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arXiv · 2201.11712

On the BBM-phenomenon in fractional Poincar\'e-Sobolev inequalities with weights

Ritva Hurri-Syrjänen·Javier C. Martínez-Perales·Carlos Pérez·Antti V. Vähäkangas
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Abstract

In this paper we unify and improve some of the results of Bourgain, Brezis and Mironescu and the weighted Poincar\'e-Sobolev estimate by Fabes, Kenig and Serapioni. More precisely, we get weighted counterparts of the Poincar\'e-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 behaviour 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 Poincar\'e inequalities.

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Ritva Hurri-Syrjänen, Javier C. Martínez-Perales, Carlos Pérez, Antti V. Vähäkangas. 2022-01-27. On the BBM-phenomenon in fractional Poincar\'e-Sobolev inequalities with weights. https://arxiv.org/abs/2201.11712

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