\left|\int_{B_r^+}UV\,dz\right|{\leqslant}\frac{r^{1+N}}{2N}H(r)+\frac{r^2}{2N}\int_{B_{r}^+} \left(|\nabla U|^2+|\nabla
V|^2\right)dz$$ whereas Lemma \[l:hardyball\] implies that, for all $r\in(0,r_0)$, $$\begin{aligned}
\label{eq:3}
\left|\int_{B_r'}huv\,dx\right|&{\leqslant}\|h\|_{L^{\infty}(B_{r_0}')}\frac{r}2\int_{B_r'}
 \frac{u^2+v^2}{|x|} \,dx\\
\notag&{\leqslant}\|h\|_{L^{\infty}(B_{r_0}')}\frac{r}{2\widetilde\Lambda}\left(\int_{B_{r}^+} \left(|\nabla U|^2+|\nabla
V|^2\right)dz\right)
+ \|h\|_{L^{\infty}(B_{r_0}')}\frac{N-1}{4\widetilde\Lambda}r^NH(r).\end{aligned}$$ From and it follows that $$\begin{aligned}
  D(r)&{\geqslant}r^{1-N} \left(\int_{B_r^+}(|\nabla U|^2+|\nabla
V|^2)dz\right)\left(1-\frac{r^2}{2N}-
        \|h\|_{L^{\infty}(B_{r_0}')}\frac{r}{2\widetilde\Lambda}\right)\\
&\qquad -rH(r)\left(\frac{r}{2N}+\|h\|_{L^{\infty}(B_{r_0}')}\frac{N-1}{4\widetilde\Lambda}\right).\end{aligned}$$ The proof of is thereby complete. Estimate follows by combination of and .

\[rem:st\] We observe that estimates and can be rewritten as $$\begin{aligned}
\label{eq:8}
&\int_{B_{r}^+} \left(|\nabla U|^2+|\nabla
V|^2\right)dz{\leqslant}D(r) r^{N-1}(1+O(r))+H(r)O(r^N),\\
\label{eq:9}
&\int_{B_{r}^+} (U^2+V^2)dz{\leqslant}\frac1N r^{N+1}D(r) (1+O(r))+H(r)O(r^{N+1}),\end{aligned}$$ as $r\to0^+$.

\[l:hprime\] We have that $H\in
  W^{1,1}_{{\rm loc}}(0,R)$ and $$\begin{aligned}
\label{H'}
 & H'(r)=2r^{-N} \int_{S_r^+}\big(U\tfrac{\partial
    U}{\partial\nu}+V\tfrac{\partial
    V}{\partial\nu}\big)
  \, dS, \quad \text{in a distributional sense and for a.e. $r\in (0,R)$},\\
 &\label{H'2} H'(r)=\frac2r D(r), \quad \text{for every } r\in (0, R).\end{aligned}$$

See the proof of [@FF Lemma 3.8].

\[l:dprime\] The function $D$ defined in belongs to $W^{1,1}_{{\rm\
      loc}}(0, R)$ and $$\begin{aligned}
\label{D'F}
  D'(r)=&\frac{2}{r^{N-1}} \int_{S_r^+}
          \left(\left|\frac{\partial U}{\partial \nu}\right|^2+
          \left|\frac{\partial V}{\partial \nu}\right|^2\right) dS
+\frac{1}{r^{N-1}} \int_{S_r^+}  UV\,dS\\
\notag&-\frac{2}{r^{N}}\int_{B_r^+}V\,\nabla U\cdot
  z\,dz-\frac{N-1}{r^{N}}\int_{B_r^+} UV\,dz\\
\notag&+\frac{N-1}{r^{N}}\int_{B_r'}huv-\frac{1}{r^{N-1}}\int_{\partial
  B_r'}huv\,dS'+2\frac1{r^{N}}\int_{B_r'}hu(x\cdot\nabla_x v)\,dx\end{aligned}$$ in a distributional sense and for a.e. $r\in (0,R)$.

For any $r\in (0,R)$ let $$\begin{aligned}
\label{I(r)}
I(r)= \int_{B_r^+} \left(|\nabla U|^2+|\nabla V|^2+UV\right)dz-
\int_{B_r'} h(x)u(x)v(x)\, dx .\end{aligned}$$ From the fact that $U,V\in  H^1(B_R^+)$ and Lemma \[l:hardyball\] it follows that $I\in
 W^{1,1}(0,R)$ and $$\label{I'(r)}
I'(r) =
\int_{S_r^+} \left(|\nabla U|^2+|\nabla V|^2+UV\right)\,dS-
\int_{\partial B_r'} h(x)u(x)v(x)\, 