Some functional inequalities for the fractional p-sub-Laplacian

TL;DR

This paper establishes new fractional Sobolev, Hardy, and Lyapunov-type inequalities for the p-sub-Laplacian on homogeneous Lie groups, providing eigenvalue estimates and advancing the understanding of fractional operators in this setting.

Contribution

It introduces novel inequalities for the fractional p-sub-Laplacian on homogeneous Lie groups, including a Lyapunov-type inequality and eigenvalue estimates.

Findings

Proved fractional Sobolev and Hardy inequalities.

Established a Lyapunov-type inequality for the fractional p-sub-Laplacian.

Derived estimates for the first eigenvalue in quasi-balls.

Abstract

In this paper we study the fractional Dirichlet p-sub-Laplacian in a Haar measurable set on homogeneous Lie groups. We prove fractional Sobolev and Hardy inequalities and we also present a Lyapunov-type inequality for the fractional p-sub-Laplacian. As a consequence of the Lyapunov-type inequality we show an estimate of the first eigenvalue in a quasi-ball for the Dirichlet fractional p-sub-Laplacian.