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

摘要

We extend the Cauchy-Riemann (or Wirtinger) operators and the Laplacian on 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 (partial derivative) over bar* for the (partial derivative) over bar -operator on extendable test forms on D (the components of (partial derivative) over bar* 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 S-p,S-1,S-0 := -Delta(p) + Id (acting weakly on the Sobolev space H-1(D)) and of Riesz's isomorphism between the Sobolev spaces H-c(1) (D)* and H-c(1)(D).