# Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights

Juan Pablo Alcon Apaza

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Juan Pablo Alcon Apaza. Local Hölder Regularity for Quasilinear Elliptic Equations with Mixed Local-Nonlocal Operators, Variable Exponents, and Weights. 2025. hal-05145081v1

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Preprint submitted on 4 Jul 2025 (v1), last revised 15 Jan 2026 (v2)

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# LOCAL HÖLDER REGULARITY FOR QUASILINEAR ELLIPTIC EQUATIONS WITH MIXED LOCAL-NONLOCAL OPERATORS, VARIABLE EXPONENTS, AND WEIGHTS

JUAN PABLO ALCON APAZA

**ABSTRACT.** We establish local boundedness and local Hölder continuity of weak solutions to the following prototype problem:

$$ -\operatorname{div}(|x|^{-2\beta}|\nabla u|^{q-2}\nabla u) + (-\Delta)^{s(\cdot, \cdot)}_{p(\cdot, \cdot), \beta}u = 0 \quad \text{in } \Omega, $$

where $\Omega \subset \mathbb{R}^n$, $n \ge 2$, is a bounded domain. The nonlocal operator is defined by

$$ (-\Delta)^{s(\cdot, \cdot)}_{p(\cdot, \cdot), \beta}u(x) := \text{P. V.} \int_{\Omega} \frac{|u(x) - u(y)|^{p(x,y)-2}(u(x) - u(y))}{|x-y|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^\beta |y|^\beta} dy. $$

Here, $p: \Omega \times \Omega \to (1, \infty)$ and $s: \Omega \times \Omega \to (0, 1)$ are measurable functions, $q := \operatorname{ess\,sup}_{\Omega \times \Omega} p$, and $0 \le \beta < n$. Our approach is analytic and relies on an adaptation of the De Giorgi–Nash–Moser theory to a mixed local–nonlocal framework with variable exponents and weights.

## CONTENTS

<table><tbody><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Preliminaries</td><td>4</td></tr><tr><td>2.1. Weighted Variable Exponent Lebesgue Spaces</td><td>5</td></tr><tr><td>2.2. Fractional Sobolev Spaces with Variable Exponent and Weight</td><td>5</td></tr><tr><td>2.3. Sobolev Inequalities</td><td>9</td></tr><tr><td>2.4. Weak Solutions</td><td>11</td></tr><tr><td>3. Energy estimates</td><td>11</td></tr><tr><td>4. Local boundedness</td><td>17</td></tr><tr><td>5. Local Hölder continuity</td><td>21</td></tr><tr><td>References</td><td>28</td></tr></tbody></table>

## 1. INTRODUCTION

In this article, we study the local boundedness and local Hölder continuity of weak solutions to the following problem:

$$ -\Delta_{q,2\beta}u + \mathcal{L}_{s(\cdot, \cdot),p(\cdot, \cdot),\beta}(u) = 0 \quad \text{in } \Omega, \quad (1.1) $$

where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with $n \ge 2$. The functions $p: \Omega \times \Omega \to (1, \infty)$ and $s: \Omega \times \Omega \to (0, 1)$ are measurable. We denote $q := \operatorname{ess\,sup}_{\Omega \times \Omega} p$, and assume $0 \le \beta < n$.

The local weighted $q$-Laplace operator is defined by

$$ -\Delta_{q,2\beta}u = -\operatorname{div}(|x|^{-2\beta}|\nabla u|^{q-2}\nabla u), $$

and the nonlocal weighted $s(x, y)$-fractional $p(x, y)$-Laplace operator is given by

$$ \mathcal{L}_{s(\cdot, \cdot), p(\cdot, \cdot), \beta}(u)(x) = \text{P. V.} \int_{\Omega} |u(x) - u(y)|^{p(x,y)-2} (u(x) - u(y)) \frac{K(x,y)}{|x|^\beta |y|^\beta} dy, $$

where P. V. denotes the principal value, and

<footnote n="2020">2020 Mathematics Subject Classification. 35B45; 35B65; 35D30; 35J92; 35R11</footnote>

<footnote>Keywords and phrases. Regularity; mixed local and nonlocal Laplacian; local boundedness; Hölder continuity; variable exponent; weight</footnote>
(H₁) K : Ω × Ω → ℝ is a measurable function satisfying K(x, y) = K(y, x), and

$$ \frac{\Lambda^{-1}}{|x-y|^{n+s(x,y)p(x,y)}} \le K(x,y) \le \frac{\Lambda}{|x-y|^{n+s(x,y)p(x,y)}}, \quad \text{a.e. } (x,y) \in \Omega \times \Omega, $$

for some $\Lambda \ge 1$.

(H₂) The functions s, p satisfy $s(x, y) = s(y, x)$ and $p(x, y) = p(y, x)$ for every $(x, y)$, and

$$ 0 < s^{-} := \essinf_{\Omega \times \Omega} s \le \esssup_{\Omega \times \Omega} s =: s^{+} < 1 $$

and

$$ 1 < p^{-} := \essinf_{\Omega \times \Omega} p \le \esssup_{\Omega \times \Omega} p =: q < \infty. $$

Over the past two decades, there has been a surge of interest in nonlocal equations of the type

$$ (-\Delta)^{s(\cdot, \cdot)} p(\cdot, \cdot) u(x) := \text{P. V.} \int_{\mathbb{R}^n} \frac{|u(x) - u(y)|^{p(x,y)-2}(u(x) - u(y))}{|x-y|^{n+s(x,y)p(x,y)}} dy = 0, \quad (1.2) $$

and related variational problems. We refer to the survey papers [42, 81] and the monographs [20, 77] for an overview of the history, developments, and applications of nonlocal problems. In the context of regularity theory, a foundational result was established by Caffarelli and Silvestre [25], who proved a Harnack inequality for the fractional Laplace equation using an extension argument. This pioneering work spurred rapid developments in the regularity theory for nonlocal equations of fractional Laplacian type, as seen in [16, 17, 24, 26, 27, 63, 64, 73, 82, 84]. Notably, Caffarelli, Chan, and Vasseur [24] adapted De Giorgi's approach to nonlocal equations.

For the fractional p-Laplace equation, Di Castro, Kuusi, and Palatucci [40, 41] established Harnack's inequality and Hölder continuity of weak solutions using De Giorgi's method. A key innovation in their work was the introduction of the *nonlocal tail*, which plays a crucial role in regularity estimates. Further developments in this direction can be found in [28, 37, 46, 58, 66, 69, 67, 68, 70, 72, 75, 78, 79].

More recently, regularity theory has been extended to nonlocal equations with nonstandard order and exponent. Bass and Kassmann [16, 17] and Silvestre [84] proved Hölder continuity and Harnack's inequality for bounded solutions to linear nonlocal equations with kernels of the form $|x - y|^{-2s(x)}$ or $\varphi(|x - y|)^{-1}$. Related results can be found in [10, 11, 65]. Additionally, De Filippis and Palatucci [39] studied nonlocal equations of double-phase type and established Hölder continuity for bounded solutions.

In [35], Cheng, Ge, and Agarwal established the global boundedness of weak solutions to nonhomogeneous $s(\cdot, \cdot)$-fractional $p(\cdot, \cdot)$-Laplace equations using De Giorgi's approach. More recently, Chaker and Kim [29] obtained local Hölder continuity for weak solutions to nonlocal problems with a variable exponent $p(\cdot, \cdot)$ and a constant order $s(\cdot, \cdot) = s$. In [80], Ok proved the local boundedness and local Hölder continuity of weak solutions to nonlocal equations with both a variable exponent $p(\cdot, \cdot)$ and a variable order $s(\cdot, \cdot)$.

The local analog of this case is the $p(x)$-Laplacian problem:

$$ -\Delta_{p(x)} u := \operatorname{div} (|\nabla u|^{p(x)-2} \nabla u) = 0, $$

which serves as a model for equations with nonstandard growth conditions. The regularity theory for this class of problems is well established; see [6, 7, 22, 23, 36, 52, 53]. In particular, [53] shows that weak solutions are locally bounded if $p(\cdot)$ is continuous, and locally Hölder continuous if $p(\cdot)$ is log-Hölder continuous.

For the mixed local and nonlocal case with $p = 2$, i.e.

$$ -\Delta u + (-\Delta)^s u = 0, \quad (1.3) $$

Foondun [56] has proved the Harnack inequality and local Hölder continuity for nonnegative solutions. Barlow-Bass-Chen-Kassmann [15] have obtained a Harnack inequality for the parabolic problem related to (1.3). Chen-Kumagai [34] has proved the Harnack inequality and local Hölder continuity for the parabolic version of (1.3). This parabolic Harnack estimate was then used to prove the elliptic Harnack inequality for (1.3) by Chen-Kim-Song-Vondraček in [33]. For more regularity results related to (1.3), we refer to Athreya-Ramachandran [9], Chen-Kim-Song [31], and Chen-Kim-Song-Vondraček [32]. The arguments in these articles combine probabilistic and analytic techniques. Moreover, the Harnack inequality is proved only for globally nonnegative solutions.

Recently, interior Sobolev regularity, the strong maximum principle, and symmetry properties—among many other qualitative features of solutions to (1.3)—have been studied by Biagi-Dipierro-Valdinoci-Vecchi [18, 19], Dipierro-Proietti Lippi-Valdinoci [48, 47], and Dipierro-Ros-Oton-Serra-Valdinoci [49]. There also exist regularity results for a nonhomogeneous analogue of (1.3). More precisely, Athreya-Ramachandran [9]
has proved a Harnack inequality using probabilistic and analytic methods, and the authors in [18, 48] have obtained boundedness as well as interior and boundary regularity results using analytic techniques.

Biagi-Dipierro-Valdinoci-Vecchi [19] have also obtained interior regularity results for a nonhomogeneous version of

$$-\Delta_p u + (-\Delta_p)^s u = 0. \qquad (1.4)$$

In [8], Antonini and Cozzi address some regularity issues for mixed local-nonlocal quasilinear operators modeled on the sum of a p-Laplacian and a fractional (s, q)-Laplacian, as in (1.4). Under suitable assumptions on the right-hand sides and the outer data, including $\partial\Omega \in C^{1,\alpha}$ and $p > sq$, they show that weak solutions of the Dirichlet problem are $C^{1,\theta}$-regular up to the boundary and establish a Hopf-type lemma for positive supersolutions.

De Filippis and Mingione [38] studied minimizers of functionals of the type

$$w \mapsto \int_{\Omega} [|Dw|^p - fw] \, dx + \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{|w(x) - w(y)|^\gamma}{|x - y|^{n+s\gamma}} \, dxdy$$

with $p, \gamma > 1 > s > 0$ and $p > s\gamma$. They proved that minimizers are locally $C^{1,\alpha}$-regular in $\Omega$ and globally Hölder continuous. For the case where $f$ is a given vector field, see [21].

Mixed local and nonlocal problems have been comprehensively studied in recent years due to their wide range of real-world applications. For instance, such operators model diffusion processes at different scales and also appear in the theory of optimal search strategies, biomathematics, animal foraging, and more; see, for example, [47] and the references therein. Another interesting application arises from the superposition of two stochastic processes with different scales, such as a classical random walk and a Lévy process; see [50] for further details.

Abdellaoui, Attar, and Bentifour [1] studied a class of nonlocal problems involving weights. They proved the existence of a weak solution to the problem

$$\begin{cases} (-\Delta)^s_{p,\beta} u = f(x, u) & \text{in } \Omega, \\ u = 0 & \text{in } \mathbb{R}^n \setminus \Omega, \end{cases}$$

for the largest class of data $f$, including $f(x, u) = f(x) \in L^1(\Omega)$. Here, $\Omega \subset \mathbb{R}^n$ is a domain containing the origin, and the operator is defined as

$$(-\Delta)^s_{p,\beta} u(x) := \mathrm{P. V.} \int_{\mathbb{R}^n} \frac{|u(x) - u(y)|^{p-2}(u(x) - u(y))}{|x - y|^{n+ps}} \frac{1}{|x|^{\beta}|y|^{\beta}} \, dy, \quad (1.5)$$

where $1 < p < n$, $0 < s < 1$, $ps < n$, and $0 \le \beta < \frac{n-ps}{2}$.

This operator (1.5) appears naturally in the context of improved Hardy inequalities. Namely, for all test functions $\phi \in C_0^\infty(\Omega)$ and for any $q < p$, we have

$$G_{s,p}(\phi) \ge C \int_{\Omega} \int_{\Omega} \frac{|v(x) - v(y)|^p}{|x - y|^{n+qs}} w(x)^{p/2} w(y)^{p/2} \, dxdy,$$

where

$$G_{s,p}(\phi) := \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{|\phi(x) - \phi(y)|^p}{|x - y|^{n+ps}} \, dxdy - \lambda_{n,p,s} \int_{\mathbb{R}^n} \frac{|\phi(x)|^p}{|x|^{ps}} \, dx,$$

$\lambda_{n,p,s}$ is the optimal Hardy constant, $w(x) = |x|^{-(n-ps)/p}$, and $v(x) = \phi(x)/w(x)$. For more details, see [2, 3, 4, 57].

We study the local boundedness of weak subsolutions to (1.1). The argument is based on an energy estimate (Lemma 3.2), the Sobolev inequality, and an iteration technique (Lemma 4.1). The local Hölder continuity of weak solutions to (1.1) is also considered. We follow the approach of Garain and Kinnunen [58] and Di Castro, Kuusi, and Palatucci [41], in which the local boundedness estimate and the logarithmic energy estimate (Lemma 3.3) play an important role.

For a measurable function $v: \Omega \to \mathbb{R}$ we define

$$T(v; x_0, r, \rho) := \underset{x \in B_\rho(x_0)}{\operatorname{ess\,sup}} \int_{\Omega \setminus B_r(x_0)} \frac{|v(y)|^{p(x,y)-1}}{|y - x_0|^{n+s(x,y)p(x,y)}} \frac{1}{|y|^{\beta}} \, dy,$$

where $B_\rho(x_0) \subset B_\rho(x_0) \subset \Omega$.

The main result of this work is the following:
**Theorem 1.1.** (Local Hölder continuity). Let $\Omega$ be a bounded domain. Assume that conditions (H$_1$) and (H$_2$) are satisfied. Also, assume that $\beta \ge 0$, $n - 2\beta - 1 + s^+ > 0$, $n - \beta > s^+\mathbf{q}$, and $(1 - s^+)(p^- - 1) \ge \frac{q}{p^+}(\mathbf{q} - p^-)$.

Let $u \in W_{\text{loc}}^{1,q}(\Omega; |x|^{-2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; |x|^{-\beta})$ be a weak solution of (1.1), satisfying $T(u; x_0, r, \rho) < \infty$ for all balls $B_r(x_0) \subset B_\rho(x_0) \subset \Omega$. Then $u$ is locally Hölder continuous in $\Omega$.

Moreover, if $B_R(x_0) \subset \Omega$ with $R \in (0, 1/2]$ is a ball such that either $0 \notin \overline{B_{3R}(x_0)}$ or $x_0 = 0$, then there exist constants $\alpha \in (0, \frac{1-s^+}{q-1})$ and $c > 0$, depending on $\text{diam}(\Omega)$, $R$, $s^-$, $s^+$, $p^-$, $\mathbf{q}$, $\Lambda$, $\beta$, and $n$, such that

$$ \underset{B_\rho(x_0)}{\text{osc}} u := \underset{B_\rho(x_0)}{\text{ess sup}} u - \underset{B_\rho(x_0)}{\text{ess inf}} u \le c \left(\frac{\rho}{r}\right)^\alpha \left[ r^{\frac{q}{q-1}} T\left(u; x_0, \frac{r}{2}, r\right)^{\frac{1}{q-1}} + \left(\int_{B_r(x_0)} |u|^q dV_{2\beta}\right)^{\frac{1}{q}} + 1 \right], $$

where $B_{2r}(x_0) \subset B_R(x_0)$, $r \in (0, 1/4]$, and $\rho \in (0, r/2]$.

If $\beta = 0$, the constants $\alpha$ and $c$ depend only on $s^-$, $s^+$, $p^-$, $\mathbf{q}$, $\Lambda$, and $n$. In the case $x_0 = 0$, the constants $\alpha$ and $c$ depend only on $s^-$, $s^+$, $p^-$, $\mathbf{q}$, $\Lambda$, $\beta$, and $n$.

Our second main result in this work is the following:

**Theorem 1.2.** (Local boundedness). Let $\Omega$ be a bounded domain. Assume conditions (H$_1$) and (H$_2$) are satisfied. Also, assume that $\beta \ge 0$ and $n - \beta - 1 + s^+ > 0$.

Let $u \in W_{\text{loc}}^{1,q}(\Omega; |x|^{-2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; |x|^{-\beta})$ be a weak solution of (1.1), satisfying $T(u; x_0, r, \rho) < \infty$ for all balls $B_r(x_0) \subset B_\rho(x_0) \subset \Omega$. Then there exists a constant $C = C(s^+, p^-, \mathbf{q}, \beta, \Lambda, n) > 0$, such that

$$ \underset{B_{r/2}(x_0)}{\text{ess sup}} u \le \delta(|x_0| + 1)^{\frac{\beta}{q-1}} r^{\frac{q}{q-1}} T\left(u_+; x_0, \frac{r}{2}, r\right)^{\frac{1}{q-1}} + C \delta^{-\frac{(q-1)\kappa}{q(\kappa-1)}} \left(\int_{B_r(x_0)} u_+^q dV_{2\beta}\right)^{\frac{1}{q}} + 1, $$

whenever $B_r(x_0) \subset \Omega$ with $r \in (0, 1/2]$ and $\delta \in (0, 1]$. Here, $\kappa$ is the exponent given in Corollary 2.10.

This article is organized as follows. Section 2 introduces key definitions and preliminary results. Section 3 is devoted to establishing the necessary energy estimates. In Sections 4 and 5, we prove the local boundedness and local Hölder continuity of weak solutions, respectively.

## 2. PRELIMINARIES

We adopt the following notation:

* $\mathbf{q} := \text{ess sup}_{\Omega \times \Omega} p$.

* For $x, y \in \mathbb{R}^n$, their Euclidean inner product is denoted by $\langle x, y \rangle$ or $x \cdot y$. For simplicity, we may occasionally write $xy$. The Euclidean norm is denoted by $|x|$.

* The Euclidean ball of radius $r$ centered at $x_0$ is denoted by $B_r(x_0) = \{x \in \mathbb{R}^n | |x - x_0| < r\}$.

* The conjugate exponent of $q > 1$ is given by $q' := \frac{q}{q-1}$.

* We define

$$ \nu_{\beta}(x) := |x|^{-\beta}, \quad \nu_{\beta}(F) := \int_{F} d\nu_{\beta}, \quad \text{and} \quad d\nu_{\beta} := |x|^{-\beta} dx, $$

where $F$ is a measurable set. The standard Lebesgue measure of $F$ is denoted by $|F|$.

