曲率下界下容量的Maz'ya不等式的逆
A converse to Maz'ya's inequality for capacities under curvature lower bound
摘要：
本文综述了Maz'ya在度量空间中将等容不等式与其对应的泛函不等式和等周不等式联系起来的一些经典不等式，并推广了Maz'ya关于$q$ -容量（$q>1$）的下界.然后，我们继续描述Buser，Bakry，Ledoux和最近的作者的结果，这表明，在适当的凸性假设下的测度度量空间，Maz'ya的不等式的容量可能会被逆转，到维数独立的数值常数：匹配的下界1-容量可以推导出的$q$ -容量配置文件。我们将这些结果推广到任意的$q > 1$和弱半凸性假设，得到了一些新的精确的半群估计。
英文摘要：
We survey some classical inequalities due to Maz'ya relating isocapacitary inequalities with their functional and isoperimetric counterparts in a measure-metric space setting, and extend Maz'ya's lower bound for the $q$-capacity ($q>1$) in terms of the 1-capacity (or isoperimetric) profile. We then proceed to describe results by Buser, Bakry, Ledoux and most recently by the author, which show that under suitable convexity assumptions on the measure-metric space, Maz'ya's inequality for capacities may be reversed, up to dimension independent numerical constants: a matching lower bound on 1-capacity may be derived in terms of the $q$-capacity profile. We extend these results to handle arbitrary $q > 1$ and weak semi-convexity assumptions, by obtaining some new delicate semi-group estimates.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
