 Dirichlet energy space

For planar Gaussian free field theory, this is the completion of real test functions $C_c^{\infty }(D)$ in the norm induced by the Dirichlet inner product, $(f,g{)}_{\nabla }=(2\pi {)}^{-1}{\int }_D\nabla f\cdot \nabla g$. On a general unbounded domain this convention must be distinguished from completion in the inhomogeneous $H^1$ norm. A conformal map identifies its energy norm with that on the unit disc by conformal invariance.