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

TL;DR

This paper establishes nearly sharp localized weighted inequalities related to Gagliardo and Sobolev seminorms, providing new characterizations of Muckenhoupt weights and applications in function spaces, with broad generality and sharpness.

Contribution

It introduces a nearly sharp localized weighted inequality involving Gagliardo and Sobolev seminorms with specific weight constants, advancing the understanding of weighted inequalities and their applications.

Findings

Derived a new characterization of Muckenhoupt weights.

Established inequalities in ball Banach function spaces.

Proved a Gagliardo--Nirenberg interpolation inequality.

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.