Abstract:
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 $H^1_c(D)$.
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