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

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Abstract

We establish an L^p_loc-existence theorem for the $\partial \bar{\partial}$-equation on a half-space of $\mathbb{C}^{n}$. The result is achieved for forms of class L^p loc as well as for those forms in the scale of W1,p 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

$\partial \bar{\partial}$-problem / distributional boundary values / extensible currents / regularizing operators

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