Preamble

Besov Estimates for Sub-elliptic Equations in the Heisenberg Group

Huimin Cheng
School of Mathematics and Statistics
Shandong Normal University
Jinan, Shandong, 250358, China
e-mail: 2995246312@qq.com

Feng Zhou*
School of Mathematics and Statistics
Shandong Normal University
Jinan, Shandong, 250358, China
e-mail: zhoufeng@u.nus.edu

February 22, 2024

Abstract

In this paper, we study weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group and investigate the regularity of these solutions. We establish horizontal Calderón-Zygmund type estimates in Besov spaces under more general assumptions on the coefficients for both homogeneous and non-homogeneous equations. This work expands the Calderón-Zygmund theory in the Heisenberg group.

Keywords: Heisenberg group; sub-elliptic equations; regularity; Besov spaces.

Mathematics Subject Classification (2020): 35R03, 35H20, 35J70.

*F. Zhou is the corresponding author.

Introduction

The main purpose of this article is to study Besov regularity of weak solutions to a class of sub-elliptic equations of the type
$$
\operatorname{div}H A(x, Xu) = 0 \quad \text{(1.1)}
$$
and
$$
\operatorname{div}_H A(x, Xu) = \operatorname{div}_H \left(|F|^{p-2}F\right) \quad \text{(1.2)}
$$
in $\Omega$, where $\Omega$ is an open and bounded subdomain in the Heisenberg group $\mathbb{H}^n = \mathbb{R}^{2n+1}$ ($n \geq 1$). We refer to (1.1) and (1.2) as the homogeneous and non-homogeneous equations, respectively. The unknown $u$ belongs to the local horizontal Sobolev space $HW^{1,p}}}(\Omega)$, which will be defined in Section 2. In both equations, the horizontal divergence operator $\operatorname{divH$ and the horizontal gradient $X$ are defined by
$$
\operatorname{div}_H F = \sumu)}^{2n} X_i F_i, \quad Xu = (X_1u, X_2u, \ldots, X_{2n-1}u, X_{2n
$$
in the distributional sense. Moreover, $A: \Omega \times \mathbb{R}^{2n} \to \mathbb{R}^{2n}$ is assumed to be a Carathéodory vector field satisfying general growth and uniform ellipticity conditions, meaning there exist constants $\nu, L, k > 0$ and $0 < \mu < 1$ such that
$$
[A(x, \xi) - A(x, \eta)] \cdot (\xi - \eta) \geq \nu \left(\mu^2 + |\xi|^2 + |\eta|^2\right)^{\frac{p-2}{2}} |\xi - \eta|^2,
$$
$$
|A(x, \xi) - A(x, \eta)| \leq L \left(\mu^2 + |\xi|^2 + |\eta|^2\right)^{\frac{p-2}{2}} |\xi - \eta|,
$$
$$
|A(x, \xi)| \leq k \left(\mu^2 + |\xi|^2\right)^{\frac{p-1}{2}}
$$
for every $\xi, \eta \in \mathbb{R}^{2n}$ and almost all $x \in \Omega$. In (1.2), $F: \Omega \to \mathbb{R}^{2n}$.

The regularity of solutions to elliptic equations in Euclidean spaces $\mathbb{R}^n$ has been well studied by Iwaniec [10], DiBenedetto and Manfredi [7]. This theory was subsequently extended to general elliptic problems; see the relevant papers [11, 12, 4, 3]. For nonlinear Calderón-Zygmund estimates in the Heisenberg group, Goldstein and Zatorska-Goldstein [8] treated the quadratic case $p = 2$. Later, $HW^{1,p}$ estimates for sub-elliptic equations on $\mathbb{H}^n$ were proved by Mingione, Zatorska-Goldstein and Zhong [14], who considered equations of the form
$$
\operatorname{div}H [b(x)a(Xu)] = \operatorname{div}_H \left(|F|^{p-2}F\right)
$$
with $b \in VMO(\Omega)$.}

Currently, research has focused on regularity estimates for weak solutions in Besov spaces in both $\mathbb{R}^n$ and $\mathbb{H}^n$ ([2, 6, 9]). Besov spaces constitute a broader class of functions compared to classical Sobolev spaces. Baisón [1] treated nonlinear elliptic equations in divergence form and obtained Besov regularity estimates for weak solutions. Clop [5] and Lyaghfouri [13] extended these Besov space results by establishing higher integrability of weak solutions.

