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|X\varphi|\leq1\}$. We use Young's inequality, (2.9), and (2.10) repeatedly.

Case 1: For the set $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|X\varphi| > 1\}$,

Case 2: For $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|X\varphi|\leq1\}$,

where we have used the facts that $2(m-1)/m < 1$.

Case 3: For $B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|X\varphi| > 1\}$, observing that $m/(m-1) > 2$, one has

Case 4: For the case of $B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|X\varphi|\leq1\}$,

where we have used the fact that $2(m-1)/m < 1$.

Combining these estimations with (3.15), we get

We apply (H3) to get (noting that $m/(m-1) < 2$)

To obtain a suitable estimate for $IV$, we need to split the domain $B_{\rho}(\xi_{0})$ into four parts: $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|X\varphi|\leq1\}, \; B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|X\varphi| > 1\}, \; B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|X\varphi > 1\}$, and $B_{\rho}(\xi_{0}) \cap\{|v/\rho|\leq1\}\cap\{|X\varphi|\leq1\}$. We use Young's inequality, (2.9), and (2.10) repeatedly.

Case 1: $B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|X\varphi| > 1\}$ and $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|X\varphi| > 1\}$, observing that $m/(m-1) > 2$,

Case 2: $B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|X\varphi|\leq1\}$ and $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|X\varphi|\leq1\}$,

Combining these estimations with (3.18), we get

where we used $\frac{m}{m-1} > 2, \frac{m(m+2)}{2(m-1)} > m+2$.

By (H1), Lemma 2.1, and (3.9), it holds that

Noting that $-1/2 < \frac{m-2}{2} < 0$, and $\phi = (1-\eta)v = 0$ due to $\eta = 1$ on $B_{\rho/2}(\xi_{0})$, we split the domain $B_{\rho}(\xi_{0})$ into four parts: $B_{\rho}(\xi_{0})\cap\{|X\phi| > 1\}\cap\{X\varphi| > 1\}, \; B_{\rho}(\xi_{0})\cap\{|X\phi|\leq1\}\cap\{X\varphi|\leq1\}, \; B_{\rho}(\xi_{0})\cap\{|X\phi| > 1\}\cap\{X\varphi|\leq1\}$, and $B_{\rho}(\xi_{0})\cap\{|X\phi|\leq1\}\cap\{X\varphi| > 1\}$. Thus by Young's inequality and the estimations in (2.9) and (2.10), there is

Substituting (3.12), (3.13), (3.15), (3.17), (3.19) and (3.21) into (3.11), we finally arrive at

where we have used $2m/(m-1) > m(m+2)/2(m-1) > m^{2}/(m-1)$ and the nondecreasing property of $\mu$.

We take $\varepsilon = [3^{m-2}\lambda {C(M, m, n)}-2a(1+3M)]/6$ with the assumption $\lambda > 2a(1+3M)/[3^{m-2}{C(M, m, n)}]$. Filling the gaps with $\theta = \frac{\frac{C(m, M)}{m-1}+[3\varepsilon +2a(1+3M){C(M, m, n)}]}{3^{m-2}\lambda {C(M, m, n)}+\frac{C(m, M)}{m-1}} < 1$ in Lemma 2.5 yields

where

The proof is completed by noting that $[m(r-1)/r(m-1)-1]Q = m/(m-1)$ and $\rho^{m/(m-1)}\leq \rho^{2(2-m)(m-1)/m}\leq\mu^{2}(\rho^{(2-m)(m-1)/m})$. □

3.2.   Approximate $\mathcal{A}$-harmonicity by linearization

In this section, we provide a linearization strategy for non-linear sub-elliptic systems (1.1). Later on, this will be the starting point for the application of the $\mathcal{A}$-harmonic approximation lemma.

Lemma 3.3. We claim that if $\rho\leq\rho_{1}^{\frac{m}{(2-m)(m-1)}}(|u_{0}|, |Xl|)$ and $\varphi\in C_{0}^{\infty}(B_{\rho}(\xi_{0}), \mathbb{R}^{N})$ with $\sup\limits_{B_{\rho}(\xi_{0})}|X\varphi|\leq1$, then there exist some constants $C_{1} = C_{1}(m, M, C_{p}, K) > 1$ such that

where we assume that $F(s, t) = K^{4/(2-m)}(s+t)(2+t)^{2}+a(1+t^{m})+b$.

Proof. A straightforward computation yields

Then, we have

where we have used the fact that $\rlap{-} \int_{B_\rho(\xi_0)}A_{i}^{\alpha}(\xi_0, u_0, Xl)X\varphi^{\alpha}d\xi = 0$.

