Proof. Take $\varphi \left( \xi \right) \in C_0^\infty \left( {{B_R}\left( {{\xi _0}} \right)} \right)$ as a cut-off function such that $0 \le \varphi \le 1,\;\varphi \equiv 1$ in ${{B_r}\left( {{\xi _0}} \right)}$, ${\rm supp}\, \varphi \subset B_\frac{R+r}{2}( \xi _0)$ and $\left| {{\nabla _H}\varphi } \right| \le \frac{c}{{R - r}}$ in ${{B_R}\left( {{\xi _0}} \right)}$. Then $(v-(v)_r)\varphi \in HW_0^{s,p}(B_R)$ and further $(v-(v)_r)\varphi\in HW_0^{s,p}( {\mathbb{H}^n})$ by zero extension. We split ${\mathbb{H}^n} \times {\mathbb{H}^n}$ into

\begin{equation*}\left( {{B_R} \times {B_R}} \right) \cup \left( {{\mathbb{H}^n}\backslash {B_R} \times {B_R}} \right) \cup \left( {{B_R} \times {\mathbb{H}^n}\backslash {B_R}} \right) \cup \left( {{\mathbb{H}^n}\backslash {B_R} \times {\mathbb{H}^n}\backslash {B_R}} \right).\end{equation*}

By virtue of lemma 2.2 and the definition of φ, we get

\begin{align*} {\left( {\int_{{B_r}} {{{\left| {v - {{\left( v \right)}_r}} \right|}^{p_s^*}}\,d\xi } } \right)^{\frac{p}{{p_s^*}}}} &\le {\left( {\int_{{\mathbb{H}^n}} {{{\left| {\left( {v - {{\left( v \right)}_r}} \right)\varphi } \right|}^{p_s^*}}\,d\xi } } \right)^{\frac{p}{{p_s^*}}}} \\ &\le c\int_{{\mathbb{H}^n}} {\int_{{\mathbb{H}^n}} {\frac{{{{\left| {\left( {v\left( \xi \right) - {{\left( v \right)}_r}} \right)\varphi \left( \xi \right) - \left( {v\left( \eta \right) - {{\left( v \right)}_r}} \right)\varphi \left( \eta \right)} \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi } d\eta } \\ &\le c\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {\left( {v\left( \xi \right) - {{\left( v \right)}_r}} \right)\varphi \left( \xi \right) - \left( {v\left( \eta \right) - {{\left( v \right)}_r}} \right)\varphi \left( \eta \right)} \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}{\mkern 1mu} d\xi d\eta } } \\ & \quad+ c \int_{{B_R}} {\int_{{H^n}\backslash {B_R}} {\frac{{{{\left| {\left( {v\left( \eta \right) - {{\left( v \right)}_r}} \right)\varphi \left( \eta \right)} \right|}^p}}}{{\left\| {{\eta ^{- 1}} \circ \xi } \right\|_{{\mathbb{H}^n}}^{Q + sp}}}{\mkern 1mu} d\xi d\eta } } \\ &=: {J_1} + {J_2}. \end{align*}

Note that

\begin{align*} {J_1}& \le c\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {v\left( \xi \right) - v\left( \eta \right)} \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi } d\eta } + c\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {\varphi \left( \xi \right) - \varphi \left( \eta \right)} \right|}^p}{{\left| {\left( {v\left( \eta \right) - {{\left( v \right)}_r}} \right)} \right|}^p}}}{{\left\| {{\eta ^{- 1}} \circ \xi } \right\|_{{\mathbb{H}^n}}^{Q + sp}}}{\mkern 1mu} d\xi d\eta } } \\ & =:c\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {v\left( \xi \right) - v\left( \eta \right)} \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi } d\eta } +J_{11}. \end{align*}

We first evaluate J ₁₁ as

\begin{align*} {J_{11}} & \le \frac{c}{{{{\left( {R - r} \right)}^p}}}\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {{v\left( \eta \right) - {{\left( v \right)}_r}} } \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + p\left( {s - 1} \right)}}}\,d\xi d\eta } } \\ & \le \frac{c}{{{{\left( {R - r} \right)}^p}}}\int_{{B_R}} {{{\left| {{v\left( \eta \right) - {{\left( v \right)}_r}} } \right|}^p}\int_{{B_{2R}}\left( \eta \right)} {\frac{1}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + p\left( {s - 1} \right)}}}\,d\xi d\eta } } \\ & \le c{\left( {\frac{R}{{R - r}}} \right)^p}{R^{- sp}}\int_{{B_R}} {{{\left| {{v\left( \eta \right) - {{\left( v \right)}_r}} } \right|}^p}\,d\eta } \\ & \le c{\left( {\frac{R}{{R - r}}} \right)^p}{R^{- sp}}\left( {\int_{{B_R}} {{{\left| {{v\left( \eta \right) - {{\left( v \right)}_R}} } \right|}^p}\,d\eta } + \int_{{B_R}} {{{\left| {{{{\left( v \right)}_R} - {{\left( v \right)}_r}} } \right|}^p}\,d\eta } } \right)\\ &\le c{\left( {\frac{R}{{R - r}}} \right)^p}\!\!{R^{- sp}}\left( {{R^{sp}}\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {{v\left( \xi \right) - v\left( \eta \right)} } \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi d\eta } } + {{\left| {{{\left( v \right)}_R} - {{\left( v \right)}_r}} \right|}^p}\left| {{B_R}} \right|} \right), \end{align*}

where in the last line we have utilized lemma 2.1. On the other hand,

\begin{align*} {\left| {{{\left( v \right)}_R} - {{\left( v \right)}_r}} \right|^p}\left| {{B_R}} \right|& = \left| {{B_R}} \right|{\left| {-\!\!\!\!\!\!\int_{{B_r}} {\left( {v - {{\left( v \right)}_R}} \right)d\xi } } \right|^p} \\ &\le \left| {{B_R}} \right|-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {v - {{\left( v \right)}_R}} \right|}^p}d\xi } \\ & \le \frac{{\left| {{B_R}} \right|}}{{\left| {{B_r}} \right|}}\int_{{B_R}} {{{\left| {v - {{\left( v \right)}_R}} \right|}^p}d\xi }\\ & \le c{\left( {\frac{R}{r}} \right)^Q}{R^{sp}}\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {{v\left( \xi \right) - v\left( \eta \right)} } \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}d\xi d\eta } }. \end{align*}

Thus

\begin{align*} {J_1}& \le c\left( {1 + {{\left( {\frac{R}{{R - r}}} \right)}^p} + {{\left( {\frac{R}{r}} \right)}^Q}{{\left( {\frac{R}{{R - r}}} \right)}^p}} \right)\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {v\left( \xi \right) - v\left( \eta \right)} \right|}^p}}}{{\left\| {{\eta ^{- 1}} \circ \xi } \right\|_{{\mathbb{H}^n}}^{Q + sp}}}{\mkern 1mu} d\xi d\eta } } \\ & \le c{\left( {\frac{R}{r}} \right)^Q}{\left( {\frac{R}{{R - r}}} \right)^p}\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {v\left( \xi \right) - v\left( \eta \right)} \right|}^p}}}{{\left\| {{\eta ^{- 1}} \circ \xi } \right\|_{{{\mathbb{H}^n}^n}}^{Q + sp}}}{\mkern 1mu} d\xi d\eta } }. \end{align*}

Moreover, for $\xi \in {{\mathbb{H}^n}\backslash {B_R}}$, $\eta \in {B_{\frac{{R + r}}{2}}}$, owing to the triangle inequality [Reference Cygan11] there holds that

\begin{align*} {\| {{\xi ^{- 1}} \circ \xi_0 } \|_{{\mathbb{H}^n}}} & \le \left( {1 + \frac{{{{\| {{\eta ^{- 1}} \circ {\xi_0}} \|}_{{\mathbb{H}^n}}}}}{{{{\| {{\xi ^{- 1}} \circ \eta } \|}_{{\mathbb{H}^n}}}}}} \right){\| {{\xi ^{- 1}} \circ \eta } \|_{{\mathbb{H}^n}}} \\ & \le \left(1 + \frac{(R+r)/2}{(R-r)/2} \right){\| {{\xi^{- 1}} \circ \eta } \|_{{\mathbb{H}^n}}} = \frac{{2R}}{{R - r}}{\| {{\xi ^{- 1}} \circ \eta } \|_{{\mathbb{H}^n}}}. \end{align*}

From this, it follows that

\begin{align*} {J_{2}}& \le c \int_{{B_{\frac{{R + r}}{2}}}} {\int_{{\mathbb{H}^n}\backslash {B_R}} {\frac{{{{\left| {{v \left( \eta \right)- {{\left( v \right)}_r}} } \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi d\eta } }\\ & \le c{\left( {\frac{R}{{R - r}}} \right)^{Q + sp}}\int_{{\mathbb{H}^n}\backslash {B_R}} {\frac{1}{{\| {{\xi ^{- 1}} \circ {\xi _0}} \|_{{\mathbb{H}^n}}^{Q + sp}}}d\xi \int_{{B_{\frac{{R + r}}{2}}}} {{{\left| {v\left( \eta \right) - {{\left( v \right)}_r}} \right|}^p}\,d\eta } } \\ & \le c\frac{{{R^Q}}}{{{{\left( {R - r} \right)}^{Q + sp}}}}\int_{{B_R}} {{{\left| {v\left( \eta \right) - {{\left( v \right)}_r}} \right|}^p}\,d\eta } \\ & \le c\frac{{{R^Q}}}{{{{\left( {R - r} \right)}^{Q + sp}}}}\left(R^{sp} + \frac{R^{Q + sp}}{r^Q}\right) \int_{B_R}\int_{B_R} \frac{|v( \xi) - v( \eta)|^p}{\|\eta ^{- 1} \circ \xi \|_{\mathbb{H}^n}^{Q + sp}}\,d\xi d\eta\\ & \le c{\left( {\frac{R}{r}} \right)^Q}{\left( {\frac{R}{{R - r}}} \right)^{Q+sp}}\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {v\left( \xi \right) - v\left( \eta \right)} \right|}^p}}}{{\left\| {{\eta ^{- 1}} \circ \xi } \right\|_{{{\mathbb{H}^n}^n}}^{Q + sp}}}{\mkern 1mu} d\xi d\eta } }, \end{align*}

the procedure of which is analogous to J ₁. Eventually, we obtain

\begin{align*} &\quad {\left( {\int_{{B_r}} {{{\left| {v - {{\left( v \right)}_r}} \right|}^{p_s^*}}\,d\xi } } \right)^{\frac{p}{{p_s^*}}}}\\ & \le c{\left( {\frac{R}{r}} \right)^Q}\left[ {\left( {\frac{R}{{R - r}}} \right)^{p}}+{\left( {\frac{R}{{R - r}}} \right)^{Q+sp}} \right]\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {v\left( \xi \right) - v\left( \eta \right)} \right|}^p}}}{{\left\| {{\eta ^{- 1}} \circ \xi } \right\|_{{{\mathbb{H}^n}^n}}^{Q + sp}}}{\mkern 1mu} d\xi d\eta }}, \end{align*}

which implies the statement.

Proof. For p < q, we first utilize the Hölder inequality to get

\begin{align*} &\quad \int_\Omega {\int_\Omega {\frac{{{{\left| {{v\left( \xi \right) - v\left( \eta \right)}} \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi d\eta } }\\ & = \int_\Omega {\int_\Omega {\frac{{{{\left| {{v\left( \xi \right) - v\left( \eta \right)} } \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{\left( {Q + tq} \right)\frac{p}{q}}}}\frac{1}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q\frac{{q - p}}{q} + (s - t)p}}}\,d\xi d\eta } } \\ & \le {\left( {\int_\Omega {\int_\Omega {\frac{{{{\left| {{v\left( \xi \right) - v\left( \eta \right)} } \right|}^q}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\,d\xi d\eta } } } \right)^{\frac{p}{q}}}{\left( {\int_\Omega {\int_\Omega {\frac{1}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + \frac{{(s - t)pq}}{{q - p}}}}}\,d\xi d\eta } } } \right)^{\frac{{q - p}}{q}}}. \end{align*}

On the other hand,

\begin{align*} \int_\Omega {\int_\Omega {\frac{1}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + \frac{{(s - t)pq}}{{q - p}}}}}\,d\xi d\eta } } & \le \int_\Omega {\int_{{B_d}\left( \eta \right)} {\frac{1}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{{\mathbb{H}}^n}}^{Q + \frac{{(s - t)pq}}{{q - p}}}}}\,d\xi d\eta } }\\ &\le Q\left| {{B_1}} \right|\int_\Omega {\int_0^d {{\rho ^{\frac{{(t - s)pq}}{{q - p}} - 1}}\,d\rho d\eta } } \nonumber\\ &= \frac{{Q\left| {{B_1}} \right|\left( {q - p} \right)}}{{(t - s)pq}}{d^{\frac{{(t - s)pq}}{{q - p}}}}\left| \Omega \right|, \end{align*}

with $d: =\mathrm{diam}\left( \Omega \right)$. The combination of preceding inequalities implies the desired display.

