Poisson extension to the upper half-plane

Core idea

For suitable boundary data $f:\mathbb R\to\mathbb C$ , its Poisson extension to $\mathbb H=\{x+iy:y>0\}$ is

u(x+iy)=(P_y*f)(x) =\frac1\pi\int_{\mathbb R}f(x-t)\frac{y}{t^2+y^2}\,dt,

where $P_y$ is the Poisson kernel. The function $u$ is harmonic in $\mathbb H$ and approaches $f$ at the boundary in the topology guaranteed by the hypotheses on $f$ .

Dirichlet problem

For bounded continuous boundary data, the Poisson extension is the unique bounded harmonic solution of the upper-half-plane Dirichlet problem. Uniqueness follows from the maximum principle after controlling behavior at infinity.

Normal derivative

When $f$ is sufficiently regular, its boundary normal derivative is described by the Dirichlet-to-Neumann operator, equivalently a Hilbert transform of (f').

References

Elias M. Stein and Rami Shakarchi, Complex Analysis, Princeton University Press, 2003. DOI record.