CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

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
Format:   HTML   LaTeX   MathJax  

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 fundamental solutions, degenerate Laplacians, Hardy inequality, H-type groups
MSC Classifications: 35H30, 26D10, 22E25 show english descriptions Quasi-elliptic equations
Inequalities involving derivatives and differential and integral operators
Nilpotent and solvable Lie groups
35H30 - Quasi-elliptic equations
26D10 - Inequalities involving derivatives and differential and integral operators
22E25 - Nilpotent and solvable Lie groups
 

© Canadian Mathematical Society, 2014 : https://cms.math.ca/