Poisson kernel of the upper half-plane

Definition

For $y>0$ , the Poisson kernel of the upper half-plane is

P_y(t)=\frac1\pi\frac{y}{t^2+y^2},\qquad t\in\mathbb R.

It is positive, integrates to $1$ , and is harmonic as a function of $t+iy\in\mathbb H$ .

Approximate identity

As $y\downarrow0$ , the family $P_y$ concentrates at the origin. Therefore $P_y*f\to f$ in standard $L^p$ regimes and at Lebesgue points. The convolution $P_y*f$ is the Poisson extension of $f$ .

Scaling

$P_y(t)=y^{-1}P_1(t/y)$ . This scaling explains the equivalent formula $\int f(x+ty)/(1+t^2)\,dt/\pi$ used in linewise extensions.

References

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