By the relationship of $T = X_{i}X_{n+i}-X_{n+i}X_{i}$, the term $I'$ can be estimated as follows:

Let

It follows that

We then obtain:

With the help of the fact that $\sup\limits_{B_{\rho}(\xi_{0})}|\varphi|\leq\rho\leq1$, we derive

For the case where $B_{1} = :B_{\rho}(\xi_{0})\cap\{|Xu-Xl|\leq1\}$, it follows, by Young's inequality and (2.9), that

and thus

On the other hand, for $B_{2} = :B_{\rho}(\xi_{0})\cap\{|Xu-Xl| > 1\}$, it follows by Young's inequality and (2.10) that

Thus, by combining these estimates and noting the definition of $F(s, t)$, we infer that

We can estimate the integrand of $III'$ in different ways depending on whether $|Xu-Xl|\leq1$ or $|Xu-Xl| > 1$.

For the first case, $|Xu-Xl|\leq1$. Applying (1.7), Lemma 2.2 (1), Hölder's inequality, and Jensen's inequality leads to

For the second case, $|Xu-Xl| > 1$. By the assumption in (H1), Lemma 2.2(1), and $\frac{2Q}{Q+2} < m < 2$, one gets

Combining the last two estimates mentioned above implies that

By employing (H3), Lemma 2.2, and Young's inequality, and noting the fact that $K(\cdot) > 1$, we deduce that

where we have used the fact that $4/(4-m) < 2, \; 1+m/2 < 2$ and $\mu(\rho)\leq1$ for $\rho\in[0, 1]$.

Using the inequality $s\mu(t)\leq s\mu(s)+t$ for $s\in[0, 1]$ and $t > 0$, we obtain:

To further estimate the term $V'$, we divide the ball $B_{\rho}(\xi_{0})$ into four parts.

Case 1: $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|Xu-Xl|\leq1\}$. By Young's inequality, the estimates of (2.9) and (2.10), and Sobolev–Poincaré inequality (2.11), it follows that

where we have used the fact that $\rho\leq\mu(\rho)\leq\mu(\sqrt{\rho})$.

Case 2: $B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|Xu-Xl|\leq1\}$. It yields

Case 3: $B_{\rho}(\xi_{0})\cap\{|v/\rho|\leq1\}\cap\{|Xu-Xl| > 1\}$. It follows

Case 4: $B_{\rho}(\xi_{0})\cap\{|v/\rho| > 1\}\cap\{|Xu-Xl| > 1\}$. It leads to

Combining these estimates above, we obtain:

Substituting (3.28), (3.29)–(3.33), and (3.35) into (3.25), we can immediately conclude that (3.23) holds. □

3.3.   Excess improvement

In this part, we apply linearization tools and $\mathcal{A}$-harmonic approximation techniques to establish improved estimates for the excess functional $\Phi$. For sake of simplicity, motivated by the form of the Caccioppoli-type inequalities, we set the following re-normalized excess functionals:

and

Lemma 3.4. Let $u\in HW^{1, m}(\Omega, \mathbb{R}^{N})\cap L^{\infty}(\Omega, \mathbb{R}^{N})$ satisfy the conditions of Theorem 1.1. Assume that the following smallness conditions are satisfied:

with $C_{2} = 8C_{1}^{2}C^{2}(m, M)C_{4}$, together with the condition

Then, we have the excess estimate for $\tau\in[\gamma, 1)$:

where $\sigma = \min\{(2-m)(m-1)/m, (m-1)/2\}$, and $K^{\ast}(s, t) = C_{7}H^{2/(m-1)^{2}(2-m)}(s, M+t)$.

Proof. For simplicity, we use the abbreviation $\Phi(\rho) = \Phi(\xi_{0}, \rho, (Xu)_{\xi_{0}, \rho})$ in what follows. For $\varepsilon > 0$ (to be determined later), we take $\delta\in(0, 1)$ and $\Upsilon\in[0, 1]$ to be the corresponding constant from the $\mathcal{A}$-harmonic approximation lemma and set

and

Noting the smallness assumptions in (3.36) and (3.37), we infer that

Then, from the definition of $\Upsilon$ and the Caccioppoli-type inequality (3.6) with $l = l_{\xi_0, \rho}$

We observe that (3.43) and (3.44) fulfill the conditions of the $\mathcal{A}$-harmonic approximation lemma, which ensures that we find an $\mathcal{A}$-harmonic function $h\in HW^{1, m}(B_{\rho}(\xi_{0}), \mathbb{R}^{N})$ such that