For the homogeneous case (1.1), we assume there exists a function $g \in L^\alpha(\Omega)$ ($0 < \alpha < 1$) such that
$$
|A(x, \xi) - A(y, \xi)| \leq d_{CC}(x, y)^\alpha (g(x) + g(y)) \left(\mu^2 + |\xi|^2\right)^{\frac{p-1}{2}}
$$
for almost every $x, y \in \Omega$ and all $\xi \in \mathbb{R}^{2n}$. Here $d_{CC}(x, y)$ denotes the CC-distance between points $x$ and $y$ in $\mathbb{H}^n$.

For the non-homogeneous case (1.2), we assume there exists a sequence of measurable non-negative functions $g_k \in L^\alpha(\Omega)$ ($k \in \mathbb{N}$, $0 < \alpha < 1$) satisfying
$$
\sum_{k=1}^\infty |g_k|{L^\alpha(\Omega)}^q < \infty \quad (1 \leq q < \infty)
$$
and
$$
|A(x, \xi) - A(y, \xi)| \leq d}(x, y)^\alpha (g_k(x) + g_k(y)) \left(\mu^2 + |\xi|^2\right)^{\frac{p-1}{2}
$$
for $\xi \in \mathbb{R}^{2n}$ and almost all $x, y \in \Omega$ such that $2^{-k} \leq d_{CC}(x, y) < 2^{-k+1}$. Following (A5), we abbreviate this as ${g_k}_k \in \ell^q(L^\alpha(\Omega))$.

By introducing the auxiliary function
$$
V(\xi) = \left(\mu^2 + |\xi|^2\right)^{\frac{p-2}{2}} \quad \text{(1.3)}
$$
with $\xi \in \mathbb{R}^{2n}$, we present the main results of this article.

Theorem 1.1. Let $0 < \alpha < 1$ and $2 \leq p < 4$. Assume that $A$ satisfies hypotheses (A1)-(A4) with $0 < \mu < 1$. If $u \in HW^{1,p}{\text{loc}}(\Omega)$ is a weak solution to (1.1), then $V(Xu) \in B^\alpha(\Omega)$ locally.

Theorem 1.2. Let $0 < \alpha < 1$, $2 \leq p < 4$, and $1 \leq q < \frac{2Q}{Q-2\alpha}$. Assume that hypotheses (A1)-(A3) and (A5) hold. If $u \in HW^{1,p}{\text{loc}}(\Omega)$ is a weak solution to (1.2) with $0 < \mu < 1$ and $|F|^{p-2}F \in B^\alpha(\Omega)$ locally.}(\Omega)$, then $V(Xu) \in B^\alpha_{2,q

See Section 2 for the definitions of $HW^{1,p}(\Omega)$ and $B^\alpha_{2,q}(\Omega)$.

The contribution of our main results is the study of a broad class of sub-elliptic equations in the Heisenberg group. Our aim is to obtain Besov regularity estimates for weak solutions. The hypotheses (A1)-(A4) (or (A5)) represent an extension of VMO conditions.

This article is organized as follows. In Section 2 we provide definitions and tools such as classical inequalities, and present two lemmas concerning reverse Hölder type inequalities for weak solutions. In Sections 3 and 4 we give the proofs of Theorem 1.1 and Theorem 1.2, respectively.

2.1 Heisenberg Group

In this section we collect basic notation and preliminaries for the Heisenberg group.

We denote by $(x, t) = (x_1, x_2, \ldots, x_{2n}, t)$ the coordinates of points in the Heisenberg group $\mathbb{H}^n$. The group structure on $\mathbb{H}^n$ is given by
$$
(x_1, x_2, \ldots, x_{2n}, t) \circ (y_1, y_2, \ldots, y_{2n}, s) = \left(x_1 + y_1, x_2 + y_2, \ldots, x_{2n} + y_{2n}, t + s + \sum_{j=1}^n (x_j y_{n+j} - x_{n+j} y_j)\right).
$$

An anisotropic dilation induces a homogeneous norm (gauge) on $(x, t)$ by
$$
|(x, t)| = \left(|x|^4 + t^2\right)^{1/4}.
$$

For $j = 1, \ldots, n$, we define the left-invariant vector fields
$$
X_j = \frac{\partial}{\partial x_j} - \frac{x_{n+j}}{2} \frac{\partial}{\partial t}, \quad X_{n+j} = \frac{\partial}{\partial x_{n+j}} + \frac{x_j}{2} \frac{\partial}{\partial t}, \quad T = \frac{\partial}{\partial t},
$$
which form a basis of the space of left-invariant vector fields on $\mathbb{H}^n$. The vector fields $X_1, X_2, \ldots, X_{2n}$ are called horizontal vector fields. The length of the horizontal gradient is then given by
$$
|Xu|^2 = \sum_{j=1}^{2n} (X_j u)^2.
$$

2.2 CC-distance and CC-Balls

By considering the well-known Carnot-Carathéodory metric with CC-distance $d_{CC}$, we define CC-balls by
$$
B_R(x_0) = {y \in \mathbb{H}^n \mid d_{CC}(x_0, y) < R}
$$
with center $x_0$ and radius $R$. Introducing the homogeneous dimension $Q = 2n + 2$, we obtain the Lebesgue measure of a CC-ball $|B_R(x_0)| \approx R^Q$.

