Anisotropic Fractional Gagliardo-Nirenberg, Weighted Caffarelli-Kohn-Nirenberg and Lyapunov-type Inequalities, and Applications to Riesz Potentials and p-sub-Laplacian Systems

Aidyn Kassymov, Michael V. Ruzhansky, Durvudkhan Suragan

2022

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DOI 10.1007/s11118-022-10029-6
Data confidence 70 medium
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Abstract

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 $\mathbb {R}^{N}$ 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.