= Solution
The convention used for planar <Gaussian free field> theory is the <Dirichlet energy space>:
$$
\boxed{H_0^1(D)=\overline{C_c^\infty(D;\mathbb R)}^{\ \|\cdot\|_\nabla},\qquad
\|f\|_\nabla^2=\frac1{2\pi}\int_D|\nabla f|^2.}
$$
It is a real <Hilbert space> with the extended <Dirichlet inner product>. A <conformal map> from the <unit disc> onto $D$ identifies the two energy completions, so local representatives can be understood through the disc's zero-boundary space.
For an arbitrary unbounded domain this is the homogeneous energy completion; it should be distinguished from completion in the inhomogeneous norm $\|f\|_{L^2}+\|\nabla f\|_{L^2}$ used in another standard definition of the <zero-boundary Sobolev space>. On bounded domains with a <Poincaré inequality>, the two norms are equivalent.
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