A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications
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| Format: | Preprint |
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2026
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| _version_ | 1866912822919692288 |
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| author | Hu, Pingxu Li, Yinqin Yang, Dachun Yuan, Wen |
| author_facet | Hu, Pingxu Li, Yinqin Yang, Dachun Yuan, Wen |
| contents | In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_1$-weight constant or with the specific $A_p$-weight constant when $p\in (1,\infty)$. As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp.
The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where $p=1$, which was pointed out to us by Emiel Lorist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09094 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications Hu, Pingxu Li, Yinqin Yang, Dachun Yuan, Wen Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 26D10, Secondary 46E35, 42B25, 42B35 In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_1$-weight constant or with the specific $A_p$-weight constant when $p\in (1,\infty)$. As applications, we further obtain a new characterization of Muckenhoupt weights and, in the framework of ball Banach function spaces, an inequality related to Gagliardo and Sobolev seminorms on cubes, a Gagliardo--Nirenberg interpolation inequality, and a Bourgain--Brezis--Mironescu formula. All these obtained results have wide generality and are proved to be (nearly) sharp. The original version of this article was published in [Adv. Math. 481 (2025), Paper No. 110537]. In this revised version, we correct an error appeared in Theorem 1.1 in the case where $p=1$, which was pointed out to us by Emiel Lorist. |
| title | A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs Primary 26D10, Secondary 46E35, 42B25, 42B35 |
| url | https://arxiv.org/abs/2601.09094 |