* For a measurable function $f : B_r(x_0) \to \mathbb{R}$, we write

$$ \int_{B_r(x_0)} f \, d\nu_{\beta} = (f)_{B_r(x_0)}, \nu_{\beta} = \frac{1}{\nu_{\beta}(B_r(x_0))} \int_{B_r(x_0)} f \, d\nu_{\beta}. $$

* For $a \in \mathbb{R}$, we set $a_+ := \max\{a, 0\}$ and $a_- := -\min\{a, 0\}$.

* We write

$$ \mathcal{A}(u, x, y) := |u(x) - u(y)|^{p(x,y)-2}(u(x) - u(y)) \quad \text{and} \quad d\mu(x, y) := \frac{1}{|x|^{\beta}|y|^{\beta}} K(x, y) dxdy. $$

* The symbols $c$, $C$, or $C_i$ (for $i \in \mathbb{N}$) denote positive constants that may vary from line to line, or even within the same line. If a constant depends on parameters $r_1, r_2, \dots$, we write it as $C(r_1, r_2, \dots)$.
2.1. **Weighted Variable Exponent Lebesgue Spaces.** In this section, we provide introductory definitions and results on weighted variable exponent Lebesgue spaces. For further details, see [44, 45, 54, 61, 71, 83, 87].

Let $V$ be a weight function on $\mathbb{R}^n$; that is, $V$ is a nonnegative and locally integrable function with respect to the Lebesgue measure.

Let $\Omega \subset \mathbb{R}^n$ be an open set, and let $p: \Omega \to [1, \infty)$ (called a *variable exponent on $\Omega$*) be a measurable function. The weighted variable exponent Lebesgue space $L^{p(x)}(\Omega; V)$ is defined by

$$ L^{p(x)}(\Omega; V) = \left\{ u : \Omega \to \mathbb{R} \mid u \text{ is measurable and } \int_{\Omega} |u|^p Vdx < \infty \right\}, \\ \|u\|_{p(x), \Omega, V} := \inf \left\{ \lambda > 0 \mid \varrho_{p(x), \Omega, V}\left(\frac{u}{\lambda}\right) \le 1 \right\}, \quad (2.1) $$

with the norm

where

$$ \varrho_{p(x), \Omega, V}(u) := \int_{\Omega} |u|^p Vdx. $$

The relationship between the modular $\varrho_{p(x), \Omega, V}(\cdot)$ and the norm $\| \cdot \|_{p(x), \Omega, V}$ is given by:

$$ \begin{aligned} & \min \left\{ \varrho_{p(x), \Omega, V}(f)^{\frac{1}{p}}, \varrho_{p(x), \Omega, V}(f)^{\frac{1}{p^+}} \right\} \\ & \le \|f\|_{p(x), \Omega, V} \le \max \left\{ \varrho_{p(x), \Omega, V}(f)^{\frac{1}{p}}, \varrho_{p(x), \Omega, V}(f)^{\frac{1}{p^+}} \right\}, \end{aligned} $$

where $p^{-} = \essinf \Omega$ and $p^{+} = \essup \Omega$.

Hölder's inequality can be written in the form:

$$ \|fg\|_{s(x), \Omega, V} \le 2\|f\|_{p(x), \Omega, V} \|g\|_{q(x), \Omega, V}, \quad (2.2) $$

where $\frac{1}{s} = \frac{1}{p} + \frac{1}{q}$; see [44, Lemma 3.2.20].

For an introduction to weighted Lebesgue and Sobolev spaces, we refer the reader to [61, 87]. Let $q \in (1, \infty)$. The weighted Lebesgue space $L^q(\Omega; V)$ is defined by

$$ L^q(\Omega; V) := \left\{ u : \Omega \to \mathbb{R} \mid u \text{ is measurable and } \int_{\Omega} |u|^q Vdx < \infty \right\}, $$

equipped with the norm

$$ \|u\|_{q, \Omega, V} := \left( \int_{\Omega} |u|^q Vdx \right)^{\frac{1}{q}}. $$

The weighted Sobolev space is defined by

$$ W^{1,q}(\Omega; V) := \{u \in L^q(\Omega; V) \mid |\nabla u| \in L^q(\Omega; V)\} $$

equipped with the norm

$$ \|u\|_{1,q,\Omega,V} := \|u\|_{q,\Omega,V} + \|\nabla u\|_{q,\Omega,V}. $$

2.2. **Fractional Sobolev Spaces with Variable Exponent and Weight.** In this subsection, we provide introductory definitions and results on weighted variable exponent fractional Sobolev spaces. For more details, see [13, 51, 62, 74].

Let $\Omega \subset \mathbb{R}^n$ be an open set and let $V: \mathbb{R}^n \to [0, \infty)$ be a weight function. Let $p: \Omega \times \Omega \to [1, \infty)$ and $s: \Omega \times \Omega \to (0, 1)$ be measurable functions such that

$$ 1 < \essinf_{\Omega \times \Omega} p \le \esssup_{\Omega \times \Omega} p < \infty, \quad p(x, y) = p(y, x), \text{ a.e. } (x, y) \in \Omega \times \Omega, \quad (2.3) $$

$$ 0 < \essinf_{\Omega \times \Omega} s \le \esssup_{\Omega \times \Omega} s < 1, \quad s(x, y) = s(y, x), \text{ a.e. } (x, y) \in \Omega \times \Omega. \quad (2.4) $$

The weighted variable exponent fractional Sobolev space is defined as

$$ W^{s(x,y),p(x,y)}(\Omega; V) := \{u \in L^{\bar{p}(x)}(\Omega; V) \mid \tilde{\varrho}_{s,p,\Omega,V}(u) < \infty\}, $$

where

$$ \tilde{\varrho}_{s,p,\Omega,V}(u) := \int_{\Omega} \int_{\Omega} \frac{|u(x) - u(y)|^{p(x,y)}}{|x - y|^{n+s(x,y)p(x,y)}} V(x)V(y)dxdy $$

and

$$ \bar{p}(x) := p(x, x). $$
We define the seminorm

$$[u]_{s,p,\Omega,\nu} := \inf \left\{ \lambda > 0 \mid \tilde{\varrho}_{s,p,\Omega,\nu} \left( \frac{u}{\lambda} \right) \le 1 \right\}.$$

Then, $W^{s(x,y),p(x,y)}(\Omega; \mathbf{V})$, endowed with the norm

$$\|u\|_{s,p,\Omega,\nu} := \|u\|_{\bar{p}(x),\Omega,\nu} + [u]_{s,p,\Omega,\nu},$$

is a Banach space; see [12, 14, 35].

We also define

$$\hat{\varrho}_{s,p,\Omega,\nu}(u) := \varrho_{\bar{p}(x),\Omega,\nu}(u) + \tilde{\varrho}_{s,p,\Omega,\nu}(u),$$

and the corresponding norm

$$|u|_{s,p,\Omega,\nu} := \inf \left\{ \lambda > 0 \mid \hat{\varrho}_{s,p,\Omega,\nu} \left( \frac{u}{\lambda} \right) \le 1 \right\}.$$

Then, the norms $\| \cdot \|_{s,p,\Omega,\nu}$ and $| \cdot |_{s,p,\Omega,\nu}$ are comparable; see [62]. The following lemma is taken from [62].

**Lemma 2.1.** Let $u \in W^{s(x,y),p(x,y)}(\Omega; \mathbf{V})$. Then:

(i) If $[u]_{s,p,\Omega,\nu} \ge 1$, then $[u]_{s,p,\Omega,\nu}^{p^-} \le \tilde{\varrho}_{s,p,\Omega,\nu}(u) \le [u]_{s,p,\Omega,\nu}^{p^+}$

(ii) If $[u]_{s,p,\Omega,\nu} \le 1$, then $[u]_{s,p,\Omega,\nu}^{p^+} \le \tilde{\varrho}_{s,p,\Omega,\nu}(u) \le [u]_{s,p,\Omega,\nu}^{p^-}$

(iii) If $|u|_{s,p,\Omega,\nu} \ge 1$, then $|u|_{s,p,\Omega,\nu}^{p^-} \le \hat{\varrho}_{s,p,\Omega,\nu}(u) \le |u|_{s,p,\Omega,\nu}^{p^+}$

(iv) If $|u|_{s,p,\Omega,\nu} \le 1$, then $|u|_{s,p,\Omega,\nu}^{p^+} \le \hat{\varrho}_{s,p,\Omega,\nu}(u) \le |u|_{s,p,\Omega,\nu}^{p^-}$

The following lemma will be used several times throughout this article.

**Lemma 2.2.** (i) If $R > 0, 0 < |y| \le R, 0 < \alpha < 1, \tilde{\beta} \ge 0$, and $n - \tilde{\beta} - \alpha > 0$, then

$$\int_{B_R(0)} \frac{1}{|x - y|^{n-\alpha} |x|^{\tilde{\beta}}} \, dx \le C \frac{R^{2\alpha}}{|y|^{\tilde{\beta}+\alpha}}, \quad (2.5)$$

where

$$C = \frac{2^{1-\alpha} \pi^{\frac{n}{2}} \Gamma\left(\frac{1-\alpha}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right) \Gamma\left(1-\frac{\alpha}{2}\right)} \left( \frac{\Gamma(n-\tilde{\beta}-\alpha)\Gamma\left(\frac{\alpha}{2}\right)}{\Gamma(n-\tilde{\beta}-\frac{\alpha}{2})} + \frac{1}{\alpha} \right).$$

(ii) If $0 < R \le |y|/2, \tilde{\beta} \ge 0$, and $n - \tilde{\beta} > \gamma > 0$, then

$$\int_{B_R(0)} \frac{1}{|x - y|^{n+\gamma} |x|^{\tilde{\beta}}} \, dx \le \frac{2^{4+\gamma} \pi^{\frac{n+1}{2}}}{3\Gamma\left(\frac{n-1}{2}\right) (n - \tilde{\beta} - \gamma)} \frac{R^{n-\tilde{\beta}-\gamma}}{|y|^n}. \quad (2.6)$$

(iii) Assume $n - \tilde{\beta} - (1 - \alpha) > 0$ and $\tilde{\beta} \ge 0$. Also, let $0 < \alpha < 1$ and $1 \le b$. Let $K \subset \mathbb{R}^n$ be a bounded measurable set. If $x \in K, x \ne 0$, and $K \subset B_r(0)$, then

$$\int_K \frac{1}{|x - y|^{n-(1-\alpha)b} |y|^{\tilde{\beta}}} \, dy \le C(r, a, b) \frac{1}{|x|^{\tilde{\beta}}}. \quad (2.7)$$

*Proof.* (i) We set $r = |x|$ and $\rho = |y|$. Then $x = rx', y = \rho y'$, where $|x'| = |y'| = 1$. Therefore,

$$\int_{B_R(0)} \frac{1}{|x - y|^{n-\alpha} |x|^{\tilde{\beta}}} \, dx = \int_0^R \frac{r^{n-1}}{r^{\tilde{\beta}} \rho^{n-\alpha}} \left( \int_{|x'|=1} \frac{1}{\left| \frac{r}{\rho} x' - y' \right|^{n-\alpha}} \, d\mathcal{H}^{n-1}(x') \right) \, dr.$$

We now set $t = \frac{r}{\rho}$; hence,

$$\int_{B_R(0)} \frac{1}{|x - y|^{n-\alpha} |x|^{\tilde{\beta}}} \, dx = \frac{1}{\rho^{\tilde{\beta}-\alpha}} \int_0^R t^{n-\tilde{\beta}-1} \left( \int_{|x'|=1} \frac{1}{|tx' - y'|^{n-\alpha}} \, d\mathcal{H}^{n-1}(x') \right) \, dt. \quad (2.8)$$

We define

$$K_{\alpha}(t) := \int_{|x'|=1} \frac{1}{|tx' - y'|^{n-\alpha}} \, d\mathcal{H}^{n-1}(x').$$

As in [55, pp. 248 - 251], we find

$$K_{\alpha}(t) = 2 \frac{\pi^{\frac{n-1}{2}}}{\Gamma\left(\frac{n-1}{2}\right)} \int_{0}^{\pi} \frac{\sin^{n-2} \theta}{(1 - 2t \cos \theta + t^2)^{\frac{n-\alpha}{2}}} \, d\theta. \quad (2.9)$$
For $t > 1$, we have

$$K_{\alpha}(t) = 2 \frac{\pi^{\frac{n-1}{2}}}{\Gamma(\frac{n-1}{2})} t^{-n+2} (t^2 - 1)^{-1+\alpha} \int_{0}^{\pi} \sin^{n-2} \theta \frac{(\sqrt{t^2 - \sin^2 \theta} + \cos \theta)^{1-\alpha}}{\sqrt{t^2 - \sin^2 \theta}} d\theta.$$

Since $\sqrt{t^2 - \sin^2 \theta} \ge |\cos \theta|,$

$$ 
\begin{aligned}
K_{\alpha}(t) &\le 2 \frac{\pi^{\frac{n-1}{2}}}{\Gamma(\frac{n-1}{2})} t^{-n+2} (t^2 - 1)^{-1+\alpha} 2^{1-\alpha} \int_{0}^{\pi} \frac{1}{|\cos \theta|^\alpha} d\theta \\
&= 2^{2-\alpha} \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n-1}{2})} t^{-n+2} (t^2 - 1)^{-1+\alpha} \frac{\Gamma(\frac{1-\alpha}{2})}{\Gamma(1-\frac{\alpha}{2})}.
\end{aligned} 
\quad (2.10) $$

Similarly, if $s < 1$,

$$ 
\begin{aligned}
K_{\alpha}(s) &= s^{-n+\alpha} K_{\alpha}\left(\frac{1}{s}\right) \\
&\le 2^{2-\alpha} \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n-1}{2})} s^{-\alpha} (1-s^2)^{-1+\alpha} \frac{\Gamma(\frac{1-\alpha}{2})}{\Gamma(1-\frac{\alpha}{2})}.
\end{aligned} 
\quad (2.11) $$

From, [59] or [5, 76],

$$ \int_{0}^{1} t^{n-\tilde{\beta}-1} K_{\alpha}(t) dt \le 2^{1-\alpha} \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n-1}{2})} \frac{\Gamma(\frac{1-\alpha}{2})}{\Gamma(1-\frac{\alpha}{2})} \frac{\Gamma(n-\tilde{\beta}-\alpha)\Gamma(\frac{\alpha}{2})}{\Gamma(n-\tilde{\beta}-\frac{\alpha}{2})} \quad (2.12) $$

and, since $t^{-\tilde{\beta}+1} \le t$ if $t \ge 1$:

$$ \int_{1}^{\frac{R}{\rho}} t^{n-\tilde{\beta}-1} K_{\alpha}(t) dt \le \frac{2^{1-\alpha}}{\alpha} \frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n-1}{2})} \frac{\Gamma(\frac{1-\alpha}{2})}{\Gamma(1-\frac{\alpha}{2})} \left(\frac{R}{\rho}\right)^{2\alpha} \quad (2.13) $$

From (2.8), (2.12), and (2.13), we obtain (2.5).

(ii) Similarly to (i), we set $r = |x|$ and $\rho = |y|$. Then $x = rx'$ and $y = \rho y'$, where $|x'| = |y'| = 1$. Then

$$ \int_{B_R(0)} \frac{1}{|x - y|^{n+\gamma} |x|^{\beta}} dx = \frac{1}{\rho^{\beta+\gamma}} \int_0^{\frac{R}{\rho}} t^{n-\beta-1} \hat{K}_{\gamma}(t) dt. \quad (2.14) $$

where $t = \frac{r}{\rho}$, and for $t > 1$,

$$ 
\begin{aligned}
\hat{K}_{\gamma}(t) &:= \int_{|x'|=1} \frac{1}{|tx' - y'|^{n+\gamma}} d\mathcal{H}^{n-1}(x') \\
&= 2 \frac{\pi^{\frac{n-1}{2}}}{\Gamma(\frac{n-1}{2})} t^{-n+2} (t^2 - 1)^{-1-\gamma} \int_{0}^{\pi} \sin^{n-2} \theta \frac{(\sqrt{t^2 - \sin^2 \theta} + \cos \theta)^{1+\gamma}}{\sqrt{t^2 - \sin^2 \theta}} d\theta.
\end{aligned} 
$$

Then, since $t^2 - 1 \ge \sqrt{t^2 - \sin^2 \theta}$ if $t \ge 2$,

$$ \hat{K}_{\gamma}(t) \le 2^{2+\gamma} \frac{\pi^{\frac{n+1}{2}}}{\Gamma(\frac{n-1}{2})} t^{-n+2} (t^2 - 1)^{-1}. $$

If $0 < s \le 1/2$, this implies $(1-s^2)^{-1} \le 3/4$, and

$$ \hat{K}_{\gamma}(s) = s^{-n-\gamma} \hat{K}_{\gamma}\left(\frac{1}{s}\right) \le \frac{2^{4+\gamma} \pi^{\frac{n+1}{2}}}{3\Gamma(\frac{n-1}{2})} s^{-\gamma}. \quad (2.15) $$
Hence,

$$ \int_{B_R(0)} \frac{1}{|x-y|^{n+\gamma}|x|^\beta} \, dx \le \frac{2^{4+\gamma}\pi^{\frac{n+1}{2}}}{3\Gamma(\frac{n-1}{2})(n-\tilde{\beta}-\gamma)} \frac{1}{|y|^{\tilde{\beta}+\gamma}} \left(\frac{R}{|y|}\right)^{n-\tilde{\beta}-\gamma}, \quad (2.16) $$

which proves (2.6).

