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}

Abstract

AbstractWe 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 W1 − 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 W1 − 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}.