Generalized Newton-Leibniz Formula and the Embedding of the Sobolev Functions with Dominating Mixed Smoothness into Hölder Spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909427373703168 |
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| author | Abdulla, Ugur G. |
| author_facet | Abdulla, Ugur G. |
| contents | It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Hölder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the Hölder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $n$-dimensional rectangle and inductive application of the Sobolev trace embedding results. The method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_09132 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Generalized Newton-Leibniz Formula and the Embedding of the Sobolev Functions with Dominating Mixed Smoothness into Hölder Spaces Abdulla, Ugur G. Functional Analysis Analysis of PDEs Classical Analysis and ODEs 46E35 It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Hölder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the Hölder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $n$-dimensional rectangle and inductive application of the Sobolev trace embedding results. The method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. |
| title | Generalized Newton-Leibniz Formula and the Embedding of the Sobolev Functions with Dominating Mixed Smoothness into Hölder Spaces |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs 46E35 |
| url | https://arxiv.org/abs/2101.09132 |