On the Poisson representation of a function harmonic in the upper half-plane (Q1409784)

The authors discuss conditions for the validity of the generalized Poisson formula for a function \(u(z)\) harmonic in the half-plane \(\operatorname{Im} z > 0\) which has the form \(u(z)= \int_{-\infty}^{+\infty} P_q (z,t) \,d \nu (t)+ \operatorname{Im} P(z)\), where \(\nu\) is a \(\sigma\)-finite Borel measure, \(P\) is a real polynomial, \(P_q(z,t)= \operatorname{Im} [(1+tz)^q \pi^{-1} (t-z)^{-1} (1+t^2)^{-q}]\). These conditions differ from known ones by weaker growth restrictions in \(\operatorname{Im} z > 0\) and stronger behavior on the real axis \(\operatorname{Im} z = 0\).