Bergman kernel

Remarks

The kernel satisfies $K(w, z) = \overline{K(z, w)}$, and $K(w, w) = \norm{K_w}^2 > 0$ when $A^2(\Omega) \ne \set{0}$. For any orthonormal basis $(e_n)$ of $A^2(\Omega)$, $K(z, w) = \sum_n e_n(z)\overline{e_n(w)}$. Example: on the unit disc the functions $e_n(z) = \sqrt{(n + 1)/\pi}\, z^n$ form an orthonormal basis, so $K(z, w) = \frac{1}{\pi}\sum_{n \ge 0} (n + 1)(z\bar w)^n = \frac{1}{\pi(1 - z\bar w)^2}$. Non-example: the function $\frac{1}{1 - z\bar w}$ is not the Bergman kernel of the disc. It is the Szegő kernel, which reproduces values for the inner product of the Hardy space $H^2$ with normalized arc length, not for the area inner product. For instance, with $w = 0$ it is the constant $1$, while $\int_{\mathbb{D}} f \cdot 1 \, dA = \pi f(0)$, not $f(0)$. If $\varphi \colon \Omega_1 \to \Omega_2$ is biholomorphic, then $K_1(z, w) = \varphi'(z) K_2(\varphi(z), \varphi(w)) \overline{\varphi'(w)}$.