If q = p, noting ${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \le \mathrm{diam}\left( \Omega \right)$ for $\xi,\eta \in \Omega$ and s < t, we can readily obtain

\begin{align*} {\left( {\int_\Omega {\int_\Omega {\frac{{{{\left| {{v\left( \xi \right) - v\left( \eta \right)} } \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi d\eta } } } \right)^{\frac{1}{p}}} &= {\left( {\int_\Omega {\int_\Omega {\frac{{{{\left| {{v\left( \xi \right) - v\left( \eta \right)} } \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tp}}}\frac{1}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{(s - t)p}}}\,d\xi d\eta } } } \right)^{\frac{1}{p}}}\\ & \le {\left( {\mathrm{diam}\left( \Omega \right)} \right)^{t - s}}{\left( {\int_\Omega {\int_\Omega {\frac{{{{\left| {{v\left( \xi \right) - v\left( \eta \right)} } \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tp}}}\,d\xi d\eta } } } \right)^{\frac{1}{p}}}. \end{align*}

Now, we complete the proof.

Proof. By the Hölder inequality and proposition 2.4, we obtain

\begin{align*} -\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{f}{{{r^s}}}} \right|}^p}\,d\xi } &\le c-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{{f - {{\left( f \right)}_r}}}{{{r^s}}}} \right|}^p}\chi_{\{f\neq0\}}\,d\xi } + c{\left| {\frac{{{{\left( f \right)}_r}}}{{{r^s}}}} \right|^p} \\ &\le c{\left( {\frac{{\left| {{\rm{supp}}\,f} \right|}}{{\left| {{B_r}} \right|}}} \right)^{\frac{{sp}}{Q}}}{\left( {-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{{f - {{\left( f \right)}_r}}}{{{r^s}}}} \right|}^{p_s^*}}\,d\xi } } \right)^{\frac{p}{{p_s^*}}}} + c{\left| {\frac{{{{\left( f \right)}_r}}}{{{r^s}}}} \right|^p}\\ &\le c\frac{{{D_1}(R,r)}}{{{r^{sp}}}}{\left( {\frac{{\left| {{\rm{supp}}\,f} \right|}}{{\left| {{B_r}} \right|}}} \right)^{\frac{{sp}}{Q}}}-\!\!\!\!\!\!\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {f\left( \xi \right) - f\left( \eta \right)} \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi } d\eta } \\ &\quad+ c{\left( {\frac{{\left| {{\rm{supp}}\,f} \right|}}{{\left| {{B_r}} \right|}}} \right)^{p - 1}}-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{f}{{{r^s}}}} \right|}^p}\,d\xi }, \end{align*}

where we used the inequality below,

\begin{equation*}{\left| {\frac{{{{\left( f \right)}_r}}}{{{r^s}}}} \right|^p} = {r^{- sp}}{\left| {-\!\!\!\!\!\!\int_{{B_r}} {f{\chi _{\left\{{f \ne 0} \right\}}}\,d\xi } } \right|^p} \le {\left( {\frac{{\left| {{\rm{supp}}\;f} \right|}}{{\left| {{B_r}} \right|}}} \right)^{p - 1}}-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{f}{{{r^s}}}} \right|}^p}\,d\xi } .\end{equation*}

On the other hand, via the Hölder inequality and proposition 2.4 again,

\begin{align*} -\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{f}{{{r^t}}}} \right|}^q}d\xi } &\le c{\left( {-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{{f - {{\left( f \right)}_r}}}{{{r^t}}}} \right|}^{p_s^*}}d\xi } } \right)^{\frac{q}{{p_s^*}}}} + c{\left| {\frac{{{{\left( f \right)}_r}}}{{{r^t}}}} \right|^q} \\ &\le c\frac{{D_1^{\frac{q}{p}}(R,r)}}{{{r^{tq}}}}{\left( {-\!\!\!\!\!\!\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {f\left( \xi \right) - f\left( \eta \right)} \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi } d\eta } } \right)^{\frac{q}{p}}}+ c{\left| {\frac{{{{\left( f \right)}_r}}}{{{r^t}}}} \right|^q}\\ &\le c\frac{{D_1^{\frac{q}{p}}(R,r)}}{{{r^{tq}}}}{\left( {-\!\!\!\!\!\!\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {f\left( \xi \right) - f\left( \eta \right)} \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi } d\eta } } \right)^{\frac{q}{p}}}\\ &\quad+ c{\left( {\frac{{\left| {{\rm{supp}}\,f} \right|}}{{\left| {{B_r}} \right|}}} \right)^{p - 1}}-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{f}{{{r^t}}}} \right|}^q}\,d\xi}, \end{align*}

where we can see that

\begin{equation*}{\left| {\frac{{{{\left( f \right)}_r}}}{{{r^t}}}} \right|^q} \le {\left( {\frac{{\left| {{\rm{supp}}\,f} \right|}}{{\left| {{B_r}} \right|}}} \right)^{q - 1}}-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{f}{{{r^t}}}} \right|}^q}\,d\xi } \le {\left( {\frac{{\left| {{\rm{supp}}\,f} \right|}}{{\left| {{B_r}} \right|}}} \right)^{p - 1}}-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{f}{{{r^t}}}} \right|}^q}\,d\xi }. \end{equation*}

We finally observe the plain relation that

\begin{equation*} -\!\!\!\!\!\!\int_{{B_r}}\left|\frac{f}{r^s}\right|^p+a_0\left|\frac{f}{r^t}\right|^q\,d\xi\le c\left(\frac{R}{r}\right)^Q-\!\!\!\!\!\!\int_{{B_R}}\left|\frac{f}{r^s}\right|^p+a_0\left|\frac{f}{r^t}\right|^q\,d\xi. \end{equation*}

In summary, we combine all the previous inequalities to arrive at the desired display.

Proof. In view of Hölder continuity of a, we have

\begin{equation*}a_{\bar R}^ + \le a_{\bar R}^ - + 4{\left[ a \right]_\alpha }{{\bar R}^\alpha } \le a_{\bar R}^ - + 8{\left[ a \right]_\alpha }{r^\alpha }.\end{equation*}

Then we by employing $tq \le sp+\alpha$, $r \le 1$ have

\begin{equation*}a_{\bar R}^ + {\left| {\frac{f}{{{r^t}}}} \right|^q} \le a_{\bar R}^ - {\left| {\frac{f}{{{r^t}}}} \right|^q} + c{r^{\alpha - tq + sp}}{\left| f \right|^{q - p}}{\left| {\frac{f}{{{r^s}}}} \right|^p}.\end{equation*}

Thus

\begin{align*} &\quad{\left[ {-\!\!\!\!\!\!\int_{{B_r}} {{{\left( {{{\left| {\frac{f}{{{r^s}}}} \right|}^p} + a_{\bar R}^ + {{\left| {\frac{f}{{{r^t}}}} \right|}^q}} \right)}^\gamma }\,d\xi } } \right]^{\frac{1}{\gamma }}}\\ &\le c\left( {1 + \| f \|_{{L^\infty }\left( {{B_r}} \right)}^{q - p}} \right){\left[ {-\!\!\!\!\!\!\int_{{B_r}} {{{\left( {{{\left| {\frac{f}{{{r^s}}}} \right|}^p} + a_{\bar R}^ - {{\left| {\frac{f}{{{r^t}}}} \right|}^q}} \right)}^\gamma }\,d\xi } } \right]^{\frac{1}{\gamma }}}\\ &\le c\left( {1 + \| f \|_{{L^\infty }\left( {{B_r}} \right)}^{q - p}} \right){\left[ {-\!\!\!\!\!\!\int_{{B_r}} {{{\left( {{{\left| {\frac{{f - {{\left( f \right)}_r}}}{{{r^s}}}} \right|}^p} + a_{\bar R}^ - {{\left| {\frac{{f - {{\left( f \right)}_r}}}{{{r^t}}}} \right|}^q}} \right)}^\gamma }\,d\xi } } \right]^{\frac{1}{\gamma }}}\\ &\quad + c\left( {1 + \| f \|_{{L^\infty }\left( {{B_r}} \right)}^{q - p}} \right)\left( {{{\left| {\frac{{{{\left( f \right)}_r}}}{{{r^s}}}} \right|}^p} + a_{\bar R}^ - {{\left| {\frac{{{{\left( f \right)}_r}}}{{{r^t}}}} \right|}^q}} \right). \end{align*}

Observe that

\begin{equation*}{\left| {\frac{{{{\left( f \right)}_r}}}{{{r^s}}}} \right|^p} + a_{\bar R}^ - {\left| {\frac{{{{\left( f \right)}_r}}}{{{r^t}}}} \right|^q} \le -\!\!\!\!\!\!\int_{{B_r}} {{{\left( {{{\left| {\frac{f}{{{r^s}}}} \right|}^p} + a_{\bar R}^ - {{\left| {\frac{f}{{{r^t}}}} \right|}^q}} \right)} }\,d\xi } .\end{equation*}

Moreover, it follows from proposition 2.4 that

\begin{align*} {\left[ {-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{{f - {{\left( f \right)}_r}}}{{{r^s}}}} \right|}^{p\gamma }}\,d\xi } } \right]^{\frac{1}{\gamma }}} &\le {\left( {-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{{f - {{\left( f \right)}_r}}}{{{r^s}}}} \right|}^{p_s^*}}\,d\xi } } \right)^{\frac{p}{{p_s^*}}}} \\ &\le \frac{{{cD_1}\left( {R,r} \right)}}{{{r^{sp}}}}-\!\!\!\!\!\!\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {f\left( \xi \right) - f\left( \eta \right)} \right|}^p}}}{{\| {{\eta ^{- 1}}\circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi d\eta } }, \end{align*}

and

\begin{align*} {\left[ {-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{{f - {{\left( f \right)}_r}}}{{{r^t}}}} \right|}^{q\gamma }}d\xi } } \right]^{\frac{1}{\gamma }}} &\le {\left( {-\!\!\!\!\!\!\int_{{B_r}} {{{\left| {\frac{{f - {{\left( f \right)}_r}}}{{{r^t}}}} \right|}^{q_t^*}}\,d\xi } } \right)^{\frac{q}{{q_t^*}}}} \\ &\le \frac{{{c{\widetilde D}_1}\left( {R,r} \right)}}{{{r^{tq}}}}-\!\!\!\!\!\!\int_{{B_R}} {\int_{{B_R}} {\frac{{{{\left| {f\left( \xi \right) - f\left( \eta \right)} \right|}^q}}}{{\| {{\eta ^{- 1}}\circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\,d\xi d\eta } }. \end{align*}