(iii) Let $x \in K \subset B_r(0)$, we have

$$ \begin{aligned} \int_K \frac{1}{|x-y|^{n-(1-a)b}|y|^\beta} \, dy &= \int_{K \setminus B_{|x|}(0)} \frac{1}{|x-y|^{n-(1-a)b}|y|^\beta} \, dy + \int_{K \cap B_{|x|}(0)} \frac{1}{|x-y|^{n-(1-a)b}|y|^\beta} \, dy \\ &\le \frac{1}{|x|^\beta} \int_{B_{2r}(y)} \frac{1}{|x-y|^{n-(1-a)b}} \, dy + \int_{B_{|x|}(0)} \frac{|x-y|^{(1-a)(b-1)}}{|x-y|^{n-(1-a)}|y|^\beta} \, dy \\ &\le \frac{|\partial B_1(0)|(2r)^{(1-a)b}}{(1-a)b|x|^\beta} + (2|x|)^{(1-a)(b-1)} \int_{B_{|x|}(0)} \frac{1}{|x-y|^{n-(1-a)}|y|^\beta} \, dy. \end{aligned} $$

Applying (2.5) in (i), we conclude (2.7). □

**Lemma 2.3.** Let $\Omega \subset \mathbb{R}^n$ be an open set. Assume that conditions (2.3) and (2.4) on $s$ and $p$ hold. Furthermore, for all measurable sets $K \in \Omega$, assume there exist constants $c_0, c_1 > 0$ such that

$$ c_0 \le \mathbf{V}(x), \quad \text{a.e. } x \in K, \quad (2.17) $$

$$ \int_K \frac{\mathbf{V}(y)}{|x-y|^{n-(1-s(x,y))p(x,y)}} \, dy \le c_1 \mathbf{V}(x), \quad \text{a.e. } x \in K \setminus \{0\}, \quad (2.18) $$

and for all $r > 0$:

$$ \operatorname*{ess\,sup}_{y \in B_r(0)} \int_{\Omega \setminus B_{2r}(0)} \frac{\mathbf{V}(x)}{|x|^{n+s(x,y)p(x,y)}} \, dx < \infty. \quad (2.19) $$

If $u \in W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}) \cap L_{\text{loc}}^{p^+}(\Omega; \mathbf{V}^2)$ and $\zeta : \Omega \to \mathbb{R}$ is a Lipschitz function with $\text{supp} \zeta \subset \Omega$, then $\zeta u \in W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}) \cap L^{p^+}(\Omega; \mathbf{V}^2)$.

*Proof.* We have

$$ |\zeta(x) - \zeta(y)| \le C_2|x - y|, \quad \forall (x, y) \in \mathbb{R}^n \times \mathbb{R}^n, \quad (2.20) $$

for some constant $C_2 > 1$.

Assume that $\text{supp} \zeta \subset B_r(0)$. Employing the identity $ab - bd = a(b - d) + d(a - b)$ for all $a, b, c, d \in \mathbb{R}$, we have

$$ \begin{aligned} & \int_{\Omega} \int_{\Omega} \frac{|\zeta(x)u(x) - \zeta(y)u(y)|^{p(x,y)}}{|x-y|^{n+s(x,y)p(x,y)}} \mathbf{V}(x)\mathbf{V}(y)dx dy \\ &= \int_{\text{supp } \zeta \cap B_{2r}(0)} \int_{\text{supp } \zeta \cap B_{2r}(0)} \frac{|\zeta(x)u(x) - \zeta(y)u(y)|^{p(x,y)}}{|x-y|^{n+s(x,y)p(x,y)}} \mathbf{V}(x)\mathbf{V}(y)dx dy \\ &\quad + 2 \int_{\text{supp } \zeta \cap B_{2r}(0)} \int_{\text{supp } \zeta \setminus B_{2r}(0)} \frac{|\zeta(y)u(y)|^{p(x,y)}}{|x-y|^{n+s(x,y)p(x,y)}} \mathbf{V}(x)\mathbf{V}(y)dx dy \\ &\le 2^{p^+-1} \int_{\text{supp } \zeta \cap B_{2r}(0)} \int_{\text{supp } \zeta \cap B_{2r}(0)} \frac{|u(x)(\zeta(x) - \zeta(y))|^{p(x,y)}}{|x-y|^{n+s(x,y)p(x,y)}} \mathbf{V}(x)\mathbf{V}(y)dx dy \\ &\quad + 2^{p^+-1} \int_{\text{supp } \zeta \cap B_{2r}(0)} \int_{\text{supp } \zeta \cap B_{2r}(0)} \frac{|\zeta(y)(u(x) - u(y))|^{p(x,y)}}{|x-y|^{n+s(x,y)p(x,y)}} \mathbf{V}(x)\mathbf{V}(y)dx dy \\ &\quad + 2 \int_{\text{supp } \zeta \cap B_r(0)} (|u(y)|^{p^+} + 1) \mathbf{V}(y) \int_{\mathbb{R}^n \setminus B_{2r}(0)} \frac{1}{|x-y|^{n+s(x,y)p(x,y)}} \mathbf{V}(x)dx dy \\ &=: I_1 + I_2 + I_3. \end{aligned} $$

By (2.20) and (2.18):

$$ I_1 \le c_1 C_2^{p^+} 2^{p^+-1} \int_{\text{supp } \zeta} (|u|^{p^+} + 1) \mathbf{V}^2 dx < \infty, \quad (2.21) $$

due to $u \in L_{\text{loc}}^{p^+}(\Omega; \mathbf{V}^2)$.
$I_2$ is finite since $u \in W^{s(x,y),p(x,y)}(\Omega; \mathbf{V})$. We also have $I_3 < \infty$ because of (2.17), (2.19), and the fact that for all $(x, y) \in (\mathbb{R}^n \setminus B_{2r}(0)) \times B_r(0)$, we obtain $|x - y| > r$, and

$$ \frac{|x|}{|x - y|} \le \frac{|x - y|}{|x - y|} + \frac{|y|}{|x - y|} < 2. $$

Therefore, we conclude the proof of the lemma.

The following result will be useful when using test functions.

**Corollary 2.4.** Let $\Omega \subset \mathbb{R}^n$ be an open set. Assume that conditions (2.3) and (2.4) on $s$ and $p$ hold; also, $n - \beta - (1 - s^-) > 0$. If $u \in W^{s(x,y),p(x,y)}(\Omega; |x|^{-\beta}) \cap L_{\text{loc}}^{p^+}(\Omega; |x|^{-2\beta})$ and $\zeta : \Omega \to \mathbb{R}$ is a Lipschitz function with supp $\zeta \subset \Omega$, then $\zeta u \in W^{s(x,y),p(x,y)}(\Omega; |x|^{-\beta}) \cap L_{\text{loc}}^{p^+}(\Omega; |x|^{-2\beta})$.

*Proof.* By Lemma 2.2 (iii), we have $\mathbf{V}(x) = |x|^{-\beta}$ satisfies (2.18) in Lemma 2.3. Hence, employing this last lemma, we conclude the proof of the corollary. $\square$

Another useful result is the following:

**Lemma 2.5.** Let $\Omega \subset \mathbb{R}^n$ be an open set. Assume that conditions (2.3) and (2.4) on $s$ and $p$ hold.

(i) Let $\mathbf{V}_0$ a weight satisfying $\mathbf{V}_0^{-1} \in L_{\text{loc}}^{\frac{1}{q-1}}(\mathbb{R}^n)$, where $q \in (1, \infty)$. If $u, v \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_0) \cap L_{\text{loc}}^{\infty}(\Omega)$, then $uv \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_0) \cap L_{\text{loc}}^{\infty}(\Omega)$.

(ii) If $u, v \in W^{s(x,y),p(x,y)}(\Omega; \mathbf{V})$, then $\max\{u, v\} \in W^{s(x,y),p(x,y)}(\Omega; \mathbf{V})$.

*Proof.* The proof of (i) follows from the observation that $W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_0) \subset W_{\text{loc}}^{1,1}(\Omega)$. The proof of (ii) is derived from the definition of the weighted variable exponent fractional Sobolev space and the identity $\max\{u, v\} = \frac{u+v+|u-v|}{2}$. $\square$

**2.3. Sobolev Inequalities.** We say $\mathbf{V}$ satisfies the doubling condition (with respect to Lebesgue measure) if

$$ \mathbf{V}(B_{2r}(x)) \le c \mathbf{V}(B_r(x)) $$

with $c$ independent of $r$ and $x$. We write $\mathbf{V} \in D_\infty$ for such $\mathbf{V}$.

For $0 < \alpha, p_0, q < \infty$, we consider pairs $\mathbf{V}_1, \mathbf{V}_2$ of weights in $D_\infty$ that satisfy

$$ \left(\frac{s}{h}\right)^{\alpha} \left(\frac{\mathbf{V}_1(B_s(y))}{\mathbf{V}_1(B_h(y))}\right)^{\frac{1}{q}} \le c \left(\frac{\mathbf{V}_2(B_s(y))}{\mathbf{V}_2(B_h(y))}\right)^{\frac{1}{p_0}}, \quad 0 < s \le h, \qquad (2.22) $$

with $c$ independent of $x, s$, and $h$. For the next proposition, we only need to assume that (2.22) holds when $\alpha = 1$ and $y \in B_{2h}(x)$ for a certain $x$.

For a measurable function $f : B_r(x_0) \to \mathbb{R}$, we define

$$ (f)_{B_r(x_0), v} := \frac{1}{\mathbf{V}(B_r(x_0))} \int_{B_r(x_0)} f \mathbf{V} dx. $$

**Proposition 2.6.** ([30, Corollary 1.4]). Let $u$ be Lipschitz continuous on a ball $B_r(x) \subset \mathbb{R}^n$, and assume that $1 < p_0 < q$, $\mathbf{V}_1 \in D_\infty$, the pair $\mathbf{V}_1, \mathbf{V}_2$ satisfies (2.22) with $\alpha = 1$ for $y \in B_{2r}(x)$, and $\mathbf{V}_2 \in A_{p_0}$. Then

$$ \left( \int_{B_r(x)} |u - (u)_{B_r(x), \mathbf{V}_1}|^q d\mathbf{V}_1 \right)^{\frac{1}{q}} \le c r \left( \int_{B_r(x)} |\nabla u|^{p_0} d\mathbf{V}_2 \right)^{\frac{1}{p_0}} $$

with $c$ independent of $u, r$, and $x$.

The following technical result will be used in this and subsequent sections.

**Lemma 2.7.** Assume that $n > \tilde{\beta} \ge 0$ and $q \in (1, \infty)$. Then:

(i) The weight $\mathbf{V}_{\tilde{\beta}}(x) = |x|^{-\tilde{\beta}}$ is doubling and belongs to the Muckenhoupt class $A_q$.

(ii) If $x_0 \in \mathbb{R}^n$ and $h \ge s > 0$, we have the estimate

$$ \frac{\mathbf{V}_{\tilde{\beta}}(B_s(x_0))}{\mathbf{V}_{\tilde{\beta}}(B_h(x_0))} \ge C(\tilde{\beta}, n) \left(\frac{s}{h}\right)^n, \qquad (2.23) $$

where

$$ \mathbf{V}_{\tilde{\beta}}(F) := \int_F d\mathbf{V}_{\tilde{\beta}} \quad \text{and} \quad d\mathbf{V}_{\tilde{\beta}} := \mathbf{V}_{\tilde{\beta}} dx, $$

for a measurable set $F \subset \mathbb{R}^n$.
*Proof.* (i) The proof can be found in one of the following references: [60, Chapter 7, Weighted Inequalities] and [85, Chapter 5, Weighted Inequalities], or [86, Chapter IX, $A_p$ Weights].

(ii) Let $h \ge s > 0$. For all $x \in B_s(x_0)$, we have

$$|x_0| + h \ge |x_0| + s > |x|.$$

Hence,

$$I := \frac{\int_{B_s(x_0)} |x|^{-\tilde{\beta}} dx}{\int_{B_h(x_0)} |x|^{-\tilde{\beta}} dx} \ge \frac{|B_s(x_0)|}{\int_{B_h(x_0)} |x|^{-\tilde{\beta}} dx (|x_0| + h)^{\tilde{\beta}}}. \quad (2.24)$$

If $3h \le |x_0|$, and noting that $|x_0| - h < |x|$ for all $x \in B_h(x_0)$, we have

$$I \ge \frac{|B_s(x_0)|}{|B_h(x_0)|} \left( \frac{|x_0| - h}{|x_0| + h} \right)^{\tilde{\beta}} \ge \frac{|B_s(x_0)|}{|B_h(x_0)|} \left( \frac{1}{2} \right)^{\tilde{\beta}}. \quad (2.25)$$

If $3h > |x_0|$ and $|x| \ge h$ for all $x \in B_h(x_0)$, then from (2.24),

$$I \ge \frac{|B_s(x_0)|}{|B_h(x_0)|} \left( \frac{h}{|x_0| + h} \right)^{\tilde{\beta}} \ge \frac{|B_s(x_0)|}{|B_h(x_0)|} \left( \frac{1}{4} \right)^{\tilde{\beta}}. \quad (2.26)$$

If $3h > |x_0|$ and $|x_1| < h$ for some $x_1 \in B_h(x_0)$, then using (2.24), we obtain

$$I \ge \frac{|B_s(x_0)|}{\int_{B_{3h}(0)} |x|^{-\tilde{\beta}} dx} \frac{1}{(|x_0| + h)^{\tilde{\beta}}} \ge \frac{(n - \tilde{\beta})|B_s(x_0)|}{|\partial B_1(0)| (3h)^n} \left( \frac{3h}{|x_0| + h} \right)^{\tilde{\beta}} \ge \frac{(n - \tilde{\beta})|B_s(x_0)|}{|\partial B_1(0)| (3h)^n} \left( \frac{3}{4} \right)^{\tilde{\beta}}. \quad (2.27)$$

From (2.24)-(2.27), we obtain

$$\frac{\nu_{\tilde{\beta}}(B_s(x_0))}{\nu_{\tilde{\beta}}(B_h(x_0))} \ge C(\tilde{\beta}, n) \left(\frac{s}{h}\right)^n.$$

□

As a consequence of Proposition 2.6 and Lemma 2.7, we obtain the following result.

**Corollary 2.8.** Let $u$ be Lipschitz continuous on a ball $B_r(x) \subset \mathbb{R}^n$, and assume that $1 < p_0 < q$, $1 - n(1/p_0 - 1/q) \ge 0$, and $n > \tilde{\beta} \ge 0$. Then

$$\left( \int_{B_r(x)} |u - (u)_{B_r(x)} \nu_{\tilde{\beta}}|^q d\nu_{\tilde{\beta}} \right)^{\frac{1}{q}} \le c r \left( \int_{B_r(x)} |\nabla u|^{p_0} d\nu_{\tilde{\beta}} \right)^{\frac{1}{p_0}}$$

with $c$ independent of $u$, $r$, and $x$.

**Proposition 2.9.** ([30, Theorem 1.5]). Let $u$ be Lipschitz continuous on a ball $B_r(x) \subset \mathbb{R}^n$, and assume that $1 < p_0 < q$, $\nu_1 \in D_\infty$, the pair $\nu_1, \nu_2$ satisfies (2.22) with $\alpha = 1$ for $y \in B_{2r}(x)$, and $\nu_2 \in A_{p_0}$. Suppose that $\text{supp } u \subset B_h(x)$. Then

$$\left( \int_{B_r(x)} |u|^q d\nu_1 \right)^{\frac{1}{q}} \le c r \left( \int_{B_r(x)} |\nabla u|^{p_0} d\nu_2 \right)^{\frac{1}{p_0}},$$

with $c$ independent of $u$, $r$, and $x$.

**Corollary 2.10.** Let $u$ be Lipschitz continuous on a ball $B_r(x) \subset \mathbb{R}^n$ with $\text{supp } u \subset B_r(x)$. Let

$$\kappa = \begin{cases} \frac{n}{n - p_0} & \text{if } 1 < p_0 < n, \\ 2 & \text{if } p_0 \ge n. \end{cases} \quad (2.28)$$

There exists a constant $c > 0$, independent of $u$, $r$, and $x$, such that

$$\left( \int_{B_r(x)} |u|^{\kappa p_0} d\nu_{\tilde{\beta}} \right)^{\frac{1}{\kappa p_0}} \le c r \left( \int_{B_r(x)} |\nabla u|^{p_0} d\nu_{\tilde{\beta}} \right)^{\frac{1}{p_0}}. \quad (2.29)$$
2.4. **Weak Solutions.** Next, we define the notion of a weak solution to (1.1).

**Definition 2.11.** Let $\Omega$ be a bounded domain, and assume conditions (H$_1$) and (H$_2$) are satisfied, and that $n - (1 - s^{-}) > \beta \ge 0$. A function $u \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$ is a weak subsolution of (1.1) if, for every nonnegative test function $\phi \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$ with $\text{supp} \phi \subset \Omega$, we have

$$ \int_{\Omega} |\nabla u|^{q-2} \nabla u \nabla \phi \, d\mathbf{V}_{2\beta} + \int_{\Omega} \int_{\Omega} \mathcal{A}(u, x, y)(\phi(x) - \phi(y)) \, d\mu(x, y) \le 0, \quad (2.30) $$

where

$$ \mathcal{A}(u, x, y) = |u(x) - u(y)|^{p(x,y)-2} (u(x) - u(y)) \quad \text{and} \quad d\mu(x, y) = \frac{1}{|x|^{\beta} |y|^{\beta}} K(x, y) \, dxdy. $$

Analogously, $u \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$ is a weak supersolution of (1.1) if the integral in (2.30) is nonnegative for every nonnegative test function $\phi \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$ with $\text{supp} \phi \subset \Omega$.

A function $u \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$ is a weak solution of (1.1) if equality holds in (2.30) for every $\phi \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$ with $\text{supp} \phi \subset \Omega$, and without any sign restriction on $\phi$.

It follows directly from Definition 2.11 that $u$ is a weak subsolution of (1.1) if and only if $-u$ is a weak supersolution of (1.1). Moreover, for any $c \in \mathbb{R}$, $u + c$ is a weak solution of (1.1) if and only if $u$ is a weak solution of (1.1).

We now discuss further structural properties of weak solutions. We denote the positive and negative parts of $a \in \mathbb{R}$ by $a_+ = \max\{a, 0\}$ and $a_- = \max\{-a, 0\}$, respectively.

**Lemma 2.12.** A function $u \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$ is a weak solution of (1.1) if and only if $u$ is both a weak subsolution and a weak supersolution of (1.1).

*Proof.* The proof follows the same argument as in [58, Lemma 2.7]. □

Next, we show that the property of being a weak subsolution is preserved under taking the positive part. As a direct consequence $u_-$ is a weak subsolution of (1.1), whenever $u$ is a weak supersolution of (1.1).

**Lemma 2.13.** Assume that $u \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$ is a weak subsolution of (1.1). Then $u_+ \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$ is a weak subsolution of (1.1).

*Proof.* The argument follows the same reasoning as in [58, Lemma 2.8], taking into account Lemma 2.5, since $u_k = \min\{ku_+, 1\} \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta}) \cap L^{\infty}(\Omega)$, for $k = 1, 2, \dots$, and for any $\ell > 0$,

$$ T_{\ell}(\phi) \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta}) \cap L^{\infty}(\Omega), $$

where $\phi \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta})$, with $\text{supp} \phi \subset \Omega$. The truncation function $T_{\ell}: \mathbb{R} \to \mathbb{R}$, $\ell > 0$, is defined by

$$ T_{\ell}(t) = \begin{cases} t & \text{if } |t| \le \ell, \\ \text{sign}(t)\ell & \text{if } |t| > \ell. \end{cases} $$

Hence, by Lemma 2.5, $u_k T_{\ell}(\phi) \in W_{\text{loc}}^{1,q}(\Omega; \mathbf{V}_{2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; \mathbf{V}_{\beta}) \cap L^{\infty}(\Omega)$.

