Levi metrics and the Romanov-Henkin kernel
There have been two distinct approaches to the study of the tangential Cauchy-Riemann operator \(\overline\partial_ b\) on the boundary \(M\) of a strictly pseudo-convex domain in \(\mathbb{C}^{n+1}\). The first, due originally to A. V. Romanov and G. M. Henkin, is based upon operators \(R_ q\) and \(R_{q+1}\) such that \(f=R_{q+1}\overline\partial_ bf+\overline\partial_ bR_ qf\) for \((0,q)\)-forms \(f\), \(1\leq q\leq n-1\). The second method is due to \textit{G. Folland} and \textit{E. Stein} [Commun. Pure Appl. Math. 27, 429-522 (1974; Zbl 0293.35012)], and involves the adjoint \(A_ b\) of \(\overline\partial_ b\), defined through a Levi metric on \(M\). The present work is devoted to showing for \(M\) with Levi metric that these two methods yield comparable results. The approximation of integral operators depends upon a scale of types that measure the degree of singularity of the kernels. It is shown that the Romanov-Henkin operator is the ``principal part'' in the sense of types of the canonical solution operator \(S\) such that \(\overline\partial_ bSf=f\) and \(A_ bSf=0\). The (kernel of the) principal part of the tangential Neumann operator is also given.
scientific article; zbMATH DE number 446299
scientific article