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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 formL   α    =   ∑   j  =  1   de2    λ  j    (  a  )∂x  j2   +   ∑   j  =  1   k    (   ∂a  j2   −  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 ,...,  λ d   ∈ (ℝ  k   )*. It is known that bounded  α -harmonic functions on  G  are precisely the “Poisson integrals” of  L ∞(ℝ  d  ) against the Poisson kernel  ν α   which is a smooth function on ℝ  d  . We prove an upper bound of  ν α   and its derivatives.

Poisson kernels on semi-direct products of abelian groups