Potential Theory on Non-Unimodular Groups

  • N. Th. Varopoulos


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].


Compact Group Haar Measure Solvable Group Brownian Bridge Convolution Semigroup 
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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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