2.3 Horizontal Sobolev Spaces and Besov Spaces

Let $L^p(\mathbb{H}^n)$ denote the Lebesgue space on the Heisenberg group. The dual space of $L^p(\mathbb{H}^n)$ is $L^{p'}(\mathbb{H}^n)$ where $\frac{1}{p} + \frac{1}{p'} = 1$. The horizontal Sobolev space with its norm is defined by
$$
HW^{1,p}(\Omega) = {u \in L^p(\Omega) : Xu \in L^p(\Omega)}, \quad |u|{HW^{1,p}(\Omega)} = |u|.} + |Xu|_{L^p(\Omega)
$$

A function $u$ belongs to $HW^{1,p}_{\text{loc}}(\Omega)$ if $u \in HW^{1,p}(\Omega_0)$ for every $\Omega_0 \Subset \Omega$.

Let the parameters satisfy $0 < \alpha < 1$, $1 \leq p < \infty$, and $1 \leq q \leq \infty$. The Besov spaces $B^\alpha_{p,q}(\Omega)$ ($\Omega \subset \mathbb{H}^n$) with their norm are defined via [16] as
$$
|u|{B^\alpha = |u|}(\Omega){L^p(\Omega)} + [u],}(\Omega)
$$
where the seminorm $[u]{B^\alpha$ is given by}(\Omega)
$$
[u]{B^\alpha}(\Omega)} = \begin{cases
\left(\displaystyle\int_{\mathbb{H}^n} \frac{|\Delta_h u|{L^p(\Omega)}^q}{|h|^{\alpha q}} \frac{dh}{|h|^Q}\right)^{1/q}, & 1 \leq q < \infty, \
\displaystyle\sup, & q = \infty.} \frac{|\Delta_h u|_{L^p(\Omega)}}{|h|^\alpha
\end{cases}
$$

In this article, we write $\Delta_h u = u(x + h) - u(x)$ for brevity.

2.4 Basic Tools

For every $\varepsilon > 0$, there exists $C(\varepsilon) > 0$ such that for all $s, t \geq 0$,
$$
st \leq \varepsilon s^p + C(\varepsilon) t^{p'}, \quad \text{(2.1)}
$$
which is the classical Young inequality, where $\frac{1}{p} + \frac{1}{p'} = 1$. In particular,
$$
ab \leq \varepsilon a^2 + C(\varepsilon) b^2. \quad \text{(2.2)}
$$

Let $B_R \Subset \mathbb{H}^n$ be a CC-ball and $f$ an integrable function on $B_R$. We define the average of $f$ over the CC-ball $B_R$ as
$$
(f){B_R} = \frac{1}{|B_R|} \int} f(x)\,dx \approx R^{-Q} \int_{B_R} f(x)\,dx. \quad \text{(2.3)
$$

We present the definition of weak solutions. If for any $\phi \in C_0^\infty(\Omega)$,
$$
\int_\Omega A(x, Xu) \cdot X\phi\,dx = \int_\Omega |F|^{p-2}F \cdot X\phi\,dx, \quad \text{(2.4)}
$$
then $u \in HW^{1,p}_{\text{loc}}(\Omega)$ is a weak solution to (1.2). Here we call $\phi$ a test function.

2.5 Reverse Hölder Type Inequalities

The higher integrability estimates for Laplace and $p$-Laplace equations are well known (see [10] and [7]). In the Heisenberg group, we have the following two results for homogeneous and non-homogeneous situations (see [14]).

