非Ahlfors正则空间中的度量拟共形与Sobolev正则性
Metric quasiconformality and Sobolev regularity in non-Ahlfors regular spaces
摘要：
给定$Q$维空间$X,Y$之间的一个同胚$f\colon X\to Y$，我们证明了$f$在合适的例外集之外满足拟共形的度量定义意味着$f$属于Sobolev类$N_{\rm{loc}}^{1,p}(X;Y)$，其中$1< p\le Q$，也意味着拟共形的几何定义的一个方向.与以前的结果不同，我们只假设Ahlfors $Q$ -正则性的逐点版本，这特别使各种加权空间被包括在理论中。出乎意料的是，我们可以应用这一点，以获得新的结果，甚至在经典的欧几里德设置。特别地，在包含卡诺群的空间中，我们能够证明Sobolev正则性$f\in N_{\rm{loc}}^{1,Q}(X;Y)$，而不需要强假设无限小畸变$h_f$属于$L^{\infty}(X)$。
英文摘要：
Given a homeomorphism $f\colon X\to Y$ between $Q$-dimensional spaces $X,Y$, we show that $f$ satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that $f$ belongs to the Sobolev class $N_{\rm{loc}}^{1,p}(X;Y)$, where $1< p\le Q$, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors $Q$-regularity, which particularly enables various weighted spaces to be included in the theory. Unexpectedly, we can apply this to obtain results that are new even in the classical Euclidean setting. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity $f\in N_{\rm{loc}}^{1,Q}(X;Y)$ without the strong assumption of the infinitesimal distortion $h_f$ belonging to $L^{\infty}(X)$.
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