= Solution
For real <test functions> $f,g\in C_c^\infty(D)$, the <Dirichlet inner product> in the normalization compatible with the logarithmic Green kernel is
$$
\boxed{(f,g)_\nabla=\frac1{2\pi}\int_D\nabla f(x)\cdot\nabla g(x)\,dx.}
$$
Its norm measures gradient energy rather than the values of the functions. A <change of variables> shows that this energy is invariant under planar <conformal maps>: the squared derivative in the transformation of the <gradient> cancels the area <Jacobian determinant>.
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