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Necessary conditions for subellipticity of the \({\bar\partial}\)-Neumann problem. (English) Zbl 0552.32017
Let \(\alpha\) be a \({\bar \partial}\)-closed form of type (p,q) with \(L^ 2\)-coefficient on a smoothly bounded domain \(\Omega\) in \({\mathbb{C}}^ n\). One of the principal methods used in the investigation of the \({\bar \partial}\)-Neumann problem (existence and regularity properties of the solution w of \({\bar \partial}u=\alpha\)) is the proof of certain a priori subelliptic estimates i.e., if U is a neighborhood of point \(z_ 0\in \partial \Omega\), a subelliptic estimate holds in U if \((1)\quad \|| u\||^ 2_{\epsilon}\leq {\mathbb{C}}(\| {\bar \partial}u\|^ 2+\| {\bar \partial}^*u\|^ 2+\| u\|^ 2)\) is valid for all \(u\in {\mathcal D}^{p,q}(U)\) where \(\|| \cdot \||^ 2_{\epsilon}\) denotes the tangential Sobolev norm of order \(\epsilon\) and \({\mathcal D}^{p,q}(U)\) the space of smooth (p,q-1) forms u. The author presents geometric conditions that must hold if a subelliptic estimate of order \(\epsilon\) is valid. One of the main results is the following theorem: Suppose that \(\Omega\) is a domain in \({\mathbb{C}}^ n\) and that \(\partial \Omega\) is smooth and pseudoconvex in a neighborhood U of \(z_ 0\). Suppose further that there is a q-dimensional complex analytic variety V passing through \(z_ 0\) such that for all \(z\in V\), z sufficently close to \(z_ 0\), \(| r(z)| \leq c| z-z_ 0|^{\eta},\) where \(\eta >0\) and r(z) is the boundary-defining function of \(\Omega\). If a subelliptic estimate of order \(\epsilon\) of the form (1) holds in U then \(\epsilon\leq 1/\eta\).
Reviewer: R.Salvi

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32T99 Pseudoconvex domains
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
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