The \(L^p\) CR Hartogs separate analyticity theorem for convex domains
The author proves an interesting generalization of the classical Hartogs theorem. This theorem states that a function of several complex variables which is separately holomorphic in each variable is continuous and hence holomorphic in its domain of definition. The author considers convex domains \(D\) with a real analytic boundary \(\partial D\), and shows that, under some additional condition on \(\partial D\), an \(L^p\) function \(f\) on \(\partial D\) is CR, and, in consequence, with holomorphic extension to all \(D\), if and only if it has holomorphic extensions on the slices \(\ell\cap D\) for all complex lines \(\ell\) parallel to the coordinate axes. Applications to represent the Szegő kernel as an iterative limit are also given.
scientific article; zbMATH DE number 6851633
scientific article