发表机构
Amherst College; The Ohio State University（阿默斯特学院; 俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文用纯度量、无导数条件刻画了超临界Sobolev空间中的连续映射，证明其等价于紧Hölder映射，并推广到一般度量空间，建立了半范数的定量比较。
AI 中文摘要
本文给出了局部超临界Sobolev空间中连续映射的纯度量且无导数刻画。具体而言，我们证明连续映射$f\colon\mathbb{R}^n\to\mathbb{R}^m$属于局部Sobolev空间$W^{1,p}_\text{loc}(\mathbb{R}^n\colon\mathbb{R}^m)$当且仅当$f$是$(p,1-n/p)$-紧Hölder映射，其中$p>n$。事实上，我们证明了以下更一般的结果：只要$X$是完备的$Q$-Ahlfors正则度量空间，且支持$p>Q$时的$p$-Poincaré不等式，$V$是任意Banach空间，紧Hölder类$CH^{p,1-Q/p}(X\colon V)$恰好由局部Hajłasz和Newtonian Sobolev空间$M^{1,p}_\text{loc} (X \colon V)$和$N^{1,p}_\text{loc}(X\colon V)$的连续代表元组成。我们还建立了相应半范数之间的定量比较。
英文摘要
In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping $f\colon\mathbb{R}^n\to\mathbb{R}^m$ lies in the local Sobolev space $W^{1,p}_\text{loc}(\mathbb{R}^n\colon\mathbb{R}^m)$ if and only if $f$ is a $(p,1-n/p)$-compactly Hölder mapping, where $p>n$. In fact, we prove the following more general result: whenever $X$ is a complete $Q$-Ahlfors regular metric space supporting a $p$-Poincaré inequality with $p>Q$, and $V$ is any Banach space, the compactly Hölder class $CH^{p,1-Q/p}(X\colon V)$ consists precisely of the continuous representatives of the local Hajłasz and Newtonian Sobolev spaces $M^{1,p}_\text{loc} (X \colon V)$ and $N^{1,p}_\text{loc}(X\colon V)$. We also establish quantitative comparisons between the corresponding seminorms.
Comments26 pages