To complete the proof of Lemma 2.13, we refer to the proof of [58, Lemma 2.8]. □

### 3. ENERGY ESTIMATES

The following energy estimate will be crucial for us.

**Lemma 3.1.** ([29, Lemma 3.4]). Let $a, b \ge 0$, $\tau_1, \tau_2 \in [0, 1]$, and $1 < p_1 \le p_2 \le p_3$. Then,

$$ |a - b|^{p_2-2} (a - b) (a\tau_1^{p_3} - b\tau_2^{p_3}) \ge \frac{1}{2} |a - b|^{p_2} (\max\{\tau_1, \tau_2\})^{p_3} - C (\max\{a, b\})^{p_2} |\tau_1 - \tau_2|^{p_2}, $$

where $C = \frac{p_3}{p_1} (2p_3)^{p_3-1}$.
**Lemma 3.2.** Let $u$ be a weak subsolution of (1.1), and define $w = (u-k)_+$ with $k \in \mathbb{R}$. Then

$$ 
\begin{aligned}
& \int_{B_r(x_0)} \psi^q |\nabla w|^q \, dV_{2\beta} + \int_{B_r(x_0)} \int_{B_r(x_0)} (\max\{\psi(x), \psi(y)\})^q |w(x) - w(y)|^{p(x,y)} \, d\mu(x, y) \\
& \le 2 [2(\mathbf{q}-1)]^{q-1} \int_{B_r(x_0)} w^q |\nabla \psi|^q \, dV_{2\beta} \\
& \quad + \frac{(2\mathbf{q})^q}{p^-} \int_{B_r(x_0)} \int_{B_r(x_0)} \max\{w(x), w(y)\}^{p(x,y)} |\psi(x) - \psi(y)|^{p(x,y)} \, d\mu(x, y) \\
& \quad + 4 \int_{\Omega \setminus B_r(x_0)} w(y)^{p(x,y)-1} w(x) \psi(x)^q \, d\mu(x, y),
\end{aligned} 
\quad (3.1) $$

whenever $B_r(x_0) \subset \Omega$ and $\psi \in C_c^\infty(B_r(x_0))$ is a nonnegative function. If $u$ is a weak supersolution of (1.1), the estimate in (3.1) holds with $w = (u-k)_-$.

*Proof.* Let $u$ be a weak subsolution of (1.1). For $w = (u-k)_+$, choose $\phi = w\psi^q$ as a test function in (2.30). We obtain

$$ 
\begin{aligned}
& 0 \ge \int_{B_r(x_0)} |\nabla u|^{q-2} \nabla u \nabla(w\psi^q) \, dV_{2\beta} + \int_{\Omega} \int_{\Omega} \mathcal{A}(u, x, y) (w(x)\psi(x)^q - w(y)\psi(y)^q) \, d\mu(x, y) \\
& =: I + J.
\end{aligned} 
$$

**Steep 1.** We have

$$ 
\begin{aligned}
I &= \int_{B_r(x_0)} |\nabla u|^{q-2} \nabla u (\psi^q \nabla w + \mathbf{q} \psi^{q-1} \nabla \psi) \, dV_{2\beta} \\
&\ge \int_{B_r(x_0)} \psi^q |\nabla w|^q \, dV_{2\beta} + \int_{B_r(x_0)} \mathbf{q} w \psi^{q-1} |\nabla u|^{q-2} \nabla u \nabla \psi \, dV_{2\beta}.
\end{aligned} 
$$

Using Young's inequality,

$$ 
\begin{aligned}
-w\psi^{q-1}|\nabla u|^{q-2}\nabla u\nabla\psi &\le \psi^q \left( \epsilon \frac{\mathbf{q}-1}{\mathbf{q}} |\nabla w|^q + \epsilon^{-q+1} \frac{1}{\mathbf{q}} \psi^{-q} w^q |\nabla \psi|^q \right) \\
&\le \epsilon \frac{\mathbf{q}-1}{\mathbf{q}} \psi^q |\nabla w|^q + \epsilon^{-q+1} \frac{1}{\mathbf{q}} w^q |\nabla \psi|^q.
\end{aligned} 
$$

Choosing $\epsilon = \frac{1}{2(\mathbf{q}-1)}$, we obtain

$$ 
\begin{aligned}
I &\ge \frac{1}{2} \int_{B_r(x_0)} \psi^q |\nabla w|^q \, dV_{2\beta} \\
& \quad - [2(\mathbf{q}-1)]^{q-1} \int_{B_r(x_0)} w^q |\nabla \psi|^q \, dV_{2\beta}.
\end{aligned} 
\quad (3.2) $$

**Steep 2.** Now we estimate $J$. We have

$$ 
\begin{aligned}
& |u(x) - u(y)|^{p(x,y)-2} (u(x) - u(y)) (w(x)\psi(x)^q - w(y)\psi(y)^q) \\
&= |u(x) - u(y)|^{p(x,y)-2} (u(x) - u(y)) \left[ (u(x) - k)_+ \psi(x)^q - (u(y) - k)_+ \psi(y)^q \right] \\
&= \begin{cases} |w(x) - w(y)|^{p(x,y)-2} (w(x) - w(y)) [w(x)\psi(x)^q - w(y)\psi(y)^q] & \text{if } u(x), u(y) > k, \\ (u(x) - u(y))^{p(x,y)-1} w(x)\psi(x)^q & \text{if } u(x) \ge k, u(y) \le k, \\ (u(y) - u(x))^{p(x,y)-1} w(y)\psi(y)^q & \text{if } u(x) \le k, u(y) \ge k, \\ 0 & \text{otherwise.} \end{cases} \\
&\ge |w(x) - w(y)|^{p(x,y)-2} (w(x) - w(y)) (w(x)\psi(x)^q - w(y)\psi(y)^q).
\end{aligned} 
$$

From Lemma 3.1, for $(x, y) \in B_r(x_0) \times B_r(x_0)$,

$$ 
\begin{aligned}
& |w(x) - w(y)|^{p(x,y)-2} (w(x) - w(y)) (w(x)\psi(x)^q - w(y)\psi(y)^q) \\
&\ge \frac{1}{2} |w(x) - w(y)|^{p(x,y)} (\max\{\psi(x), \psi(y)\})^q \\
& \quad - \frac{\mathbf{q}}{p^-} (2\mathbf{q})^{q-1} (\max\{w(x), w(y)\})^{p(x,y)} |\psi(x) - \psi(y)|^{p(x,y)},
\end{aligned} 
\quad (3.3) $$
If $(x, y) \in (\Omega \setminus B_r(x_0)) \times B_r(x_0)$, then

$$
\begin{align}
& |w(x) - w(y)|^{p(x,y)-2} (w(x) - w(y)) (w(x)\psi(x)^q - w(y)\psi(y)^q) \nonumber \\
& = -|w(x) - w(y)|^{p(x,y)-2} (w(x) - w(y))w(y)\psi(y)^q \tag{3.4} \\
& \geq -w(x)^{p(x,y)-1}w(y)\psi(y)^q. \nonumber
\end{align}
$$

Similarly, if $(x, y) \in B_r(x_0) \times (\Omega \setminus B_r(x_0))$, then

$$
\begin{align}
& |w(x) - w(y)|^{p(x,y)-2} (w(x) - w(y)) (w(x)\psi(x)^q - w(y)\psi(y)^q) \nonumber \\
& = |w(x) - w(y)|^{p(x,y)-2} (w(x) - w(y))w(x)\psi(x)^q \tag{3.5} \\
& \geq -w(y)^{p(x,y)-1}w(x)\psi(x)^q. \nonumber
\end{align}
$$

Hence,

$$
\begin{align}
J \ge & \frac{1}{2} \int_{\Omega} \int_{\Omega} |w(x) - w(y)|^{p(x,y)} (\max \{\psi(x), \psi(y)\})^q \, d\mu(x, y) \nonumber \\
& - \frac{\mathbf{q}(2\mathbf{q})^{q-1}}{p^{-}} \int_{\Omega} \int_{\Omega} (\max \{w(x), w(y)\})^{p(x,y)} |\psi(x) - \psi(y)|^{p(x,y)} \, d\mu(x, y) \tag{3.6} \\
& - 2 \int_{\Omega \setminus B_r(x_0)} \int_{B_r(x_0)} w(y)^{p(x,y)-1} w(x) \psi(x)^q \, d\mu(x, y) \nonumber
\end{align}
$$

By (3.2) - (3.6), we obtain (3.1). In the case of a weak supersolution, the estimate in (3.1) follows by applying
the obtained result to $-u$. $\square$

Recall that for a measurable function $v : \Omega \to \mathbb{R}$, we define

$$
T(v; x_0, r, \rho) := \underset{x \in B_{\rho}(x_0)}{\operatorname{ess \, sup}} \int_{\Omega \setminus B_r(x_0)} \frac{|v(y)|^{p(x,y)-1}}{|y - x_0|^{n+s(x,y)p(x,y)}} \frac{1}{|y|^{\beta}} \, dy,
$$

where $B_r(x_0) \subset B_{\rho}(x_0) \subset \Omega$.

Next, we obtain a logarithmic energy estimate.

**Lemma 3.3.** Suppose that $n - 2\beta - 1 + s^- > 0$, $\beta \ge 0$, and $n - \beta > s^+\mathbf{q}$. Furthermore, let $d > 0$ and $0 < 2r \le \frac{R}{2}$. Assume that $u$ is a weak supersolution of (1.1) such that $u \ge 0$ in $B_R(x_0) \subset \Omega$.

If $0 \notin B_{3R}(x_0)$ and $0 < R_0 \le |x_0|/2$, then there exists a constant $c_1 = c_1(s^-, s^+, p^-, \mathbf{q}, \beta, \Lambda, n) > 0$ such that

$$
\begin{align}
& \frac{1}{\nabla_{2\beta} (B_r(x_0))} \int_{B_r(x_0)} |\nabla \ln(u+d)|^q d\nabla_{2\beta} \nonumber \\
& \le c_1 \left( \frac{|x_0|+1}{R_0} \right)^\beta \max \{1, d^{p^--q}\} \left( r^{-s^+q} + \frac{R_0^{n-s^+q}}{|x_0|^n} \right) \tag{3.7} \\
& \phantom{\le} + c_1 r^{-q} + c_1(|x_0|+1)^\beta d^{1-q} T(u_-, x_0, R, 3r/2). \nonumber
\end{align}
$$

If $x_0 = 0$, then there exists a constant $c_2 = c_2(s^-, p^-, s^+, \mathbf{q}, \beta, \Lambda, n) > 0$ such that

$$
\begin{align}
& \int_{B_r(x_0)} |\nabla \ln(u+d)|^q d\nabla_{2\beta} \tag{3.8} \\
& \le c_2 \max \{1, d^{p^--q}\} r^{n-s^+q-2\beta} + c_2 r^{n-q-2\beta} + c_2 d^{1-q} r^{n-\beta} T(u_-, x_0, R, 3r/2), \nonumber
\end{align}
$$

where $c_2 = c_2(s^-, p^-, s^+, \mathbf{q}, \beta, \Lambda, n) > 0$.

Proof. Let $\psi \in C_c^\infty(B_{3r/2}(x_0))$ be such that $0 \le \psi \le 1$ in $B_{3r/2}(x_0)$, $\psi = 1$ in $B_r(x_0)$, and $|\nabla\psi| \le C_1/r$ in $B_{3r/2}(x_0)$. By choosing $\phi = (u+d)^{1-q}\psi^q$ as a test function in (2.30), we obtain

$$
\begin{align*}
0 \le & \int_{B_{2r}(x_0)} \int_{B_{2r}(x_0)} \mathcal{A}(u, x, y) \left[ (u(x) + d)^{1-q}\psi(x)^q - (u(y) + d)^{1-q}\psi(y)^q \right] d\mu(x, y) \\
& + 2 \int_{\Omega \setminus B_{2r}(x_0)} \int_{B_{2r}(x_0)} \mathcal{A}(u, x, y) (u(x) + d)^{1-q}\psi(x)^q d\mu(x, y) \\
& + \int_{B_{2r}(x_0)} |\nabla u|^{q-2} \nabla u \nabla [(u+d)^{1-q}\psi^q] d\nabla_{2\beta} \\
& = I_1 + I_2 + I_3.
\end{align*}
$$
Estimate of $I_1$: If $u(x) > u(y)$,

$$ egin{aligned} & |u(x) - u(y)|^{p(x,y)-2}(u(x) - u(y)) \left[ \frac{\psi(x)^q}{(u(x)+d)^{q-1}} - \frac{\psi(y)^q}{(u(y)+d)^{q-1}} \right] \\ & \le \frac{(u(x)-u(y))^{p(x,y)-1}}{(u(x)+d)^{q-1}} (\psi(x)^q - \psi(y)^q) \\ & \le \left( \frac{u(x)-u(y)}{u(x)+d} \right)^{p(x,y)-1} \frac{1}{(u(x)+d)^{q-p(x,y)}} |\psi(x)^q - \psi(y)^q| \\ & \le \max\{1, d^{p-q}\} |\psi(x)^q - \psi(y)^q|, \end{aligned} \quad (3.9) $$

and a similar estimate is obtained when $u(x) < u(y)$.

If $0 \notin B_{3R}(x_0)$, then, since

$$ |\psi(x)^q - \psi(y)^q| \le \left( \sup_{B_{3r/2}(x_0)} \psi^{q-1} |\nabla \psi| \right) |x-y| \le C_1 r^{-1} |x-y|, \quad (3.10) $$

from (3.9), we have

$$ egin{aligned} I_1 &\le \frac{C_1^q \Lambda \max\{1, d^{p-q}\}}{(|x_0| - 2r)^\beta} r^{-q} \int_{B_{2r}(x_0)} \frac{1}{|y|^\beta} \int_{B_{4r}(y)} |x-y|^{-n+(1-s^+)q} \, dx dy \\ &= \frac{2^{2(1-s^+)q} |\partial B_1(0)| C_1^q \Lambda \max\{1, d^{p-q}\} (|x_0| + 2r)^\beta}{(1-s^+)q (|x_0| - 2r)^\beta} r^{-s^+q} \mathbf{V}_{2\beta} (B_{2r}(x_0)) \end{aligned} \quad (3.11) $$
$$ \le c(\beta, n) \frac{2^{2(1-s^+)q} C_1^q \Lambda \max\{1, d^{p-q}\} (2)^\beta}{(1-s^+)q} r^{-s^+q} \mathbf{V}_{2\beta} (B_{3r/2}(x_0)), $$

because $6r \le |x_0|$, and by Lemma 2.7 (ii).

If $x_0 = 0$, then employing Lemma 2.2 (i), we get

$$ egin{aligned} I_1 &\le C_2 r^{-q} \int_{B_{2r}(0)} \frac{1}{|y|^\beta} \int_{B_{2r}(0)} \frac{1}{|x-y|^{n-(1-s^+)}} \frac{1}{|x|^\beta} |x-y|^{(1-s^+) (q-1)} \, dxdy \\ &\le C_2 C_3 r^{-q+(1-s^+) (q-1)} \int_{B_{2r}(0)} \frac{(2r)^{2(1-s^+)}}{|y|^{2\beta+(1-s^+)}} \, dy \end{aligned} \quad (3.12) $$
$$ = C_2 C_3 \frac{2^{n-2\beta+1-s^+} |\partial B_1(0)|}{[n-2\beta-(1-s^+)]} r^{n-2\beta-s^+ q}, $$

where $C_2 = C_1^q \Lambda \max\{1, d^{p-q}\}$ and $C_3 = C_3(s^+, q, \beta, n) > 0$.

Estimate of $I_2$: We follow the lines of the proof of [41, p 1290, Lemma 1.3].

Notice that when $y \in B_R(x_0)$, we have $u(y) \ge 0$, and so

$$ egin{aligned} \frac{(u(x) - u(y))_+^{p(x,y)-1}}{(d + u(x))^{q-1}} &= \frac{(u(x) - u(y))_+^{p(x,y)-1}}{(d + u(x))^{p(x,y)-1}} (d + u(x))^{p(x,y)-q} \end{aligned} \quad (3.13) $$
$$ \le \max\{1, d^{p-q}\}, $$

for all $x \in B_{2r}(x_0)$ and $y \in B_R(x_0)$.

