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

Jin, Yongyang ; Zhang, Genkai

Canadian journal of mathematics, Tome 62 (2010) no. 5, pp. 1116-1130

Résumé

Let $\mathbb{G}$ be a step-two nilpotent group of $\text{H}$ -type with Lie algebra $\mathfrak{G}\,=\,V\,\oplus \,\text{t}$ . We define a class of vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ on $\mathbb{G}$ depending on a real parameter $k\,\ge \,1$ , and we consider the corresponding $p$ -Laplacian operator ${{L}_{p,\,k}}u\,=\,di{{v}_{X}}\left( {{\left| {{\nabla }_{X}}u \right|}^{p-2}}{{\nabla }_{X}}u \right)$ . For $k\,=\,1$ the vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ 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 $X$ .

Détail

DOI : 10.4153/CJM-2010-033-9

Mots-clés : 35H30, 26D10, 22E25, fundamental solutions, degenerate Laplacians, Hardy inequality, H-type groups