On Wirtinger derivations, the adjoint of the operator , and applications

Abstract

We extend the Cauchy– Riemann (or Wirtinger) operators and the Laplacian on to zero-degree currents on a (possibly singular) Riemann subdomain of a complex space (without recourse to resolution of singularities). The former extension gives rise to an adjoint operator for the -operator on extendable test forms on (the components of 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 (acting weakly on the Sobolev space ) and of Riesz's isomorphism between the Sobolev spaces and .