In this paper, we establish weighted higher order exponential type inequalities in the geodesic space $(X,d,\mu )$ by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

中文翻译：

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度量空间和应用中的加权高阶指数类型不等式

在本文中，我们建立了测地空间中的加权高阶指数型不等式 $（X，d，\mu ）$通过提出抽象的高阶庞加莱不等式。在非加权情况下，这些也是新的。作为应用程序，我们在测地环境中获得了加权的Trudinger定理，并为分层Lie组上定义的Folland-Stein型Sobolev空间中的函数提供了加权的高阶指数类型估计。还给出了连通齐次空间中的高阶指数型不等式。