Merging the last four inequalities leads to

\begin{align*} &\quad {\left[ {-\!\!\!\!\!\!\int_{{B_r}} {{{\left( {{{\left| {\frac{f}{{{r^s}}}} \right|}^p} + a_{\bar R}^ + {{\left| {\frac{f}{{{r^t}}}} \right|}^q}} \right)}^\gamma }\,d\xi } } \right]^{\frac{1}{\gamma }}} \\ &\le c\left( {1 + \| f \|_{{L^\infty }\left( {{B_r}} \right)}^{q - p}} \right)\left( {\frac{{{D_1}\left( {R,r} \right)}}{{{r^{sp}}}} + \frac{{{{\widetilde D}_1}\left( {R,r} \right)}}{{{r^{tq}}}}} \right)\\ &\qquad\cdot-\!\!\!\!\!\!\int_{{B_R}} {\int_{{B_R}} {\left( {\frac{{{{\left| {f\left( \xi \right) - f\left( \eta \right)} \right|}^p}}}{{\| {{\eta ^{- 1}}\circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}} + a_{\bar R}^ - \frac{{{{\left| {f\left( \xi \right) - f\left( \eta \right)} \right|}^q}}}{{\| {{\eta ^{- 1}}\circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}} \right)\,d\xi d\eta } } \\ &\quad + c\left( {1 + \| f \|_{{L^\infty }\left( {{B_r}} \right)}^{q - p}} \right)-\!\!\!\!\!\!\int_{{B_R}} {\left( {{{\left| {\frac{f}{{{r^s}}}} \right|}^p} + a_{\bar R}^ - {{\left| {\frac{f}{{{r^t}}}} \right|}^q}} \right)\,d\xi } \\ & \le c\left( {1 + \| f \|_{{L^\infty }\left( {{B_r}} \right)}^{q - p}} \right)\left( {\frac{{{D_1}\left( {R,r} \right)}}{{{r^{sp}}}} + \frac{{{{\widetilde D}_1}\left( {R,r} \right)}}{{{r^{tq}}}}} \right)-\!\!\!\!\!\!\int_{{B_R}} {\int_{{B_R}} {\frac{{H\left( {\xi ,\eta ,\left| {f\left( \xi \right) - f\left( \eta \right)} \right|} \right)}}{{\| {{\eta ^{- 1}}\circ \xi } \|_{{\mathbb{H}^n}}^Q}}\,d\xi d\eta } } \\ &\quad + c\left( {1 + \| f \|_{{L^\infty }\left( {{B_r}} \right)}^{q - p}} \right)-\!\!\!\!\!\!\int_{{B_R}} {\left( {{{\left| {\frac{f}{{{r^s}}}} \right|}^p} + a_{\bar R}^ - {{\left| {\frac{f}{{{r^t}}}} \right|}^q}} \right)\,d\xi } . \end{align*}

We now finish the proof.

Proof. By $v \in {\mathcal A}(\Omega)$, proposition 2.4 and (1.8), we get $v \in {L^q}\left( {{B_{3r/2}}} \right)$ in (i). Thus, we just consider condition (ii). By the definition of $\rho \left( {v \varphi ;\mathbb{H}^n} \right)$, we have

(3.1)\begin{align} \rho \left( {v \varphi ;{\mathbb{H}^n}} \right) &= 2\int_{{\mathbb{H}^n}\backslash {B_{3r/2}}} {\int_{{B_{3r/2}}} {H\left( {\xi ,\eta ,\left| {v \left( \xi \right)\varphi \left( \xi \right) - v \left( \eta \right)\varphi \left( \eta \right)} \right|} \right)\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} }\nonumber \\ &\quad + \int_{{B_{3r/2}}} {\int_{{B_{3r/2}}} {H\left( {\xi ,\eta ,\left| {v \left( \xi \right)\varphi \left( \xi \right) - v \left( \eta \right)\varphi \left( \eta \right)} \right|} \right)\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} } \nonumber \\ &=: 2{I_1} + {I_2}. \end{align}

Owing to $\varphi \in HW_0^{1,\infty }\left( {{B_r}} \right)$, we find

(3.2)\begin{align} {I_1} &\le {\left( {{{\| \varphi \|}_{{L^\infty }\left( {{B_r}} \right)}} + 1} \right)^q}\int_{{\mathbb{H}^n}\backslash {B_{3r/2}}} {\int_{{B_r}} {\left( {\frac{{{{\left| {v \left( \xi \right)} \right|}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}} + {{\| a \|}_{{L^\infty }}}\frac{{{{\left| {v \left( \xi \right)} \right|}^q}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}} \right)\;d\xi d\eta } } \nonumber\\ &\le c{\left( {{{\| \varphi \|}_{{L^\infty }\left( {{B_r}} \right)}} + 1} \right)^q}\left( {{r^{- sp}}\int_{{B_r}} {{{\left| {v \left( \xi \right)} \right|}^p}\;d\xi } + {{\| a \|}_{{L^\infty }}}{r^{- tq}}\int_{{B_r}} {{{\left| {v \left( \xi \right)} \right|}^q}\;d\xi } } \right) \lt \infty . \end{align}

The term I ₂ is estimated as

(3.3)\begin{align} {I_2} &\le c\int_{B_{3r/2}} {\int_{B_{3r/2}} {H\left( {\xi ,\eta ,\left| {\left( {v \left( \xi \right) - v \left( \eta \right)} \right)\varphi \left( \eta \right)} \right|} \right)\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} }\nonumber \\ &\quad + c\int_{B_{3r/2}} {\int_{B_{3r/2}} {H\left( {\xi ,\eta ,\left| {v \left( \xi \right)\left( {\varphi \left( \xi \right) - \varphi \left( \eta \right)} \right)} \right|} \right)\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} } \nonumber\\ &\le c{\left( {{{\| \varphi \|}_{{L^\infty }\left( {{B_r}} \right)}} + 1} \right)^q}\int_{B_{3r/2}} {\int_{B_{3r/2}} {H\left( {\xi ,\eta ,\left| {v \left( \xi \right)} \right|} \right)\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} } \nonumber\\ &\quad + c\| {{\nabla _H}\varphi } \|_{{L^\infty }\left( {{B_r}} \right)}^p\int_{B_{3r/2}} {{{\left| {v \left( \xi \right)} \right|}^p}\int_{B_{3r}} {\frac{{d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + (s - 1)p}}}} } \;d\xi \nonumber\\ &\quad + c\| {{\nabla _H}\varphi } \|_{{L^\infty }\left( {{B_r}} \right)}^q{\| a \|_{{L^\infty }}}\int_{B_{3r/2}} {{{\left| {v \left( \xi \right)} \right|}^q}\int_{B_{3r}} {\frac{{d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + (t - 1)p}}}} } \;d\xi \nonumber\\ &\le c{\left( {{{\| \varphi \|}_{{L^\infty }\left( {{B_r}} \right)}} + 1} \right)^q}\rho \left( {v ;{B_{2r}}} \right) + c\| {{\nabla _H}\varphi } \|_{{L^\infty }\left( {{B_r}} \right)}^p{r^{\left( {1 - s} \right)p}}\int_{{B_{2r}}} {{{\left| {v \left( \xi \right)} \right|}^p}} \;d\xi \nonumber\\ &\quad + c\| {{\nabla _H}\varphi } \|_{{L^\infty }\left( {{B_r}} \right)}^q{\| a \|_{{L^\infty }}}{r^{\left( {1 - t} \right)q}}\int_{{B_{2r}}} {{{\left| {v \left( \xi \right)} \right|}^q}} \;d\xi \nonumber\\ & \lt \infty . \end{align}

Thus, it follows $\rho \left( {v \varphi ;{\mathbb{H}^n}} \right) \lt \infty$ by combining (3.2), (3.3) with (3.1).

Proof. We just consider the estimate for $w_+$, since the estimate for $w_-$ can be proved similarly. By lemma 3.1, it follows that ${w_+}{\phi ^q} \in {\mathcal{A}}(\Omega )$ from $u \in {\mathcal{A}}(\Omega )$ and $\phi \in C_0^\infty \left( {{B_r}} \right) \subset HW_0^{1,\infty} \left( {{B_r}} \right)$, so we can take the testing function $\varphi={w_+}{\phi ^q}$ in (1.7). Then we have

(3.6)\begin{align} 0 &= \int_{{B_r}} {\int_{{B_r}} {\Bigg[\frac{{{J _p}( {u( \xi) - u( \eta)})( {{w_ + }( \xi ){\phi^q}( \xi) - {w_ + }( \eta ){\phi^q}( \eta)})}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}} } \nonumber\\ &\quad + a(\xi ,\eta )\frac{{{J _q}( {u( \xi) - u( \eta )} )( {{w_ + }( \xi ){\phi^q}( \xi ) - {w_ + }( \eta){\phi^q}( \eta)})}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\Bigg]\,d\xi d\eta \nonumber\\ &\quad +2{\int _{{\mathbb{H}^n}\backslash {B_r}}}\int_{{B_r}} \Bigg[\frac{{{J _p}( {u( \xi ) - u( \eta)}){w_ + }( \xi){\phi^q}( \xi)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\nonumber\\ &\quad + a( \xi ,\eta)\frac{{{J _q}( {u( \xi) - u( \eta)}){w_ + }( \xi){\phi^q}( \xi)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\Bigg]\,d\xi d\eta \nonumber\\ &=: J_1+J_2. \end{align}

We first estimate J ₁. Since J ₁ is symmetry for ξ and η, we may suppose without loss of generality that ${u\left( \xi \right) \ge u\left( \eta \right)}$. Then for $l \in \{p,q\}$, it yields

\begin{align*} &\quad {J _l}\left( {u\left( \xi \right) - u\left( \eta \right)} \right)\left( {{w_+ }\left( \xi \right){\phi^q}\left( \xi \right) - {w_+ }\left( \eta \right){\phi^q}\left( \eta \right)} \right) \\ & = \left\{\begin{array}{l} {\left( {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right)^{l - 1}}\left( {{w_+ }\left( \xi \right){\phi^q}\left( \xi \right) - {w_+ }\left( \eta \right){\phi^q}\left( \eta \right)} \right),\quad \text{if } u\left( \xi \right) \ge u\left( \eta \right) \ge k\\ {\left( {u\left( \xi \right) - u\left( \eta \right)} \right)^{l - 1}}{w_+ }\left( \xi \right){\phi^q}\left( \xi \right),\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \;\text{if } u\left( \xi \right) \ge k \ge u\left( \eta \right)\\ 0,\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\;\;\;\quad\text{if } k \ge u\left( \xi \right) \ge u\left( \eta \right) \end{array} \right.\\ &\ge {J _l}\left( {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right)\left( {{w_+ }\left( \xi \right){\phi^q}\left( \xi \right) - {w_+ }\left( \eta \right){\phi^q}\left( \eta \right)} \right). \end{align*}

Moreover,

\begin{align*} &\quad {w_+ }\left( \xi \right){\phi^q}\left( \xi \right) - {w_+ }\left( \eta \right){\phi^q}\left( \eta \right) \\ & = \frac{{{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)}}{2}\left( {{\phi^q}\left( \xi \right) + {\phi^q}\left( \eta \right)} \right) + \frac{{{w_+ }\left( \xi \right) + {w_+ }\left( \eta \right)}}{2}\left( {{\phi^q}\left( \xi \right) - {\phi^q}\left( \eta \right)} \right), \end{align*}

which implies

\begin{align*} &\quad {J _l}\left( {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right)\left( {{w_+ }\left( \xi \right){\phi^q}\left( \xi \right) - {w_+ }\left( \eta \right){\phi^q}\left( \eta \right)} \right) \\ &\ge {| {{w_+ }( \xi) - {w_+ }( \eta)}|^l}\frac{{{\phi^q}( \xi) + {\phi^q}( \eta)}}{2} - {| {{w_+ }( \xi) - {w_+ }( \eta)}|^{l - 1}}\frac{{{w_+ }( \xi) + {w_+ }( \eta)}}{2}| {{\phi^q}( \xi) - {\phi^q}( \eta)}|. \end{align*}

