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.

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Let$D$be a relatively compact domain in$C$^{2}with smooth connected boundary ∂$D$. A compact set$K⊂∂D$is called removable if any continuous CR function defined on ∂$D/K$admits a holomorphic extension to$D$. If$D$is strictly pseudoconvex, a theorem of B. Jöricke states that any compact$K$contained in a smooth totally real disc$S⊂∂D$is removable. In the present article we show that this theorem is true without any assumption on pseudoconvexity.