For $1\leq p\leq\frac{3}{2}$, the weighted Sobolev inequality

(2.20)

$$(\int_{M}|u|^{2p}(A^{-1/4}+|\vec{\mu}|_{a})^{2p-4}d\text{Vol})^{1/p}\leq C\int_{M}|\nabla u|^{2}d\text{Vol}$$

holds for $T^{2}$-invariant functions on $(\mathbb{C}^{3},g^{(2)})$. The constant here depends only on the scale invariant ellipticity bound (2.11)