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Potential Theory on Non-Unimodular Groups

  • N. Th. Varopoulos

Abstract

Let G be a connected Lie group and let X 1,... X k be left invariant fields (i.e. X f g = (X f) g , f g (x) = f(gx)) that generate the Lie algebra. We can consider then \(\Delta = - X_1^2 - X_2^2 \cdots - X_k^2 \) and T t = e−tΔ which is a convolution semigroup since it commutes with the left action of G. It follows that if we denote by d g the left Haar measure of G, we have \(T_t f(x) = \int\limits_G {f(y)\phi _t (y^{ - 1} x)d^l y} \), cf. [1]. The behaviour of ϕ t as t → ∞ when G is unimodular, i.e. when D g = dg (= the right Haar measure up to multiplicative constant), is well understood, cf. [1], [2].

Keywords

Compact Group Haar Measure Solvable Group Brownian Bridge Convolution Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • N. Th. Varopoulos
    • 1
  1. 1.Université Paris 6Paris Cedex 05France

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