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 ${\mathbb{G}}^{n+m}$ 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 $1\le i\le n$ 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 $X={\partial }_x$ and $Y_{\alpha }=|x{|}^{\alpha }{\partial }_y$ satisfies $\mathrm{MCP}(K,N)$ if and only if $K\le 0$ and $N\ge N_{\alpha }$, 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 $\alpha \ge 2$, 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 $\alpha =1$.

27 pages, 1 figure