Estimates for the Szegö and Henkin kernels in Hardy classes on certain unbounded weakly pseudoconvex domains
The smooth, unbounded, pseudoconvex domains in question are of the form \[ \Omega=\{(z_ 1,z_ 2)\in C^ 2:\;\text{Im} z_ 2>\varphi(z_ 2)\}, \] where \(\varphi\) is a subharmonic and non-harmonic polynomial of degree \(m\). The first task is the definition of Hardy spaces \({\mathcal H}^ p\) on \(\partial\Omega\). Two definitions (first in terms of atoms and non- isotropic balls in the sense of \textit{A. Nagel}, \textit{E. M. Stein} and \textit{S. Wainger} [Acta Math. 155, 103-147 (1985; Zbl 0578.32044)] and secondly through an appropriate notion of maximal function as in \textit{R. A. Macias} and \textit{C. Segovia} [Adv. Math. 33, 271-309 (1979; Zbl 0431.46019)] and \textit{A. Uchiyama} [Trans. Am. Math. Soc. 262, 579-592 (1980; Zbl 0503.46020)]) are shown to be equivalent. This is not easy and it occupies 10 pages of \S2, Local \({\mathcal H}^ p\) spaces on \(\partial\Omega\). \S3, Estimates for the Szegö kernel in Hardy spaces, is devoted to a lengthy proof that the Szegö projection extends to a bounded operator from \({\mathcal H}^ p(\partial\Omega)\) to the local Hardy spaces \({\mathcal H}_{\text{loc}}^ p(\partial\Omega)\) for \(p_ m<p\leq 1\), where the index \(p_ m\) depends upon the degree \(m\) of \(\varphi\). \S4, Estimates for the Henkin kernel in Hardy spaces, deals with weakly pseudoconvex domains \(H_ m=\{(z_ 1,z_ 2)\in C^ 2\): \(\text{Im} z_ 2>| z_ 1|^{2m}\}\) and proves that the Henkin projection extends to a bounded operator from \(L^ p(\partial H_ m)\) to \(L^ p(\partial H_ m)\) for \(1<p<\infty\), using methods in the book of \textit{E. M. Stein}, Singular integrals and nilpotent Lie groups (in preparation). The reviewer has not checked the delicate estimates involved in the present work.
scientific article; zbMATH DE number 55689
scientific article