\int_{B_r^+}
hV_t(\nabla U\cdot z)\,dz.\end{aligned}$$ Hence, taking into account Lemma \[l:Poincare\], $$\begin{aligned}
 \label{eq:*}
  \left|\int_{B_r'}\!\!hu\,x\cdot
  \nabla_x v\,dx\right|\!\leq\!{\rm const\,} \!\!\left(
r\sqrt{r^N H(r)B(r)}
+r\!\!\int_{B_r^+}(|\nabla U|^2+|\nabla V|^2)\,dz\!+\!
\int_{S_r^+}(U^2+V^2)\,dS
\right)\end{aligned}$$ for some ${\rm const\,>}0$ independent of $r$. In a similar way we obtain that $$\begin{aligned}
  \left|\int_{B_r'}hv\,x\cdot
  \nabla_x u\,dx\right|\leq{\rm const\,} \left(
r\sqrt{r^N H(r)B(r)}
+r\int_{B_r^+}(|\nabla U|^2+|\nabla V|^2)\,dz+
\int_{S_r^+}(U^2+V^2)\,dS
\right).\end{aligned}$$ As a consequence, in view of we conclude that $$\label{eq:17}
\frac{\left|-r\int_{\partial B_r'}huv\,dS'+2\int_{B_r'}hu\,x\cdot
  \nabla_x v\,dx\right|} {\int_{S_r^+}
    (U^2+V^2)\,dS}{\leqslant}\mathcal N(r)O(1)+\sqrt{\frac{B(r)}{r^NH(r)}}\,O(r)+O(1)$$ as $r\to0^+$.

Inserting - into the proof of the lemma follows.

Inspired by [@FF-edinb Lemma 5.9], in the following lemma we estimate $B$ in terms of the derivative $D'$.

\[l:stima\_B\] Let $B$ be defined in . Then there exist $C_1,C_2,\bar
r>0$ such that $$B(r){\leqslant}2 r^{N-1}D'(r)+C_1 r^{N-2} (D(r)+C_2 H(r)) \quad\text{and}\quad
D(r)+C_2H(r){\geqslant}0\quad\text{for all }r\in
(0,\bar r).$$

From the definition of $D$ (see ) we have that $$\label{eq:21}
  D'(r)=r^{1-N}B(r)-(N-1)r^{-1}D(r)+r^{1-N}\int_{S_r^+}UV\,dS-r^{1-N}\int_{\partial B_r'}huv\,dS'.$$ From it follows that $$\int_{\partial B_r'}huv\,dS'=\frac1r\int_{B_r'}hu\,x\cdot
  \nabla_x v\,dx+\frac1r \int_{B_r'}hv\,x\cdot
  \nabla_x u\,dx+\frac1r \int_{B_r'}uv(Nh+x\cdot
  \nabla h)\,dx.$$ By and we deduce that, for every $\e>0$, there exists $C_\e>0$ such that $$\begin{aligned}
\bigg|\frac1r\int_{B_r'}&hu\,x\cdot
  \nabla_x v\,dx\bigg|\\
&{\leqslant}{\varepsilon}B(r)+C_{\varepsilon}r^N H(r)+O(1) \int_{B_{r}^+} \left(|\nabla U|^2+|\nabla
V|^2\right)dz+O(1)r^{N-1}H(r)\\
&{\leqslant}{\varepsilon}B(r)+O(1)r^{N-1}H(r)+O(1)r^{N-1}D(r)\quad\text{as }r\to0^+.\end{aligned}$$ An analogous estimate holds for the term $\frac1r \int_{B_r'}hv\,x\cdot
  \nabla_x u\,dx$, whereas implies that $$\frac1r \int_{B_r'}uv(Nh+x\cdot
  \nabla h)\,dx =O(1)r^{N-1}H(r)+O(1)r^{N-1}D(r) \quad\text{as }r\to0^+.$$ Therefore we conclude that $$\label{eq:20}
  