Lemma 2.1. Let $u \in HW^{1,p}(\Omega)$ with $2 < p < 4$ be a weak solution to (1.1) under hypotheses (A1)-(A4). There exists a constant $c(n, p, \nu, k, L)$, independent of $\mu$, the solution $u$, and the vector field $A(x, \nabla u)$, such that the following inequality holds for any CC-ball $B_R \Subset \Omega$:
$$
\left(\frac{1}{|B_R|} \int_{B_R} |Xu|^p\,dx\right)^{1/p} \leq c \left(\frac{1}{|B_{2R}|} \int_{B_{2R}} (\mu + |Xu|)^p\,dx\right)^{1/p}. \quad \text{(2.5)}
$$

Lemma 2.2. Let $u \in HW^{1,p}(\Omega)$ with $2 < p < 4$ be a weak solution to equation (1.2). Assume that (A1)-(A3) and (A5) hold. If $F \in L^q_{\text{loc}}(\Omega)$, then $Xu \in L^q_{\text{loc}}(\Omega)$, where $q \in (p, \infty)$. Moreover, there exists a positive constant $C(n, p, \nu, L, q, a)$ such that
$$
\left(\frac{1}{|B_R|} \int_{B_R} |Xu|^q\,dx\right)^{1/q} \leq C \left(\frac{1}{|B_{2R}|} \int_{B_{2R}} (\mu + |Xu|)^p\,dx\right)^{1/p} + C \left(\frac{1}{|B_{2R}|} \int_{B_{2R}} |F|^q\,dx\right)^{1/q} \quad \text{(2.6)}
$$
for any CC-ball $B_R \Subset \Omega$.

3 Proof of Theorem 1.1

In this section we present the proof of Theorem 1.1. Inspired by [5], for the vector field $A(x, \xi)$ appearing in (1.2), we introduce for $\xi \in \mathbb{R}^{2n}$ and a CC-ball $B \subset \Omega$:
$$
A_B(\xi) = \frac{1}{|B|} \int_B A(x, \xi)\,dx, \quad \text{(3.1)}
$$
and define
$$
V(x, B) = \sup_{\xi \in \mathbb{R}^{2n}} \frac{|A(x, \xi) - A_B(\xi)|}{\left(\mu^2 + |\xi|^2\right)^{\frac{p-1}{2}}}, \quad \text{(3.2)}
$$
where $B \subset \Omega$ is a CC-ball and $x \in \Omega$. It follows that if $A: \Omega \times \mathbb{R}^{2n} \to \mathbb{R}^{2n}$ is a Carathéodory vector field such that (A1)-(A4) hold, then $A$ is locally uniformly in VMO, that is,
$$
\lim_{R \to 0} \sup_{c(B) \in K, r(B) < R} \frac{1}{|B|} \int_B V(x, B)\,dx = 0, \quad \text{(3.3)}
$$
where $K \Subset \Omega$, and $c(B)$ and $r(B)$ denote the center and radius of the CC-ball $B$, respectively.

To prove Theorem 1.1, we note that there exists a constant $\widehat{C} > 0$ such that
$$
\widehat{C}^{-1} \left(\mu^2 + |\xi|^2 + |\eta|^2\right)^{\frac{p-2}{2}} |\xi - \eta|^2 \leq |V(\xi) - V(\eta)|^2 \leq \widehat{C} \left(\mu^2 + |\xi|^2 + |\eta|^2\right)^{\frac{p-2}{2}} |\xi - \eta|^2, \quad \text{(3.4)}
$$
for any $\xi, \eta \in \mathbb{R}^{2n}$ with $|\xi - \eta| \neq 0$.

We are now ready to present the proof.

Proof of Theorem 1.1. Let $B_{3R} \Subset \Omega$ and choose a test function $\phi = \Delta_{-h}(\eta^2 \Delta_h u)$ for (1.1), where $\eta \in C_0^\infty(B_{3R})$ is a cutoff function satisfying $0 \leq \eta(x) \leq 1$, $\eta(x) \equiv 1$ for $x \in B_R$, $\eta(x) \equiv 0$ for $x \in B_{3R} \setminus B_{2R}$, and $|X\eta| \leq C/R$.

Testing (1.1) with this $\phi$ yields
$$
\int_\Omega [A(x+h, Xu(x+h)) - A(x+h, Xu)] \cdot \eta^2 \Delta_h Xu\,dx = G_1 + G_2 + G_3 + G_4,
$$
where
$$
G_1 = \int_\Omega [A(x+h, Xu(x+h)) - A(x+h, Xu)] \cdot \eta^2 \Delta_h Xu\,dx,
$$
$$
G_2 = \int_\Omega [A(x, Xu) - A(x+h, Xu)] \cdot \eta^2 \Delta_h Xu\,dx,
$$
$$
G_3 = \int_\Omega [A(x+h, Xu) - A(x+h, Xu(x+h))] \cdot 2\eta X\eta \Delta_h u\,dx,
$$
$$
G_4 = \int_\Omega [A(x, Xu) - A(x+h, Xu)] \cdot 2\eta X\eta \Delta_h u\,dx.
$$

