Convolution powers on groups of polynomial volume growth

Abstract

We give certain estimates concerning the asymptotic behavior of convolution powers of measures on discrete groups and connected Lie groups of polynomial volume growth. We also give similar estimates for the heat kernels associated to left invariant subLaplacians on connected Lie groups of polynomial volume growth, and to second order elliptic differential operators with quasiperiodic coefficients on ℝn. Using these results, we can obtain a simple proof of the LP-boundedness of the associated Riesz transform operators.