Poisson kernels on semi-direct products of abelian groups

Abstract

Let G be a semi direct product G = Rd × Rk. On G we consider a class of second order left-invariant differential operators of the form Lα = Σj=1 d e 2 λ j (a) ∂xj 2 + Σ j = 1 k (∂ a j 2 - 2 αj ∂aj) , where a ϵ Rk and λ1,⋯, λd ? (Rk )∗. It is known that bounded α-harmonic functions on G are precisely the "Poisson integrals" of L∞(Rd) against the Poisson kernel να which is a smooth function on Rd. We prove an upper bound of να and its derivatives.