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Bounded integral operators on strictly \(q\)-convex domains in \(\mathbb C^n\). (Beschränkte Integraloperatoren auf streng \(q\)-konvexen Gebieten in \(\mathbb C^n\).) (German) Zbl 0578.32005

Let \(G\) be a bounded domain in \(\mathbb C^n\) with \(C^\infty\) smooth boundary, strictly \(q\)-convex in the sense of A. Andreotti and H. Grauert [Bull. Soc. Math. Fr. 90, 193–259 (1962; Zbl 0106.05501)]. For each \(r\geq q-1\), the author constructs a linear integral operator \(T_r\) on continuously differentiable bounded \((0,r)\)-forms \(\gamma\) on \(G\) with \(\bar \partial\gamma\) also bounded on \(G\), such that
\[ \gamma ={\bar \partial}T_r(\gamma) + T_{r+1}({\bar \partial}\gamma)\quad\text{for}\quad q\leq r\leq n, \]
\[ \gamma =T_r(\gamma) + T_{r+1}({\bar \partial}\gamma)\quad\text{for}\quad r=q-1, \] and \[ \sup | T_r(\gamma)| \leq C (\sup | \gamma | + \sup | {\bar \partial}\gamma |), \] where \(C\) is a constant depending only on \(r\) and \(G\).
The construction uses kernels of Bochner-Martinelli, Cauchy-Fantappié, and the author [Bonn. Math. Schr. 133, 152 S. (1981; Zbl 0504.32002)].
Reviewer: Theodore J. Barth

MSC:

32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
32F10 \(q\)-convexity, \(q\)-concavity
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
45P05 Integral operators
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