The measure contraction property on Grushin spaces

Abstract: We determine the sharp measure contraction exponents of two families of Grushin-type metric measure spaces. The radial Grushin space $\mathbb{G}<sup>{n+m}$ is $\mathbb{R}<sup>{n}\times\mathbb{R}<sup>{m}$, 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 $\mathbb{G}<sup>{n+m}$ satisfies $\operatorname{MCP}(K,N)$ if and only if $N\geq n+4m$ and $K\leq 0$. We also show that, for $α\geq1$, the $α$-Grushin plane generated by $X=\partial_x$ and $Y_α=|x|<sup>α\partial_y$ satisfies $\operatorname{MCP}(K,N)$ if and only if $K\leq0$ and $N\geq N_α$, where [ N_α:= 1+\max_{L>1} \frac{(2α+1)L}{(L-1)^{2α+1}+1}. ] This resolves the conjecture posed in arXiv:2010.16350 and, for integer $α\geq2$, 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 $α=1$.