Moreover, when $y \in \Omega \setminus B_R(x_0)$,

$$ egin{aligned} & \frac{(u(x) - u(y))_+^{p(x,y)-1}}{(d + u(x))^{q-1}} \\ &\le 2^{p(x,y)-1} \left[ \frac{u(x)^{p(x,y)-1}}{(d + u(x))^{p(x,y)-1}} (d + u(x))^{p(x,y)-q} + \frac{(u_-(y))^{p(x,y)-1}}{(d + u(x))^{q-1}} \right] \end{aligned} \quad (3.14) $$
$$ \le 2^{q-1} \left[ \max\{1, d^{p-q}\} + d^{1-q} (u_-(y))^{p(x,y)-1} \right], $$

for all $x \in B_{2r}(x_0)$.
We have supp $\psi \subset B_{3r/2}(x_0)$. From (3.13) and (3.14), we see that

$$
\begin{align*}
I_2 \le & 2 \int_{B_{2r}(x_0)} \int_{B_R(x_0) \setminus B_{2r}(x_0)} K(x, y)(u(x) - u(y))_+^{p(x,y)-1} (d+u(x))^{1-q} \psi(x)^q \frac{1}{|x|^\beta |y|^\beta} \, dydx \\
& + 2 \int_{B_{2r}(x_0)} \int_{\Omega \setminus B_R(x_0)} K(x, y)(u(x) - u(y))_+^{p(x,y)-1} (d+u(x))^{1-q} \psi(x)^q \frac{1}{|x|^\beta |y|^\beta} \, dydx \\
\le & 2\Lambda \max \{1, d^{p^--q}\} \left\{ \underbrace{\int_{B_{3r/2}(x_0)} \int_{\Omega \setminus B_{2r}(x_0)} \frac{1}{|x-y|^{n+s^+q}} \frac{1}{|x|^\beta |y|^\beta} \, dydx}_{I_{2,1}} \right. \\
& + 2^{q-1} \underbrace{\int_{B_{3r/2}(x_0)} \int_{\Omega \setminus B_{2r}(x_0)} \left( \frac{1}{|x-y|^{n+s^-p^-}} + \frac{1}{|x-y|^{n+s^+q}} \right) \frac{1}{|x|^\beta |y|^\beta} \, dydx}_{I_{2,2}} \\
& \left. + 2^{1-q} \Lambda d^{1-q} \underbrace{\int_{B_{3r/2}(x_0)} \int_{\Omega \setminus B_R(x_0)} \frac{(u_-(y))^{p(x,y)-1}}{|x-y|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^\beta |y|^\beta} \, dydx}_{I_{2,3}} \right\}
\end{align*}
$$

If $0 \notin \overline{B_{3R}(x_0)}$ and $0 < R_0 \le |x_0|/2$, then

$$
I_{2,1} \le \frac{1}{R_0^\beta} \int_{B_{3r/2}(x_0)} \frac{1}{|x|^\beta} \int_{\mathbb{R}^n \setminus (B_{2r}(x_0) \cup B_{R_0}(0))} \frac{4^{n+s^+q}}{|x_0 - y|^{n+s^+q}} \, dydx \\
+ \int_{B_{3r/2}(x_0)} \frac{1}{|x|^\beta} \int_{B_{R_0}(0)} \frac{4^{n+s^+q}}{|x_0 - y|^{n+s^+q}} \frac{1}{|y|^\beta} \, dydx. \tag{3.15}
$$

In the estimate above, we used the fact that for $(x, y) \in B_{3r/2}(x_0) \times \mathbb{R}^n \setminus B_{2r}(x_0)$, it holds that $|x - y| \ge r/2$, and moreover,

$$
\frac{|y - x_0|}{|y - x|} \le 1 + \frac{|x - x_0|}{|x - y|} \le 1 + \frac{3}{2} \frac{1}{|y|} = 4. \tag{3.16}
$$

By Lemma 2.2 (ii), we deduce

$$
\begin{align}
& \frac{1}{R_0^\beta} \int_{\mathbb{R}^n \setminus B_{2r}(x_0)} \frac{1}{|x_0 - y|^{n+s^+q}} \, dy + \int_{B_{R_0}(0)} \frac{1}{|x_0 - y|^{n+s^+q}} \frac{1}{|y|^\beta} \, dy \tag{3.17} \\
\le{}& C_4(s^+, \mathbf{q}, \beta, n) \frac{(2r)^{-s^+\mathbf{q}}}{R_0^\beta} \frac{\left(|x_0| + \frac{3r}{2}\right)^\beta}{|x|^\beta} + C_4 \frac{R_0^{n-s^+\mathbf{q}-\beta}}{|x_0|^n} \frac{\left(|x_0| + \frac{3r}{2}\right)^\beta}{|x|^\beta}, \nonumber
\end{align}
$$

because

$$
\frac{|x_0| + \frac{3r}{2}}{|x|} > 1 \quad \text{in} \quad B_{3r/2}(x_0). \tag{3.18}
$$

From (3.15) and (3.17), it follows that

$$
\begin{align}
I_{2,1} \le C_5(s^+, \mathbf{q}, \beta, n) \frac{\left(|x_0| + \frac{3r}{2}\right)^\beta}{R_0^\beta} (2r)^{-s^+\mathbf{q}} \mathbf{V}_{2\beta} (B_{3r/2}(x_0)) \tag{3.19} \\
\qquad + C_5 \frac{\left(|x_0| + \frac{3r}{2}\right)^\beta R_0^{n-s^+\mathbf{q}-\beta}}{|x_0|^n} \mathbf{V}_{2\beta} (B_{3r/2}(x_0)). \nonumber
\end{align}
$$
Now we estimate $I_{2,1}$ when $x_0 = 0$. From (3.16),

$$ 
\begin{aligned}
I_{2,1} & \le 2^{2(n+s^+\mathbf{q})} \int_{B_{3r/2}(0)} \frac{1}{|x|\beta} \int_{\mathbb{R}^n \setminus B_{2r}(0)} \frac{1}{|y|^{n+s^+\mathbf{q}+\beta}} \, dydx \\
& \le C_6(s^+, \mathbf{q}, \beta, n) r^{n-s^+\mathbf{q}-2\beta}.
\end{aligned} 
\quad (3.20) $$

Similarly to what we did for $I_{2,1}$, for $I_{2,2}$ we have

$$ I_{2,2} \le \begin{cases} C_7 \mathbf{V}_{2\beta} (B_{3r/2}(x_0)) \left( \frac{|x_0|+1}{R_0} \right)^\beta \left( r^{-s^+\mathbf{q}} + \frac{R_0^{n-s^+\mathbf{q}}}{|x_0|^n} \right) & \text{if } 0 \notin \overline{B_{3R}(x_0)}, \\ C_7 r^{n-s^+\mathbf{q}-2\beta} & \text{if } x_0 = 0, \end{cases} \quad (3.21) $$

where $C_7 = C_7(s^-, s^+, p^-, \mathbf{q}, \beta, n) > 0$.

For $I_{2,3}$, using (3.16) and (3.18), we obtain

$$ I_{2,3} \le \left( |x_0| + \frac{3r}{2} \right)^\beta \mathbf{V}_{2\beta} (B_{3r/2}(x_0)) T(u_-, x_0, R, 3r/2). \quad (3.22) $$

Therefore, from (3.19), (3.21), and (3.22),

$$ I_2 \le C_8 \max \{1, d^{p^-q}\} \mathbf{V}_{2\beta} (B_{3r/2}(x_0)) \left( \frac{|x_0|+1}{R_0} \right)^\beta \left( r^{-s^+q} + \frac{R_0^{n-s^+q}}{|x_0|^n} \right) + C_8 d^{1-q} (|x_0|+1)^\beta \mathbf{V}_{2\beta} (B_{3r/2}(x_0)) T(u_-, x_0, R, 3r/2) \quad \text{if } 0 \notin \overline{B_{3R}(x_0)}, \quad (3.23) $$

where $C_8 = C_8(s^-, s^+, p^-, \mathbf{q}, \beta, \Lambda, n) > 0$.

And, if $x_0 = 0$, combining (3.20), (3.21), and (3.22), we obtain

$$ I_2 \le C_9 \max \{1, d^{p^-q}\} r^{n-2\beta-s^+q} + C_9 d^{1-q} r^{n-\beta} T(u_-, x_0, R, 3r/2), \quad (3.24) $$

where $C_9 = C_9(s^-, s^+, p^-, \mathbf{q}, \beta, \Lambda, n) > 0$.

Estimate of $I_3$: We have

$$ 
\begin{aligned}
I_3 &= \int_{B_{2r}(x_0)} |\nabla u|^{q-2} \nabla u \nabla [(u+d)^{1-q} \psi^q] \, d\mathbf{V}_{2\beta} \\
&\le (1-q) \int_{B_{2r}(x_0)} |\nabla \ln(u+d)|^q \psi^q \, d\mathbf{V}_{2\beta} + C_1 q \int_{B_{2r}(x_0)} |\nabla u|^{q-1} (u+d)^{1-q} \psi^{q-1} \frac{1}{r} \, d\mathbf{V}_{2\beta}.
\end{aligned} 
$$

Employing Young's inequality:

$$ |\nabla u|^{q-1} (u+d)^{1-q} \psi^{q-1} \frac{1}{r} \le \frac{q-1}{q} \epsilon |\nabla u|^{q} (u+d)^{-q} \psi^{q} + \frac{1}{q} \epsilon^{-q+1} \frac{1}{r^q}. $$

Choosing $\epsilon = \frac{1}{2C_1}$, we obtain

$$ I_3 \le -\frac{q-1}{2} \int_{B_r(x_0)} |\nabla \ln(u+d)|^q d\mathbf{V}_{2\beta} + C_1 (2C_1)^{q-1} \frac{1}{r^q} \mathbf{V}_{2\beta}(B_{3r/2}(x_0)). \quad (3.25) $$

Hence, using (3.11), (3.23), (3.25), and Lemma 2.7 (ii), the estimate (3.7) follows. Furthermore, employing (3.12), (3.24), and (3.25), we obtain (3.8). $\square$

As a consequence of Lemma 3.3, we have the following result.

**Corollary 3.4.** Suppose that $n - 2\beta - 1 + s^- > 0$, $\beta \ge 0$, and $n - \beta > s^+\mathbf{q}$. Furthermore, let $d > 0$ and $0 < 2r \le R \le 1/2$. Assume that $u$ is a weak supersolution of (1.1) such that $u \ge 0$ in $B_R(x_0) \subset \Omega$. Let $a, d > 0$ and $b > 1$, and define

$$ v = \min \left\{ \left( \ln \left( \frac{a+d}{u+d} \right) \right)_+ , \ln b \right\}. $$
If $0 \notin \overline{B_{3R}(x_0)}$ and $0 < R_0 \le |x_0|/2$, then there exists a constant $c_1 = c_1(s^-, p^-, s^+, \mathbf{q}, \beta, \Lambda, n) > 0$ such that

$$ \int_{B_r(x_0)} |v - (v)_{B_r(x_0), v_{2\beta}}|^q dV_{2\beta} \\ \le c_1 \max \{1, d^{p^--q}\} \left(\frac{|x_0|+1}{R_0}\right)^\beta \left(r^{q-s^+q} + \frac{R_0^{n-s^+q}}{|x_0|^n} r^q\right) \\ + c_1 + c_1 d^{1-q}(|x_0|+1)^\beta r^q T(u_-; x_0, R, 3r/2). \quad (3.26) $$

If $x_0 = 0$, then there exists a constant $c_2 = c_2(s^-, p^-, s^+, \mathbf{q}, \beta, \Lambda, n) > 0$ such that

$$ \int_{B_r(x_0)} |v - (v)_{B_r(x_0), v_{2\beta}}|^q dV_{2\beta} \\ \le c_2 \max \{1, d^{p^--q}\} r^{q-s^+q} + c_2 + c_2 d^{1-q} r^{q+\beta} T(u_-; x_0, R, 3r/2), \quad (3.27) $$

where $c_2 = c_2(s^-, p^-, s^+, \mathbf{q}, \beta, \Lambda, n) > 0$.

*Proof.* By Corollary 2.8, we have

$$ \int_{B_r(x_0)} |v - (v)_{B_r(x_0), v_{2\beta}}|^q dV_{2\beta} \\ \le C_1(\beta, \mathbf{q}, n) \frac{1}{V_{2\beta}(B_r(x_0))} r^q \int_{B_r(x_0)} |\nabla v|^q dV_{2\beta} \quad \text{if } 0 \notin \overline{B_{3R}(x_0)}, \quad (3.28) $$

and

$$ \int_{B_r(x_0)} |v - (v)_{B_r(x_0), v_{2\beta}}|^q dV_{2\beta} \\ \le C_2(\beta, \mathbf{q}, n) r^{q-n+2\beta} \int_{B_r(x_0)} |\nabla v|^q dV_{2\beta} \quad \text{if } x_0 = 0. \quad (3.29) $$

Since $v$ is a truncation of the sum of a constant and $\ln(u+d)$, it holds that

$$ \int_{B_r(x_0)} |\nabla v|^q dV_{2\beta} \le \int_{B_r(x_0)} |\nabla \ln(u+d)|^q dV_{2\beta}. \quad (3.30) $$

The estimates in (3.26) and (3.27) follow by applying Lemma 3.3 together with (3.28) - (3.30). $\square$

## 4. LOCAL BOUNDEDNESS

We apply the following lemma from real analysis. For the proof of Lemma 4.1, see [43, Lemma 4.1].

**Lemma 4.1.** Let $(Y_j)_{j=0}^\infty$ be a sequence of positive real numbers such that $Y_0 \le c_0^{-1/\beta} b^{-1/\beta^2}$ and $Y_{j+1} \le c_0 b^j Y_j^{1+\beta}$, $j = 0, 1, 2, \dots$, for some constants $c_0, b > 1$ and $\beta > 0$. Then $\lim_{j \to \infty} Y_j = 0$.

We now show that weak subsolutions of (1.1) are locally bounded. The statement of this result, as given in the introduction, is as follows:

**Theorem 1.2.** (Local boundedness). Let $\Omega$ be a bounded domain. Assume conditions (H$_1$) and (H$_2$) are satisfied. Also, assume that $\beta \ge 0$ and $n - \beta - 1 + s^+ > 0$.

Let $u \in W_{\text{loc}}^{1,q}(\Omega; |x|^{-2\beta}) \cap W^{s(x,y)}(x,y) \cap W^{(x,y),(p(x,y),\Omega)}; |x|^{-\beta}$ be a weak solution of (1.1), satisfying $T(u; x_0, r, \rho) < \infty$ for all balls $B_r(x_0) \subset B_\rho(x_0) \subset \Omega$. Then there exists a constant $C = C(s^+, p^-, \mathbf{q}, \beta, \Lambda, n) > 0$, such that

$$ \ess \sup_{B_{r/2}(x_0)} u \le \delta(|x_0| + 1)^{\frac{\beta}{q-1}} r^{\frac{\mathbf{q}}{q-1}} T(u_+; x_0, \frac{r}{2}, r)^{\frac{1}{q-1}} + C \delta^{-\frac{(q-1)\kappa}{q(\kappa-1)}} \left( \int_{B_r(x_0)} u_+^q dV_{2\beta} \right)^{\frac{1}{q}} + 1, \quad (4.1) $$

whenever $B_r(x_0) \subset \Omega$ with $r \in (0, 1/2]$ and $\delta \in (0, 1]$. Here, $\kappa$ is the exponent given in Corollary 2.10.

*Proof.* Let $B_r(x_0) \subset \Omega$ with $r \in (0, 1]$. For $j = 0, 1, 2, \dots$, we define

$$ r_j := \frac{r}{2}(1 + 2^{-j}), \quad \tilde{r}_j := \frac{r_j + r_{j+1}}{2}, \quad B_j := B_{r_j}(x_0), \quad \tilde{B}_j := B_{\tilde{r}_j}(x_0). $$

Let $(\psi_j)_{j=0}^\infty \subset C_c^\infty(\tilde{B}_j)$ be a sequence of cutoff functions such that $0 \le \psi_j \le 1$ in $\tilde{B}_j$, $\psi_j = 1$ in $B_{j+1}$, and $|\nabla \psi_j| \le \frac{C_1(n)2^{j+3}}{r}$, with $C_1 > 1$, for all $j = 0, 1, 2, \dots$.
For $j = 0, 1, 2, \dots$ and $k, \tilde{k} \ge 0$, we define

$$k_j := k + (1 - 2^{-j})\tilde{k}, \quad \tilde{k}_j := \frac{k_j + k_{j+1}}{2}, \quad w_j := (u - k_j)_+, \quad \tilde{w}_j := (u - \tilde{k}_j)_+.$$

From Lemma 2.7 (ii),

$$ \left(\frac{\tilde{k}}{2^{j+2}}\right)^{\frac{q(\kappa-1)}{\kappa}} \left(\int_{B_{j+1}} w_{j+1}^{\mathbf{q}} \, d\mathbf{V}_{2\beta}\right)^{\frac{1}{\kappa}} = (k_{j+1} - \tilde{k}_j)^{\frac{q(\kappa-1)}{\kappa}} \left(\int_{B_{j+1}} w_{j+1}^{\mathbf{q}} \, d\mathbf{V}_{2\beta}\right)^{\frac{1}{\kappa}} \quad (4.2) \\ \le \left[\left(\frac{3}{2}\right)^n C_2(\beta, n)\right]^{\frac{1}{\kappa}} \left(\int_{\tilde{B}_j} |\tilde{w}_j \psi_j|^{\mathbf{q}\kappa} \, d\mathbf{V}_{2\beta}\right)^{\frac{1}{\kappa}}, $$

where $\kappa$ is given by (2.28).

By the Sobolev inequality in Corollary 2.10, with $C_3 = C_3(\beta, \mathbf{q}, n) > 0$, we obtain

$$ \left(\int_{\tilde{B}_j} |\tilde{w}_j \psi_j|^{\mathbf{q}\kappa} \, d\mathbf{V}_{2\beta}\right)^{\frac{1}{\kappa}} \le C_3 \frac{\mathbf{V}_{2\beta}(B_j)}{\mathbf{V}_{2\beta}(\tilde{B}_j)} \tilde{r}_j^{\mathbf{q}} \int_{B_j} |\nabla (\tilde{w}_j \psi_j)|^{\mathbf{q}} \, d\mathbf{V}_{2\beta} \quad (4.3) \\ \le C_4(\beta, \mathbf{q}, n) \left( \underbrace{r^{\mathbf{q}} \int_{B_j} \tilde{w}_j^{\mathbf{q}} |\nabla \psi_j|^{\mathbf{q}} \, d\mathbf{V}_{2\beta}}_{I_1} + \underbrace{r^{\mathbf{q}} \int_{B_j} \psi_j^{\mathbf{q}} |\nabla \tilde{w}_j|^{\mathbf{q}} \, d\mathbf{V}_{2\beta}}_{I_2} \right), $$

since by Lemma 2.7 (ii), $\frac{r_j}{\tilde{r}_j} = \frac{2^{j+2}+4}{2^{j+2}+3} \le \frac{4}{3}$, and $\tilde{r}_j \le r$.

*Estimate of $I_1$: Using the properties of $\psi_j$, we have*

$$ I_1 = r^{\mathbf{q}} \int_{B_j} \tilde{w}_j^{\mathbf{q}} |\nabla \psi_j|^{\mathbf{q}} \, d\mathbf{V}_{2\beta} \le 2^{3\mathbf{q}} C_1^{\mathbf{q}} 2^{j\mathbf{q}} \int_{B_j} w_j^{\mathbf{q}} \, d\mathbf{V}_{2\beta}. \quad (4.4) $$

*Estimate of $I_2$: By Lemma 3.2, we obtain*

$$ \begin{aligned} I_2 &= r^{\mathbf{q}} \int_{B_j} \psi_j^{\mathbf{q}} |\nabla \tilde{w}_j|^{\mathbf{q}} \, d\mathbf{V}_{2\beta} \\ &\le C_5(\mathbf{q}) r^{\mathbf{q}} \left( \int_{B_j} \tilde{w}_j^{\mathbf{q}} |\nabla \psi_j|^{\mathbf{q}} \, d\mathbf{V}_{2\beta} \right. \\ &\qquad \left. + \frac{1}{\mathbf{V}_{2\beta}(B_j)} \int_{B_j} \int_{B_j} \max\{\tilde{w}_j(x), \tilde{w}_j(y)\}^{p(x,y)} |\psi_j(x) - \psi_j(y)|^{p(x,y)} \, d\mu(x,y) \right. \\ &\qquad \left. + \frac{1}{\mathbf{V}_{2\beta}(B_j)} \int_{\Omega \setminus B_j} \int_{B_j} \tilde{w}(y)^{p(x,y)-1} \tilde{w}(x) \psi_j(x)^{\mathbf{q}} \, d\mu(x,y) \right) \\ &= J_1 + J_2 + J_3 \end{aligned} \quad (4.5) $$

*Estimates of $J_1$ and $J_2$: To estimate $J_2$, proceeding similarly to estimate (4.5) in [41, p. 1292], we obtain:*

$$ \begin{aligned} J_2 &\le \frac{C_5 r^{\mathbf{q}} \Lambda}{\mathbf{V}_{2\beta}(B_j)} \int_{B_j} \int_{B_j} (\tilde{w}_j(x) + \tilde{w}_j(y))^{p(x,y)} \frac{|\psi_j(x) - \psi_j(y)|^{p(x,y)}}{|x-y|^{n+s(x,y)p(x,y)} |x|^{\beta} |y|^{\beta}} \, dxdy \\ &\le \frac{2^{\mathbf{q}} C_5 r^{\mathbf{q}} \Lambda}{\mathbf{V}_{2\beta}(B_j)} \int_{B_j} \int_{B_j \cap \{u > \tilde{k}_j\}} \tilde{w}_j(x)^{p(x,y)} \frac{|\psi_j(x) - \psi_j(y)|^{p(x,y)}}{|x-y|^{n+s(x,y)p(x,y)} |x|^{\beta} |y|^{\beta}} \, dxdy \\ &\le \frac{2^{2\mathbf{q}+1} C_5 C_1^{\mathbf{q}} 2^{j\mathbf{q}} \Lambda}{\mathbf{V}_{2\beta}(B_j)} \int_{B_j} \int_{B_j} \left[ w_j(x)^{\mathbf{q}} + \left(2^{j+2} \frac{w_j(x)}{\tilde{k}}\right)^{\mathbf{q}} \right] \frac{1}{|x-y|^{n-(1-s(x,y))p(x,y)} |x|^{\beta} |y|^{\beta}} \, dxdy, \end{aligned} $$

because

$$ \frac{w_j}{\tilde{k}_j - k_j} \ge 1 \quad \text{in} \quad B_j \cap \{u > \tilde{k}_j\}, $$
and we use the estimates $|\psi_j(x) - \psi_j(y)| \le (\sup_{\tilde{B}_j} |\nabla \psi_j|) |x-y|$, $\tilde{k}_j - k_j = \frac{\tilde{k}}{2^{j+2}}$, and the inequality $a^{p(x,y)} \le a^{q+1}$ for all $a \ge 0$ and $(x, y) \in \Omega \times \Omega$.

