On the isomorphism of Sobolev-type classes on metric spaces
type: Item
instance of: scholarly article
title: On the isomorphism of Sobolev-type classes on metric spaces
zbMATH Open document ID: 1481.46031
DOI: 10.1134/S0037446621040133
author: N. E. Zubov
published in: Siberian Mathematical Journal
publication date: 2021-09-27T00:00:00Z
review text: 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 …
reviewed by: Thomas Zürcher
Mathematics Subject Classification ID: 46E36
Mathematics Subject Classification ID: 30L05
zbMATH DE Number: 7401167
zbMATH Keywords: metric spaces
zbMATH Keywords: Sobolev-type functions
zbMATH Keywords: isomorphism
MaRDI profile type: Publication
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full work available at URL: s0037446621040133
OpenAlex ID: W3217785725
description: scientific article; zbMATH DE number 7401167