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

Citations:

Zbl 0106.05501; Zbl 0504.32002
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