Hyperbolic Poisson kernel on the disk

Definition

For $z\in\mathbb D$ and $b\in S^1=\partial\mathbb D$ , the hyperbolic Poisson kernel is

P_b(z)=\frac{1-|z|^2}{|z-b|^2}.

For fixed $b$ , it is positive and tends to infinity as $z$ approaches $b$ non-tangentially.

Boundary representation

If $F$ is suitable boundary data on $S^1$ , its harmonic extension to the disk is obtained by integrating $F(b)P_b(z)$ against normalized angular measure. This is the disk-model analogue of the upper-half-plane Poisson kernel.

Spectral powers

Complex powers $P_b(z)^{1/2+ir}$ are hyperbolic plane waves and solve the Laplace eigenvalue equation with spectral parameter $r$ .

References

Sigurdur Helgason, Groups and Geometric Analysis, AMS, 2000. Publisher record.