A Unified Hölder Lebesgue Framework for Caffarelli Kohn Nirenberg Inequalities
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918152489664512 |
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| author | Dong, Mengxia |
| author_facet | Dong, Mengxia |
| contents | We develop a unified Hölder Lebesgue scale \(X^p\) and its weighted, higher order variants \(X^{k,p,a}\) to extend the Caffarelli Kohn Nirenberg (CKN) inequality beyond the classical Lebesgue regime. Within this framework we prove a two parameter interpolation theorem that is continuous in the triplet \((k,1/p,a)\) and bridges integrability and regularity across the Lebesgue Hölder spectrum. As a consequence we obtain a generalized CKN inequality on bounded punctured domains \(Ω\subset\mathbb{R}^n\setminus\{0\}\); the dependence of the constant on \(Ω\) is characterized precisely by the (non)integrability of the weights at the origin. At the critical endpoint \(p=n\) we establish a localized, weighted Brezis Wainger type bound via Trudinger Moser together with a localized weighted Hardy lemma, yielding an endpoint CKN inequality with a logarithmic loss. Sharp constants are not pursued; rather, we prove existence of constants depending only on the structural parameters and coarse geometry of \(Ω\). Several corollaries, including a unified Hardy--Sobolev inequality, follow from the same interpolation mechanism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_00949 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Unified Hölder Lebesgue Framework for Caffarelli Kohn Nirenberg Inequalities Dong, Mengxia Analysis of PDEs Functional Analysis We develop a unified Hölder Lebesgue scale \(X^p\) and its weighted, higher order variants \(X^{k,p,a}\) to extend the Caffarelli Kohn Nirenberg (CKN) inequality beyond the classical Lebesgue regime. Within this framework we prove a two parameter interpolation theorem that is continuous in the triplet \((k,1/p,a)\) and bridges integrability and regularity across the Lebesgue Hölder spectrum. As a consequence we obtain a generalized CKN inequality on bounded punctured domains \(Ω\subset\mathbb{R}^n\setminus\{0\}\); the dependence of the constant on \(Ω\) is characterized precisely by the (non)integrability of the weights at the origin. At the critical endpoint \(p=n\) we establish a localized, weighted Brezis Wainger type bound via Trudinger Moser together with a localized weighted Hardy lemma, yielding an endpoint CKN inequality with a logarithmic loss. Sharp constants are not pursued; rather, we prove existence of constants depending only on the structural parameters and coarse geometry of \(Ω\). Several corollaries, including a unified Hardy--Sobolev inequality, follow from the same interpolation mechanism. |
| title | A Unified Hölder Lebesgue Framework for Caffarelli Kohn Nirenberg Inequalities |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2510.00949 |