We estimate each $G_i$ ($1 \leq i \leq 4$). By (A1), it is clear that
$$
G_1 \geq \nu \int_\Omega \left(\mu^2 + |Xu(x+h)|^2 + |Xu|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2 \eta^2\,dx. \quad \text{(3.5)}
$$

For $G_2$, using (A4) and (2.2), we obtain
$$
|G_2| \leq C \int_\Omega |h|^\alpha (g(x) + g(x+h)) \left(\mu^2 + |Xu|^2\right)^{\frac{p-1}{2}} |\Delta_h Xu| \eta^2\,dx
$$
$$
\leq \varepsilon \int_\Omega \left(\mu^2 + |Xu|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2 \eta^2\,dx + C(\varepsilon) |h|^{2\alpha} \int_\Omega (g(x) + g(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx, \quad \text{(3.6)}
$$
where $\varepsilon > 0$ will be chosen later. By (A2) and (2.2), we deduce that
$$
|G_3| \leq C \int_\Omega \left(\mu^2 + |Xu|^2 + |Xu(x+h)|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu| \eta |X\eta| |\Delta_h u|\,dx
$$
$$
\leq \varepsilon \int_\Omega \left(\mu^2 + |Xu|^2 + |Xu(x+h)|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2 \eta^2\,dx + C(\varepsilon) \int_\Omega \left(\mu^2 + |Xu|^2 + |Xu(x+h)|^2\right)^{\frac{p-2}{2}} |X\eta|^2 |\Delta_h u|^2\,dx. \quad \text{(3.7)}
$$

Applying the Lagrange Mean Value Theorem, we obtain
$$
\int_\Omega \left(\mu^2 + |Xu|^2 + |Xu(x+h)|^2\right)^{\frac{p-2}{2}} |X\eta|^2 |\Delta_h u|^2\,dx \leq C |h|^2 \int_{B_{2R+|h|}} \left(\mu^2 + 2|Xu|^2\right)^{\frac{p-2}{2}} |Xu|^2\,dx \leq C |h|^2 \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx. \quad \text{(3.8)}
$$

To estimate $G_4$, hypothesis (A4) and (2.2) give
$$
|G_4| \leq C \int_\Omega |h|^\alpha (g(x) + g(x+h)) \left(\mu^2 + |Xu|^2\right)^{\frac{p-1}{2}} \eta |X\eta| |\Delta_h u|\,dx
$$
$$
\leq \varepsilon \int_\Omega \left(\mu^2 + |Xu|^2\right)^{\frac{p-2}{2}} \eta^2 |\Delta_h u|^2\,dx + C(\varepsilon) |h|^{2\alpha} \int_\Omega (g(x) + g(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx. \quad \text{(3.9)}
$$

Here we note that
$$
\int_\Omega \left(\mu^2 + |Xu|^2\right)^{\frac{p-2}{2}} \eta^2 |\Delta_h u|^2\,dx \leq C |h|^2 \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx. \quad \text{(3.10)}
$$

Combining the estimates for $G_i$ and choosing $\varepsilon$ sufficiently small, we obtain
$$
\int_\Omega \left(\mu^2 + |Xu(x+h)|^2 + |Xu|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2 \eta^2\,dx \leq C |h|^{2\alpha} \int_\Omega (g(x) + g(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx + C |h|^2 \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx. \quad \text{(3.11)}
$$

By the definition of $V$ and (3.4), we have
$$
|\Delta_h V|^2 \leq C \left(\mu^2 + |Xu(x+h)|^2 + |Xu|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2. \quad \text{(3.12)}
$$

Integrating both sides of (3.12) over $B_R$ and applying the properties of $\eta$, we get
$$
\int_{B_R} |\Delta_h V|^2\,dx \leq C \int_\Omega \left(\mu^2 + |Xu(x+h)|^2 + |Xu|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2 \eta^2\,dx
$$
$$
\leq C |h|^{2\alpha} \int_\Omega (g(x) + g(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx + C |h|^2 \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx. \quad \text{(3.13)}
$$

Dividing both sides of (3.13) by $|h|^{2\alpha}$, we obtain
$$
\int_{B_R} \frac{|\Delta_h V|^2}{|h|^{2\alpha}}\,dx \leq C \int_\Omega (g(x) + g(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx + C |h|^{2-2\alpha} \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx =: P_1 + P_2. \quad \text{(3.14)}
$$

Finally, we show that $P_i$ is bounded for each $i$. By Lemma 2.1, we have $|Xu|^p \in L^t(\Omega)$ for some $t > 1$. In particular, $|Xu|^p \in L^{\frac{Q}{Q-2\alpha}}(\Omega)$. Choosing $0 < |h| < \delta < R$ and using (A4), we obtain
$$
P_1 \leq C |g|{L^\alpha(\Omega)}^2 \left|\left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\right| < \infty.}{Q-2\alpha}}(\Omega)
$$

Since $u \in HW^{1,p}{\text{loc}}(\Omega)$, we have $P_2 < \infty$. It follows that
$$
\sup\,dx < \infty,} \int_{B_R} \frac{|\Delta_h V|^2}{|h|^{2\alpha}
$$
that is, $V(Xu) \in B^\alpha_{2,\infty}(\Omega)$ locally.

