A sharp localized weighted inequality related to Gagliardo and Sobolev seminorms and its applications

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.