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Mat. Sb., 2020, Volume 211, Number 6, Pages 40–94 (Mi msb9199)  

This article is cited in 1 scientific paper (total in 1 paper)

Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$

S. K. Vodopyanova, A. I. Tyulenevb*

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: Let $S\subset\mathbb R^n$ be a nonempty closed set such that for some $d\in[0,n]$ and $\varepsilon>0$ the $d$-Hausdorff content $\mathscr H^d_\infty(S\cap Q(x,r))\geq\varepsilon r^d$ for all cubes $Q(x,r)$ with centre $x\in S$ and edge length $2r\in(0,2]$. For each $p>\max\{1,n-d\}$ we give an intrinsic characterization of the trace space $W_p^1(\mathbb R^n)|_S$ of the Sobolev space $W_p^1(\mathbb R^n)$ to the set $S$. Furthermore, we prove the existence of a bounded linear operator $\operatorname{Ext}\colon W_p^1(\mathbb R^n)|_S\to W_p^1(\mathbb R^n)$ such that $\operatorname{Ext}$ is the right inverse to the standard trace operator. Our results extend those available in the case $p\in(1,n]$ for Ahlfors-regular sets $S$.
Bibliography: 36 titles.

Keywords: Sobolev spaces, Whitney problem, traces, extension operators.

Funding Agency Grant Number
Ministry of Science and Higher Education of the Russian Federation 0314-2019-0006
The research of S. K. Vodopyanov was carried out within the framework of a state assignment of the Ministry of Education and Science of the Russian Federation for the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (project no. 0314-2019-0006).

* Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm9199

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English version:
Sbornik: Mathematics, 2020, 211:6, 786–837

Bibliographic databases:

UDC: 517.518
MSC: 46E35, 28A78, 28A25
Received: 27.11.2018 and 14.02.2020

Citation: S. K. Vodopyanov, A. I. Tyulenev, “Sobolev $W^1_p$-spaces on $d$-thick closed subsets of $\mathbb R^n$”, Mat. Sb., 211:6 (2020), 40–94; Sb. Math., 211:6 (2020), 786–837

Citation in format AMSBIB
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\by S.~K.~Vodopyanov, A.~I.~Tyulenev
\paper Sobolev $W^1_p$-spaces on~$d$-thick closed subsets of $\mathbb R^n$
\jour Mat. Sb.
\yr 2020
\vol 211
\issue 6
\pages 40--94
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\crossref{https://doi.org/10.4213/sm9199}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=4104775}
\elib{https://elibrary.ru/item.asp?id=45186168}
\transl
\jour Sb. Math.
\yr 2020
\vol 211
\issue 6
\pages 786--837
\crossref{https://doi.org/10.1070/SM9199}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85094592409}


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    This publication is cited in the following articles:
    1. A. I. Tyulenev, “Almost Sharp Descriptions of Traces of Sobolev Spaces on Compacta”, Math. Notes, 110:6 (2021), 976–980  mathnet  crossref  crossref
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