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arXiv · 2601.09094

A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications

Pingxu Hu·Yinqin Li·Dachun Yang·Wen Yuan
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Abstract

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.

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Pingxu Hu, Yinqin Li, Dachun Yang, Wen Yuan. 2026-01-14. A Sharp Localized Weighted Inequality Related to Gagliardo and Sobolev Seminorms and Its Applications. https://arxiv.org/abs/2601.09094

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