Asymptotic Behavior of Poisson Kernels on NA Groups |
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Authors: | Dariusz Buraczewski Ewa Damek Andrzej Hulanicki |
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Affiliation: | 1. Institute of Mathematics, Wroc?aw University , Wroc?aw , Poland dbura@math.uni.wroc.pl;3. Institute of Mathematics, Wroc?aw University , Wroc?aw , Poland |
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Abstract: | On a Lie group S = NA, that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, a probability measure μ is considered and treated as a distribution according to which transformations s ∈ S acting on N = S/A are sampled. Under natural conditions, formulated some over thirty years ago, there is a μ-invariant measure m on N. Properties of m have been intensively studied by a number of authors. The present article deals with the situation when μ(A) = ?(s t ∈ A), where ?+ ? t → s t ∈ S is the diffusion on S generated by a second order subelliptic, hypoelliptic, left-invariant operator on S. In this article the most general operators of this kind are considered. Precise asymptotic for m at infinity and for the Green function of the operator are given. To achieve this goal a pseudodifferential calculus for operators with coefficients of finite smoothness is formulated and applied. |
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Keywords: | Heat semigroup Invariant measure Nilpotent Lie group Subelliptic operator |
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