$ L^p $-theory for the $ {\partial }\overline{\partial} $-equation and isomorphisms results

Abstract

We establish an $ L^{p}_{\rm loc} $-existence theorem for the $ {\partial }\overline{\partial} $-equation on a half-space of $ \mathbb C^n $. The result is achieved for forms of class $ L^{p}_{\rm loc} $ as well as for those forms in the scale of $ W^{1, p}_{\rm loc} $-Sobolev spaces and admitting distributional boundary values. Some isomorphisms and regularity results in relation to de Rham, Bott–Chern, and Aeppli cohomology groups are moreover obtained.

Keywords

• $ {\partial }\overline{\partial} $-problem
• distributional boundary values
• extensible currents
• regularizing operators