Poisson kernel

Summary

The Poisson kernel is the integral kernel that produces a harmonic function in a disc from its values on the boundary circle. On the unit disc it is $P(z, \zeta) = (1 - \abs{z}^2)/\abs{\zeta - z}^2$ for $\abs{z} < 1$ and $\abs{\zeta} = 1$. Integrating a continuous function on the circle against it solves the Dirichlet problem for the disc. Similar kernels exist for the upper half-plane and for balls in $\R^n$. They are central in harmonic analysis and in the theory of Hardy spaces.