By scaling analysis we may assume $A\sim 1$. Let $u$ be a $T^{2}$-invariant function with $\int_{M}|\nabla u|^{2}=1$, so descends to a function on the base $\mathbb{R}^{4}$. Since the weighted Sobolev inequality holds on Euclidean $\mathbb{R}^{4}$ (by an interpolation of standard Sobolev inequality and Hardy inequality),

$$(\int_{\mathbb{R}^{4}}|u|^{2p}(1+|\vec{\mu}|_{a})^{2p-4}d\text{Vol}_{a})^{1/p}\leq C\int_{\mathbb{R}^{4}}|\nabla_{g_{a}}u|^{2}d\text{Vol}_{a}\leq C\int_{M}|\nabla_{g^{(2)}}u|^{2}d\text{Vol}\leq C,$$

where the second inequality is easily seen using the model metric in Section 2.3. The LHS in this inequality is uniformly equivalent to the LHS in (2.20) except in the region $\{\text{dist}_{g_{a}}(\cdot,\mathfrak{D})\leq 1\}$. So we are left to prove

$$(\int_{\text{dist}(\cdot,\mathfrak{D})\lesssim 1}|u|^{2p}(1+|\vec{\mu}|_{a})^{2p-4}d\text{Vol})^{1/p}\leq C.$$

For $x\in\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$, Sobolev inequality on bounded balls imply

$$(\int_{B(x,1)}|u-\bar{u}(x)|^{2p})^{1/p}\leq C\int_{B(x,2)}|\nabla u|^{2},\quad\bar{u}(x)=\text{Vol}(B(x,1))^{-1}\int_{B(x,1)}u$$

Furthermore we can find a point $x^{\prime}$ with $\text{dist}(x,x^{\prime})\leq 3$, $\text{dist}(x^{\prime},\mathfrak{D})\gtrsim 2$, and by Sobolev inequality

$$(\int_{B(x^{\prime},1)}|u-\bar{u}(x^{\prime})|^{2p})^{1/p}\leq C\int_{B(x,5)}|\nabla u|^{2},\quad\bar{u}(x^{\prime})=\text{Vol}(B(x^{\prime},1))^{-1}\int_{B(x^{\prime},1)}u.$$

By Poincaré inequality

$$|\bar{u}(x)-\bar{u}(x^{\prime})|^{2}\leq\int_{B(x,5)}|\nabla u|^{2}.$$

Combining these,

$$(\int_{B(x,1)}|u|^{2p})^{1/p}\leq C(\int_{B(x^{\prime},1)}|u|^{2p})^{1/p}+C\int_{B(x,5)}|\nabla u|^{2}.$$

Multiplying this inequality by $(1+|\vec{\mu}|_{a}(x))^{(2p-4)/p}$, and summing over $x\in\mathfrak{D}$, we obtain

$$\begin{split}&(\int_{\text{dist}(\cdot,\mathfrak{D})\lesssim 1}|u|^{2p}(1+|\vec{\mu}|_{a})^{2p-4}d\text{Vol})^{1/p}\\ \leq&C(\int_{\text{dist}(\cdot,\mathfrak{D})\gtrsim 1}|u|^{2p}(1+|\vec{\mu}|_{a})^{2p-4}d\text{Vol})^{1/p}+C\int|\nabla u|^{2}\leq C\end{split}$$

as required. ∎