Anisotropic fractional Gagliardo-Nirenberg, weighted Caffarelli-Kohn-Nirenberg and Lyapunov-type inequalities, and applications to Riesz potentials and $p$-sub-Laplacian systems

TL;DR

This paper establishes new fractional inequalities on homogeneous Lie groups, including Gagliardo-Nirenberg, Caffarelli-Kohn-Nirenberg, and Lyapunov-type inequalities, with applications to eigenvalue estimates for Riesz potentials and p-sub-Laplacian systems.

Contribution

It introduces novel fractional inequalities on homogeneous Lie groups and applies them to eigenvalue estimation for Riesz potentials and p-sub-Laplacian systems.

Findings

New Lyapunov inequality for Riesz potential in classical setting

Two-sided eigenvalue estimates for Riesz potential

Lyapunov inequality for fractional p-sub-Laplacian systems

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