On the BBM-Phenomenon in Fractional Poincaré–Sobolev Inequalities with Weights
Hurri-Syrjänen, R., Martínez-Perales, J. C., Pérez, C., & Vähäkangas, A. V. (2023). On the BBM-Phenomenon in Fractional Poincaré–Sobolev Inequalities with Weights. International Mathematics Research Notices, 2023(20), 17205-17244. https://doi.org/10.1093/imrn/rnac246
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International Mathematics Research NoticesDate
2023Copyright
© 2023 Oxford University Press
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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This work was supported by the Basque Government [BERC 2018-2021 program to J.M., IT1247-19 project and BERC 2018-2021 program to C.P.]; the Spanish State Research Agency through BCAM Severo Ochoa excellence accreditation [SEV-2017-2018 to J.M., SEV-2017-0718 to C.P., project PID2020-113156GB-I00/AEI /10.13039/501100011033 to C.P.]; and the “HAPDE”. The authors acknowledge the support to this pro ...
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