Centered densities on Lie groups of polynomial volume growth

Abstract

We study the asymptotic behavior of the convolution powers φ*n =φ*φ*⋯φ* of a centered density φ on a connected Lie group G of polynomial volume growth. The main tool is a Harnack inequality which is proved by using ideas from Homogenization theory and by adapting the method of Krylov and Safonov. Applying this inequality we prove that the positive φ-harmonic functions are constant. We also characterise the φ-harmonic functions which grow polynomially. We give Gaussian estimates for φ*n as well as for the differences ∂zφ*n(g) = φ*n (gz) - φ*n (g) and ∂1φ*n = φ*n+1 - φ*n. We give estimates, similar to the ones given by the classical Berry-Esseen theorem, for φ*n and ∂z φ*n. We use these estimates to study the associated Riesz transforms.