Abstract
In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse Hölder inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov spaces and multiplier theorems for both spectral and Fourier multipliers in Besov spaces on graded Lie groups. In particular, we give a number of sufficient conditions for the boundedness of Fourier multipliers in Besov spaces.
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Acknowledgements
We would like to thank the referee for his/her comments allowing us to improve the final version of this manuscript. We also thank Julio Delgado for his remarks.
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The second author was supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02. The authors were supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations.
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Cardona, D., Ruzhansky, M. Littlewood-Paley Theorem, Nikolskii Inequality, Besov Spaces, Fourier and Spectral Multipliers on Graded Lie Groups. Potential Anal 60, 965–1005 (2024). https://doi.org/10.1007/s11118-023-10076-7
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DOI: https://doi.org/10.1007/s11118-023-10076-7
Keywords
- Nikolskii inequality
- Besov spaces
- Littlewood-Paley theorem
- Fourier multipliers
- Spectral multipliers
- Graded Lie groups
- Nilpotent Lie groups