The boundary regularity of the solution of the \({\bar\partial}\)-equation in the product of strictly pseudoconvex domains (Q1071139)

This paper deals with the Cauchy-Riemann equation in the product of two copies of a strictly pseudoconvex domain \({\mathfrak D}\). Let \({\mathfrak D}\) be a strictly pseudoconvex domain in \({\mathbb{C}}^ n\). The author proves that for every \({\bar \partial}\)-closed differential (0,q)-form f, (q\(\geq 1)\), with coefficients of class \({\mathcal C}^{\infty}({\mathfrak D}\times{\mathfrak D})\), and continuous in the set \(\bar {\mathfrak D}\times{\mathfrak D}\bar {\;}-\Delta ({\mathfrak D})\) (\(\Delta({\mathfrak D})\) is the diagonal in \({\mathfrak D}\times {\mathfrak D}.)\), the equation \({\bar \partial}u=f\) admits a solution u with the same boundary properties. As an application, the author proves: Set \(Q:=(\bar {\mathfrak D}\times \bar {\mathfrak D})-\{(z,z); z\in \partial D\},\) \((A_ Q)_ 0({\mathfrak D}\times{\mathfrak D})\) the space of all functions which are holomorphic in \({\mathfrak D}\times{\mathfrak D}\), continuous in Q and vanish on \(\Delta\) (\({\mathfrak D})\). Let \(g_ 1,...,g_ N\in (A_ Q)_ 0({\mathfrak D}\times{\mathfrak D})\) satisfy the following poperties: \((i) \{(z,s)\in Q; g_ 1(z,s)=...=g_ N(z,s)=0\}=\Delta ({\mathfrak D});\) (ii) for every \(z\in {\mathfrak D}\), the germs at (z,z) of the functions \(g_ i\) \((i=1,...,N)\) generate the ideal of germs at (z,z) of holomorphic functions which vanish on \(\Delta\) (\({\mathfrak D})\). Then for every \(f\in (A_ Q)_ 0({\mathfrak D}\times{\mathfrak D})\) there exist functions \(f_ 1,...,f_ N\in A_ Q({\mathfrak D}\times{\mathfrak D})\) such that \(f=\sum^{N}_{i=1}g_ if_ i.\)