Cauchy kernels in strictly pseudoconvex domains and an application to a Mergolyan type approximation problem
Let D be a bounded, strictly pseudoconvex domain in \({\mathbb{C}}^ n\). Khenkin and Ramírez constructed a kernel \(H(\zeta\),z), \(\zeta\in bD\), \(z\in D\) which is holomorphic in z and reproduces holomorphic functions in D from its values on the boundary. In the first part of the paper the authors construct a kernel \(C(\zeta\),z), which is an extension to \(D\times D\) of \(H(\zeta\),z) such that the Koppelman formulas for arbitrary (p,q)-forms hold for \(C(\zeta\),z). The kernel \(C(\zeta\),z) enjoys a well known property of Cauchy's kernel in one variable: the integral of \(| C(\zeta,z)|\) over an Euclidean ball is comparable to the radius. In the second part these results are applied to the following approximation problem: Consider functions f satisfying the equations \(\partial^ 2f/\partial \bar z_ i\partial \bar z_ j=0,\quad i,j=1,...,n,\) in D (or, which is the same, \(f=h_ 0+\Sigma^{n}_{j=1}h_ j\bar z_ j\) with \(h_ j\) holomorphic in D, 0\(\leq j\leq n)\). Then every continuous function on \(\bar D\) satisfying these equations in D is the uniform limit on \(\bar D\) of functions satisfying them in a (fixed) neighborhood of \(\bar D\).
scientific article; zbMATH DE number 3855476