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

Published online by Cambridge University Press:  20 November 2018

Yongyang Jin*
Affiliation:
Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310032, China
Genkai Zhang*
Affiliation:
Mathematical Sciences, Chalmers University of Technology and Mathematical Sciences, Göteborg University, Göteborg, Sweden
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Abstract

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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 $X$.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2010

Footnotes

The first author’s research was supported by NNSF of China (10871180), NSF of Zhejiang province (Y6090359, Y6090383) and Department of Education, Zhejiang (Z200803357) The second author was supported by the Swedish Research Council and a STINT Institutional Grant.

References

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