Abstract
An extended version of the Maz’ya–Shaposhnikova theorem on the limit as \(s\rightarrow 0^+\) of the Gagliardo–Slobodeckij fractional seminorm is established in the Orlicz space setting. Our result holds in fractional Orlicz–Sobolev spaces associated with Young functions satisfying the \(\Delta _2\)-condition, and, as shown by counterexamples, it may fail if this condition is dropped.
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Acknowledgements
We wish to thank the referees for their careful reading of the paper and for their valuable comments.
Funding
This research was partly funded by: (1) Research Project 201758MTR2 of the Italian Ministry of Education, University and Research (MIUR) Prin 2017 “Direct and inverse problems for partial differential equations: theoretical aspects and applications”; (2) GNAMPA of the Italian INdAM – National Institute of High Mathematics (grant number not available); (3) Grant P201-18-00580S of the Czech Science Foundation; (4) Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - GZ 2047/1, Projekt-ID 390685813.
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Communicated by Mieczyslaw Mastylo.
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Alberico, A., Cianchi, A., Pick, L. et al. On the Limit as \(s\rightarrow 0^+\) of Fractional Orlicz–Sobolev Spaces. J Fourier Anal Appl 26, 80 (2020). https://doi.org/10.1007/s00041-020-09785-z
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DOI: https://doi.org/10.1007/s00041-020-09785-z