Potential Theory on Non-Unimodular Groups

N. Th. Varopoulos
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N. Th. Varopoulos: Université Paris 6

A chapter in Harmonic Analysis and Discrete Potential Theory, 1992, pp 215-221 from Springer

Abstract: 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 ℓ 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 ℓ 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 (search for similar items in EconPapers)
Date: 1992
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DOI: 10.1007/978-1-4899-2323-3_17

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