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Convolution in perfect Lie groups

Published online by Cambridge University Press:  11 February 2016

YVES BENOIST  and
NICOLAS DE SAXCÉ
YVES BENOIST
Affiliation:
CNRS, Université Paris-Sud Bat.425, 91405 Orsay, France. e-mail: yves.benoist@math.u-psud.fr
NICOLAS DE SAXCÉ
Affiliation:
CNRS, Université Paris-Nord, LAGA, 93430 Villetaneuse, France. e-mail: desaxce@math.univ-paris13.fr
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Abstract

Let G be a connected perfect real Lie group. We show that there exists α < dim G and p ∈ $\mathbb{N}$* such that if μ is a compactly supported α-Frostman Borel measure on G, then the pth convolution power μ*p is absolutely continuous with respect to the Haar measure on G, with arbitrarily smooth density. As an application, we obtain that if A ⊂ G is a Borel set with Hausdorff dimension at least α, then the p-fold product set Ap contains a non-empty open set.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 1 , July 2016 , pp. 31 - 45
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

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