双重度量测度空间上的Absolutely continuous映射
Absolutely continuous mappings on doubling metric measure spaces
摘要：
根据Malcolm的多元绝对连续函数的定义，我们考虑了$Q$ -在双倍度量测度空间$X$和Banach空间$V$之间的绝对连续映射$f\colon X\to V$.研究了这些映射与$p\ge Q$的Sobolev映射$f\in N^{1,p}(X;V)$之间的关系.特别地，Ahlfors $Q$-正则空间上的局部$Q$-绝对连续映射是$N^{1,Q}_{\rm{loc}}(X;V)$中的连续映射，并且如果$V$满足Radon-Nikodym性质，则几乎处处可微。相反，虽然连续的Sobolev映射$f\in N^{1,Q}_{\rm{loc}}(X;V)$通常不是局部$Q$ -绝对连续的，但是如果进一步假设$f$是伪单调的，则该含义成立。满足松弛拟共形条件的伪单调映射也是$Q$ -绝对连续的.
英文摘要：
Following Malý's definition of absolutely continuous functions of several variables, we consider $Q$-absolutely continuous mappings $f\colon X\to V$ between a doubling metric measure space $X$ and a Banach space $V$. The relation between these mappings and Sobolev mappings $f\in N^{1,p}(X;V)$ for $p\ge Q$ is investigated. In particular, a locally $Q$-absolutely continuous mapping on an Ahlfors $Q$-regular space is a continuous mapping in $N^{1,Q}_{\rm{loc}}(X;V)$, as well as differentiable almost everywhere in terms of Cheeger derivatives provided $V$ satisfies the Radon-Nikodym property. Conversely, though a continuous Sobolev mapping $f\in N^{1,Q}_{\rm{loc}}(X;V)$ is generally not locally $Q$-absolutely continuous, this implication holds if $f$ is further assumed to be pseudomonotone. It follows that pseudomonotone mappings satisfying a relaxed quasiconformality condition are also $Q$-absolutely continuous.
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。
