Abstract
In this paper we prove the fractional Gagliardo-Nirenberg inequality on homogeneous Lie groups. Also, we establish weighted fractional Caffarelli-Kohn-Nirenberg inequality and Lyapunov-type inequality for the Riesz potential on homogeneous Lie groups. The obtained Lyapunov inequality for the Riesz potential is new already in the classical setting of \(\mathbb {R}^{N}\). As an application, we give two-sided estimate for the first eigenvalue of the Riesz potential. Also, we obtain Lyapunov inequality for the system of the fractional p-sub-Laplacian equations and give an application to estimate its eigenvalues.
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Acknowledgements
This research was partially funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP09258745). Also, the authors were supported in parts by the EPSRC grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151. The authors are also supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021).
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Kassymov, A., Ruzhansky, M. & Suragan, D. Anisotropic Fractional Gagliardo-Nirenberg, Weighted Caffarelli-Kohn-Nirenberg and Lyapunov-type Inequalities, and Applications to Riesz Potentials and p-sub-Laplacian Systems. Potential Anal 59, 1971–1994 (2023). https://doi.org/10.1007/s11118-022-10029-6
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DOI: https://doi.org/10.1007/s11118-022-10029-6
Keywords
- Fractional Gagliardo-Nirenberg inequality
- Fractional Caffarelli-Kohn-Nirenberg inequality
- Fractional Lyapunov-type inequality
- Homogeneous Lie group