Since

\begin{align*} \left| {{\phi^q}\left( \xi \right) - {\phi^q}\left( \eta \right)} \right| &\le q\left( {{\phi^{q - 1}}\left( \xi \right) + {\phi^{q - 1}}\left( \eta \right)} \right)\left| {\phi\left( \xi \right) - \phi\left( \eta \right)} \right| \\ & \le c\left( q \right){\left( {{\phi^q}\left( \xi \right) +{\phi^q}\left( \eta \right)} \right)^{\frac{{q - 1}}{q}}}\left| {\phi \left( \xi \right) - \phi\left( \eta \right)} \right|, \end{align*}

from lemma 3.2, we use Young’s inequality, $0\le \phi\le 1$ and ${\frac{{q - 1}}{q}} \gt 0$ to deduce that

\begin{align*} &\quad {\left| {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right|^{l - 1}}\frac{{{w_+ }\left( \xi \right) + {w_+ }\left( \eta \right)}}{2}\left| {{\phi^q}\left( \xi \right) - {\phi^q}\left( \eta \right)} \right| \\ &\le c\left( q \right){\left| {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right|^{l - 1}}\left( {{w_+ }\left( \xi \right) + {w_+ }\left( \eta \right)} \right){\left( {{\phi^q}\left( \xi \right) + {\phi^q}\left( \eta \right)} \right)^{\frac{{l - 1}}{l} + \frac{{q - l}}{{ql}}}}\left| {\phi\left( \xi \right) - \phi\left( \eta \right)} \right|\\ &\le \varepsilon {\left| {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right|^l}\left( {{\phi^q}\left( \xi \right) + {\phi^q}\left( \eta \right)} \right) \\ &\quad+ c\left( {\varepsilon ,q} \right){\left( {{\phi^q}\left( \xi \right) + {\phi^q}\left( \eta \right)} \right)^{\frac{{q - l}}{q}}}{\left| {\phi\left( \xi \right) -\phi\left( \eta \right)} \right|^l}{\left( {{w_+ }\left( \xi \right) + {w_+ }\left( \eta \right)} \right)^l}\\ & \le \varepsilon {\left| {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right|^l}\left( {{\phi^q}\left( \xi \right) + {\phi^q}\left( \eta \right)} \right) + c\left( {\varepsilon ,q} \right){\left| {\phi\left( \xi \right) -\phi\left( \eta \right)} \right|^l}{\left( {{w_+ }\left( \xi \right) + {w_+ }\left( \eta \right)} \right)^l}. \end{align*}

Then, by choosing ɛ small enough, we have

\begin{align*} & {J _l}\left( {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right)\left( {{w_+ }\left( \xi \right){\phi^q}\left( \xi \right) - {w_+ }\left( \eta \right){\phi^q}\left( \eta \right)} \right) \\ \ge & {\left| {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right|^l}\frac{{{\phi^q}\left( \xi \right) + {\phi^q}\left( \eta \right)}}{4} - c{\left| {\phi( \xi) -\phi\left( \eta \right)} \right|^l}{\left( {{w_+ }\left( \xi \right) + {w_+ }\left( \eta \right)} \right)^l}. \end{align*}

Thus, we get

(3.7)\begin{align} {J_1} \ge&\int_{{B_r}} {\int_{{B_r}} {\Bigg[ {\frac{{{{| {{w_ + }( \xi) - {w_ + }( \eta)}|}^p}(\phi^q(\xi) + \phi^q( \eta))/4 - c{{| {\phi( \xi) - \phi( \eta)}|}^p}{{( {{w_ + }( \xi ) + {w_ + }( \eta)})}^p}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}} } } \nonumber \\ &+ a( {\xi ,\eta })\frac{{{\left| {{w_ + }\left( \xi \right) - {w_ + }\left( \eta \right)} \right|}^q}(\phi^q(\xi) +\phi^q(\eta))/4 }{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\nonumber\\ & - \frac{c{{\left| {\phi( \xi) -\phi( \eta)} \right|}^q}{\left( {{w_ + }\left( \xi \right) + {w_ + }\left( \eta \right)} \right)}^q}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}} \Bigg]d\xi d\eta \nonumber \\ \ge& {\int_{{B_r}} {H\left( {\xi ,\eta ,\left| {{w_+ }\left( \xi \right) - {w_+ }\left( \eta \right)} \right|} \right)\left( {\phi^q\left( \xi \right) + {\phi^q}\left( \eta \right)} \right)\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} } \nonumber\\ &-c\int_{{B_r}} {\int_{{B_r}} {H\left( {\xi ,\eta ,\left| {\phi( \xi) -\phi( \eta)} \right|\left( {{w_+ }\left( \xi \right) + {w_+ }\left( \eta \right)} \right)} \right)\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} } . \end{align}

Now we estimate J ₂. Note that

(3.8)\begin{equation} {J _l}\left( {u\left( \xi \right) - u\left( \eta \right)} \right){w_+ }\left( \xi \right) \ge - w_+ ^{l - 1}\left( \eta \right){w_+ }\left( \xi \right). \end{equation}

In fact, when $u\left( \xi \right) \ge u\left( \eta \right)$, it easy to see that the inequality (3.8) holds. When $u\left( \xi \right) \lt u\left( \eta \right)$ and $u\left( \xi \right)\le k$, ${w_+ }\left( \xi \right)=0$, the inequality (3.8) also holds. When $k \lt u\left( \xi \right) \lt u\left( \eta \right)$,

\begin{align*} {J _l}( {u( \xi) - u( \eta)}){w_+ }( \xi) = - {| {{w_+ }( \xi) - {w_+ }( \eta)}|^{l - 1}}{w_+ }( \xi) \ge - w_+ ^{l - 1}( \eta){w_+ }( \xi). \end{align*}

Thus, we apply (3.8) and (3.4) to get

(3.9)\begin{align} J_2 &= 2{\int _{{\mathbb{H}^n}\backslash {B_r}}}\int_{{B_r}} \Bigg[\frac{{{J _p}( {u( \xi ) - u( \eta )}){w_ + }( \xi){\phi^q}( \xi )}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}+ a( {\xi ,\eta })\frac{{{J _q}( {u( \xi) - u( \eta )} ){w_ + }( \xi){\phi^q}( \xi)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\Bigg]\nonumber\\ & \quad \times\,d\xi d\eta \nonumber\\ &\ge - c{\int _{{\mathbb{H}^n}\backslash {B_r}}}\int_{{B_r}} {\left[ {\frac{{w_+ ^{p - 1}( \eta ){w_+ }( \xi){\phi^q}( \xi)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{sp}}} + a(\xi ,\eta)\frac{{w_+ ^{q - 1}( \eta){w_+ }( \xi){\phi^q}( \xi)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq}}}} \right]\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} \nonumber \\ &\ge - c\left( {\mathop {\sup }\limits_{\xi \in {\rm{supp}}\;\phi} \int_{{\mathbb{H}^n}\backslash {B_r}} {h( {\xi ,\eta ,{w_+ }( \eta )})\frac{d\eta }{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} } \right)\int_{{B_r}} {{w_\pm }( \xi){\phi^q}( \xi)\,d\xi }. \end{align}

Combining (3.6), (3.7) with (3.9), we get (3.5).

Proof. Let us give some notations as below,

\begin{equation*}{H_\rho }\left( {\xi ,\eta ,\tau } \right) = \frac{{{\tau ^p}}}{{{\rho ^{sp}}}} + a\left( {\xi ,\eta } \right)\frac{{{\tau ^q}}}{{{\rho ^{tq}}}},\;\;\;{h_\rho}\left( {\xi ,\eta ,\tau } \right) = \frac{{{\tau ^{p - 1}}}}{{{\rho^{sp}}}} + a\left( {\xi ,\eta } \right)\frac{{{\tau ^{q - 1}}}}{{{\rho^{tq}}}}\end{equation*}

and

\begin{equation*}{G_\rho }\left( \tau \right) = \frac{{{\tau ^p}}}{{{\rho ^{sp}}}} + a_\rho ^ + \frac{{{\tau ^q}}}{{{\rho ^{tq}}}},\;\;\;{g_\rho }\left( \tau \right) = \frac{{{\tau ^{p - 1}}}}{{{\rho ^{sp}}}} + a_\rho ^ + \frac{{{\tau ^{q - 1}}}}{{{\rho ^{tq}}}},\end{equation*}

with $a_\rho ^ + : = \mathop {\sup }\limits_{{B_\rho } \times {B_\rho }} a\left( {\cdot , \cdot } \right)$ and $\tau \ge 0$.

Consider a cut-off function $\phi\in C_0^\infty \left( {{B_{\frac{{3r}}{2}}}\left( {{\xi _0}} \right)} \right)$ satisfying

\begin{equation*} 0 \le \phi\le 1, \quad \phi\equiv 1 \text{in} B_r \quad \text{and} \quad \left| {{\nabla _H}\phi} \right| \le \frac{c}{r} \text{in} {{B_{\frac{{3r}}{2}}}}. \end{equation*}

Taking the test function $\varphi \left( \xi \right): = \frac{{\phi^q\left( \xi \right)}}{{{g_{2r}}\left( {u\left( \xi \right) + d} \right)}}$, we have from the weak formulation that

(4.1)\begin{align} 0 &= \int_{{B_{2r}}} {\int_{{B_{2r}}} {\Bigg[\frac{{{J_p}(u( \xi) - u( \eta))}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}} } \left( {\frac{{{\phi^q}( \xi)}}{{{g_{2r}}(\overline{u}(\xi))}} - \frac{{\phi^q( \eta)}}{{{g_{2r}}( {\overline{u}( \eta)} )}}} \right)\nonumber \\ & \quad+ a( \xi ,\eta )\frac{{{J_q}( u( \xi) - u( \eta))}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\left( {\frac{{\phi^q( \xi)}}{{{g_{2r}}( \bar u( \xi))}} - \frac{{\phi^q( \eta)}}{{{g_{2r}}(\bar u( \eta))}}}\right)\Bigg]\,d\xi d\eta \nonumber \\ &\quad +2{\int _{{\mathbb{H}^n}\backslash {B_{2r}}}}\int_{{B_{2r}}} {\left[\frac{{{J_p}( u( \xi) - u( \eta ))}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}} + a(\xi ,\eta )\frac{{{J_q}( u( \xi) - u( \eta))}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\right]\frac{\phi^q(\xi)}{g_{2r}(\overline{u}(\xi))}\,d\xi d\eta } \nonumber \\ &=:I_1+I_2, \end{align}

with $\bar u: = u + d$.

In what follows, we deal with I ₁ in the case $\bar u\left( \xi \right) \ge \bar u\left( \eta \right)$ that is divided into two subcases:

(4.2)\begin{equation} \bar u\left( \xi \right) \ge \bar u\left( \eta \right)\ge \frac{1}{2}\bar u\left( \xi \right), \end{equation}

and

(4.3)\begin{equation} \bar u\left( \xi \right) \ge 2\bar u\left( \eta \right). \end{equation}

If (4.2) occurs, we first observe that

(4.4)\begin{align} &\quad \frac{{{\phi ^q}\left( \xi \right)}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}} - \frac{{{\phi^q}\left( \eta \right)}}{{{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}}\nonumber \\ &\le \frac{c{{\phi^{q - 1}}\left( \xi \right)\sup_{B_{3R/2}} \left| {{\nabla _H}\phi} \right|{{\| {{\eta ^{- 1}} \circ \xi } \|}_{{\mathbb{H}^n}}}}}{{{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}} \nonumber \\ &+{\phi^q}\left( \xi \right)\int_0^1 {\frac{d}{{d\sigma }}\left( {g_{2r}^{- 1}\left( {\sigma \bar u\left( \xi \right) + (1 - \sigma )\bar u\left( \eta \right)} \right)} \right)\,d\sigma } \nonumber \\ & \le \frac{c{{\phi^{q - 1}}\left( \xi \right){r^{- 1}}{{\| {{\eta ^{- 1}} \circ \xi } \|}_{{\mathbb{H}^n}}}}}{{{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}} - \frac{{\left( {p - 1} \right){\phi^q}\left( \xi \right)\left( {\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}{{{2^q}{G_{2r}}\left( {\bar u\left( \eta \right)} \right)}}, \end{align}

where the first inequality holds naturally when $\phi(\xi) \le \phi(\eta)$. Here, we have used (4.2) and

\begin{align*} \int_0^1 {\frac{d}{{d\sigma }}\left( {g_{2r}^{- 1}\left( {\sigma \bar u\left( \xi \right) + (1 - \sigma )\bar u\left( \eta \right)} \right)} \right)\,d\sigma } &\ge \frac{{\left( {p - 1} \right)\left( {\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}{{{G_{2r}}\left( {\bar u\left( \xi \right)} \right)}} \nonumber\\ &\ge \frac{{\left( {p - 1} \right)\left( {\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}{{{2^q}{G_{2r}}\left( {\bar u\left( \eta \right)} \right)}}, \end{align*}

the details of which can be found in [Reference Byun, Ok and Song4]. Then, combining (4.4) and Young’s inequality yields

