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# Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups

Published:2010-01-26
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
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## 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 $L_{p,k} u= \operatorname{div}_X (|\nabla_{X} u|^{p-2} \nabla_X u)$. 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 $X$.
 Keywords: fundamental solutions, degenerate Laplacians, Hardy inequality, H-type groups
 MSC Classifications: 35H30 - Quasi-elliptic equations 26D10 - Inequalities involving derivatives and differential and integral operators 22E25 - Nilpotent and solvable Lie groups