The measure contraction property on Grushin spaces

We determine the sharp measure contraction exponents of two families of Grushin-type metric measure spaces. The radial Grushin space ${\mathbb{G}}^{n+m}$ is ${\mathbb{R}}^n\times {\mathbb{R}}^m$, equipped with Lebesgue measure and generated by $X_i={\mathrm{\partial }}_{x_i}$ and $Y_j=\mathrm{\mid }x\mathrm{\mid }{\mathrm{\partial }}_{y_j}$, for $1\le i\le n$ and $1\le j\le m$. We prove that ${\mathbb{G}}^{n+m}$ satisfies $\mathrm{MCP}(K,N)$ if and only if $N\ge n+4m$ and $K\le 0$. We also show that, for $\alpha \ge 1$, the $\alpha$-Grushin plane generated by $X={\mathrm{\partial }}_x$ and $Y_{\alpha }=\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }{\mathrm{\partial }}_y$ satisfies $\mathrm{MCP}(K,N)$ if and only if $K\le 0$ and $N\ge N_{\alpha }$, where $$N_{\alpha }:=1+\underset{L>1}{\max }\frac{(2\alpha +1)L}{(L-1{)}^{2\alpha +1}+1}\mathrm{.}$$ This resolves the conjecture posed in arXiv:2010.16350 and, for integer $\alpha \ge 2$, provides the first examples of real-analytic sub-Riemannian structures with noninteger curvature exponent. Both results recover the known curvature exponent $5$ of the classical Grushin plane, corresponding respectively to $n=m=1$ and $\alpha =1$.