(4.5)\begin{align} F\left( {\xi ,\eta } \right)&:= \left( {\frac{{{J_p}\left( {u\left( \xi \right) - u\left( \eta \right)} \right)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{sp}}} + a\left( {\xi ,\eta } \right)\frac{{{J_q}\left( {u\left( \xi \right) - u\left( \eta \right)} \right)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq}}}} \right)\left( {\frac{{{\phi^q}\left( \xi \right)}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}} - \frac{{{\phi^q}( \eta)}}{{{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}}} \right)\nonumber \\ &\le \frac{{c{\phi^{q - 1}}\left( \xi \right){r^{- 1}}{{\| {{\eta ^{- 1}} \circ \xi } \|}_{{\mathbb{H}^n}}}\bar u\left( \eta \right)}}{{{G_{2r}}\left( {\bar u\left( \eta \right)} \right)}}\nonumber\\ &\left( {\frac{{{{\left| {\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right|}^{p - 1}}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{sp}}} + a\left( {\xi ,\eta } \right)\frac{{{{\left| {\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right|}^{q - 1}}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq}}}} \right)\nonumber \\ &\quad - \frac{{\left( {p - 1} \right){\phi^q}\left( \xi \right)H\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}{{{2^q}{G_{2r}}\left( {\bar u\left( \eta \right)} \right)}} \nonumber \\ &\le \frac{{\varepsilon {\phi^{\frac{{\left( {q - 1} \right)p}}{{p - 1}}}}\left( \xi \right){{\left| {\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right|}^p}}}{{{G_{2r}}\left( {\bar u\left( \eta \right)} \right)\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{sp}}} + a\left( {\xi ,\eta } \right)\frac{{\varepsilon {\phi^q}\left( \xi \right){{\left| {\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right|}^q}}}{{{G_{2r}}\left( {\bar u\left( \eta \right)} \right)\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq}}} \nonumber \\ &\quad - \frac{{\left( {p - 1} \right){\phi^q}\left( \xi \right)H\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}{{{2^q}{G_{2r}}\left( {\bar u\left( \eta \right)} \right)}}\nonumber \\ &\quad + c\left( \varepsilon \right)\frac{{{r^{- p}}\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^p{{\left| {\bar u\left( \eta \right)} \right|}^p}}}{{{G_{2r}}\left( {\bar u\left( \eta \right)} \right)\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{sp}}} + c\left( \varepsilon \right)a_{2r}^ + \frac{{{r^{- q}}\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^q{{\left| {\bar u\left( \eta \right)} \right|}^q}}}{{{G_{2r}}\left( {\bar u\left( \eta \right)} \right)\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq}}}\nonumber \\ &\le - \frac{{\left( {p - 1} \right){\phi^q}\left( \xi \right)H\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}{{{2^{q + 1}}{G_{2r}}\left( {\bar u\left( \eta \right)} \right)}} + c\frac{{{r^{p\left( {s - 1} \right)}}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{p\left( {s - 1} \right)}}} + c\frac{{{r^{q\left( {t - 1} \right)}}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{q\left( {t - 1} \right)}}}, \end{align}

where ɛ was chosen as $\frac{{p - 1}}{{{2^{q + 1}}}}$, $\frac{{\left( {q - 1} \right)p}}{{p - 1}} \gt q$ and c > 0 is independent of a. We proceed to evaluate ${{G_{2r}}\left( {\bar u\left( \eta \right)} \right)}$. For $\xi,\eta \in B_{2r}$, recalling the Hölder continuity of a, we get

\begin{equation*}a_{2r}^ + = a_{2r}^ + - a\left( {\xi ,\eta } \right) + a\left( {\xi ,\eta } \right) \le 2{\left[ a \right]_\alpha }{\left( {4r} \right)^\alpha } + a\left( {\xi ,\eta } \right).\end{equation*}

Thus this implies by the facts that $r\le 1$ and $tq \le sp+\alpha$ that

(4.6)\begin{align} {G_{2r}}\left( {\bar u\left( \eta \right)} \right) & \le \frac{{{{\bar u}^p}\left( \eta \right)}}{{{{\left( {2r} \right)}^{sp}}}} + 2{\left[ a \right]_\alpha }{\left( {4r} \right)^\alpha }\frac{{{{\bar u}^q}\left( \eta \right)}}{{{{\left( {2r} \right)}^{tq}}}} + a\left( {\xi ,\eta } \right)\frac{{{{\bar u}^q}\left( \eta \right)}}{{{{\left( {2r} \right)}^{tq}}}}\nonumber \\ & \le \left( {1 + 8{{\left[ a \right]}_\alpha }{r^{\alpha + sp - tq}}\| u \|_{{L^\infty }\left( {\Omega '} \right)}^{q - p}} \right)\frac{{{{\bar u}^p}\left( \eta \right)}}{{{{\left( {2r} \right)}^{sp}}}} + a\left( {\xi ,\eta } \right)\frac{{{{\bar u}^q}\left( \eta \right)}}{{{{\left( {2r} \right)}^{tq}}}}\nonumber\\ & \le c\left( {1 + \| u \|_{{L^\infty }\left( {\Omega '} \right)}^{q - p}} \right){H_{2r}}\left( {\xi ,\eta ,\bar u\left( \eta \right)} \right). \end{align}

Next, we will obtain an estimate on $\log \bar u$. It is easy to find

\begin{align*} \log \frac{{\bar u\left( \xi \right)}}{{\bar u\left( \eta \right)}}= \int_0^1 {\frac{{\bar u\left( \xi \right) - \bar u\left( \eta \right)}}{{\bar u\left( \eta \right) + \sigma \left( {\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}}\, d\sigma \le \frac{(\bar u( \xi)-\bar u(\eta))/\|\eta ^{-1}\circ \xi \|_{{\mathbb{H}^n}}^s} {\bar u(\eta)/(2r)^s} \frac{\|\eta ^{- 1}\circ \xi\|_{{\mathbb{H}^n}}^s}{(2r)^s}, \end{align*}

so, by the monotonicity of the function $f(\tau)=(\tau^p+a(\xi,\eta)\tau^q\|\eta ^{-1}\circ \xi \|_{\mathbb{H}^n}^{-(t-s)q})/\tau$ with $\tau\ge0$,

(4.7)\begin{align} \log \frac{\bar u(\xi)}{u(\eta)}&\le\frac{\|\eta ^{- 1} \circ \xi \|_{\mathbb{H}^n}^s}{(2r)^s}\left[\frac{\left(\frac{\bar u(\xi)-\bar u(\eta)}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^s}\right)^p+a(\xi,\eta)\left(\frac{\bar u(\xi)-\bar u(\eta)}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^s}\right)^q\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{-(t-s)q}}{\left(\frac{\bar u(\eta)}{(2r)^s}\right)^p+a(\xi,\eta)\left(\frac{\bar u(\eta)}{(2r)^s}\right)^q\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{-(t-s)q}}+ 1\right] \nonumber\\ &\le \frac{{cH\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}{{{H_{2r}}\left( {\xi ,\eta ,\bar u\left( \eta \right)} \right)}} + \frac{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^s}}{{{{\left( {2r} \right)}^s}}}, \end{align}

where we need to note ${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \le 4r$. It follows from (4.5)–(4.7) that

\begin{equation*}F\left( {\xi ,\eta } \right) \le - \frac{{{\phi^q}\left( \xi \right)}}{{cK}}\log \frac{{\bar u\left( \xi \right)}}{{\bar u\left( \eta \right)}} + \frac{{c\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^s}}{{{{\left( {2r} \right)}^s}}} + \frac{{c\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{p(1 - s)}}}{{{{\left( {2r} \right)}^{p(1 - s)}}}} + \frac{{c\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{q(1 - t)}}}{{{{\left( {2r} \right)}^{q(1 - t)}}}}.\end{equation*}

Second, we in the case (4.3) tackle the integral I ₁. Applying lemma 3.2 and the relation $\bar u\left( \xi \right) \ge 2\bar u\left( \eta \right)$, we could derive

\begin{align*} \frac{{{\phi^q}\left( \xi \right)}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}} - \frac{{{\phi^q}( \eta)}}{{{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}}&\le \frac{{{\phi^q}\left( \xi \right) - {\phi^q}\left( \eta \right)}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}} + {\phi ^q}\left( \eta \right)\left( {\frac{1}{{{g_{2r}}\left( {2\bar u\left( \eta \right)} \right)}} - \frac{1}{{{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}}} \right)\\ &\le\frac{{\varepsilon {\phi^q}\left( \eta \right) + c\left( \varepsilon \right){{\left| {\phi\left( \xi \right) - \phi\left( \eta \right)} \right|}^q}}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}} - \frac{{{2^{p - 1}} - 1}}{{{2^{p - 1}}}}\frac{{\phi ^q( \eta)}}{{{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}}\\ &\le\frac{{c{{\left| {\phi( \xi) - \phi( \eta)} \right|}^q}}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}} - \frac{{\left( {{2^{p - 1}} - 1} \right){\phi^q}\left( \eta \right)}}{{{2^p}{g_{2r}}\left( {\bar u\left( \eta \right)} \right)},} \end{align*}

with $\varepsilon = \frac{{{2^{p - 1}} - 1}}{{{2^p}}}$. Thereby, it holds that

\begin{align*} F\left( {\xi ,\eta } \right)& \le \frac{{ch\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right){{| {\phi( \xi) -\phi( \eta)}|}^q}}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}} - \frac{{h\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right){\phi ^q}\left( \eta \right)}}{{c{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}} \\ & \le \frac{{c{{\left( {2r} \right)}^{- q}}\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^qh\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}} - \frac{{h\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right){\phi^q}\left( \eta \right)}}{{cK{h_{2r}}\left( {\xi ,\eta ,\bar u\left( \eta \right)} \right)}}. \end{align*}

Here $F\left( {\xi ,\eta } \right)$ is the same as that in (4.5) and the estimate for ${{g_{2r}}\left( {\bar u\left( \eta \right)} \right)}$ is similar to (4.6). Moreover, via $\bar u\left( \xi \right) \ge 2\bar u\left( \eta \right)\ge 0$ in $B_{2r}$,

\begin{align*} \frac{{h\left( {\xi ,\eta ,\bar u( \xi) - \bar u( \eta)} \right)}}{{{g_{2r}}( {\bar u( \xi)})}} &\le \frac{\frac{|\bar u(\xi)-\bar u(\eta)|^{p-1}}{\|\eta^{-1} \circ \xi\|_{\mathbb{H}^n}^{sp}}+a(\xi,\eta)\frac{|\bar u(\xi)-\bar u(\eta)|^{q-1}}{\|\eta^{-1} \circ \xi\|_{\mathbb{H}^n}^{tq}}}{\frac{|\bar u(\xi)-\bar u(\eta)|^{p-1}}{(2r)^{sp}}+a_{2r}^ +\frac{|\bar u(\xi)-\bar u(\eta)|^{q-1}}{(2r)^{tq}}} \\ &\le \frac{{{{\left( {2r} \right)}^{sp}}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{sp}}} + \frac{{{{\left( {2r} \right)}^{tq}}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq}}}, \end{align*}

and further

\begin{equation*}F\left( {\xi ,\eta } \right) \le \frac{{c\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{q - sp}}}{{{{\left( {2r} \right)}^{q - sp}}}} + \frac{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{q\left( {1 - t} \right)}}}{{{{\left( {2r} \right)}^{q\left( {1 - t} \right)}}}} - \frac{{h\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right){\phi ^q}\left( \eta \right)}}{{cK{h_{2r}}\left( {\xi ,\eta ,\bar u\left( \eta \right)} \right)}}.\end{equation*}

