A new approach to Sobolev spaces in metric measure spaces

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• 作者：Tomas Sjö ; din ^{tomas.sjodin@liu.se" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 46E35 ; secondary ; 30L99 ; 31E05
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：142
• 期：Complete
• 页码：194-237
• 全文大小：788 K

文摘

Let (X,d_X,μ)$(X,d_X,\mu )$ be a metric measure space where X$X$ is locally compact and separable and a6148e8d08478d" title="Click to view the MathML source">μ$\mu$ is a Borel regular measure such that 0<μ(B(x,r))<∞$0<\mu \left(B(x,r)\right)<\infty$ for every ball B(x,r)$B(x,r)$ with center x∈X$x\in X$ and radius r>0$r>0$. We define X$\mathcal{X}$ to be the set of all positive, finite non-zero regular Borel measures with compact support in X$X$ which are dominated by a6148e8d08478d" title="Click to view the MathML source">μ$\mu$, and M=X∪{0}$\mathcal{M}=\mathcal{X}\cup \left\{0\right\}$. By introducing a kind of mass transport metric d_M$d_{\mathcal{M}}$ on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for functions 46a169af" title="Click to view the MathML source">F:X→R$F:\mathcal{X}\to \mathbb{R}$, and then for functions f:X→[−∞,∞]$f:X\to [-\infty ,\infty ]$ by identifying them with the unique element F_f:X→R$F_f:\mathcal{X}\to \mathbb{R}$ defined by the mean-value integral: In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space Rⁿ${\mathbb{R}}^n$ with Lebesgue measure.