Poisson kernels on semi-direct products of abelian groups

Abstract

Abstract Let G be a semi direct product G = ℝ d ⋊ ℝ k . On G we consider a class of second order left-invariant differential operators of the form L α = ∑ j = 1 d e 2 λ j ( a ) ∂ x j 2 + ∑ j = 1 k ( ∂ a j 2 − 2 α j ∂ a j ) , ${{\cal L}_\alpha } = \sum\limits_{j = 1}^d {\mkern 1mu} e{{\mkern 1mu} ^{2{\lambda _j}(a)}}\partial _{{x_j}}^2 + \sum\limits_{j = 1}^k {(\partial _{{a_j}}^2 - 2{\alpha _j}{\partial _{{a_j}}})},$ where a ∈ ℝ k and λ 1,..., λd ∈ (ℝ k )*. It is known that bounded 𝓛α-harmonic functions on G are precisely the “Poisson integrals” of L∞(ℝ d ) against the Poisson kernel να which is a smooth function on ℝ d . We prove an upper bound of να and its derivatives.