4 Proof of Theorem 1.2

For the non-homogeneous case, we need the following lemma.

Lemma 4.1. Let $A: \Omega \times \mathbb{R}^{2n} \to \mathbb{R}^{2n}$ be a Carathéodory vector field such that (A1)-(A3) and (A5) hold. Then $A$ is locally uniformly in VMO, that is,
$$
\lim_{R \to 0} \sup_{c(B) \in K, r(B) < R} \frac{1}{|B|} \int_B V(x, B)\,dx = 0, \quad \text{(4.1)}
$$
where $V(x, B)$ is given in (3.2), $K \Subset \Omega$, and $c(B)$ and $r(B)$ denote the center and radius of the CC-ball $B$, respectively.

Proof. Given a point $x \in \Omega$, let $A_k(x) = {y \in \Omega : 2^{-k} \leq d_{CC}(x, y) < 2^{-k+1}}$. We obtain
$$
\frac{1}{|B|} \int_B V(x, B)\,dx \leq \frac{1}{|B|} \int_B \sup_{\xi \in \mathbb{R}^{2n}} \frac{|A(x, \xi) - A(y, \xi)|}{\left(\mu^2 + |\xi|^2\right)^{\frac{p-1}{2}}}\,dy\,dx
$$
$$
\leq \frac{C(Q, \alpha)}{|B|} \sum_{k} \int_{B \cap A_k(x)} d_{CC}(x, y)^\alpha (g_k(x) + g_k(y))\,dy\,dx.
$$

By Hölder's inequality, we acquire
$$
\frac{1}{|B|} \int_{B \cap A_k(x)} (g_k(x) + g_k(y))^\alpha\,dy\,dx \leq C(Q, \alpha, q) |B|^{-\frac{\alpha}{Q}} |g_k|_{L^\alpha(B)}.
$$

Choosing $r > 0$ small enough and observing that $x \mapsto |g_k|{L^\alpha(B_r(x))}$ is continuous on the set ${x \in \Omega : \text{dist}(x, \partial\Omega) > r}$, we find that for each $x_r \in K$ with $r$ sufficiently small,
$$
|g_k| r \to 0.} \to 0 \quad \text{as
$$

Each of the limits on the right-hand side equals zero, which completes the proof.

With the help of the preceding lemma, we can now prove Theorem 1.2.

Proof of Theorem 1.2. Assume that $B_{3R} \Subset \Omega$ and choose a test function $\phi = \Delta_{-h}(\eta^2 \Delta_h u)$ for (1.2), where $\eta \in C_0^\infty(\Omega)$ is a cutoff function satisfying $0 \leq \eta(x) \leq 1$, $\eta(x) \equiv 1$ for $x \in B_R$, $\eta(x) \equiv 0$ for $x \in B_{3R} \setminus B_{2R}$, and $|X\eta| \leq C/R$.

According to the definition of weak solution and our choice of test function, we obtain
$$
\int_\Omega [A(x+h, Xu(x+h)) - A(x+h, Xu)] \cdot \eta^2 \Delta_h Xu\,dx
$$
$$
+ \int_\Omega [A(x, Xu) - A(x+h, Xu)] \cdot \eta^2 \Delta_h Xu\,dx
$$
$$
+ \int_\Omega [A(x+h, Xu) - A(x+h, Xu(x+h))] \cdot 2\eta X\eta \Delta_h u\,dx
$$
$$
+ \int_\Omega [A(x, Xu) - A(x+h, Xu)] \cdot 2\eta X\eta \Delta_h u\,dx
$$
$$
= \int_\Omega |F|^{p-2}F \cdot 2\eta X\eta \Delta_h u\,dx + \int_\Omega |F|^{p-2}F \cdot \eta^2 \Delta_h Xu\,dx,
$$
which we denote as $G_1 + G_2 + G_3 + G_4 = G_5 + G_6$. \quad \text{(4.2)}

We have already estimated the terms $G_1$ through $G_4$ in the proof of Theorem 1.1. Thus it remains to estimate $G_5$ and $G_6$.

