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

摘要

We consider a domain Ω with Lipschitz boundary, which is relatively compact in ann-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 flatCR manifolds of arbitrary codimension.