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1. CJM 2010 (vol 62 pp. 1116)
Degenerate p-Laplacian Operators and Hardy Type Inequalities on
H-Type Groups 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 Categories:35H30, 26D10, 22E25 |
2. CJM 2007 (vol 59 pp. 1301)
Strichartz Inequalities for the Wave Equation with the Full Laplacian on the Heisenberg Group We prove dispersive and Strichartz inequalities for the solution of the wave
equation related to the full
Laplacian on the Heisenberg group, by means of Besov spaces defined by a
Littlewood--Paley
decomposition related to the spectral resolution of the full Laplacian.
This requires a careful
analysis due also to the non-homogeneous nature of the full Laplacian.
This result has to be compared to a previous one by Bahouri, G\'erard
and Xu concerning the solution of the wave equation related to
the Kohn Laplacian.
Keywords:nilpotent and solvable Lie groups, smoothness and regularity of solutions of PDEs Categories:22E25, 35B65 |
3. CJM 2004 (vol 56 pp. 963)
A Berry-Esseen Type Theorem on Nilpotent Covering Graphs We prove an estimate for the speed of convergence of the
transition probability for a symmetric random walk
on a nilpotent covering graph.
To obtain this estimate, we give a complete proof of
the Gaussian bound for the gradient of the Markov kernel.
Categories:22E25, 60J15, 58G32 |