Grushin空间上的测度收缩性质
The measure contraction property on Grushin spaces
AI总结：
本文确定了Grushin型度量测度空间的锐利测度收缩指数，解决了相关猜想，并首次给出具有非整数曲率指数的实解析子黎曼结构例子。
AI中文摘要：
我们确定了两个Grushin型度量测度空间的锐利测度收缩指数。径向Grushin空间$\mathbb{G}^{n+m}$是$\mathbb{R}^{n}\times\mathbb{R}^{m}$，配备Lebesgue测度，由$X_i=\partial_{x_i}$和$Y_j=|x|\partial_{y_j}$生成，其中$1\leq i\leq n$且$1\leq j\leq m$。我们证明$\mathbb{G}^{n+m}$满足$\operatorname{MCP}(K,N)$当且仅当$N\geq n+4m$且$K\leq 0$。我们还证明，对于$\alpha\geq1$，由$X=\partial_x$和$Y_\alpha=|x|^\alpha\partial_y$生成的$\alpha$-Grushin平面满足$\operatorname{MCP}(K,N)$当且仅当$K\leq0$且$N\geq N_\alpha$，其中\\[ N_\alpha:= 1+\max_{L>1} \frac{(2\alpha+1)L}{(L-1)^{2\alpha+1}+1}. \\] 这解决了arXiv:2010.16350中提出的猜想，并且对于整数$\alpha\geq2$，提供了具有非整数曲率指数的实解析子黎曼结构的首批例子。两个结果都恢复了经典Grushin平面的已知曲率指数$5$，分别对应于$n=m=1$和$\alpha=1$。
英文摘要：
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 $α\geq1$, the $α$-Grushin plane generated by $X=\partial_x$ and $Y_α=|x|^α\partial_y$ 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 $α=1$.
