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

We establish an $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p$-existence theorem for the $\mathrm{\partial }\overline{\mathrm{\partial }}$-equation on a half-space of ${\mathbb{C}}^n$. The result is achieved for forms of class $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^p$ as well as for those forms in the scale of $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,p}$-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.