On Wirtinger derivations, the adjoint of the operator $\overline\partial$, and applications

We extend the Cauchy–Riemann (or Wirtinger) operators and the Laplacian on ${\mathbb C}^m$ to zero-degree currents on a (possibly singular) Riemann subdomain $D$ of a complex space (without recourse to resolution of singularities). The former extension gives rise to an adjoint operator $\overline\partial^*$ for the $\overline\partial$-operator on extendable test forms on $D$ (the components of $\overline\partial^*$ are the Wirtinger derivations). By means of the Wirtinger derivations, we generalize Gunning's theorem on the Cauchy–Riemann criterion (in the weak sense) for locally integrable functions to zero-degree currents on a complex space. To prove this result, we first give a generalization of Weyl's lemma to a Helmholtz operator. In the case of a continuous (resp. Lipschitzian) zero-degree current, we give a characterization of ‘weak holomorphy’ in terms of a local mean-value property (resp. an Euler operation). Wirtinger derivations also enable us to give explicit representations of the Green operator for the modified Laplacian ${\mathcal S}_{p,1,0}:= - \triangle_{p} + \mathrm{Id}$ (acting weakly on the Sobolev space $H^{-1}(D)$) and of Riesz's isomorphism between the Sobolev spaces $H^1_c(D)^*$ and $H^1_c(D)$.