Abstract
Embeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. In particular, besides recovering standard embeddings for classical fractional Sobolev spaces, novel results are derived in borderline situations where the latter fail. For instance, limiting embeddings of Pohozhaev-Trudinger-Yudovich type into exponential spaces are offered. The equivalence of Gagliardo-Slobodeckij norms in fractional Orlicz-Sobolev spaces to norms defined via Littlewood-Paley decompositions, oscillations, or Besov type difference quotients is established as well. This equivalence, of independent interest, is a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.
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Open access funding provided by Università degli Studi di Firenze within the CRUI-CARE Agreement. This research was partly funded by: (a) 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”; (b) GNAMPA of the Italian INdAM – National Institute of High Mathematics (grant number not available).
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Breit, D., Cianchi, A. Inclusion Relations Among Fractional Orlicz-Sobolev Spaces and a Littlewood-Paley Characterization. Potential Anal (2024). https://doi.org/10.1007/s11118-024-10136-6
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DOI: https://doi.org/10.1007/s11118-024-10136-6
Keywords
- Fractional Orlicz-Sobolev spaces
- Embeddings
- Convolution inequalities
- Equivalent norms
- Littlewood-Paley decomposition