The measure contraction property on Grushin spaces

RESUMO

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 $X_i=\partial_{x_i}$ and $Y_j=|x|\partial_{y_j}$, for $1\leq i\leq n$ and $1\leq j\leq m$. We prove that $\mathbb{G}^{n+m}$ satisfies $\operatorname{MCP}(K,N)$ if and only if $N\geq n+4m$ and $K\leq 0$. We also show that, for $\alpha\geq1$, the $\alpha$-Grushin plane generated by $X=\partial_x$ and $Y_\alpha=|x|^\alpha\partial_y$ satisfies $\operatorname{MCP}(K,N)$ if and only if $K\leq0$ and $N\geq N_\alpha$, where \[ N_\alpha := 1+\max_{L>1} \frac{(2\alpha+1)L}{(L-1)^{2\alpha+1}+1}. \] This resolves the conjecture posed in arXiv:2010.16350 and, for integer $\alpha\geq2$, 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 $\alpha=1$.