Global solvability and regularity for ∂̄ on an annulus between two weakly pseudoconvex domains

Abstract

Let M 1 {M_1} and M 2 {M_2} be two bounded pseudo-convex domains in C n {{\mathbf {C}}^n} with smooth boundaries such that M ¯ 1 ⊂ M 2 {\overline M _1} \subset {M_2} . We consider the Cauchy-Riemann operators ∂ ¯ \overline \partial on the annulus M = M 2 ∖ M ¯ 1 M = {M_2}\backslash {\overline M _1} . The main result of this paper is the following: Given a ∂ ¯ \overline \partial -closed ( p , q ) (p,q) form α \alpha , 0 > q > n 0 > q > n , which is C ∞ {C^\infty } on M ¯ \overline M and which is cohomologous to zero on M M , there exists a ( p , q − 1 ) (p,q - 1) form u u which is C ∞ {C^\infty } on M ¯ \overline M such that ∂ ¯ u = α \overline \partial u = \alpha .