Isoperimetric Inequality for degenerate elliptic operators of Grushin type

Let $n,m\ge 1$, $\alpha\in(0,1)$, and $\beta\ge 0$. For the Grushin-type operator $$L=-\nabla_x\!\cdot\!\bigl(|x|^{2\alpha}\nabla_x\bigr)+|x|^{2\beta}\Delta_y \qquad \text{on } \mathbb R^n\times \mathbb R^m,$$ we prove the isoperimetric inequality on the associated Grushin space. Equivalently, if $$Q=\frac{n+m(\beta+1-\alpha)}{1-\alpha},$$ then $$|\Omega|^{\frac{Q-1}{Q}}\le C\,P(\Omega)$$ for every smooth bounded domain $\Omega\subset \mathbb R^{n+m}$.