On the changes of the sign of a harmonic function in the half-plane (Q2759172)

The authors study the question whether the sign of the Poisson integral \(\text{sgn}(P(f))(x+\varepsilon)\) in \({\mathbb R}^{2}_{+}\) converges as \(\varepsilon\rightarrow 0\), where \(x\) is on the boundary. According to the Fatou theorem \(P(f)\) converges to its boundary value within the non-tangential cone, also the sign converges at the points where \(f\) is non zero. The main theorem states that if \(f\in\text{Lip}_{1}\) then the sign of the Poisson integral converges almost everywhere. It is also shown that, if one has less regularity the convergence may fail.