A note on the kernel of the \({\bar \partial}\)-Neumann operator on strongly pseudo-convex domains
The paper offers a wonderful relationship between a special Heisenberg coordinate system and a normalized Leir metric on a strongly pseudo convex domain in \({\mathbb{C}}^ n\). This connection is then viewed on its relationship to the \({\bar \partial}\)-Neumann operator. The \({\bar \partial}\)-Neumann problem, \[ (1.1)\quad \square u=({\bar \partial}{\bar \partial}^*+{\bar \partial}^*{\bar \partial})u=f, \] \[ (1.2)\quad u\in domain({\bar \partial}^*),\quad {\bar \partial}u\in domain({\bar \partial}^*), \] in a pseudo-convex domain, \(\Omega \subset {\mathbb{C}}^{n+1}\), has attracted many researchers in recent years. Consequently there is a large literature developing the problem in many mathematical settings. This note requires a basic understanding of these settings. It begins with a \({\bar \partial}_ b\) complex. This can be defined as a partially complex manifold in a real \({\mathbb{C}}^{\infty}\) manifold, M, of dimension \(2n+1\), \(n=1,2,..\). together with a subbundle, \(T_{1,0}\), of the complex tangent bundle, \({\mathbb{C}}TM\). It satisfies (a) \(\dim_{{\mathbb{C}}} T_{1,0}=n\), (b) \(T_{1,0}\cap \bar T_{1,0}=\{\emptyset \}\), (c) \(T_{1,0}\) is integrable in the sense of Frobenius, i.e. \(z_ 1.z_ 2\) sections of \(T_{1,0}\) then so is their Lie bracket, \([z_ 1,z_ 2]\) and (d) the line bundle \((T_{1,0}\oplus \bar T_{1,0})^{\perp}\subset {\mathbb{C}}T^*M\). This definition is implicit in definition 2.1 given in the paper. Assuming the reader is familiar with the general theory, then a Levi metric is defined as \[ ds^ 2={\mathcal K}(z)\sum^{n}_{i,j=1}<z_ i,z_ j>_ L w_ i\otimes \bar w_ j \] having several additional requirements but the \((z_ 1,...,z_ n)\) forms a basics for the subbundle, \(T_{1,0}\) in the appropriate setting. The solution, u, in problem (1.1) is given by an operator, N, relating it to f. The principal result is relating special Heisenberg coordinates to a normalized Levi metric and its connection to the operator, N.
scientific article; zbMATH DE number 4096045
scientific article