# zbMATH — the first resource for mathematics

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

##### MSC:
 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators 32T99 Pseudoconvex domains 35N15 $$\overline\partial$$-Neumann problems and formal complexes in context of PDEs
Full Text: