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Subelliptic estimates for the $${\bar \partial}$$-Neumann problem on pseudoconvex domains. (English) Zbl 0627.32013
This article gives a necessary and sufficient condition for the $${\bar \partial}$$-Neumann problem on a bounded pseudoconvex domain $$\Omega$$ in $${\mathbb{C}}^ n$$ with smooth boundary $$b\Omega =\{r=0\}$$ to satisfy local subelliptic estimate at a boundary point $$z_ 0\in b\Omega$$. The condition is given by $$D_ q(z_ 0)<\infty$$, where q is the degree of the antiholomorphic part of the space of forms under consideration and $$D_ q(z_ 0)=\sup_{V^ q} \{\tau (V^ q,z_ 0)\}.$$ Here the supremum is taken over germs $$V^ q$$ of q-dimensional subvarieties passing through $$z_ 0$$ and lying outside $$\Omega$$, $$\tau (V^ q,z_ 0)$$ denotes the maximum value of $$ord_ 0(r\circ \gamma^ k_ S)/ord_ 0(\gamma^ k_ S)$$ among the one-dimensional components $$V^ q_{S,k}=\{z=\gamma^ k_ S(t):({\mathbb{C}},0)\to ({\mathbb{C}}^ n,z_ 0)\}$$, $$k=1,...,P$$ of the intersection $$V^ q\cap S$$ by an $$(n-q+1)-$$ dimensional plane S in general position. This result gives a final answer to the problem initiated and studied by J. J. Kohn.
Reviewer: A.Kaneko

##### MSC:
 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 32T99 Pseudoconvex domains 35N15 $$\overline\partial$$-Neumann problems and formal complexes in context of PDEs 65H10 Numerical computation of solutions to systems of equations 35J55 Systems of elliptic equations, boundary value problems (MSC2000)
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