Explicit formulas for the Szegő kernel on certain weakly pseudoconvex domains

Abstract

Abstract. The objective of this paper is to determine the Szegö kernel of the domain 3 = {(z, f, w) e C+m+1 ; Qmw > ||z||2 + |K||2"} explicitly in closed form.The purpose of this paper is to determine the Szegö kernel explicitly for a new class of weakly pseudoconvex domains. This class of domains is given by 3 = {(z, C , w) £ Cn+m+x ; ^smw> \\z\\2 + \\ Ç \\2p} in the complex space Cn+m+x. Here z £ C", ( £ Cm, w £ C, n and m are integers, n > 0, m > 1, and p is any positive number. We use the notation ||z||2 = z • z ^ £"=i |z;|2 and || C \\2p = ( C • If = (££, I C,\2)P ■ The unbounded domain 3 is weakly pseudoconvex with degenerate Levi form whenever p > 1. Furthermore the domain 2 is not "decoupled" when m > 1. The definition of decoupled domains can be found, for example, in [Mc]. Decoupled domains were considered recently, for example, by [DT], [Mc].There are relatively few classes of domains with smooth boundary where the Szegö kernel is explicitly known in closed form. We consider here a class of weakly pseudoconvex domains with smooth boundary. In particular, we do not assume that the defining function is decoupled. The authors believe that this is one of the few examples of such domains where the Szegö kernel is computed explicitly. See also the work of Christ [C].Recent interest in explicit formulas for the Szegö and Bergman kernels is motivated by the surprising effectiveness of these formulas. We illustrate this point by two cases. The striking discovery of Christ and Geller [CG] that the Szegö kernel of certain weakly pseudoconvex domains is not analytic off the diagonal was derived from Nagel's [N] explicit formula. In [M2], an explicit formula allowed Machedon to test if the Szegö kernel is a singular integral with respect to a certain nonisotropic metric.