Regularity of the \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\overline{\partial }}$\end{document} ‐equation and Liouville's theorem for pseudoconcave compacts in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${{\mathbb {C}}{\mathbb {P}}^n}$\end{document}

We prove regularity results for the \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\overline{\partial }$\end{document} ‐equation for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {O}(-m)$\end{document} ‐valued forms in the Sobolev spaces W 1 − ε on pseudoconcave domains with \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}^2$\end{document} boundary in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {C}}{\mathbb {P}}^n$\end{document} . We also show that the holomorphic sections of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {O}(-m)$\end{document} of class W 1 − ε vanish on a pseudoconcave domain with \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {C}^2$\end{document} boundary in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathbb {C}}{\mathbb {P}}^n$\end{document} .