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Uniform estimates for the Cauchy-Riemann equation on \(q\)-concave wedges. (English) Zbl 0796.32008

Coupet, B. (ed.) et al., Colloque d’analyse complexe et géométrie. Marseille, France, 13-17 janvier 1992. Paris: Société Mathématique de France, Astérisque. 217, 151-182 (1993).
This paper is devoted to the study of the Cauchy Riemann equation on \(q\)- concave wedges. Let \(\varphi_ 1, \dots, \varphi_ N\) be real \(C^ 2\)- functions on a domain \(G \subset\mathbb{C}^ n\) with nonempty common zero locus \(E\), such that \(d \varphi_ 1 \wedge \cdots \wedge d \varphi_ N \neq 0\) in \(G\) and for \(\lambda = (\lambda_ 1, \dots, \lambda_ N) \in R^ N_ +\) with \(\lambda_ 1+\cdots+\lambda_ N=1\), the Levi form of \[ \lambda_ 1 \varphi_ 1+ \cdots + \lambda_ N \varphi_ N \] has at least \(q+1\) positive eigenvalues in \(G\).
Set \(D = \cap_{j=1}^ N \{z \in G:\varphi_ j (z)>0\}\). The main theorem states that for each \(x \in E\) there exists a ball \(B_ R(x)\) such that the equation \(\overline \partial u=f\) is solvable on \(D \cap B_ R^{(x)}\) with uniform estimates for each continuous \(\overline \partial\)-closed \((n,r)\)-form \(f\) on \(E\) \((1 \leq r \leq q-N)\) satisfying the estimate \[ | f(x) | \leq \bigl( \text{dist} (x,\partial D) \bigr)^{-\beta}, \quad x \in D. \]
For the entire collection see [Zbl 0782.00057].

MSC:

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