Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case
Let (X, d, µ) be a doubling metric measure space with doubling dimension γ, i. e. for any balls B(x, R) and B(x, r), r < R, following inequality holds µ(B(x, R)) ≤ aµ (R/r)γµ(B(x, r)) for some positive constants γ and aµ. Hajłasz – Sobolev space Mpα(X) can be defined upon such general structure....
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| Format: | Article |
| Language: | Belarusian |
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Belarusian State University
2020-03-01
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| Series: | Журнал Белорусского государственного университета: Математика, информатика |
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| Online Access: | https://journals.bsu.by/index.php/mathematics/article/view/1139 |
| _version_ | 1818065387646877696 |
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| author | Sergey A. Bondarev |
| author_facet | Sergey A. Bondarev |
| author_sort | Sergey A. Bondarev |
| collection | DOAJ |
| description | Let (X, d, µ) be a doubling metric measure space with doubling dimension γ, i. e. for any balls B(x, R) and B(x, r), r < R, following inequality holds µ(B(x, R)) ≤ aµ (R/r)γµ(B(x, r)) for some positive constants γ and aµ. Hajłasz – Sobolev space Mpα(X) can be defined upon such general structure. In the Euclidean case Hajłasz – Sobolev space coincides with classical Sobolev space when p > 1, α = 1. In this article we discuss inclusion of functions from Hajłasz – Sobolev space Mpα(X) into the space of continuous functions for p ≤ 1 in the critical case γ = α p. More precisely, it is shown that any function from Hajłasz – Sobolev class Mpα(X), 0 < p ≤ 1, α > 0, has a continuous representative in case of uniformly perfect space (X, d, µ). |
| first_indexed | 2024-12-10T14:51:05Z |
| format | Article |
| id | doaj.art-13f1955bc1f147ecb8d0d1fa5f2c0c93 |
| institution | Directory Open Access Journal |
| issn | 2520-6508 2617-3956 |
| language | Belarusian |
| last_indexed | 2024-12-10T14:51:05Z |
| publishDate | 2020-03-01 |
| publisher | Belarusian State University |
| record_format | Article |
| series | Журнал Белорусского государственного университета: Математика, информатика |
| spelling | doaj.art-13f1955bc1f147ecb8d0d1fa5f2c0c932022-12-22T01:44:26ZbelBelarusian State UniversityЖурнал Белорусского государственного университета: Математика, информатика2520-65082617-39562020-03-01161210.33581/2520-6508-2020-1-6-121139Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical caseSergey A. Bondarev0Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, BelarusLet (X, d, µ) be a doubling metric measure space with doubling dimension γ, i. e. for any balls B(x, R) and B(x, r), r < R, following inequality holds µ(B(x, R)) ≤ aµ (R/r)γµ(B(x, r)) for some positive constants γ and aµ. Hajłasz – Sobolev space Mpα(X) can be defined upon such general structure. In the Euclidean case Hajłasz – Sobolev space coincides with classical Sobolev space when p > 1, α = 1. In this article we discuss inclusion of functions from Hajłasz – Sobolev space Mpα(X) into the space of continuous functions for p ≤ 1 in the critical case γ = α p. More precisely, it is shown that any function from Hajłasz – Sobolev class Mpα(X), 0 < p ≤ 1, α > 0, has a continuous representative in case of uniformly perfect space (X, d, µ).https://journals.bsu.by/index.php/mathematics/article/view/1139analysis on metric measure spacessobolev spaces |
| spellingShingle | Sergey A. Bondarev Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case Журнал Белорусского государственного университета: Математика, информатика analysis on metric measure spaces sobolev spaces |
| title | Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case |
| title_full | Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case |
| title_fullStr | Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case |
| title_full_unstemmed | Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case |
| title_short | Inclusion of Hajłasz – Sobolev class Mpα(X) into the space of continuous functions in the critical case |
| title_sort | inclusion of hajlasz sobolev class mpα x into the space of continuous functions in the critical case |
| topic | analysis on metric measure spaces sobolev spaces |
| url | https://journals.bsu.by/index.php/mathematics/article/view/1139 |
| work_keys_str_mv | AT sergeyabondarev inclusionofhajłaszsobolevclassmpaxintothespaceofcontinuousfunctionsinthecriticalcase |