On Choquet integrals and Sobolev type inequalities
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916388242718720 |
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| author | Harjulehto, Petteri Hurri-Syrjänen, Ritva |
| author_facet | Harjulehto, Petteri Hurri-Syrjänen, Ritva |
| contents | We consider integrals in the sense of Choquet with respect to the $δ$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<δ\le n$. In particular, for these functions we prove Sobolev inequalities in the limiting case $p=δ/n$ and in the case $p>δ$, here $p$ is the integrability exponent of the absolute value of the gradient of any given function. The results complement previously known Poincaré-Sobolev and Morrey inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_09964 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Choquet integrals and Sobolev type inequalities Harjulehto, Petteri Hurri-Syrjänen, Ritva Analysis of PDEs Functional Analysis 46E35, 31C15 (Primary), 26B35, 26D10 ( Secondary) We consider integrals in the sense of Choquet with respect to the $δ$-dimensional Hausdorff content for continuously differentiable functions defined on open, connected sets in the Euclidean $n$-space, $n\geq 2$, $0<δ\le n$. In particular, for these functions we prove Sobolev inequalities in the limiting case $p=δ/n$ and in the case $p>δ$, here $p$ is the integrability exponent of the absolute value of the gradient of any given function. The results complement previously known Poincaré-Sobolev and Morrey inequalities. |
| title | On Choquet integrals and Sobolev type inequalities |
| topic | Analysis of PDEs Functional Analysis 46E35, 31C15 (Primary), 26B35, 26D10 ( Secondary) |
| url | https://arxiv.org/abs/2311.09964 |