Now we obtain an estimate on $\log \frac{{\bar u\left( \xi \right)}}{{\bar u\left( \eta \right)}}$ under (4.3). Notice $\bar u(\xi)\le 2(\bar u( \xi) - \bar u( \eta))$. we get

\begin{align*} &\quad\log \frac{\bar u(\xi)}{\bar u( \eta)}\\ &\le \frac{c\left( (\bar u(\xi) - \bar u( \eta))/\|\eta ^{- 1} \circ \xi\|_{\mathbb{H}^n}^s\right)^{p - 1}}{(\bar u(\eta)/(2r)^s)^{p - 1}} \frac{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{s\left( {p - 1} \right)}}}{{{{\left( {2r} \right)}^{s\left( {p - 1} \right)}}}}\\ &\le c\frac{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{s\left( {p - 1} \right)}}}{{{{\left( {2r} \right)}^{s\left( {p - 1} \right)}}}}\left[\frac{\left(\frac{\bar u(\xi)-\bar u(\eta)}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^s}\right)^{p-1}+a(\xi,\eta)\left(\frac{\bar u(\xi)-\bar u(\eta)}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^s}\right)^{q-1}\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{-(t-s)q}}{\left(\frac{\bar u(\eta)}{(2r)^s}\right)^{p-1}+a(\xi,\eta)\left(\frac{\bar u(\eta)}{(2r)^s}\right)^{q-1}\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{-(t-s)q}}+ 1\right]\\ &\le \frac{{ch\left( {\xi ,\eta ,\bar u\left( \xi \right) - \bar u\left( \eta \right)} \right)}}{{{h_{2r}}\left( {\xi ,\eta ,\bar u\left( \eta \right)} \right)}} + \frac{{c\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{s\left( {p - 1} \right)}}}{{{{\left( {2r} \right)}^{s\left( {p - 1} \right)}}}}, \end{align*}

where the fact ${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \le 4r$ was utilized. Noting $q\ge p$ and ${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \le 4r$ again,

\begin{align*} F\left( {\xi ,\eta } \right) \le - \frac{{{\phi^q}\left( \xi \right)}}{{cK}}\log \frac{{\bar u\left( \xi \right)}}{{\bar u\left( \eta \right)}} + \frac{{c\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{p\left( {1 - s} \right)}}}{{{{\left( {2r} \right)}^{p\left( {1 - s} \right)}}}} + \frac{{c\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{q(1 - t)}}}{{{{\left( {2r} \right)}^{q(1 - t)}}}} + \frac{{c\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{s(p - 1)}}}{{{{\left( {2r} \right)}^{s(p - 1)}}}}. \end{align*}

At this moment, for $\bar u\left( \xi \right) \ge \bar u\left( \eta \right)$, the integral I ₁ is evaluated as

(4.8)\begin{align} I_1 &\le -\frac{1}{cK}\int_{B_{2r}}\int_{B_{2r}}\min\left\{\phi^q(\xi),\phi^q(\eta)\right\}\left|\log\frac{\bar u(\xi)}{\bar u(\eta)}\right| \frac{d\xi d\eta }{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^Q}\nonumber \\ & + c\int_{B_{2r}}\int_{B_{2r}}\Bigg[\frac{\|\eta^{-1} \circ \xi \|_{\mathbb{H}^n}^{p - sp}}{r^{p(1 - s)}}\nonumber \\ &+ \frac{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{q(1 - t)}}{r^{q(1 - t)}} + \frac{\|\eta^{-1} \circ \xi\|_{\mathbb{H}^n}^{s(p - 1)}}{r^{s(p - 1)}} + \frac{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^s}{r^s}\Bigg]\frac{d\xi d\eta}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^Q} \nonumber \\ & \le -\frac{1}{cK}\int_{B_{2r}}\int_{B_{2r}} \left|\log\frac{\bar u(\xi)}{\bar u(\eta)}\right|\frac{d\xi d\eta }{\|\eta^{-1}\circ \xi\|_{\mathbb{H}^n}^Q}+ cr^Q, \end{align}

where

\begin{align*} \int_{{B_{2r}}} {\int_{{B_{2r}}} {\frac{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^l}}{{{r^l}}}\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} } & \le \int_{{B_{2r}}} {\int_{{B_{4r}(\eta)}} {\frac{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^l}}{{{r^l}}}\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}} } \\ &\le \frac{c}{r^l}\int_{{B_{2r}}}\int^{4r}_0\rho^{l-1}\,d\rho d\eta \le c{r^Q}. \end{align*}

Furthermore, if $\bar u\left( \xi \right) \lt \bar u\left( \eta \right)$, the same estimate still holds true through exchanging the roles of ξ and η.

For the second contribution I ₂ in (4.1), we first observe that if $\eta \in {B_R}$, then ${( {u\left( \xi \right) - u\left( \eta \right)} )_ + }$ $\le u\left( \xi \right) + d$ by $u\left( \eta \right) \ge 0$, and that if $\eta \in {\mathbb{H}^n}\backslash {B_R}$, then ${\left( {u\left( \xi \right) - u\left( \eta \right)} \right)_ + } \le u\left( \xi \right) + u_-\left( \eta \right) \le \bar u\left( \xi \right) + u_-\left( \eta \right)$. From this and ${\rm supp}\, \phi\subset {B_{\frac{{3r}}{2}}}$, we can evaluate I ₂ as

(4.9)\begin{align} I_2 &\le 2{\int _{{B_R}\backslash {B_{2r}}}}\int_{{B_{\frac{{3r}}{2}}}} {\left[\frac{{\left( {u\left( \xi \right) - u\left( \eta \right)} \right)_ + ^{p - 1}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}} + a\left( {\xi ,\eta } \right)\frac{{\left( {u\left( \xi \right) - u\left( \eta \right)} \right)_ + ^{q - 1}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\right]\frac{{d\xi d\eta }}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}}} \nonumber \\ &\quad + 2{\int _{{\mathbb{H}^n}\backslash {B_R}}}\int_{{B_{\frac{{3r}}{2}}}} {\left[\frac{{\left( {u\left( \xi \right) - u\left( \eta \right)} \right)_ + ^{p - 1}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}} + a\left( {\xi ,\eta } \right)\frac{{\left( {u\left( \xi \right) - u\left( \eta \right)} \right)_ + ^{q - 1}}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + tq}}}\right]\frac{{d\xi d\eta }}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}}}\nonumber \\ & \le {\int _{{\mathbb{H}^n}\backslash {B_{2r}}}}\int_{{B_{\frac{{3r}}{2}}}} {\frac{c{h\left( {\xi ,\eta ,\bar u\left( \xi \right)} \right)}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)\| {{\eta ^{- 1}} \circ \xi }\|_{{\mathbb{H}^n}}^Q}}\,d\xi d\eta } + {\int _{{\mathbb{H}^n}\backslash {B_R}}}\int_{{B_{\frac{{3r}}{2}}}}\nonumber\\ &{\frac{c{h\left( {\xi ,\eta ,u\left( \eta \right)} \right)}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}}\,d\xi d\eta } \nonumber \\ &= :{I_{21}} + {I_{22}}. \end{align}

We now intend to control precisely the term $\frac{{h\left( {\xi ,\eta ,\bar u\left( \xi \right)} \right)}}{{{g_{2r}}\left( {\bar u\left( \xi \right)} \right)}}$ by some constants. In view of the condition (1.6), there holds that, for $\xi \in B_{2r}$ and $\eta \in \mathbb{H}^n$,

(4.10)\begin{align} a\left( {\xi ,\eta } \right)& \le a\left( {\xi ,\eta } \right) - a\left( {\xi ,\xi } \right) + a_{2r}^ + \nonumber \\ & \le {\left( {2{{\| a \|}_{{L^\infty }}}} \right)^{1 - \frac{{tq - sp}}{\alpha }}}{\left| {a\left( {\xi ,\eta } \right) - a\left( {\xi ,\xi } \right)} \right|^{\frac{{tq - sp}}{\alpha }}} + a_{2r}^ + \nonumber \\ & \le c\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq - sp} + a_{2r}^ + . \end{align}

This indicates

\begin{align*} {I_{21}} & \le c{\int _{{\mathbb{H}^n}\backslash {B_{2r}}}}\int_{{B_{\frac{{3r}}{2}}}} {\frac{{\frac{{{{\bar u}^{p - 1}}\left( \xi \right) + {{\bar u}^{q - 1}}\left( \xi \right)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{sp}}} + a_{2r}^ + \frac{{{{\bar u}^{q - 1}}\left( \xi \right)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq}}}}}{{\frac{{{{\bar u}^{p - 1}}\left( \xi \right)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{sp}}}\frac{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{sp}}}{{{{\left( {2r} \right)}^{sp}}}} + a_{2r}^ + \frac{{{{\bar u}^{q - 1}}\left( \xi \right)}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq}}}\frac{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{tq}}}{{{{\left( {2r} \right)}^{tq}}}}}}\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi } \|_{\mathbb{H}{^n}}^Q}}} \\ & \le cK{\int _{{\mathbb{H}^n}\backslash {B_{2r}}}}\int_{{B_{\frac{{3r}}{2}}}} {\frac{(r/2)^{sp}}{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\xi d\eta,} \end{align*}

by virtue of ${\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \gt \frac{r}{2}$. For $\xi \in {B_{\frac{{3r}}{2}}}$ and $\eta \in \mathbb{H}^n\backslash {B_{2r}}$, via the triangle inequality,

(4.11)\begin{align} {\| {{\eta ^{- 1}} \circ {\xi _0}} \|_{{\mathbb{H}^n}}} & \le\left( {1 + \frac{{{{\| {{\xi ^{- 1}} \circ {\xi _0}} \|}_{{\mathbb{H}^n}}}}}{{{{\| {{\eta ^{- 1}} \circ \xi } \|}_{{\mathbb{H}^n}}}}}} \right){\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} \nonumber\\ & \le \left( 1 + \frac{3r/2}{r/2} \right){\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}} = 4{\| {{\eta ^{- 1}} \circ \xi }\|_{{\mathbb{H}^n}}}, \end{align}

Thus by [Reference Manfredini, Palatucci, Piccinini and Polidoro31, Lemma 2.6],

(4.12)\begin{equation} {I_{21}} \le cK\left| {{B_{\frac{{3r}}{2}}}} \right|{\int _{{\mathbb{H}^n}\backslash {B_{2r}}}}\frac{{{r^{sp}}}}{{\| {{\eta ^{- 1}} \circ {\xi _0}} \|_{{\mathbb{H}^n}}^{Q + sp}}}\,d\eta \le cK{r^Q}. \end{equation}

Let us proceed to examine I ₂₂. With the aid of (4.10), (4.11) and $u(\xi)\ge 0$ in ${{B_{\frac{{3r}}{2}}}}$,

(4.13)\begin{align} I_{22} & \le c{\int _{{\mathbb{H}^n}\backslash {B_R}}}\int_{{B_{\frac{{3r}}{2}}}} {\left( {\frac{{u_ - ^{p - 1}( \eta) + u_ - ^{q - 1}( \eta)}}{{\| {{\eta ^{- 1}} \circ \xi }\|_{{\mathbb{H}^n}}^{Q + sp}}} + a_{2r}^ + \frac{{u_ - ^{q - 1}( \eta )}}{{\| {{\eta ^{- 1}} \circ \xi }\|_{{\mathbb{H}^n}}^{Q + tq}}}} \right){g^{- 1}}( d)\,d\xi d\eta } \nonumber \\ & \le c{r^Q}{g^{- 1}}( d){\int _{{\mathbb{H}^n}\backslash {B_R}}}\left( {\frac{{u_ - ^{p - 1}( \eta) + u_ - ^{q - 1}( \eta)}}{{\| {{\eta ^{- 1}} \circ {\xi _0}}\|_{{\mathbb{H}^n}}^{Q + sp}}} + a_{2r}^ + \frac{{u_ - ^{q - 1}(\eta)}}{{\| {{\eta ^{- 1}} \circ {\xi _0}}\|_{{\mathbb{H}^n}}^{Q + tq}}}} \right)\,d\eta\nonumber \\ & \le c{r^{Q+sp}}{d^{1 - p}}{\int _{{\mathbb{H}^n}\backslash {B_R}}}\frac{{u_ - ^{p - 1}( \eta) + u_ - ^{q - 1}( \eta)}}{{\| {{\eta ^{- 1}} \circ {\xi _0}}\|_{{\mathbb{H}^n}}^{Q + sp}}}d\eta \nonumber \\ &\quad + c{r^{Q + tq}}{d^{1 - q}}{\int _{{\mathbb{H}^n}\backslash {B_R}}}\frac{{u_ - ^{q - 1}( \eta)}}{{\| {{\eta ^{- 1}} \circ {\xi _0}}\|_{{\mathbb{H}^n}}^{Q + tq}}}\,d\eta, \end{align}

where we notice $\eta \in {\mathbb{H}^n}\backslash {B_R} \subset {\mathbb{H}^n}\backslash {B_{2r}}$.

