Degenerate p-Laplacian operators on H-type groups and applications to Hardy type inequalities

TL;DR

This paper studies a class of degenerate p-Laplacian operators on H-type groups, deriving their fundamental solutions and establishing Hardy inequalities, extending analysis on sub-Riemannian structures with applications to inequalities.

Contribution

It introduces a new class of vector fields on H-type groups, derives their fundamental solutions, and establishes Hardy inequalities for these operators.

Findings

Fundamental solutions for the degenerate p-Laplacian operators were obtained.

Hardy type inequalities associated with these operators were established.

The results extend analysis on sub-Riemannian geometries and potential applications.

Abstract

Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \text{div}_X (|\na_{X} u|^{p-2} \na_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$, for $k=2$ and $\mathbb G$ being the Heisenberg group they are introduced by Greiner \cite{Greiner-cjm79}. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.