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The evolution and Poisson kernels on nilpotent meta-abelian groups

100%
Penney R., Urban R.
Studia Mathematica
|
2013
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tom 219
|
nr 1
69-96
EN
Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to $ℝ^{k}$, k>1. We consider a class of second order left-invariant differential operators on S of the form $ℒ_{α} = L^{a} + Δ_{α}$, where $α ∈ ℝ^{k}$, and for each $a ∈ ℝ^{k}, L^a$ is left-invariant second order differential operator on N and $Δ_{α} = Δ - ⟨α,∇⟩$, where Δ is the usual Laplacian on $ℝ^{k}$. Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively) we obtain an upper estimate for the transition probabilities of the evolution on N generated by $L^{σ(t)}$, where σ is a continuous function from [0,∞) to $ℝ^{k}$. We also give an upper bound for the Poisson kernel for $ℒ_{α}$.
2
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Estimates for the Poisson kernel on higher rank NA groups

100%
Penney R., Urban R.
Colloquium Mathematicum
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2010
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tom 118
|
nr 1
259-281
EN
We obtain an estimate for the Poisson kernel for the class of second order left-invariant differential operators on higher rank NA groups.
3
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Unbounded harmonic functions on homogeneous manifolds of negative curvature

100%
Penney R., Urban R.
Colloquium Mathematicum
|
2002
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tom 91
|
nr 1
99-121
EN
We study unbounded harmonic functions for a second order differential operator on a homogeneous manifold of negative curvature which is a semidirect product of a nilpotent Lie group N and A = ℝ⁺. We prove that if F is harmonic and satisfies some growth condition then F has an asymptotic expansion as a → 0 with coefficients from 𝓓'(N). Then we single out a set of at most two of these coefficients which determine F. Then using asymptotic expansions we are able to prove some theorems answering partially the following question. Is a given harmonic function the Poisson integral of "something" from the boundary N?
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