Merging (4.8), (4.9), (4.12), (4.13) with (4.1) arrives eventually at the desired estimate with the positive constant c depending upon $n,p,q,s,t,\alpha,[a]_{\alpha} $ and $\| a \|_{L^\infty }$.

Proof. Notice that, since w is a truncation of $\log (u+d)$,

\begin{align*} -\!\!\!\!\!\!\int_{{B_r}} {| {w - {( w)}_r}|\,d\eta } & \le -\!\!\!\!\!\!\int_{{B_r}} {\left| {-\!\!\!\!\!\!\int_{{B_r}} {( {w( \eta) - w( \xi)})\,d\xi } } \right|\,d\eta } \\ & \le -\!\!\!\!\!\!\int_{{B_r}} {-\!\!\!\!\!\!\int_{{B_r}} {| {w( \xi ) - w( \eta)}|\,d\xi } d\eta } \\ & \le -\!\!\!\!\!\!\int_{{B_r}} {-\!\!\!\!\!\!\int_{{B_r}} {\frac{{\left| {\log \left( {u\left( \xi \right) + d} \right) - \log \left( {u\left( \eta \right) + d} \right)} \right|}}{{{\| {{\eta ^{- 1}} \circ \xi } \|_{{\mathbb{H}^n}}^Q}/{{{\left( {2r} \right)}^Q}}}}\,d\xi } d\eta } \\ & \le -\!\!\!\!\!\!\int_{{B_r}} {\int_{{B_r}} {\left| {\log \frac{{u\left( \xi \right) + d}}{{u\left( \eta \right) + d}}} \right|\frac{{d\xi d\eta }}{{\| {{\eta ^{- 1}} \circ \xi }\|_{{\mathbb{H}^n}}^Q}}} }. \end{align*}

Then the desired result is a plain consequence of lemma 4.1.

Proof. Argue by induction. The conclusion is obvious for j = 0 and then assume it holds true for $i\le j$. Now we show this claim for j + 1. Let us notice the simple fact that either

(4.14)\begin{equation} \left|2B_{j+1}\cap\left\{u\ge \inf_{B_j}u+\omega(r_j)/2\right\}\right|\ge\frac{1}{2}|2B_{j+1}|, \end{equation}

or

(4.15)\begin{equation} \left|2B_{j+1}\cap\left\{u \lt \inf_{B_j}u+\omega(r_j)/2\right\}\right|\ge\frac{1}{2}|2B_{j+1}|. \end{equation}

Define

\begin{equation*} u_j=\begin{cases}u-\inf_{B_j}u, &\text{if (4.14) occurs},\\[2mm] \sup_{B_j}u-u, &\text{if (4.15) occurs}. \end{cases} \end{equation*}

Obviously, $u_j\ge0$ in B_j and

(4.16)\begin{equation} |2B_{j+1}\cap\{u_j\geq\omega(r_j)/2\}|\ge\frac{1}{2}|2B_{j+1}|. \end{equation}

Moreover, u_j is a weak solution to (1.1) such that

(4.17)\begin{equation} \sup_{B_i}|u_j|\le \omega(r_i) \quad\text{for any } i\in\{0,1,2,\cdots,j\}. \end{equation}

Now we set an auxiliary function

\begin{equation*} w:=\min\left\{\left[\log\left(\frac{\omega(r_j)/2+d}{u_j+d}\right)\right]_+,k\right\} \quad\text{with } k \gt 0. \end{equation*}

Applying corollary 4.2 derives

(4.18)\begin{align} &\quad-\!\!\!\!\!\!\int_{2B_{j+1}}|w-(w)_{2B_{j+1}}|\,d\xi \nonumber\\ &\leq CK^2\left(1+d^{1-p}r^{sp}_{j+1}\int_{\mathbb{H}^n\setminus B_j}\frac{|u_j|^{p-1}+|u_j|^{q-1}}{\|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}^{Q+sp}}\,d\xi+d^{1-q}r^{tq}_{j+1}\int_{\mathbb{H}^n\setminus B_j}\frac{|u_j|^{q-1}}{\|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}^{Q+tq}}\,d\xi\right), \end{align}

with K defined as in lemma 4.1. We evaluate the second integral at the right-hand side. By means of (4.17) and the definition of $\omega(r_0)$,

(4.19)\begin{align} &\quad r^{tq}_{j+1}\int_{\mathbb{H}^n\setminus B_j}\frac{|u_j|^{q-1}}{\|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}^{Q+tq}}\,d\xi \nonumber\\ &=r^{tq}_j\sum^j_{i=1}\int_{B_{i-1}\setminus B_i}\frac{|u_j|^{q-1}}{\|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}^{Q+tq}}\,d\xi +r^{tq}_j\int_{\mathbb{H}^n\setminus B_0}\frac{|u_j|^{q-1}}{\|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}^{Q+tq}}\,d\xi \nonumber\\ &\leq\sum^j_{i=1}\omega(r_{i-1})^{q-1}\left(\frac{r_j}{r_i}\right)^{tq}+Cr^{tq}_j\int_{\mathbb{H}^n\setminus B_0}\frac{|u|^{q-1}+(\sup_{B_0}|u|)^{q-1}}{\|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}^{Q+tq}}\,d\xi \nonumber\\ &\leq C\sum^j_{i=1}\left(\frac{r_j}{r_i}\right)^{tq}\omega(r_{i-1})^{q-1} \nonumber\\ &\le C\frac{4^{tq-\beta(q-1)}}{(tq-\beta(q-1))\log4}\sigma^{-\beta(q-1)}\omega(r_j)^{q-1}, \end{align}

where we used the fact that $\beta \lt \frac{sp}{q-1}\left(\le \frac{tq}{q-1}\right)$. Analogously,

(4.20)\begin{align} r^{sp}_j\int_{\mathbb{H}^n\setminus B_j}\frac{|u_j|^{p-1}+|u_j|^{q-1}}{\|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}^{Q+sp}}\,d\xi &\le C(1+\|u\|^{q-p}_{L^\infty(\Omega')})\sum^j_{i=1}\left(\frac{r_j}{r_i}\right)^{sp}\omega(r_{i-1})^{p-1} \nonumber\\ &\le CN\sigma^{-\beta(p-1)}\omega(r_j)^{p-1}, \end{align}

with $\beta \lt \frac{sp}{q-1}\left(\le \frac{sp}{p-1}\right)$, where $N:=1+\|u\|^{q-p}_{L^\infty(\Omega')}$ and the derivation of $\|u\|^{q-p}_{L^\infty(\Omega')}$ is from the term $|u_j|^{q-1}$, and C > 0 depends on $n,p,s$ and the difference of $\frac{sp}{p-1}$ and β. Combining (4.19), (4.20) with (4.18) and remembering $\frac{r_{j+1}}{r_j}=\sigma$, we get

\begin{align*} &\quad -\!\!\!\!\!\!\int_{2B_{j+1}}|w-(w)_{2B_{j+1}}|\,d\xi \\ &\leq CK^2\left(1+Nd^{1-p}\sigma^{sp-\beta(p-1)}\omega(r_j)^{p-1}+d^{1-q}\sigma^{tq-\beta(q-1)}\omega(r_j)^{q-1}\right), \end{align*}

where C depends on $n,p,q,s,t$ and the difference of β and $\frac{tq}{q-1}$, and $\frac{sp}{p-1}$.

In what follows, picking

\begin{equation*} d:=\sigma^{\frac{sp}{q-1}-\beta}\omega(r_j), \end{equation*}

and recalling $\omega(r_j)=\sigma^{j\beta}\omega(r_0)$, we find

\begin{align*} &\quad-\!\!\!\!\!\!\int_{2B_{j+1}}|w-(w)_{2B_{j+1}}|\,d\xi \\ &\leq CK^2\left[1+N\sigma^{\left({\frac{sp}{q-1}-\beta}\right)(1-p)+\left({\frac{sp}{p-1}-\beta}\right)(p-1)} +\sigma^{\left({\frac{sp}{q-1}-\beta}\right)(1-q)+\left({\frac{tq}{q-1}-\beta}\right)(q-1)}\right] \leq CN^3, \end{align*}

where C depends on $n,p,q,s,t,\alpha,[a]_\alpha,\|a\|_{L^\infty}$ and the difference of β and $\frac{tq}{q-1}$, and $\frac{sp}{p-1}$. Here we need to utilize the definition of K as in lemma 4.1, and $\omega(r_j)\le 2\|u\|_{L^\infty(\Omega')}$. From the last inequality,

\begin{equation*} \frac{|2B_{j+1}\cap\{w=k\}|}{|2B_{j+1}|}\le \frac{CN^3}{k}. \end{equation*}

We refer to [Reference Di Castro, Kuusi and Palatucci14, page 1296] for the details. By taking

\begin{equation*} k=\log\left(\frac{\omega(r_j)/2+\varepsilon\omega(r_j)}{3\varepsilon\omega(r_j)}\right)=\log\left(\frac{1/2+\varepsilon}{3\varepsilon}\right)\approx\log\frac{1}{\varepsilon}, \end{equation*}

with $\varepsilon:=\sigma^{\frac{sp}{q-1}-\beta}$, it holds that

(4.21)\begin{equation} \frac{|2B_{j+1}\cap\{u_j\leq2\varepsilon\omega(r_j)\}|}{|2B_{j+1}|}\le \frac{CN^3}{k}\le \frac{C_{\rm log}N^3}{\log\frac{1}{\sigma}} \end{equation}

for the constant $C_{\rm log} \gt 0$ depending on $n,p,q,s,t,\alpha,[a]_\alpha,\|a\|_{L^\infty}$ and β.

At this moment, we are going to perform a suitable iteration. For each $i=0,1,\cdots$, let

\begin{equation*} \rho_i=r_{j+1}+2^{-i}r_{j+1}, \quad \hat{\rho_i}=\frac{\rho_i+3\rho_{i+1}}{4}, \quad \tilde{\rho_i}=\frac{3\rho_i+\rho_{i+1}}{4}, \end{equation*}

and the corresponding balls

\begin{equation*} B^i=B_{\rho_i}, \quad \hat{B}^i=B_{\hat{\rho_i}}, \quad \tilde{B}^i=B_{\tilde{\rho_i}}. \end{equation*}

Then take the cut-off functions $\psi_i\in C^\infty_0(\tilde{B}^i)$ such that

\begin{equation*} 0\le \psi_i\le 1, \quad \psi_i\equiv 1 \text{in } \hat{B}^{i} \quad\text{and} \quad |\nabla_H\psi_i|\le 2^{i+2}r^{-1}_{j+1}. \end{equation*}

Besides, set

\begin{equation*} k_i=(1+2^{-i})\varepsilon\omega(r_j), \quad w_i=(k_i-u_j)_+, \end{equation*}

and

\begin{equation*} A_i=\frac{|B^i\cap \{u_j\leq k_i\}|}{|B^i|}=\frac{|B^i\cap \{w_j\ge0\}|}{|B^i|}. \end{equation*}

Observe the apparent facts that

\begin{equation*} r_{j+1}\leq \rho_{i+1} \lt \hat{\rho}_i \lt \tilde{\rho}_i \lt \rho_i\le2r_{j+1},\quad 0\le w_i\le k_i\le2\varepsilon\omega(r_j), \end{equation*}

and denote

\begin{equation*} a^+_{j+1}:=\sup_{B_{2r_{j+1}}\times B_{2r_{j+1}}}a(\cdot,\cdot),\ \ \ a^-_{j+1}:=\inf_{B_{2r_{j+1}}\times B_{2r_{j+1}}}a(\cdot,\cdot),\ \ \ \overline{G}(\tau):=\frac{\tau^p}{r^{sp}_{j+1}}+a^+_{j+1}\frac{\tau^q}{r^{tq}_{j+1}}. \end{equation*}

