= Green-kernel expansion in the Dirichlet space
{c}
Fix a bounded planar domain $D$ and explicitly choose the <Dirichlet form> normalization
$$
(f,g)_\nabla=\frac1{2\pi}\int_D\nabla f\cdot\nabla g\,dx.
$$
Its Dirichlet <Green function of the Laplacian> satisfies $-\Delta_xG_D(x,y)=2\pi\delta_y(x)$. Let $(e_n)$ be an <orthonormal basis> of the corresponding <Sobolev space> $H_0^1(D)$. The formal expansion
$$
G_D(x,y)=\sum_{n\geq1}e_n(x)e_n(y)
$$
is interpreted as a weak kernel identity. Concretely, for smooth compactly supported <test functions> $\phi,\psi$,
$$
\sum_n\left(\int_De_n\phi\right)\left(\int_De_n\psi\right)
=\iint_{D\times D}\phi(x)G_D(x,y)\psi(y)\,dx\,dy.
$$
The identity follows by applying <Parseval's identity> to the two Riesz representatives for the integration functionals. It is not a claim that arbitrary <Sobolev space> basis <functions> have pointwise values everywhere or that the series converges on the singular diagonal.
For a finite measure $\rho$ with finite <Green energy>, the same Hilbert-space identity gives $\sum_n(\int e_n\,d\rho)^2=\iint G_D(x,y)\rho(dx)\rho(dy)$. This is the square-summability condition for a <Finite-Green-energy measure pairing with a Gaussian free field>. An alternative energy normalization rescales both the basis and the Green kernel; specifying the factor prevents an unnoticed $2\pi$ mismatch.<Gaussian free fields for mathematicians by Scott Sheffield>[Section 2, Hilbert-space basis construction; the energy has been explicitly rescaled here]{ref}
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