If $x_0 \notin B_{3r}(0)$, then $0 \notin \overline{B_r(x_0)}$, and

$$
\begin{align}
J_2 &\le C_6(\mathbf{q}, \Lambda) \frac{2^{2j\mathbf{q}}}{\min\{\tilde{k}\mathbf{q}, 1\}(|x_0| - r)^\beta} \int_{B_j} w_j(x)^{\mathbf{q}} |x|^\beta \left[ \int_{B_{2r_j}(x)} \frac{1}{|x-y|^{n-(1-s^+)p^-}} dy \right] d\mathbf{V}_{2\beta}(x) \nonumber \\
&\le C_7(s^+, p^-, \mathbf{q}, \Lambda, n) \frac{2^{2j\mathbf{q}}(|x_0| + r)^\beta}{\min\{\tilde{k}\mathbf{q}, 1\}(|x_0| - r)^\beta} r_j^{(1-s^+)p^-} \int_{B_j} w_j^{\mathbf{q}} d\mathbf{V}_{2\beta} \nonumber \\
&\le C_7 \frac{2^{2j\mathbf{q}} 2^\beta}{\min\{\tilde{k}\mathbf{q}, 1\}} r_j^{(1-s^+)p^-} \int_{B_j} w_j^{\mathbf{q}} d\mathbf{V}_{2\beta}. \tag{4.6}
\end{align}
$$

If $x_0 \in B_{3r}(0)$, then from $|x - y| < 2r_j$ in $B_j$, we have

$$
J_2 \le C_6(\mathbf{q}, \Lambda) \frac{2^{2j\mathbf{q}} (2r_j)^{(1-s^+)(p^- - 1)}}{\min\{\tilde{k}\mathbf{q}, 1\}} \int_{B_j} w_j(x)^{\mathbf{q}} |x|^\beta \left[ \int_{B_{2r_j}(x)} \frac{1}{|x-y|^{n-(1-s^+)\beta} |y|^\beta} dy \right] d\mathbf{V}_{2\beta}(x). \quad (4.7)
$$

If $x \in B_j \setminus \{0\}$, then by Lemma 2.2 (i):

$$
\begin{align}
& \int_{B_{2r_j}(x)} \frac{1}{|x-y|^{n-(1-s^+)}|y|^\beta} dy \nonumber \\
&= \int_{B_{2r_j}(x) \setminus B_{2|x|}(0)} \frac{1}{|x-y|^{n-(1-s^+)}|y|^\beta} dy + \int_{B_{2|x|}(0)} \frac{1}{|x-y|^{n-(1-s^+)}|y|^\beta} dy \tag{4.8} \\
&\le C_8(s^+, \beta, n) \frac{1}{|x|^\beta} (r_j^{1-s^+} + |x|^{1-s^+}) \le C_9(s^+, \beta, n) \frac{1}{|x|^\beta} r_j^{1-s^+}. \nonumber
\end{align}
$$

If $x=0$, then

$$
\int_{B_{2r_j}(x)} \frac{1}{|x-y|^{n-(1-s^+)}|y|^\beta} dy \le C_{10}(s^+, \beta, n) r_j^{1-s^+-\beta}. \quad (4.9)
$$

From (4.7)-(4.9), we obtain

$$
J_2 \le C_{11} \frac{2^{2j\mathbf{q}}}{\min\{\tilde{k}\mathbf{q}, 1\}} \int_{B_j} w_j^{\mathbf{q}} d\mathbf{V}_{2\beta}, \quad \text{if } x_0 \in B_{3r}(0) \quad (4.10)
$$

where $C_{11} := C_{11}(s^+, p^-, \mathbf{q}, \beta, \Lambda, n) > 0$.

To estimate $J_1$, we use the estimate for $I_1$ in (4.4). Considering (4.6) and (4.10), we get

$$
J_i \le C_{12} \frac{2^{2j\mathbf{q}}}{\min\{\tilde{k}\mathbf{q}, 1\}} \int_{B_j} w_j^{\mathbf{q}} d\mathbf{V}_{2\beta}, \quad (4.11)
$$

for $i = 1, 2$, where $C_{12} := C_{12}(s^+, p^-, \mathbf{q}, \beta, \Lambda, n) > 0$.

Estimate of $J_3$: We observe that $w_j(x)^q \ge (\tilde{k}_j - k_j)^{q-1} \tilde{w}_j(x)$, and

$$
\frac{|y - x_0|}{|x - y|} \le \frac{|y - x| + |x - x_0|}{|x - y|} \le 1 + \frac{\tilde{r}_j}{r_j - \tilde{r}_j} \le 2^{j+4},
$$

for $y \in \mathbb{R}^n \setminus B_j$ and $x \in \tilde{B}_j \supset \text{supp} \psi_j$.
For any $\delta \in (0, 1]$, we have

$$
\begin{align*}
J_3 &\le \frac{C_5 \Lambda r^q}{\mathbf{V}2\beta(B_j)} \int_{\Omega \setminus B_j} \int_{B_j} \tilde{w}_j(y) p^{(x,y)-1} \tilde{w}_j(x) \psi_j(x)^q \frac{1}{|x-y|^{n+s(x,y)} p(x,y) |x|^{\beta} |y|^{\beta}} \, dx dy \\
&\le C_{13} 2^{j(n+\mathbf{q}s^+)} r^q \int_{B_j} |x|^{\beta} \tilde{w}_j \, d\mathbf{V}_{2\beta} \cdot \text{ess sup}_{x \in B_{r_j}(x_0)} \int_{\Omega \setminus B_{r_j}(x_0)} \frac{\tilde{w}_j(y) p^{(x,y)-1}}{|x_0 - y|^{n+s(x,y)} p(x,y) |y|^{\beta}} \, dy \\
&\le C_{13} (|x_0| + r_j)^\beta 2^{j(n+\mathbf{q}s^+)} r^q T(w_0; x_0, \frac{r}{2}, r) \int_{B_j} \frac{w_j^q}{(\tilde{k}_j - k_j)^{q-1}} \, d\mathbf{V}_{2\beta} \\
&\le C_{14} (|x_0| + 1)^\beta \frac{2^{j(n+\mathbf{q}s^+ + \mathbf{q}-1)}}{\tilde{k}^{q-1}} r^q T(w_0; x_0, \frac{r}{2}, r) \int_{B_j} w_j^q \, d\mathbf{V}_{2\beta} \\
&\le C_{14} 2^{j(n+\mathbf{q}s^+ + \mathbf{q}-1)} \delta^{1-q} \int_{B_j} w_j^q \, d\mathbf{V}_{2\beta}, \tag{4.12}
\end{align*}
$$

where $C_i = C_i(s^+, q, \Lambda, n) > 0$ for $i = 13, 14$, and

$$
\tilde{k} \geq \delta(|x_0| + 1)^{\frac{\beta}{q-1}} r^{\frac{q}{q-1}} T(w_0; x_0, \frac{r}{2}, r)^{\frac{1}{q-1}} + 1
$$

with $\delta \in (0, 1]$.

By applying (4.11) and (4.12) to (4.5), we obtain

$$
I_2 \le C_{15} 2^{2j(n+\mathbf{q}s^+ + \mathbf{q})} \delta^{1-\mathbf{q}} \int_{B_j} w_j^{\mathbf{q}} \, d\mathbf{V}_{2\beta} \quad (4.13)
$$

where $C_{15} := C_{15}(s^{+}, p^{-}, \mathbf{q}, \beta, \Lambda, n) > 1$, and $\delta \in (0, 1]$.

We now proceed to complete the proof of the theorem.

Inserting (4.2), (4.4), and (4.13) into (4.3), we have

$$
\frac{1}{\tilde{k}} \left( \int_{B_{j+1}} |w_{j+1}|^{\mathbf{q}} \, d\mathbf{V}_{2\beta} \right)^{\frac{1}{\mathbf{q}}} \le C_{16} 2^{j \frac{2(n+\mathbf{q}s^+ + \mathbf{q}-1)}{\mathbf{q}}} \kappa \delta^{\frac{1-\mathbf{q}}{\mathbf{q}}} \kappa \frac{1}{\tilde{k}^{\kappa}} \left( \int_{B_j} w_j^{\mathbf{q}} \, d\mathbf{V}_{2\beta} \right)^{\frac{\kappa}{\mathbf{q}}},
$$

where $C_{16} = C_{16}(s^{+}, p^{-}, \mathbf{q}, \beta, \Lambda, n) > 1$.

Set

$$
Y_j := \left( \int_{B_j} w_j^q \, d\mathbf{V}_{2\beta} \right)^{\frac{1}{q}}
$$

and

$$
\tilde{k} = \delta(|x_0| + 1)^{\frac{\beta}{q-1}} r^{\frac{\mathbf{q}}{q-1}} T(w_0; x_0, \frac{r}{2}, r)^{\frac{1}{q-1}} + c_0^{\frac{1}{\gamma}} b^{\frac{1}{\gamma^2}} \left( \int_{B_r(x_0)} w_0^q \, d\mathbf{V}_{2\beta} \right)^{\frac{1}{q}} + 1,
$$

where

$$
c_0 = C_{16} \delta^{\frac{1-q}{q}} \kappa, \quad b = 2^{\left[ \frac{2(n+qs^++q-1)}{q} + \frac{\kappa-1}{\kappa} \right]} \kappa \quad \text{and} \quad \gamma = \kappa - 1.
$$

Hence,

$$
\frac{Y_{j+1}}{\tilde{k}} \leq c_0 b^j \left(\frac{Y_j}{\tilde{k}}\right)^{\kappa}.
$$

Moreover, from the definition of $\tilde{k}$, we have

$$
\frac{Y_0}{\tilde{k}} \leq c_0^{-\frac{1}{\gamma}} b^{-\frac{1}{\gamma^2}}.
$$

Thus, by Lemma 4.1, we obtain $Y_j \to 0$ as $j \to \infty$, which implies

$$
\operatorname{ess\,sup}_{B_{r/2}(x_0)} u \leq k + \tilde{k},
$$

which gives (4.1) by choosing $k = 0$.

□
5. LOCAL HÖLDER CONTINUITY

The following local Hölder continuity result (as stated in the introduction) for weak solutions of (1.1) follows
from Lemma 5.1 below.

**Theorem 1.1.** (Local Hölder continuity). Let $\Omega$ be a bounded domain. Assume that conditions $(H_1)$ and $(H_2)$ are satisfied. Also, assume that $\beta \ge 0$, $n - 2\beta - 1 + s^+ > 0$, $n - \beta > s^+\mathbf{q}$, and $(1 - s^+)(p^- - 1) \ge \frac{q}{p^-}(\mathbf{q} - p^-)$.

Let $u \in W_{\text{loc}}^{1,q}(\Omega; |x|^{-2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; |x|^{-\beta})$ be a weak solution of (1.1), satisfying $T(u; x_0, r, \rho) < \infty$ for all balls $B_r(x_0) \subset B_\rho(x_0) \subset \Omega$. Then $u$ is locally Hölder continuous in $\Omega$.

Moreover, if $B_R(x_0) \subset \Omega$ with $R \in (0, 1/2]$ is a ball such that either $0 \notin \overline{B_{3R}(x_0)}$ or $x_0 = 0$, then there exist constants $\alpha \in (0, \frac{1-s^+}{q-1})$ and $c > 0$, depending on $\text{diam}(\Omega)$, $R$, $s^-$, $s^+$, $p^-$, $\mathbf{q}$, $\Lambda$, $\beta$, and $n$, such that

$$
\operatorname{osc}_{B_{\rho}(x_0)} u := \ess \sup_{B_{\rho}(x_0)} u - \ess \inf_{B_{\rho}(x_0)} u \le c \left(\frac{\rho}{r}\right)^{\alpha} \left[ r^{\frac{q}{q-1}} T\left(u; x_0, \frac{r}{2}, r\right)^{\frac{1}{q-1}} + \left(\int_{B_r(x_0)} |u|^q \, dV_{2\beta}\right)^{\frac{1}{q}} + 1 \right],
$$

where $B_{2r}(x_0) \subset B_R(x_0)$, $r \in (0, 1/4]$, and $\rho \in (0, r/2]$.

If $\beta = 0$, the constants $\alpha$ and $c$ depend only on $s^-$, $s^+$, $p^-$, $\mathbf{q}$, $\Lambda$, and $n$. In the case $x_0 = 0$, the constants $\alpha$ and $c$ depend only on $s^-$, $s^+$, $p^-$, $\mathbf{q}$, $\Lambda$, $\beta$, and $n$.

We prove the next result by arguing similarly to the proof of [58, Theorem 5.1] and [41, Lemma 5.1].

Choosing $\delta(|x_0| + 1)^{\frac{\beta}{q-1}} = 1$ in Theorem 1.2, we obtain

$$
\ess \sup_{B_{r/2}(x_0)} |u| \le r^{\frac{q}{q-1}} T \left( u; x_0, \frac{r}{2}, r \right)^{\frac{1}{q-1}} + C \left( \int_{B_r(x_0)} |u|^q \, dV_{2\beta} \right)^{\frac{1}{q}} + 1, \quad (5.1)
$$

and

$$
\operatorname{osc}_{B_{r/2}(x_0)} u \le 2r^{\frac{q}{q-1}} T \left( u; x_0, \frac{r}{2}, r \right)^{\frac{1}{q-1}} + 2C \left( \int_{B_r(x_0)} |u|^q \, dV_{2\beta} \right)^{\frac{1}{q}} + 2, \quad (5.2)
$$

where $C = C(s^+, p^-, \mathbf{q}, \beta, \Lambda, n) > 0$.

**Lemma 5.1.** Let $\Omega$ be a bounded domain. Assume that conditions $(H_1)$ and $(H_2)$ are satisfied. Also assume that $\beta \ge 0$, $n - 2\beta - 1 + s^+ > 0$, $n - \beta > s^+\mathbf{q}$, and $(1 - s^+)(p^- - 1) \ge \frac{q}{p^-}(\mathbf{q} - p^-)$.

Let $u \in W_{\text{loc}}^{1,q}(\Omega; |x|^{-2\beta}) \cap W^{s(x,y),p(x,y)}(\Omega; |x|^{-\beta})$ be a weak solution of (1.1) satisfying $T(u; x_0, r, \rho) < \infty$ for all balls $B_r(x_0) \subset B_\rho(x_0) \subset \Omega$. Suppose that $B_R(x_0) \subset \Omega$ is a ball such that either $0 \notin \overline{B_{3R}(x_0)}$ or $x_0 = 0$.

Then there exist constants $\eta \in (0, 1/4]$ and $\alpha \in (0, \frac{1-s^+}{q-1})$, depending only on $\text{diam}(\Omega)$, $R$, $s^-$, $s^+$, $p^-$, $\mathbf{q}$, $\Lambda$, $\beta$, and $n$, such that the following holds:

Assume that $0 < 2r \le R \le 1/2$. Define

$$
r_j := \eta^j \frac{r}{2} \quad \text{and} \quad B_j := B_{r_j}(x_0), \quad j = 0, 1, 2, \dots
$$

Set

$$
\frac{1}{2}\omega(r_0) := r^{\frac{q}{q-1}} T \left( u; x_0, \frac{r}{2}, r \right)^{\frac{1}{q-1}} + C \left( \int_{B_r(x_0)} |u|^q \, dV_{2\beta} \right)^{\frac{1}{q}} + 1, \quad (5.3)
$$

where $C$ is the constant from (5.2).

For each $j = 1, 2, \dots$, let

$$
\omega(r_j) := \left(\frac{r_j}{r_0}\right)^{\alpha} \omega(r_0). \qquad (5.4)
$$

Then, for all $j = 0, 1, 2, \dots$,

$$
\operatorname{osc}_{B_j} u \le \omega(r_j). \tag{5.5}
$$

If $\beta = 0$, the constants $\eta$ and $\alpha$ depend only on $s^-$, $s^+$, $p^-$, $\mathbf{q}$, $\Lambda$, and $n$. In the case $x_0 = 0$, the constants $\eta$ and $\alpha$ depend only on $s^-$, $s^+$, $p^-$, $\mathbf{q}$, $\Lambda$, $\beta$, and $n$.

*Proof.* By Lemmas 2.12 and 2.13, $u_+$ and $(-u)_+$ are weak subsolutions of (1.1).