With the help of Caccioppoli inequality (lemma 3.3), we derive

(4.22)\begin{align} &\quad-\!\!\!\!\!\!\int_{\hat{B}^i}\int_{\hat{B}^i}\frac{H(\xi,\eta,|w_i(\xi)-w_i(\eta)|)}{\|\eta^{-1}\circ\xi\|^Q_{\mathbb{H}^n}}\,d\xi d\eta \nonumber\\ &\le C-\!\!\!\!\!\!\int_{B^i}\int_{B^i}\frac{H(\xi,\eta,(w_i(\xi)+w_i(\eta))|\psi_i(\xi)-\psi_i(\eta)|)}{\|\eta^{-1}\circ\xi\|^Q_{\mathbb{H}^n}}\,d\xi d\eta \nonumber\\ &\quad +C-\!\!\!\!\!\!\int_{B^i}w_i\psi_i^q\,d\xi\left(\sup_{\eta\in \tilde{B}^i}\int_{\mathbb{H}^n\setminus B^i}\frac{h(\xi,\eta,w_i(\xi))}{\|\eta^{-1}\circ\xi\|^Q_{\mathbb{H}^n}}\,d\xi\right) \nonumber\\ &=:J_1+J_2. \end{align}

Via the definition of w_i and ψ_i, J ₁ is evaluated as

(4.23)\begin{align} J_1&\leq C\frac{2^{ip}k_i^p}{r_{j+1}^p}\int_{B^i\cap \{u_j\le k_i\}}-\!\!\!\!\!\!\int_{B^i}\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{-Q+(1-s)p}\,d\xi d\eta \nonumber\\ &\quad+Ca^+_{j+1}\frac{2^{iq}k_i^q}{r_{j+1}^q}\int_{B^i\cap \{u_j\le k_i\}}-\!\!\!\!\!\!\int_{B^i}\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}^{-Q+(1-t)q}\,d\xi d\eta \nonumber\\ &\leq C2^{iq}\overline{G}(k_i)A_i, \end{align}

and moreover, we have

\begin{equation*} -\!\!\!\!\!\!\int_{B^i}w_i\psi_i^q\,d\xi\le Ck_iA_i. \end{equation*}

As for the nonlocal integral in J ₂, we first note that if $\eta\in\tilde{B}^i$ and $\xi\in\mathbb{H}^n\setminus B^i$, then

\begin{equation*} \|\xi_0^{-1}\circ\xi\|_{\mathbb{H}^n}\le\left(1+\frac{\|\xi_0^{-1}\circ\eta\|_{\mathbb{H}^n}}{\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}}\right)\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n} \le2^{i+4}\|\eta^{-1}\circ\xi\|_{\mathbb{H}^n}. \end{equation*}

Furthermore, $w_i\le k_i\le2\varepsilon\omega(r_j)$ in B_j (by $u_j\ge0$ in B_j), and $w_i\le k_i+|u|$ in $\mathbb{H}^n\setminus B_j$. In a similar way to treat I ₂ in the proof of lemma 4.1, by applying (4.19), (4.20), the definition of ɛ and $B_{j+1}\subset B^i$ we derive

\begin{align*} &\quad \sup_{\eta\in \tilde{B}^i}\int_{\mathbb{H}^n\setminus B^i}\frac{h(\xi,\eta,w_i(\xi))}{\|\eta^{-1}\circ\xi\|^Q_{\mathbb{H}^n}}\,d\xi \\ &\leq \sup_{\eta\in \tilde{B}^i}\int_{\mathbb{H}^n\setminus B^i}\frac{w_i^{p-1}+w_i^{q-1}}{\|\eta^{-1}\circ\xi\|^{Q+sp}_{\mathbb{H}^n}} +a^+_{j+1}\frac{w_i^{q-1}}{\|\eta^{-1}\circ\xi\|^{Q+tq}_{\mathbb{H}^n}}\,d\xi\\ &\leq C2^{i(Q+sp+tq)}\int_{\mathbb{H}^n\setminus B_{j+1}}\frac{w_i^{p-1}+w_i^{q-1}}{\|\xi_0^{-1}\circ\xi\|^{Q+sp}_{\mathbb{H}^n}} +a^+_{j+1}\frac{w_i^{q-1}}{\|\xi_0^{-1}\circ\xi\|^{Q+tq}_{\mathbb{H}^n}}\,d\xi\\ &\leq C2^{i(Q+sp+tq)}\int_{\mathbb{H}^n\setminus B_{j}}\frac{|u_j|^{p-1}+|u_j|^{q-1}}{\|\xi_0^{-1}\circ\xi\|^{Q+sp}_{\mathbb{H}^n}}+ a^+_{j+1}\frac{|u_j|^{q-1}}{\|\xi_0^{-1}\circ\xi\|^{Q+tq}_{\mathbb{H}^n}}\,d\xi\\ &\quad+ C2^{i(Q+sp+tq)}\int_{\mathbb{H}^n\setminus B_{j+1}}\frac{k_i^{p-1}+k_i^{q-1}}{\|\xi_0^{-1}\circ\xi\|^{Q+sp}_{\mathbb{H}^n}}+ a^+_{j+1}\frac{k_i^{q-1}}{\|\xi_0^{-1}\circ\xi\|^{Q+tq}_{\mathbb{H}^n}}\,d\xi\\ &\leq C2^{i(Q+sp+tq)}\left(\frac{N\omega(r_j)^{p-1}}{r_j^{sp}\sigma^{\beta(p-1)}}+a^+_{j+1}\frac{\omega(r_j)^{q-1}}{r_j^{tq}\sigma^{\beta(q-1)}} +\frac{k_i^{p-1}+k_i^{q-1}}{r_{j+1}^{sp}}+a^+_{j+1}\frac{k_i^{q-1}}{r_{j+1}^{tq}}\right) \\ &\leq C2^{i(Q+sp+tq)}\left(\frac{Nk_i^{p-1}}{\varepsilon^{p-1}r_j^{sp}\sigma^{\beta(p-1)}}+a^+_{j+1}\frac{k_i^{q-1}}{\varepsilon^{q-1}r_j^{tq}\sigma^{\beta(q-1)}} +\frac{Nk_i^{p-1}}{r_{j+1}^{sp}}+a^+_{j+1}\frac{k_i^{q-1}}{r_{j+1}^{tq}}\right) \\ &\leq CN2^{i(Q+sp+tq)}\left(\frac{\sigma^{sp-\frac{sp(p-1)}{q-1}}k_i^{p-1}}{r_{j+1}^{sp}}+a^+_{j+1}\frac{\sigma^{tq-sp}k_i^{q-1}}{r_{j+1}^{tq}} +\frac{\overline{G}(k_i)}{k_i}\right) \\ &\le CN2^{i(Q+sp+tq)}\frac{\overline{G}(k_i)}{k_i}. \end{align*}

Therefore,

(4.24)\begin{equation} J_2\le CN2^{i(Q+sp+tq)}\overline{G}(k_i)A_i. \end{equation}

On the other hand, making use of lemma 2.8 with $u:=w_i$ yields that

(4.25)\begin{align} &\quad A^\frac{1}{\gamma}_{i+1}\overline{G}(k_i-k_{i+1})\nonumber\\ &\le\left(-\!\!\!\!\!\!\int_{B^{i+1}}\left(\left|\frac{w_i}{r^s_{j+1}}\right|^p+a^+_{j+1}\left|\frac{w_i}{r^t_{j+1}}\right|^q\right)^\gamma\,d\xi\right) ^\frac{1}{\gamma} \nonumber\\ &\leq CN\left(\frac{D_1(\hat{\rho}_i,\rho_{i+1})}{r_{j+1}^{sp}}+\frac{\widetilde{D}_1(\hat{\rho}_i,\rho_{i+1})}{r_{j+1}^{tq}}\right) -\!\!\!\!\!\!\int_{\hat{B}^i}\int_{\hat{B}^i}\frac{H(\xi,\eta,|w_i(\xi)-w_i(\eta)|)}{\|\eta^{-1}\circ\xi\|^Q_{\mathbb{H}^n}}\,d\xi d\eta \nonumber\\ &\quad+CN-\!\!\!\!\!\!\int_{\hat{B}^i}\left|\frac{w_i}{r^s_{j+1}}\right|^p +a^-_{j+1}\left|\frac{w_i}{r^t_{j+1}}\right|^q\,d\xi. \end{align}

Thanks to the definitions of $D_1,\widetilde{D}_1$ and $\hat{\rho}_i,\rho_{i+1}$, we from $\hat{\rho}_i\approx\rho_{i+1}\approx r_{j+1}$ and $\hat{\rho}_i-\rho_{i+1}=2^{-i-3}r_{j+1}$ calculate

\begin{equation*} \frac{D_1(\hat{\rho}_i,\rho_{i+1})}{r_{j+1}^{sp}}\le C2^{i(Q+sp+p)}, \quad \frac{\widetilde{D}_1(\hat{\rho}_i,\rho_{i+1})}{r_{j+1}^{tq}}\le C2^{i(Q+tq+q)}. \end{equation*}

It is easy to obtain

(4.26)\begin{equation} -\!\!\!\!\!\!\int_{\hat{B}^i}\left|\frac{w_i}{r^s_{j+1}}\right|^p +a^-_{j+1}\left|\frac{w_i}{r^t_{j+1}}\right|^q\,d\xi\le C-\!\!\!\!\!\!\int_{{B}^i}\overline{G}(w_i)\,d\xi\le C\overline{G}(k_i)A_i. \end{equation}

It follows from (4.22)–(4.26) that

\begin{equation*} \begin{split} A^\frac{1}{\gamma}_{i+1}\overline{G}(2^{-i-1}\varepsilon\omega(r_j))&=A^\frac{1}{\gamma}_{i+1}\overline{G}(k_i-k_{i+1})\\ &\le CN^22^{i2(Q+2q)}\overline{G}(k_i)A_i\\ &\le CN^2 2^{i2(Q+2q)}\overline{G}(\varepsilon\omega(r_j))A_i, \end{split} \end{equation*}

and further

\begin{equation*} A_{i+1}\le CN^{2\gamma}2^{i2(Q+3q)\gamma}A^\gamma_i, \end{equation*}

where $\gamma=\min\left\{\frac{p^*_s}{p},\frac{q^*_t}{q}\right\} \gt 1$ and C depends on $\mathrm{\textbf{data}}$ and β.

Now if A ₀ fulfils

(4.27)\begin{align} A_0=\frac{|2B_{j+1}\cap\{u_j\leq 2\varepsilon\omega(r_j)\}|}{|2B_{j+1}|}\leq (CN^{2\gamma})^{-\frac{1}{\gamma-1}}2^{-\frac{2\gamma(Q+3q)}{(\gamma-1)^2}}=:\mu, \end{align}

then by lemma 3.4 we deduce $A_i\rightarrow0$ as $i\rightarrow\infty$. This means

\begin{equation*} u_j\ge\varepsilon\omega(r_j) \quad\text{a.e. in} \ B_{j+1}, \end{equation*}

which together with (4.17) leads to

\begin{equation*} \mathop{\rm osc}\limits_{B_{j+1}}u\le (1-\varepsilon)\omega(r_j)=(1-\varepsilon)\sigma^{-\beta}\omega(r_{j+1}). \end{equation*}

Finally, choosing $\beta\in\left(0,\frac{sp}{q-1}\right)$ small enough such that

\begin{equation*} \sigma^\beta\geq 1-\varepsilon=1-\sigma^{\frac{sp}{q-1}-\beta}, \end{equation*}

then $\mathrm{osc}_{B_{j+1}}u\leq\omega(r_{j+1})$, and β depends on $\mathrm{\textbf{data}}$ and $\|u\|_{L^\infty(\Omega')}$. Indeed, due to (4.21), it yields that

\begin{equation*} A_0\le \frac{C_{\rm log}N^3}{\log\frac{1}{\sigma}}\leq \mu, \end{equation*}

by picking $\sigma\le \mathrm{exp}\left(-\frac{C_{\rm log}N^3}{\mu}\right)$. Then, we select $\sigma=\min\left\{\frac{1}{4},\mathrm{exp}\left(-\frac{C_{\rm log}N^3}{\mu}\right)\right\}$ to ensure the condition (4.27) does hold true. Now we finish the proof.