Abstract view
Degenerate pLaplacian Operators and Hardy Type Inequalities on
HType Groups


Published:20100126
Printed: Oct 2010
Yongyang Jin,
Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310032, China
Genkai Zhang,
Mathematical Sciences, Chalmers University of Technology and Mathematical Sciences, Göteborg University, Göteborg, Sweden
Abstract
Let $\mathbb G$ be a steptwo nilpotent group of Htype 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 $L_{p,k} u= \operatorname{div}_X (\nabla_{X} u^{p2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. 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 $X$.