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The \(\bar \partial\)-problem with support conditions on some weakly pseudoconvex domains. (English) Zbl 1078.32023

Consider a relative compact domain \(\Omega\) with Lipschitz boundary in an \(n\)-dimensional Kähler manifold \((X,\omega)\). Assume that \(\Omega\) satisfies the \(\log \delta\)-pseudoconvexity” condition there exists a smooth bounded function \(\phi\) on \(X\) such that \(i\partial \overline \partial (-\log \delta + \phi) \geq \omega\) in \(\Omega\), where \(\delta \) is the distance function to the boundary \(\partial \Omega\) with respect to \(\omega\).
In the paper under review, the author shows that the \(\overline \partial\)-equation with exact support in \(\Omega\) admits a solution in bidegree \((p,q)\), \(1\leq q \leq n-1\). Moreover, the author shows that the range of \(\overline \partial\) acting on smooth \((p,n-1)\)-forms with support in \(\overline \Omega\) is closed. Then the author applies the main result of this paper to obtain the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on the boundaries of weakly pseudoconvex domain in Stein manfolds and the solvability of the tangential Cauchy-Riemann equations for currents on Levi-flat CR manfolds of arbitrary codimension.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators
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