Applying (2.2), we get
$$
|G_5| \leq C \int_\Omega |\Delta_h (|F|^{p-2}F)| |\Delta_h u| \eta\,dx
$$
$$
\leq C |h|^{2\alpha} \int_\Omega |F|^{2(p-1)}\,dx + C \int_\Omega |\Delta_h u|^2 \eta^2\,dx.
$$

By the Lagrange Mean Value Theorem, the second term can be controlled by
$$
\int_\Omega |\Delta_h u|^2 \eta^2\,dx \leq C |h|^2 \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx. \quad \text{(4.3)}
$$

For the estimate of $G_6$, we have
$$
|G_6| \leq C \int_\Omega |\Delta_h (|F|^{p-2}F)| |\Delta_h Xu| \eta^2\,dx
$$
$$
\leq C |h|^{2\alpha} \int_\Omega |F|^{2(p-1)}\,dx + \varepsilon \int_\Omega |\Delta_h Xu|^2 \eta^2\,dx. \quad \text{(4.4)}
$$

Similarly, one obtains
$$
\int_\Omega |\Delta_h Xu|^2 \eta^2\,dx \leq \mu^{p-2} \int_\Omega |\Delta_h Xu|^2 \eta^2\,dx \leq \int_\Omega \left(\mu^2 + |Xu(x+h)|^2 + |Xu|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2 \eta^2\,dx. \quad \text{(4.5)}
$$

Combining the estimates for $G_i$, we have
$$
\left(\nu - 2\varepsilon\right) \int_\Omega \left(\mu^2 + |Xu(x+h)|^2 + |Xu|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2 \eta^2\,dx
$$
$$
\leq C |h|^{2\alpha} \int_\Omega (g_k(x) + g_k(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx + C |h|^{2\alpha} \int_\Omega |F|^{2(p-1)}\,dx + C |h|^2 \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx. \quad \text{(4.6)}
$$

Choosing $\varepsilon = \nu/4$, we obtain
$$
\int_\Omega \left(\mu^2 + |Xu(x+h)|^2 + |Xu|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2 \eta^2\,dx
$$
$$
\leq C |h|^{2\alpha} \int_\Omega (g_k(x) + g_k(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx + C |h|^{2\alpha} \int_\Omega |F|^{2(p-1)}\,dx + C |h|^2 \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx. \quad \text{(4.7)}
$$

Using (1.3) and (3.4), we have
$$
|\Delta_h V|^2 \leq C \left(\mu^2 + |Xu(x+h)|^2 + |Xu|^2\right)^{\frac{p-2}{2}} |\Delta_h Xu|^2. \quad \text{(4.8)}
$$

From (4.7), it follows that
$$
\int_{B_R} |\Delta_h V|^2\,dx \leq C |h|^2 \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx + C |h|^{2\alpha} \int_\Omega (g_k(x) + g_k(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx + C |h|^{2\alpha} \int_\Omega |F|^{2(p-1)}\,dx. \quad \text{(4.9)}
$$

Dividing both sides of (4.9) by $|h|^{2\alpha}$ and applying the properties of $\eta$, we derive
$$
\int_{B_R} \frac{|\Delta_h V|^2}{|h|^{2\alpha}}\,dx \leq C |h|^{2-2\alpha} \int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx + C \int_\Omega (g_k(x) + g_k(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx + C \int_\Omega |F|^{2(p-1)}\,dx. \quad \text{(4.10)}
$$

Taking the $1/2$ power, we obtain
$$
\left(\int_{B_R} \frac{|\Delta_h V|^2}{|h|^{2\alpha}}\,dx\right)^{1/2} \leq C |h|^{1-\alpha} \left(\int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx\right)^{1/2}
$$
$$
+ C \left(\int_\Omega (g_k(x) + g_k(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx\right)^{1/2} + C \left(\int_\Omega |F|^{2(p-1)}\,dx\right)^{1/2}. \quad \text{(4.11)}
$$

