论文标题
在Lipschitz同源班的模量上
On the Moduli of Lipschitz Homology Classes
论文作者
论文摘要
我们定义了一种基于$ l^p $ - 积聚可测量的差分形式的Lipschitz表面的模量$ \ peripatatorName {dmod} _p $，从而推广了Aikawa和ohtsuka的矢量模量。我们表明，该模量满足同源二元定理，其中Hölder共轭指数$ P，q \ in（1，\ infty）$，每个相对Lipschitz $ k $ k $ c $ c $ c $ c $具有独特的dualipchitz $（n-k）$ - 同源级$ C'$ $ \ operatoTorname {dmod} _p^{1/p}（c）\ propatatorName {dmod} _q^{1/q} {1/q}（c'）= 1 $和$ c $ c $ c $ c $ c $ c $ c $ c $ c'$ to to 1。as $ \ operate operatial coptical coltical clangorane { $ \ operatorname {mod} _p $，我们立即恢复了估计$ \ peripatorName {mod} _p^{1/p} {1/p}（c）\ operatorname {mod} _q^{1/q}（c'}（c'c'）\ leq 1 $，在freedman和freedman and y lohvans中出现的工作。我们的理论是在Lipschitz Riemannian歧管的一般环境中提出的，尽管我们的结果在平滑的环境中也是新的。我们还根据Lipschitz $ K $链的集成在Lipschitz歧管上的封闭式和精确的Sobolev形式的表征。
We define a type of modulus $\operatorname{dMod}_p$ for Lipschitz surfaces based on $L^p$-integrable measurable differential forms, generalizing the vector modulus of Aikawa and Ohtsuka. We show that this modulus satisfies a homological duality theorem, where for Hölder conjugate exponents $p, q \in (1, \infty)$, every relative Lipschitz $k$-homology class $c$ has a unique dual Lipschitz $(n-k)$-homology class $c'$ such that $\operatorname{dMod}_p^{1/p}(c) \operatorname{dMod}_q^{1/q}(c') = 1$ and the Poincaré dual of $c$ maps $c'$ to 1. As $\operatorname{dMod}_p$ is larger than the classical surface modulus $\operatorname{Mod}_p$, we immediately recover a more general version of the estimate $\operatorname{Mod}_p^{1/p}(c) \operatorname{Mod}_q^{1/q}(c') \leq 1$, which appears in works by Freedman and He and by Lohvansuu. Our theory is formulated in the general setting of Lipschitz Riemannian manifolds, though our results appear new in the smooth setting as well. We also provide a characterization of closed and exact Sobolev forms on Lipschitz manifolds based on integration over Lipschitz $k$-chains.
