On asymptotic expansion of the harmonic Berezin transform on the half-space

We give a complete asymptotic expansion of the Berezin transform associated to the Bergman space of harmonic functions on the half-space H ≔ { ( x , y ) ∈ R n + 1 ; x ∈ R n , y > 0 } that are square-integrable with respect to the measure y α d x d y ( α ≥ 0 ) as α → ∞ .