Geometric characterization of generalized Hajłasz-Sobolev embedding domains
In this article, the authors study the embedding properties of Hajłasz-Sobolev spaces with generalized smoothness on Euclidean domains, whose regularity is described via a smoothness weight function ϕ:[0,∞)→[0,∞)\phi :\left[0,\infty )\to \left[0,\infty ). Given any bounded domain with the slice prop...
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| Language: | English |
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De Gruyter
2025-03-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2025-0077 |
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| author | Li Ziwei Yang Dachun Yuan Wen |
| author_facet | Li Ziwei Yang Dachun Yuan Wen |
| author_sort | Li Ziwei |
| collection | DOAJ |
| description | In this article, the authors study the embedding properties of Hajłasz-Sobolev spaces with generalized smoothness on Euclidean domains, whose regularity is described via a smoothness weight function ϕ:[0,∞)→[0,∞)\phi :\left[0,\infty )\to \left[0,\infty ). Given any bounded domain with the slice property, the authors prove that it is a generalized Hajłasz-Sobolev embedding domain if and only if it is a generalized FF-weak cigar domain, where FF is a modulus of continuity related to the weight function ϕ\phi of the generalized Hajłasz-Sobolev spaces under consideration. Comparing with the classical Hajłasz-Soblev spaces, one main difficulty in dealing with generalized Hajłasz-Sobolev spaces lies in that both its smoothness weight function ϕ\phi and the related modulus of continuity FF have no explicit expression. To overcome this, the authors introduce and use some key indices to accurately describe the increasing or the decreasing behavior of both ϕ\phi and FF. Besides the classical Hajłasz-Sobolev spaces, this result can be applied to many other nontrivial spaces such as Hajłasz-Sobolev spaces with logarithmic smoothness and is of wide generality. |
| format | Article |
| id | doaj-art-314644711dcf4783a2b4e967db72efc2 |
| institution | OA Journals |
| issn | 2191-950X |
| language | English |
| publishDate | 2025-03-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj-art-314644711dcf4783a2b4e967db72efc22025-08-20T02:17:13ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2025-03-0114124125710.1515/anona-2025-0077Geometric characterization of generalized Hajłasz-Sobolev embedding domainsLi Ziwei0Yang Dachun1Yuan Wen2Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, The People’s Republic of ChinaLaboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, The People’s Republic of ChinaLaboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, The People’s Republic of ChinaIn this article, the authors study the embedding properties of Hajłasz-Sobolev spaces with generalized smoothness on Euclidean domains, whose regularity is described via a smoothness weight function ϕ:[0,∞)→[0,∞)\phi :\left[0,\infty )\to \left[0,\infty ). Given any bounded domain with the slice property, the authors prove that it is a generalized Hajłasz-Sobolev embedding domain if and only if it is a generalized FF-weak cigar domain, where FF is a modulus of continuity related to the weight function ϕ\phi of the generalized Hajłasz-Sobolev spaces under consideration. Comparing with the classical Hajłasz-Soblev spaces, one main difficulty in dealing with generalized Hajłasz-Sobolev spaces lies in that both its smoothness weight function ϕ\phi and the related modulus of continuity FF have no explicit expression. To overcome this, the authors introduce and use some key indices to accurately describe the increasing or the decreasing behavior of both ϕ\phi and FF. Besides the classical Hajłasz-Sobolev spaces, this result can be applied to many other nontrivial spaces such as Hajłasz-Sobolev spaces with logarithmic smoothness and is of wide generality.https://doi.org/10.1515/anona-2025-0077hajłasz-sobolev embeddinggeneralized smoothnesscigar domainself-improvingprimary 46e35secondary 46e3642b3530l99 |
| spellingShingle | Li Ziwei Yang Dachun Yuan Wen Geometric characterization of generalized Hajłasz-Sobolev embedding domains Advances in Nonlinear Analysis hajłasz-sobolev embedding generalized smoothness cigar domain self-improving primary 46e35 secondary 46e36 42b35 30l99 |
| title | Geometric characterization of generalized Hajłasz-Sobolev embedding domains |
| title_full | Geometric characterization of generalized Hajłasz-Sobolev embedding domains |
| title_fullStr | Geometric characterization of generalized Hajłasz-Sobolev embedding domains |
| title_full_unstemmed | Geometric characterization of generalized Hajłasz-Sobolev embedding domains |
| title_short | Geometric characterization of generalized Hajłasz-Sobolev embedding domains |
| title_sort | geometric characterization of generalized hajlasz sobolev embedding domains |
| topic | hajłasz-sobolev embedding generalized smoothness cigar domain self-improving primary 46e35 secondary 46e36 42b35 30l99 |
| url | https://doi.org/10.1515/anona-2025-0077 |
| work_keys_str_mv | AT liziwei geometriccharacterizationofgeneralizedhajłaszsobolevembeddingdomains AT yangdachun geometriccharacterizationofgeneralizedhajłaszsobolevembeddingdomains AT yuanwen geometriccharacterizationofgeneralizedhajłaszsobolevembeddingdomains |