On the isomorphism of Sobolev-type classes on metric spaces
As the definition of Sobolev spaces involves the Lebesgue integral, the elements of the space are usually considered equal if they differ on a set of measure zero. However, when considering the spaces as a whole, the situation might be different. The author is interested in properties of domains such that the corresponding Sobolev spaces agree. In this paper, domains are considered in metric spaces with mild regularity assumptions on the measure, and the Sobolev spaces are the Hajłasz spaces. Let us recall the corresponding definitions; the setting is the one where \((X,d,\mu)\) is a metric measure space. Given a \(\mu\)\nobreakdash-measurable function \(u\colon X\to \overline{\mathbb{R}}\), then a function \(g\colon X\to [0,\infty)\) is called \textit{admissible} if there exists a set \(N\subset X\) such that \(\mu(N)=0\) and \[ \lvert u(x)-u(y)\rvert\leq d(x,y)(g(x)+g(y)) \] for all points \(x\)~and~\(y\) in~\(X\setminus N\). The symbol~\(D(u)\) stands for the set of all admissible functions for~\(u\). Assuming \(p\geq 1\), the set \(D_p(u)\) is the intersection of~\(D(u)\) with~\(L_p(X,\mu)\). Finally, \begin{align*} S_p^1 & =\{u\colon X\to \overline{\mathbb{R}}\mid D_p(u)\not=\emptyset\}, \\ M_p^1(X,d,\mu) & =\{u\in L_p(X,\mu)\mid u\in S^1_p(X,d,\mu)\}. \end{align*} The spaces are equipped with seminorms \[ \lVert u\mid S_p^1\rVert=\inf_{g\in D_p(u)}\lVert g\mid L_p\rVert \] and \[ \lVert u\mid M_p^1\rVert=\lVert u\mid L_p\rVert+\lVert u\mid S_p^1\rVert. \] Let us now define when two metric measure space are equivalent. Let \((X,d,\mu)\) be a metric measure space and \(Y\subset X\). The spaces \((X,d,\mu)\)~and~\((Y,d,\mu)\) are called \emph{\(S^1_p\)\nobreakdash-equivalent} if the restriction operator~\(\theta\colon S_p^1(X,d,\mu)\to S_p^1(Y,d,\mu)\) given by \(\theta u=u|_Y\) is an isomorphism of the Sobolev-type vector spaces. The main result states that the spaces \((X,d,\mu)\)~and~\((Y,d,\mu)\), where \(Y\subset X\) and \((X,d,\mu)\) is locally homogeneous and \(\mu\) satisfies the doubling condition, are equivalent if and only if the symmetric difference of \(X\)~and~\(Y\) has measure zero. In the second part of the paper, bounded domains~\(G\) in~\(\mathbb{R}^n\) are considered.
scientific article; zbMATH DE number 7401167