From (5.2), it follows that (5.5) holds true for $j = 0$. Suppose (5.5) holds for all $i = 0, \dots, j$ for some $j \in \{0, 1, 2, \dots\}$. To prove (5.5) for $j + 1$, we proceed by induction.
We observe that either

$$ \frac{\mathbf{V}_{2\beta}\left(B_{2r_{j+1}}(x_0) \cap \left\{u \ge \operatorname{ess\,inf}_{B_j} u + \frac{\omega(r_j)}{2}\right\}\right)}{\mathbf{V}_{2\beta}(B_{2r_{j+1}}(x_0))} \ge \frac{1}{2}, \qquad (5.6) $$

or

$$ \frac{\mathbf{V}_{2\beta}\left(B_{2r_{j+1}}(x_0) \cap \left\{u \le \operatorname{ess\,inf}_{B_j} u + \frac{\omega(r_j)}{2}\right\}\right)}{\mathbf{V}_{2\beta}(B_{2r_{j+1}}(x_0))} \ge \frac{1}{2} \qquad (5.7) $$

holds. Define

$$ u_j := \begin{cases} u - \operatorname{ess\,inf}_{B_j} u & \text{if (5.6) holds,} \\ \omega(r_j) - \left(u - \operatorname{ess\,inf}_{B_j} u\right) & \text{if (5.7) holds.} \end{cases} \qquad (5.8) $$

Then $u_j$ is a weak solution of (1.1). Also, by the induction hypothesis in both cases, $u_j \ge 0$ in $B_j$, and

$$ \operatorname{ess\,sup}_{B_j} |u_j| \le 2\omega(r_i), \quad \forall i \in \{0, \dots, j\}. \qquad (5.9) $$

Furthermore, from (5.1), we obtain

$$ \begin{aligned} |u_j| &\le |u| + \omega(r_j) + \operatorname{ess\,sup}_{B_j} |u| \\ &\le |u| + \omega(r_j) + \frac{1}{2}\omega(r_0) \quad \text{in } \Omega. \end{aligned} \qquad (5.10) $$

We also have

$$ \frac{\mathbf{V}_{2\beta}\left(B_{2r_{j+1}}(x_0) \cap \left\{u_j \ge \frac{\omega(r_j)}{2}\right\}\right)}{\mathbf{V}_{2\beta}(B_{2r_{j+1}}(x_0))} \ge \frac{1}{2}. \qquad (5.11) $$

Set

$$ \epsilon := \eta^{\frac{q}{q-1}-\alpha}. $$

Then the following estimate holds:

$$ \begin{aligned} [\epsilon\omega(r_j)]^{p^-} r_{j+1}^{(1-s^+)q} &= \left[\eta^{\frac{q}{q-1}-\alpha} \left(\frac{r_j}{r_0}\right)^{\alpha} \omega(r_0)\right]^{p^-} r_{j+1}^{(1-s^+)q} \\ &\le \eta^{j[-\alpha(\mathbf{q}-p^-)+(1-s^+)\mathbf{q}]} \eta^{-\frac{(\mathbf{q}-p^-)\mathbf{q}}{q-1}+\alpha(\mathbf{q}-p^-)+(1-s^+)\mathbf{q}} \left(\frac{r}{2}\right)^{(1-s^+)\mathbf{q}} \quad (5.12) \\ &\le 2^{-(1-s^+)\mathbf{q}} \le 1, \end{aligned} $$

provided that $(1-s^+) (\mathbf{q}-1) \ge \mathbf{q}-p^-$ and $(1-s^+) \mathbf{q} \ge \alpha (\mathbf{q}-p^-)$.

In the case where $\frac{(1-s^+)\mathbf{q}}{q-1} \ge \alpha$, we observe that

$$ \begin{aligned} [\epsilon\omega(r_j)]^{1-q} \left(\frac{r_{j+1}}{r_j}\right)^q r_j^{(1-s^+)q} &= \left[\eta^{\frac{q}{q-1}-\alpha} \left(\frac{r_j}{r_0}\right)^{\alpha} \omega(r_0)\right]^{1-q} \left(\frac{r_{j+1}}{r_j}\right)^q r_j^{(1-s^+)q} \quad (5.13) \\ &\le \eta^{j[-\alpha(\mathbf{q}-1)+(1-s^+)\mathbf{q}]} \eta^{-\mathbf{q}+\alpha(\mathbf{q}-1)+\mathbf{q}} \le 1. \end{aligned} $$

If $(1-s^+) (\mathbf{q}-1) \ge \frac{\mathbf{q}}{p^-} (\mathbf{q}-p^-)$ and $(1-s^+) p^- \ge \alpha (\mathbf{q}-p^-)$, then we also have:

$$ \begin{aligned} [\epsilon\omega(r_j)]^{p^-} r_{j+1}^{(1-s^+)p^-} &= \left[\eta^{\frac{q}{q-1}-\alpha} \left(\frac{r_j}{r_0}\right)^{\alpha} \omega(r_0)\right]^{p^-} r_{j+1}^{(1-s^+)p^-} \\ &\le \eta^{j[-\alpha(\mathbf{q}-p^-)+(1-s^+)p^-]} \eta^{-\frac{(\mathbf{q}-p^-)\mathbf{q}}{q-1}+\alpha(\mathbf{q}-p^-)+(1-s^+)p^-} \left(\frac{r}{2}\right)^{(1-s^+)p^-} \quad (5.14) \\ &\le 2^{-(1-s^+)\mathbf{q}} \le 1. \end{aligned} $$

Step 1: Let $\epsilon = \eta^{\frac{q}{q-1}-\alpha}$ with

$$ \frac{1-s^+}{\mathbf{q}-1} \ge \alpha > 0 \quad \text{and} \quad (1-s^+) (p^- - 1) \ge \frac{\mathbf{q}}{p^-} (\mathbf{q}-p^-). $$
We claim that

$$ \frac{\nu_{2\beta} \left( B_{2r_{j+1}}(x_0) \cap \{u_j \le 2\epsilon\omega(r_j)\} \right)}{\nu_{2\beta} \left( B_{2r_{j+1}}(x_0) \right)} \le \frac{\hat{C}}{\ln \frac{1}{\eta}}, \qquad (5.15) $$

for some constant $\hat{C} = \hat{C}(\text{diam}(\Omega), R, s^-, s^+, p^-, \mathbf{q}, \Lambda, n) > 0$. Furthermore, if $\beta = 0$, the constant $\hat{C}$ depends only on $s^-, s^+, p^-, \mathbf{q}, \Lambda$, and $n$. In the case $x_0 = 0$, the constant $\hat{C}$ depends only on $s^-, s^+, p^-, \mathbf{q}, \Lambda, \beta$, and $n$.

We set

$$ \lambda := \ln \left( \frac{\frac{\omega(r_j)}{2} + \epsilon\omega(r_j)}{3\epsilon\omega(r_j)} \right) = \ln \left( \frac{\frac{1}{2} + \epsilon}{3\epsilon} \right) \ge \frac{1}{2} \ln \frac{1}{\epsilon} \qquad (5.16) $$

and define

$$ \Theta := \min \left\{ \left( \ln \left( \frac{\omega(r_j)}{2} + \epsilon\omega(r_j)} \right) \right)_+, \lambda \right\}. $$

By (5.11), we have

$$ \begin{aligned} \lambda &= \frac{1}{\nu_{2\beta} \left( B_{2r_{j+1}}(x_0) \cap \left\{ u_j \ge \frac{\omega(r_j)}{2} \right\} \right)} \int_{B_{2r_{j+1}}(x_0) \cap \left\{ u_j \ge \frac{\omega(r_j)}{2} \right\}} \lambda \, d\nu_{2\beta} \\ &= \frac{1}{\nu_{2\beta} \left( B_{2r_{j+1}}(x_0) \cap \left\{ u_j \ge \frac{\omega(r_j)}{2} \right\} \right)} \int_{B_{2r_{j+1}}(x_0) \cap \{\Theta=0\}} \lambda \, d\nu_{2\beta} \qquad (5.17) \\ &\le \frac{2}{\nu_{2\beta} \left( B_{2r_{j+1}}(x_0) \right)} \int_{B_{2r_{j+1}}(x_0)} (\lambda - \Theta) \, d\nu_{2\beta} \\ &= 2 \left( \lambda - (\Theta)_{B_{2r_{j+1}}(x_0), \nu_{2\beta}} \right), \end{aligned} $$

where $(\Theta)_{B_{2r_{j+1}}(x_0), \nu_{2\beta}} = \int_{B_{2r_{j+1}}(x_0)} \Theta \, d\nu_{2\beta}$.

Integrating (5.17) over the set $\{B_{2r_{j+1}}(x_0) \cap \Theta = \lambda\}$, we get

$$ \frac{\nu_{2\beta} \left( B_{2r_{j+1}}(x_0) \cap \{\Theta = \lambda\} \right)}{\nu_{2\beta} \left( B_{2r_{j+1}}(x_0) \right)} \lambda \le \frac{2}{\nu_{2\beta} \left( B_{2r_{j+1}}(x_0) \right)} \int_{B_{2r_{j+1}}(x_0)} \left| \Theta - (\Theta)_{B_{2r_{j+1}}(x_0), \nu_{2\beta}} \right| d\nu_{2\beta}. \qquad (5.18) $$

Applying Corollary 3.4 with $r = 2r_{j+1}$, $R = r_j$, $a = \frac{\omega(r_j)}{2}$, $d = \epsilon\omega(r_j)$, and $b = e^\lambda$, and using (5.12), we obtain $d^{p^-} \mathbf{q} r_{j+1}^{(1-s^+) \mathbf{q}} \le 1$, and

$$ \begin{aligned} & \int_{B_{2r_{j+1}}(x_0)} \left| \Theta - (\Theta)_{B_{2r_{j+1}}(x_0), \nu_{2\beta}} \right|^q d\nu_{2\beta} \\ & \le C_1 \left[ (\epsilon\omega(r_j))^{1-q} \left( \frac{r_{j+1}}{r_j} \right)^q r_j^{qT} (u_j; x_0, r_j, 3r_{j+1}) + 1 \right], \end{aligned} \qquad (5.19) $$

provided $0 \notin \overline{R_{3R}(x_0)}$, where $C_1 = C_1(\text{diam}(\Omega), R, s^-, s^+, p^-, \mathbf{q}, \Lambda, n) > 0$.

In the case $x_0 = 0$, we obtain

$$ \begin{aligned} & \int_{B_{2r_{j+1}}(x_0)} \left| \Theta - (\Theta)_{B_{2r_{j+1}}(x_0), \nu_{2\beta}} \right|^q d\nu_{2\beta} \\ & \le C_2 \left[ (\epsilon\omega(r_j))^{1-q} \left( \frac{r_{j+1}}{r_j} \right)^q r_j^{q+\beta} T(u_j; x_0, r_j, 3r_{j+1}) + 1 \right], \end{aligned} \qquad (5.20) $$

where $C_2 = C_2(s^-, s^+, p^-, \mathbf{q}, \Lambda, n) > 0$

Following the lines of the estimate (5.6) in [41, pp. 1294 - 1295], we claim:

$$ r_j^q T(u_j; x_0, r_j, \ell) \le C_3 \left( \eta^{-\alpha(\mathbf{q}-1)} \omega(r_j)^{\mathbf{q}-1} + r_j^{(1-s^+)\mathbf{q}} \right) \quad \text{if} \quad 0 \notin \overline{B_{3R}(x_0)}, \qquad (5.21) $$

$$ r_j^{q+\beta} T(u_j; x_0, r_j, \ell) \le C_4 \left( \eta^{-\alpha(\mathbf{q}-1)} \omega(r_j)^{\mathbf{q}-1} + r_j^{(1-s^+)\mathbf{q}} \right) \quad \text{if} \quad x_0 = 0, \qquad (5.22) $$

with $3r_{j+1} \le \ell \le r/2$, where $C_3 = C_3(\text{diam}(\Omega), R, s^-, s^+, p^-, \mathbf{q}, \Lambda, n) > 0$ and $C_4 = C_4(s^-, s^+, p^-, \mathbf{q}, \Lambda, n) > 0$.
Now we prove statements (5.21) and (5.22), which appear below between (5.23) and (5.31):

$$
\begin{equation}
\begin{aligned}
r_j^q T(u_j; x_0, r_j, \ell) \le r_j^q & \sum_{i=1}^j \int_{B_{i-1} \setminus B_i} \frac{|u_j(x)|^{q-1} + 1}{|x-x_0|^{n+s^+ q} |x|^{\beta}} \frac{1}{|x|^{\beta}} \, dx \\
& + \operatorname*{ess\,sup}_{y \in B_{\ell}(x_0)} r_j^q \int_{\Omega \setminus B_0} \frac{|u_j(x)|^{p(x,y)-1}}{|x-x_0|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^{\beta}} \, dx.
\end{aligned}
\tag{5.23}
\end{equation}
$$

If $0 \notin \overline{B_{3R}(x_0)}$, then $2R < |x|$ in $B_R(x_0)$. By (5.9), we have

$$
\begin{equation}
\begin{aligned}
r_j^q \int_{B_{i-1} \setminus B_i} \frac{|u_j(x)|^{q-1} + 1}{|x-x_0|^{n+s^+ q} |x|^{\beta}} \frac{1}{|x|^{\beta}} \, dx
&\le \frac{|\mathbb{S}^{n-1}| 2^{q-1}}{s^+ q (2R)^{\beta}} \omega(r_{i-1})^{q-1} \left(\frac{r_j}{r_i}\right)^q r_i^q r_i^{-s^+ q} \\
&\quad + \frac{|\mathbb{S}^{n-1}|}{s^+ q (2R)^{\beta}} r_j^q \eta^{-s^+ q} r_{i-1}^{-s^+ q},
\end{aligned}
\tag{5.24}
\end{equation}
$$

$$
\text{due to } r_i^{-s^+ q - \beta} - r_{i-1}^{-s^+ q - \beta} = \frac{1 - \eta^{s^+ q + \beta}}{\eta^{s^+ q + \beta}} r_{i-1}^{-s^+ q - \beta}.
$$

Furthermore, from (5.10) and (5.3), we have

$$
\begin{align}
& r_j^q \int_{\Omega \setminus B_0} \frac{|u_j(x)|^{p(x,y)-1}}{|x-x_0|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^{\beta}} \, dx \nonumber \\
&\le 3^{q-1} r_j^q \int_{\Omega \setminus B_0} \frac{|u|^{p(x,y)-1}}{|x-x_0|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^{\beta}} + \frac{\omega(r_j)^{p(x,y)-1} + (\omega(r_0)/2)^{p(x,y)-1}}{|x-x_0|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^{\beta}} \, dx \tag{5.25} \\
&\le 3^{q-1} r_j^q \left( r^{-\frac{q}{q-1}} \frac{1}{2} \omega(r_0) \right)^{q-1} + 3^{q-1} r_j^q \left[ \omega(r_j)^{q-1} + \left( \frac{1}{2} \omega(r_0) \right)^{q-1} + 2 \right] \nonumber \\
&\quad \cdot \int_{\mathbb{R}^n \setminus B_0} \left( \frac{1}{|x-x_0|^{n+s^- p^-}} + \frac{1}{|x-x_0|^{n+s^+ q}} \right) \frac{1}{|x|^{\beta}} \, dx \quad \text{if } y \in B_{\ell}(x_0). \nonumber
\end{align}
$$

By employing the same argument as in (3.17), we obtain, for (5.25),

$$
\begin{equation}
\begin{aligned}
r_j^q \int_{\Omega \setminus B_0} \frac{|u_j(x)|^{p(x,y)-1}}{|x-x_0|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^{\beta}} \, dx
&\le C_5 \left(\frac{r_j}{r_1}\right)^q r_1^q r_0^{-q} \omega(r_0)^{q-1} \\
&\quad + C_5 \left(\frac{r_j}{r_1}\right)^q r_1^q r_0^{-s^+q} \left[\omega(r_0)^{q-1} + 1\right]
\end{aligned}
\tag{5.26}
\end{equation}
$$

$$
\text{where } C_5 = C_5(\operatorname{diam}(\Omega), R, s^{-}, s^{+}, p^{-}, \mathbf{q}, \beta, n) > 0.
$$

Hence, from (5.23), (5.24), and (5.26), we have

$$
r_j^q T(u_j; x_0, r_j, \ell) \le C_6 \sum_{i=1}^j \left[ \left( \frac{r_j}{r_i} \right)^q \omega(r_{i-1})^{q-1} + r_j^q \eta^{-s^+ q} r_{i-1}^{-s^+ q} \right] \quad \text{if } 0 \notin \overline{B_{3R}(x_0)}, \quad (5.27)
$$

$$
\text{where } C_6 = C_6(\operatorname{diam}(\Omega), R, s^{-}, s^{+}, p^{-}, \mathbf{q}, \beta, n) > 0.
$$

On the other hand, following a similar argument to the one used to obtain (5.27), we get

$$
r_j^{q+\beta} T(u_j; x_0, r_j, \ell) \le C_7 \sum_{i=1}^j \left[ \left( \frac{r_j}{r_i} \right)^q \omega(r_{i-1})^{q-1} + r_j^{q+\beta} \eta^{-s^+ q - \beta} r_{i-1}^{-s^+ q - \beta} \right] \quad \text{if } x_0 = 0, \quad (5.28)
$$

$$
\text{where } C_7 = C_7(s^{-}, s^{+}, p^{-}, \mathbf{q}, \beta, n) > 0.
$$

We now proceed to estimate the right-hand side of (5.28):

$$
\begin{align}
\sum_{i=1}^{j} r_j^{q+\beta} r_{i-1}^{-s^+ q - \beta} &= r_j^{q+\beta} r_0^{-(s^+ q + \beta)} \frac{\eta^{-j(s^+ q + \beta)} - 1}{\eta^{-(s^+ q + \beta)} - 1} \nonumber \\
&= r_j^{q+\beta} r_{j-1}^{-(s^+ q + \beta)} \frac{1 - \eta^{j(s^+ q + \beta)}}{1 - \eta^{s^+ q + \beta}} \tag{5.29} \\
&\le \frac{4^{s^+ q + \beta}}{(\ln 4)(s^+ q + \beta)} \eta^{s^+ q + \beta} r_j^{(1-s^+)q}. \nonumber
\end{align}
$$

Here, we use the inequality $(1 - \eta^t)\sigma^t \ge \sigma^t - 1 \ge (\ln \sigma)t$, which holds if $0 < \eta \le 1/\sigma \le 1$ and $t \ge 0$.
We also have

$$
\begin{align*}
& \sum_{i=1}^{j} \left(\frac{r_j}{r_i}\right)^q \omega(r_{i-1})^{q-1} \\
&= \omega(r_0)^{q-1} \left(\frac{r_j}{r_0}\right)^{\alpha(q-1)} \sum_{i=1}^{j} \left(\frac{r_{i-1}}{r_i}\right)^{\alpha(q-1)} \left(\frac{r_j}{r_i}\right)^{q-\alpha(q-1)} \\
&= \omega(r_j)^{q-1} \eta^{-\alpha(q-1)} \sum_{i=0}^{j-1} \eta^i [\mathbf{q} - \alpha(\mathbf{q}-1)] \tag{5.30} \\
&\leq \omega(r_j)^{q-1} \frac{\eta^{-\alpha(q-1)}}{1 - \eta^{q-\alpha(q-1)}} \\
&\leq \frac{4^{q-\alpha(q-1)}}{(\ln 4)[\mathbf{q} - \alpha(\mathbf{q}-1)]} \eta^{-\alpha(q-1)} \omega(r_j)^{q-1} \\
&\leq \frac{4^q}{(\ln 4)(\mathbf{q}-1)} \eta^{-\alpha(q-1)} \omega(r_j)^{q-1}.
\end{align*}
$$

From (5.28), (5.29), and (5.30), we conclude the proof of (5.22), namely,

$$
r_j^{q+\beta} T(u_j; x_0, r_j, \ell) \le C_4 (\eta^{-\alpha(q-1)} \omega(r_j)^{q-1} + r_j^{(1-s^+)q}) \quad \text{if } x_0 = 0. \quad (5.31)
$$

Similarly, we prove (5.21).

