标题： 特殊不可逆Finsler流形上的尖锐Hardy不等式和谱间隙不等式 显示英文标题

标题： Sharp Hardy and spectral gap inequalities on special irreversible Finsler manifolds

摘要： 在可逆Finsler设置中，各种类型的Hardy不等式的精确性已经被很好地理解；而无限可逆性意味着这些函数不等式的失效，参见Kristály、Huang和Zhao [Trans. Am. Math. Soc., 2020]。 然而，在具有有限可逆性的不可逆流形的情况下，关于类型Hardy不等式的精确性没有证据。 事实上，我们不知道有任何特定的例子表明精确性仍然存在。 在本文中，我们提出了两个这样的例子，涉及两种著名的不等式：经典的/加权Hardy不等式（假设非正的旗曲率）和McKean类型的谱间隙估计（假设强负的旗曲率）。 在这两种情况下，我们提供了一族Finsler度量测度流形，在这些流形上这些不等式是精确的。 我们还建立了某些充分条件，这些条件保证了这些空间上更复杂的类型Hardy不等式的精确性。 我们相关的技术工具是Riccati对方法（用于证明Hardy不等式）的Finsler扩展，这也启发了我们构造的主要思想。 显示英文摘要

摘要： The sharpness of various Hardy-type inequalities is well-understood in the reversible Finsler setting; while infinite reversibility implies the failure of these functional inequalities, cf. Krist\'aly, Huang, and Zhao [Trans. Am. Math. Soc., 2020]. However, in the remaining case of irreversible manifolds with finite reversibility, there is no evidence on the sharpness of Hardy-type inequalities. In fact, we are not aware of any particular examples where the sharpness persists. In this paper we present two such examples involving two celebrated inequalities: the classical/weighted Hardy inequality (assuming non-positive flag curvature) and the McKean-type spectral gap estimate (assuming strong negative flag curvature). In both cases, we provide a family of Finsler metric measure manifolds on which these inequalities are sharp. We also establish some sufficient conditions, which guarantee the sharpness of more involved Hardy-type inequalities on these spaces. Our relevant technical tool is a Finslerian extension of the method of Riccati pairs (for proving Hardy inequalities), which also inspires the main ideas of our constructions.