With the help of Lemma 2.2,

Next, we proceed to estimate the right-hand side of (3.46), by decomposing $B_{\theta\rho}(\xi_{0})$ into a set with

and

Then, by Lemma 2.2 (1) and Hölder's inequality, we obtain:

where we have used the term

Since $V(\zeta)$ is monotone increasing in $\zeta$, it follows that, from (3.46),

Now it remains for us to estimate $E$, noting that

Note that the smallness conditions in (3.36) and (3.37) imply that $C_{4}\Upsilon^{2}\leq1$ with $C_{4} = \max\{8C_{0}, (2\theta)^{-Q}\}$, where we have assumed $4C_{1}^{2}C^{2}(m, M)C_{4}\leq1$, which is no restriction. By applying the priori estimate for constant coefficients sub-elliptic systems, we have the following:

The Caccioppoli-type inequality applied to $B_{\theta\rho}(\xi_{0})$ with $u_{0} = u_{\xi_{0}, \rho}$, $Xl = (Xu)_{\xi_{0}, \rho}+\Upsilon(Xh)_{(\xi_{0}, 2\theta\rho)}$, and $\theta\in(0, 1/4]$ yields

where

By Lemma 2.2, one gets

To estimate the right-hand side, we employ (3.45) to infer that

Using Lemma 2.2 and the Sobolev–Poincaré-type inequality in Lemma 2.3 leads to

where we assume that $C_{5} = 4CC_{0}C_{p}^{4}$.

Now, we are in the position to estimate $\rlap{-} \int_{B_{2\theta\rho}(\xi_{0})}\left|\frac{\omega-\Upsilon h(\xi)}{2\theta\rho}\right|^{2}\mathrm{d}\xi$. Since $\left|V\left(\frac{\omega-\Upsilon h}{\rho}\right)\right|^{2}$ is bounded almost everywhere (3.45), we denote its upper bound by $M_{1}$. Lemma 2.2 (1) implies that

and

Hence, we have

Furthermore, it leads to

Using Lemmas 2.2–2.4 and (3.45) yields

Noting that $\Upsilon(Xh)_{\xi_{0}, 2\theta\rho}\leq2\sqrt{2}\sqrt{C_{0}}\Upsilon\leq1$ and the definition of $H(\cdot)$, we obtain:

and

where we have used the fact $\mu^{2}(\rho^{(m-1)/2})\geq\mu^{2}(\sqrt{\rho})$ and the nondecreasing property of $\mu$.

Combining all the above estimates with ($3.52$) and letting $\varepsilon = \theta^{Q+4}$, we arrive at

where $\sigma = \min\{(2-m)(m-1)/m, (m-1)/2\}$ and $C_{6}$ depends only on $Q, \; N, \; m, \; M, \; \lambda$, and $C_{p}$. For any given $\tau\in(\gamma, 1)$, choosing $\theta\in(0, \frac{1}{4})$ suitably such that $C_{3}C_{6}\theta^{2}\leq\theta^{2\tau}$, we easily find (note the definition of $\gamma$)

where the constant $C_{7}$ has the same dependencies as $C_{6}$ and$(2-m)(m-1)/m\leq3-2\sqrt{2} < 1/2, K^{\ast}(s, t) = C_{7}H^{2/(m-1)^{2}(2-m)}(s, M+t)$. □

For $T > 0$, we find $\Phi_{0}(T) > 0$ (depending on $Q, N, \lambda, L, \tau$ and $\omega$) such that

and

With $\Phi_{0}(T)$ from (3.64) and (3.65), we choose $\rho_{0}(T)\in(0, 1]$ (depending on $Q, N, \lambda, L, \tau, \omega, \eta$ and $\kappa$) such that

where $K_{0}(T): = K^{\ast}(2T, 2T)$.

The rest of the process to obtain Theorem 1.1 is very similar to that in ^[12]. We omit it here.

Author contributions

Beibei Chen: formal analysis, methodology, writing–original draft; Jialin Wang: formal analysis, methodology, writing–original draft, funding acquisition, supervision, writing–review and editing; Dongni Liao: formal analysis, methodology, funding acquisition, supervision, writing–review and editing. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors express their gratitude to the referees for their insightful comments and valuable suggestions. This research is supported by the National Natural Science Foundation of China (No. 12061010), Jiangxi Provincial Natural Science Foundation (No. 20242BAB26003), and the Science and Technology Planning Project of Jiangxi Province (No. GJJ2201204).

Conflict of interest

The authors declare no competing interests.