 Solution

The convention used for planar Gaussian free field theory is the Dirichlet energy space: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 $\Vert f{\Vert }_{L^2}+\Vert \nabla f{\Vert }_{L^2}$ used in another standard definition of the zero-boundary Sobolev space. On bounded domains with a Poincaré inequality, the two norms are equivalent.