ABSTRACT
In this chapter, we define the Green’s function G(z, a) associated to a bounded domain Ω with C∞ smooth boundary, and we show how this function is related to the Bergman kernel of the domain. For a fixed point a ∈ Ω, the Green’s function is defined via G(z, a) = − ln |z − a| + ua(z) where ua(z) is the harmonic function of z on Ω that solves the Dirichlet problem with boundary data equal to ln |z − a|. In view of Theorem 14.2,G(z, a) is a harmonic function of z on Ω−{a} that extends C∞ smoothly up to bΩ, that vanishes on bΩ, and that has the property that G(z, a) + ln |z − a| is bounded near a (and so it has a removable singularity at a). It is a consequence of the maximum principle that these properties characterizeG(z, a). In fact, the condition that G(z, a) extend C∞ smoothly to the boundary is much stronger than necessary. It is enough to know only that G(z, a) extends continuously to the boundary. The maximum principle also yields that G(z, a) > 0 if z, a ∈ Ω and z 6= a.
