Abstract
Abstract In this paper, we unify and improve some of the results of Bourgain, Brezis, and Mironescu and the weighted Poincaré–Sobolev estimate by Fabes, Kenig, and Serapioni. More precisely, we get weighted counterparts of the Poincaré–Sobolev-type inequality and also of the Hardy type inequality in the fractional case under some mild natural restrictions. A main feature of the results we obtain is the fact that we keep track of the behavior of the constants involved when the fractional parameter approaches to $1$. Our main method is based on techniques coming from harmonic analysis related to the self-improving property of generalized Poincaré inequalities.
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