We now proceed to complete Step 1:

By (5.19) and (5.21), if $0 \notin \overline{B_{3R}(x_0)}$, then

$$
\begin{align}
& \int_{B_{2r_{j+1}}(x_0)} \left| \Theta - (\Theta)_{B_{2r_{j+1}}(x_0), \nu_{2\beta}} \right|^q \mathrm{d}\mathbf{V}_{2\beta} \nonumber \\
& \le C_1 C_3 [\epsilon \omega(r_j)]^{1-q} \left( \frac{r_{j+1}}{r_j} \right)^q \left( \eta^{-\alpha(\mathbf{q}-1)} \omega(r_j)^{\mathbf{q}-1} + r_j^{(1-s^+)\mathbf{q}} \right) \tag{5.32} \\
& \le 2C_1 C_3, \nonumber
\end{align}
$$

due to (5.13).

Similarly, from (5.20) and (5.22), we obtain

$$
\begin{align}
& \int_{B_{2r_{j+1}}(x_0)} \left| \Theta - (\Theta)_{B_{2r_{j+1}}(x_0), \nu_{2\beta}} \right|^q \mathrm{d}\mathbf{V}_{2\beta} \nonumber \\
& \le C_2 C_4 [\epsilon \omega(r_j)]^{1-q} \left( \frac{r_{j+1}}{r_j} \right)^q \left( \eta^{-\alpha(\mathbf{q}-1)} \omega(r_j)^{\mathbf{q}-1} + r_j^{(1-s^+)\mathbf{q}} \right) \tag{5.33} \\
& \le 2C_2 C_4 \quad \text{if } x_0 = 0. \nonumber
\end{align}
$$

Using these two inequalities, together with (5.16), (5.18), (5.19), and (5.20), we obtain (5.15).

Step 2: We now use an iteration argument to obtain (5.5) for $i = j + 1$. To this end, for every $i = 0, 1, 2, \dots$, let

$$
\rho_i := (1 + 2^{-i}) r_{j+1}, \quad \hat{\rho}_i := \frac{\rho_i + \rho_{i+1}}{2}, \quad B^i := B_{\rho_i}(x_0), \quad \hat{B}^i := B_{\rho_i}(x_0).
$$

Recalling that $\epsilon = \eta^{\frac{q}{q-1}-\alpha}$, define $k_i := (1 + 2^{-i})\epsilon\omega(r_j)$ and

$$
A^i := B^i \cap \{u_j \le k_i\}, \quad i = 0, 1, 2, \dots
$$

Let $w_i := (k_i - u_j)_+$. Consider a sequence $(\psi_i)_{i=0}^\infty \subset C_c^\infty(\hat{B}^i)$ such that $0 \le \psi_i \le 1$ in $\hat{B}^i$, $\psi_i = 1$ in $B^{i+1}$, and
$|\nabla \psi_i| \le \frac{C_8 2^i}{\rho_i} \le \frac{C_8 2^i}{r_{j+1}}$ in $\hat{B}^i$, with $C_8 = C_8(n) > 1$.
Applying Corollary 2.10, we obtain

$$
\begin{align}
(k_i - k_{i+1})^q \left( \frac{\nu_{2\beta} (A^{i+1})}{\nu_{2\beta} (B^{i+1})} \right)^{\frac{1}{\kappa}} &\le \left( \int_{B^{i+1}} w_i^{\kappa q} \, d\nu_{2\beta} \right)^{\frac{1}{\kappa}} \le C_9 \left( \int_{B^i} w_i^{\kappa q} \psi_i^{\kappa q} \, d\nu_{2\beta} \right)^{\frac{1}{\kappa}} \tag{5.34} \\
&\le C_{10} \rho_i^q \int_{B^i} |\nabla (w_i \psi_i)|^q \, d\nu_{2\beta} \nonumber \\
&\le 2^{2q-1} C_{10} (I+J), \nonumber
\end{align}
$$

where $C_h = C_h(\beta, \mathbf{q}, n) > 0$ for $h = 9, 10$, and

$$
I := r_{j+1}^{\mathbf{q}} \int_{B^i} w_i^{\mathbf{q}} |\nabla \psi_i|^{\mathbf{q}} \, d\nu_{2\beta} \quad \text{and} \quad J := r_{j+1}^{\mathbf{q}} \int_{B^i} |\nabla w_i|^{\mathbf{q}} \psi_i^{\mathbf{q}} \, d\nu_{2\beta}.
$$

Estimate of $I$: Since $u_j \ge 0$ in $B_j$, we have

$$
w_i \le k_i \le 2\epsilon\omega(r_j) \quad \text{in } B_j, \tag{5.35}
$$

and $B^i \subset B_j$. Thus,

$$
I = r_{j+1}^{\mathbf{q}} \int_{B^i} w_i^{\mathbf{q}} |\nabla \psi_i|^q d\nu_{2\beta} \le 2^q C_8^q [\epsilon \omega(r_j)]^q 2^{iq} \frac{\nu_{2\beta}(A^i)}{\nu_{2\beta}(B^i)}. \quad (5.36)
$$

Estimate of $J$: By Lemma 3.2, there exists a constant $C_{11} = C_{11}(p^{-}, \mathbf{q}, \Lambda) > 0$ such that

$$
r_{j+1}^{\mathbf{q}} \int_{B^i} |\nabla w_i|^{\mathbf{q}} \psi_i^{\mathbf{q}} \, d\nu_{2\beta} \le C_{11} (J_1 + J_2 + J_3) \quad (5.37)
$$

where

$$
J_1 := r_{j+1}^{\mathbf{q}} \int_{B^i} w_i^{\mathbf{q}} |\nabla \psi_i|^{\mathbf{q}} \, d\nu_{2\beta},
$$
$$
J_2 := r_{j+1}^{\mathbf{q}} \int_{B^i} \int_{B^i} \max\{w_i(x), w_i(y)\}^{p(x,y)} \frac{|\psi_i(x) - \psi_i(y)|^{p(x,y)}}{|x-y|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^{\beta}|y|^{\beta}} \, dxdy
$$

and

$$
J_3 := \sup_{x \in \hat{B}^i} r_{j+1}^{\mathbf{q}} \int_{\Omega \setminus B^i} \frac{w_i(y)^{p(x,y)-1}}{|x-y|^{n+s(x,y)p(x,y)}} \frac{1}{|y|^{\beta}} \, dy \cdot \int_{B^i} |x|^{\beta} w_i \psi_i^{\mathbf{q}} \, d\nu_{2\beta}(x).
$$

Here, note that $|x| \le \rho_i + |x_0| \le 2r_{j+1} + |x_0|$ in $B^i$.

From (5.36), we get

$$
J_1 \le C_{12}(\mathbf{q}, n) [\epsilon\omega(r_j)]^q 2^{iq} \nu_{2\beta}(A^i). \quad (5.38)
$$

Next, we estimate $J_2$. Assume $0 \notin \bar{B}_{3R}(x_0)$. Using (5.35) and the fact that $|\psi_i(x) - \psi_i(y)| \le (\sup_{\hat{B}^i} |\nabla \psi_i|)|x-y|$ in $B^i$, we obtain

$$
\begin{align}
J_2 &\le r_{j+1}^{\mathbf{q}} \int_{B^i} \int_{B^i} (w_i(x) + w_i(y))^{p(x,y)} \frac{|\psi_i(x) - \psi_i(y)|^{p(x,y)}}{|x-y|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^{\beta}|y|^{\beta}} \, dxdy \nonumber \\
&\le \frac{2^{q+1} C_8^{q} 2^{iq}}{(2R)^{2\beta}} \int_{B^i \cap \{u_j \le k_i\}} \int_{B^i} \frac{[2\epsilon\omega(r_j)]^{p^-} + [2\epsilon\omega(r_j)]^q}{|x-y|^{n-(1-s^+)p^-}} \, dxdy \nonumber \\
&\le C_{13} 2^{iq} \left\{ r_{j+1}^{(1-s^+)p^-} [2\epsilon\omega(r_j)]^{p^-} + [2\epsilon\omega(r_j)]^q \right\} \nu_{2\beta}(B^i \cap \{u_j \le k_i\}), \tag{5.39}
\end{align}
$$

since $B^i \subset B_{2\rho_i}(y)$ for all $y \in B^i$. Here, $C_{13} = C_{13}(R, s^+, p^-, \mathbf{q}, \beta, n) > 0$. In the case $x_0 = 0$, we obtain a similar estimate as in (5.39), using the argument from (4.8)

To estimate $J_3$, note that

$$
\operatorname{ess\,sup}_{x \in \hat{B}^i} r_{j+1}^{\mathbf{q}} \int_{\Omega \setminus B^i} \frac{w_i(y)^{p(x,y)-1}}{|x-y|^{n+s(x,y)p(x,y)}} \frac{1}{|y|^{\beta}} \, dy \le 2^{(i+4)(n+s^+\mathbf{q})} r_{j+1}^{\mathbf{q}} T(w_i; x_0, r_{j+1}, 3r_{j+1})
$$

which follows from the inequality

$$
\begin{align*}
\frac{1}{|y-x|} &= \frac{1}{|y-x_0|} \frac{|y-x_0|}{|y-x|} \le \frac{1}{|y-x_0|} \left( 1 + \frac{|x-x_0|}{|y-x|} \right) \\
&\le \frac{1}{|y-x_0|} \left( 1 + \frac{\hat{\rho}_i}{\rho_i - \hat{\rho}_i} \right) \le \frac{2^{i+4}}{|y-x_0|},
\end{align*}
$$
where $x \in \hat{B}^i$ and $y \in \mathbb{R}^n \setminus B^i$, and from the facts that $B_{r_{j+1}}(x_0) = B_{j+1} \subset B^i$ and $\hat{B}^i \subset B_{3r_{j+1}}(x_0)$.
Recalling (5.22) and (5.35), together with the fact that

$$B_{3r_{j+1}}(x_0) \subset B_{r_j}(x_0) \quad \text{and} \quad w_i \le |u_j| + 2\epsilon\omega(r_j) \text{ in } \Omega,$$

we further obtain, if $x_0 = 0$:

$$
\begin{align*}
& r_{j+1}^{\mathbf{q}} T(w_i; x_0, r_{j+1}, 3r_{j+1}) \\
&\le \underset{y \in B_{r_j}(x_0)}{\text{ess sup}} r_{j+1}^{\mathbf{q}} \int_{B_j \setminus B_{j+1}} \frac{w_i(x)^{p(x,y)-1}}{|x-x_0|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^{\beta}} \mathrm{d}x \\
&\quad + \underset{y \in B_{r_j}(x_0)}{\text{ess sup}} 2^{\mathbf{q}-1} r_{j+1}^{\mathbf{q}} \int_{\Omega \setminus B_j} \frac{[2\epsilon\omega(r_j)]^{p(x,y)-1}}{|x-x_0|^{n+s(x,y)p(x,y)}} \frac{1}{|x|^{\beta}} \mathrm{d}x + 2^{\mathbf{q}-1} r_{j+1}^{\mathbf{q}} T(u_i; x_0, r_j, 3r_{j+1}) \\
&\le C_{14}(s^{-}, s^{+}, p^{-}, \mathbf{q}, \beta, n) r_{j+1}^{\mathbf{q}} \{[\epsilon\omega(r_j)]^{\mathbf{q}-1} + [\epsilon\omega(r_j)]^{p^{-}-1}\} \\
&\quad \cdot \left[ r_{j+1}^{-s^{+}\mathbf{q}-\beta} + \int_{\Omega \setminus B_j} \left( \frac{1}{|x-x_0|^{n+s^{-}p^{-}}} + \frac{1}{|x-x_0|^{n+s^{+}\mathbf{q}}} \right) \frac{1}{|x|^{\beta}} \mathrm{d}x \right] \\
&\quad + C_{14} r_{j+1}^{\mathbf{q}} r_j^{-\mathbf{q}-\beta} \{\eta^{-\mathbf{q}} [\epsilon\omega(r_j)]^{\mathbf{q}-1} + r_j^{(1-s^{+})\mathbf{q}}\} \\
&\le C_{15}(s^{-}, s^{+}, p^{-}, \mathbf{q}, \beta, n) r_{j+1}^{-\beta} \{[\epsilon\omega(r_j)]^{\mathbf{q}-1} + [\epsilon\omega(r_j)]^{p^{-}-1} r_{j+1}^{(1-s^{+})\mathbf{q}} + \left(\frac{r_{j+1}}{r_j}\right)^{\mathbf{q}} r_j^{(1-s^{+})\mathbf{q}}\}.
\end{align*}
$$

Hence, employing also (5.14), if $x_0 = 0$, we have:

$$
\begin{align}
J_3 &\le C_{15} 2^{(i+4)(n+s^+\mathbf{q})+1} r_{j+1}^{-\beta} (2r_{j+1})^{\beta} \left\{ [\epsilon\omega(r_j)]^{\mathbf{q}-1} + \left(\frac{r_{j+1}}{r_j}\right)^{\mathbf{q}} r_j^{(1-s^+)\mathbf{q}} \right\} \int_{B^i} w_i \psi_i^{\mathbf{q}} \, d\mathbf{V}_{2\beta} \tag{5.40} \\
&\le C_{15} 2^{(i+4)(n+s^+\mathbf{q})+1} \left\{ [\epsilon\omega(r_j)]^{\mathbf{q}} + \left(\frac{r_{j+1}}{r_j}\right)^{\mathbf{q}} r_j^{(1-s^+)\mathbf{q}} [\epsilon\omega(r_j)] \right\} \mathbf{V}_{2\beta} (B^i \cap \{u_j \le k_i\}). \nonumber
\end{align}
$$

For the case $0 \notin B_{3R}(x_0)$, using the estimates in (3.17) and (5.21), we obtain a result similar to (5.40).

We now proceed to complete the induction argument, and thus conclude the proof of this lemma.

Recalling from (5.13) and (5.14) that

$$[\epsilon\omega(r_j)]^{1-q} \left(\frac{r_{j+1}}{r_j}\right)^q r_j^{(1-s^+)q} \le 1 \quad \text{and} \quad [\epsilon\omega(r_j)]^{p^-} r_{j+1}^{(1-s^+)p^-} \le 1,$$

then by (5.38), (5.39), and (5.40), we conclude that

$$J = r_{j+1}^{\mathbf{q}} \int_{B^i} |\nabla w_i|^{\mathbf{q}} \psi_i^{\mathbf{q}} \, d\mathbf{V}_{2\beta} \le C_{16} [\epsilon\omega(r_j)]^{\mathbf{q}} 2^{i(n+\mathbf{q})} \frac{\mathbf{V}_{2\beta}(A^i)}{\mathbf{V}_{2\beta}(B^i)}, \quad (5.41)$$

where $C_{16} = C_{16}(\text{diam}(\Omega), R, s^{-}, s^{+}, p^{-}, \mathbf{q}, \beta, \Lambda, n) > 0$.

Therefore, from (5.34), (5.36), and (5.41), we get

$$(k_i - k_{i+1})^{\mathbf{q}} \left( \frac{\mathbf{V}_{2\beta}(A^{i+1})}{\mathbf{V}_{2\beta}(B^{i+1})} \right)^{\frac{1}{\kappa}} \le C_{17} 2^{i(n+\mathbf{q})} [\epsilon\omega(r_j)]^{\mathbf{q}} \frac{\mathbf{V}_{2\beta}(A^i)}{\mathbf{V}_{2\beta}(B^i)} \quad (5.42)$$

where $C_{17} = C_{17}(\text{diam}(\Omega), R, s^{-}, s^{+}, p^{-}, \mathbf{q}, \beta, \Lambda, n) > 0$.

Let

$$Y_i = \frac{\mathbf{V}_{2\beta}(A^i)}{\mathbf{V}_{2\beta}(B^i)}, \quad i = 0, 1, 2, \dots$$

Noting that $k_i - k_{i+1} = 2^{-i-1}\epsilon\omega(r_j)$, we get

$$Y_{i+1} \le 2^{\mathbf{q}} C_{17} 2^{i(2\mathbf{q}+n)\kappa} Y_i^{\kappa}$$

By (5.15) in Steep 1, we have

$$Y_0 \le \frac{\hat{C}(\text{diam}(\Omega), R, s^{-}, s^{+}, p^{-}, \mathbf{q}, \beta, \Lambda, n)}{\ln \frac{1}{\eta}}.$$
Let

$$c_0 = 2^q C_{17}, \quad b = 2^{(2q+n)\kappa}, \quad \gamma = \kappa - 1 \quad \text{and} \quad \eta_1 = c_0^{-\frac{1}{\gamma}} b^{-\frac{1}{\gamma^2}}$$

By choosing $\eta = \frac{1}{2} \min\{\frac{1}{4}, e^{-\frac{\hat{c}}{\eta_1}}\}$, we ensure $Y_0 \le \eta_1$. Thus, by Lemma 4.1, we deduce that $\lim_{i \to \infty} Y_i = 0$, and therefore, $u_j \ge \epsilon\omega(r_j)$ in $B_{j+1}$. Using the definition of $u_j$ from (5.8), we obtain

$$\underset{B_{j+1}}{\operatorname{osc}} u \le (1 - \epsilon)\omega(r_j) = (1 - \epsilon)\eta^{-\alpha}\omega(r_{j+1}) \le \omega(r_{j+1}), \quad (5.43)$$

where we have chosen $\alpha \in (0, \frac{1-s^+}{q-1}]$ small enough such that

$$\eta^{\alpha} \ge 1 - \epsilon = 1 - \eta^{\frac{q}{q-1} - \alpha}.$$

Thus, inequality (5.43) proves the induction estimate (5.5) for $i = j + 1$. Hence, the result follows. $\square$

**Author Contributions:** All authors contributed equally to the writing, review, and editing of this work. All authors have read and approved the final version of the manuscript.

**Funding:** This work was supported by CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico, Grant 153232/2024-2.

**Data Availability:** No data were used in the research described in this article.

**Declarations**

**Conflict of Interest:** The authors declare that there is no conflict of interest.

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