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\(\mathcal{C}^k\)-regularity for the \(\bar\partial\)-equation with a support condition. (English) Zbl 1458.32034

Summary: Let \(D\) be a \(\mathcal{C}^d\) \(q\)-convex intersection, \(d\geq 2\), \(0\leq q\leq n-1\), in a complex manifold \(X\) of complex dimension \(n\), \(n\geq 2\), and let \(E\) be a holomorphic vector bundle of rank \(N\) over \(X\). In this paper, \(\mathcal{C}^k\)-estimates, \(k=2,3,\dots,\infty\), for solutions to the \(\bar\partial\)-equation with small loss of smoothness are obtained for \(E\)-valued \((0,s)\)-forms on \(D\) when \(n-q\leq s\leq n\). In addition, we solve the \(\bar\partial\)-equation with a support condition in \(\mathcal{C}^k\)-spaces. More precisely, we prove that for a \(\bar\partial\)-closed form \(f\) in \(\mathcal{C}_{0,q}^{k}(X\setminus D,E)\), \(1\leq q\leq n-2\), \(n\geq 3\), with compact support and for \(\varepsilon\) with \(0<\varepsilon <1\) there exists a form \(u\) in \(\mathcal{C}_{0,q-1}^{k-\varepsilon}(X\setminus D,E)\) with compact support such that \(\bar\partial u=f\) in \(X\setminus\overline D\). Applications are given for a separation theorem of Andreotti-Vesentini type in \(\mathcal{C}^k\)-setting and for the solvability of the \(\bar\partial\)-equation for currents.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32F10 \(q\)-convexity, \(q\)-concavity
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