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

Abstract In this paper we prove the fractional Gagliardo-Nirenberg inequality on homogeneous Lie groups. Also, establish weighted Caffarelli-Kohn-Nirenberg and Lyapunov-type for Riesz potential The obtained Lyapunov is new already in classical setting of $\mathbb {R}^{N}$ ℝ N . As an application, give two-sided estimate first eigenvalue potential. obtain system p -sub-Laplacian equations application to its eigenvalues.