Restricting to $B_\delta$ with $0 < |h| < \delta$ and taking the $L^q$ norm with respect to the measure $\frac{dh}{|h|^Q}$, we obtain
$$
\left(\int_{B_\delta} \left(\int_{B_R} \frac{|\Delta_h V|^2}{|h|^{2\alpha}}\,dx\right)^{q/2} \frac{dh}{|h|^Q}\right)^{1/q}
$$
$$
\leq C \left(\int_{B_\delta} |h|^{(1-\alpha)q} \left(\int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx\right)^{q/2} \frac{dh}{|h|^Q}\right)^{1/q}
$$
$$
+ C \left(\int_{B_\delta} \left(\int_\Omega (g_k(x) + g_k(x+h))^2 \left(\mu^2 + |Xu|^2\right)^{\frac{p}{2}}\,dx\right)^{q/2} \frac{dh}{|h|^Q}\right)^{1/q}
$$
$$
+ C \left(\int_{B_\delta} \left(\int_\Omega |F|^{2(p-1)}\,dx\right)^{q/2} \frac{dh}{|h|^Q}\right)^{1/q} =: P_1 + P_2 + P_3. \quad \text{(4.12)}
$$

We shall show that each $P_i$ ($1 \leq i \leq 3$) is bounded. Since $|F|^{p-2}F \in B^\alpha_{2,q}(\Omega)$ and $1 \leq q < \frac{2Q}{Q-2\alpha}$, we have $|F|^{p-2}F \in L^{\frac{Q}{Q-2\alpha}}(\Omega)$. By Lemma 2.2, we get $|Xu|^{p-2}Xu \in L^{\frac{2Q}{Q-2\alpha}}(\Omega)$, which implies $|Xu|^p \in L^{\frac{Q}{Q-2\alpha}}(\Omega)$.

To estimate $P_1$, we write the $L^q$ norm in polar coordinates. There is no harm in assuming $\delta = 1$, so $h \in B_1 \cap \mathbb{R}^{2n}$ is equivalent to $h = r\xi$ with $0 \leq r < 1$ and $\xi$ in the unit sphere $S^{2n-1}$. Let $d\sigma(\xi)$ be the surface measure on $S^{2n-1}$. Setting $r_k = 2^{-k}$, we estimate $P_1$ as
$$
P_1 \leq C \sum_{k=0}^\infty \int_{r_{k+1}}^{r_k} \int_{S^{2n-1}} |(g_k(\cdot + r\xi) + g_k(\cdot))(\mu^2 + |Xu|^2)^{\frac{p}{2}}|{L^1(B\,d\sigma(\xi)\,dr.})}^{q/2} r^{Q-1
$$

Since $|Xu|^p \in L^{\frac{Q}{Q-2\alpha}}(\Omega)$ and $g_k \in L^\alpha(\Omega)$, we have
$$
|(g_k(\cdot + r\xi) + g_k(\cdot))(\mu^2 + |Xu|^2)^{\frac{p}{2}}|{L^1(B \leq |g_k(\cdot + r\xi) + g_k(\cdot)|}){L^\alpha(B|})} |(\mu^2 + |Xu|^2)^{\frac{p}{2}{L^{\frac{Q}{Q-2\alpha}}(B.})
$$

Moreover,
$$
|g_k(\cdot + r\xi) + g_k(\cdot)|{L^\alpha(B \leq |g_k|}){L^\alpha(B + |g_k|}){L^\alpha(B \leq 2|g_k|}){L^\alpha(\widetilde{B}_R)}
$$
for each $\xi \in S^{2n-1}$ and $r} \leq r \leq r_k$, where $\widetilde{BR = B$. Therefore,
$$
P_1 \leq C |(\mu^2 + |Xu|^2)^{\frac{p}{2}}|{L^{\frac{Q}{Q-2\alpha}}(B |{g_k}})k| < \infty.}_R))
$$

In the Heisenberg group, a direct calculation gives
$$
\int_{B_\delta \cap \mathbb{H}^n} |h|^{(1-\alpha)q-Q}\,dx = C(\alpha, q, Q) \int_0^\delta \rho^{(1-\alpha)q-1}\,d\rho < \infty,
$$
provided $(1-\alpha)q > 0$, which holds by our assumptions.

Since $u \in HW^{1,p}(\Omega)$, we deduce that
$$
P_2 \leq C \left(\int_{B_\delta} |h|^{(1-\alpha)q-Q}\,dh\right)^{1/q} \left(\int_{B_{2R+|h|}} (\mu + |Xu|)^p\,dx\right)^{1/2} < \infty.
$$

Finally, because $|F|^{p-2}F \in B^\alpha_{2,q}(\Omega)$, we have
$$
P_3 = C \left||F|^{p-2}F\right|{L^q(\frac{dh}{|h|^Q}; L^2(B < \infty.}))
$$

Therefore, we complete the proof of Theorem 1.2.

Acknowledgments

The authors are supported by the National Natural Science Foundation of China (NNSF Grant No. 12001333) and Shandong Provincial Natural Science Foundation (Grant No. ZR2020QA005).

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