The $\bar \partial $ -problem with support conditions on some weakly pseudoconvex domainswith support conditions on some weakly pseudoconvex domains

Abstract

We consider a domain Ω with Lipschitz boundary, which is relatively compact in an n-dimensional Kähler manifold and satisfies some “logδ-pseudoconvexity” condition. We show that the $\bar \partial $ -equation with exact support in ω admits a solution in bidegrees (p, q), 1≤q≤n−1. Moreover, the range of $\bar \partial $ acting on smooth (p, n−1)-forms with support in $\bar \Omega $ is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi flat CR manifolds of arbitrary codimension.